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   "released": "October 9, 2012",
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    "thompson-calculus-made-easy-1914/eq-3914dd9c18",
    "thompson-calculus-made-easy-1914/eq-7d41bcaf3c",
    "thompson-calculus-made-easy-1914/eq-b4e5b7cb08",
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    "thompson-calculus-made-easy-1914/eq-3233f32bc3",
    "thompson-calculus-made-easy-1914/eq-a614f9d5d0",
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   "number": "X",
   "title": "Geometrical Meaning of Differentiation",
   "name": "Thompson 1914, ch. X: Geometrical Meaning of Differentiation",
   "pages": [
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   "number": "XI",
   "title": "Maxima and Minima",
   "name": "Thompson 1914, ch. XI: Maxima and Minima",
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    "thompson-calculus-made-easy-1914/eq-5ecac923e2",
    "thompson-calculus-made-easy-1914/eq-f06c657a66",
    "thompson-calculus-made-easy-1914/eq-ae47a5c558",
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    "thompson-calculus-made-easy-1914/eq-a28f2dc210",
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    "thompson-calculus-made-easy-1914/eq-800ec7f740",
    "thompson-calculus-made-easy-1914/eq-4ebf478238",
    "thompson-calculus-made-easy-1914/eq-dbb06339ed",
    "thompson-calculus-made-easy-1914/eq-402756a770",
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    "thompson-calculus-made-easy-1914/eq-02ea1873db",
    "thompson-calculus-made-easy-1914/eq-2f86321752",
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    "thompson-calculus-made-easy-1914/eq-725d6b5fbb",
    "thompson-calculus-made-easy-1914/eq-5656f64a41",
    "thompson-calculus-made-easy-1914/eq-93ef987bb2",
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   "number": "XII",
   "title": "Curvature of Curves",
   "name": "Thompson 1914, ch. XII: Curvature of Curves",
   "pages": [
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    "121"
   ],
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   "number": "XIII",
   "title": "Other Useful Dodges",
   "name": "Thompson 1914, ch. XIII: Other Useful Dodges",
   "pages": [
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    "134"
   ],
   "concepts": [
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    "concept/calculus",
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   "number": "XIV",
   "title": "On true Compound Interest and the Law of Organic Growth",
   "name": "Thompson 1914, ch. XIV: On true Compound Interest and the Law of Organic Growth",
   "pages": [
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   "id": "thompson-calculus-made-easy-1914/ch-xv",
   "number": "XV",
   "title": "How to deal with Sines and Cosines",
   "name": "Thompson 1914, ch. XV: How to deal with Sines and Cosines",
   "pages": [
    "165",
    "175"
   ],
   "concepts": [
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    "concept/cosecant",
    "concept/cosine",
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    "concept/inverse-function",
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   "id": "thompson-calculus-made-easy-1914/ch-xvi",
   "number": "XVI",
   "title": "Partial Differentiation",
   "name": "Thompson 1914, ch. XVI: Partial Differentiation",
   "pages": [
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    "182"
   ],
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    "concept/differential-equation",
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    "method/differentiation",
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    "thompson-calculus-made-easy-1914/x-624a1e39ea",
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    "thompson-calculus-made-easy-1914/eq-37ba606fc7",
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    "thompson-calculus-made-easy-1914/eq-746f2d64f2",
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    "thompson-calculus-made-easy-1914/eq-1339629dbb",
    "thompson-calculus-made-easy-1914/eq-dd75bd9097",
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   "number": "XVII",
   "title": "Integration",
   "name": "Thompson 1914, ch. XVII: Integration",
   "pages": [
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   "number": "XVIII",
   "title": "Integrating as the Reverse of Differentiating",
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   "number": "XIX",
   "title": "On Finding Areas by Integrating",
   "name": "Thompson 1914, ch. XIX: On Finding Areas by Integrating",
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   "title": "Dodges, Pitfalls, and Triumphs",
   "name": "Thompson 1914, ch. XX: Dodges, Pitfalls, and Triumphs",
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   "number": "Epilogue and Apologue",
   "title": "Epilogue and Apologue",
   "name": "Thompson 1914, Epilogue and Apologue",
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   "title": "Table of Standard Forms",
   "name": "Thompson 1914, Table of Standard Forms",
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    "theorem/product-rule-for-differentiation",
    "theorem/quotient-rule",
    "theorem/quotient-rule-for-differentiation",
    "theorem/sum-rule-for-differentiation",
    "theorem/sum-rule-for-integration"
   ],
   "excerpts": [
    "thompson-calculus-made-easy-1914/x-624276693a",
    "thompson-calculus-made-easy-1914/x-eb8f544e3b",
    "thompson-calculus-made-easy-1914/x-7805b97462",
    "thompson-calculus-made-easy-1914/x-342917ba59",
    "thompson-calculus-made-easy-1914/x-6f701edc38",
    "thompson-calculus-made-easy-1914/x-3fa780aaf0"
   ],
   "equations": [
    "thompson-calculus-made-easy-1914/eq-a46e74cc91",
    "thompson-calculus-made-easy-1914/eq-cd7d495dd0",
    "thompson-calculus-made-easy-1914/eq-716bfee8b4",
    "thompson-calculus-made-easy-1914/eq-3d360f534f",
    "thompson-calculus-made-easy-1914/eq-b745cc87f3",
    "thompson-calculus-made-easy-1914/eq-180be9d000",
    "thompson-calculus-made-easy-1914/eq-e1186b8e51",
    "thompson-calculus-made-easy-1914/eq-a87f82215e",
    "thompson-calculus-made-easy-1914/eq-13d61cb689",
    "thompson-calculus-made-easy-1914/eq-e35c85441c",
    "thompson-calculus-made-easy-1914/eq-9d7bc865d0",
    "thompson-calculus-made-easy-1914/eq-57d855071e",
    "thompson-calculus-made-easy-1914/eq-8815df15f1",
    "thompson-calculus-made-easy-1914/eq-7403babbc9",
    "thompson-calculus-made-easy-1914/eq-48ff44bf6a",
    "thompson-calculus-made-easy-1914/eq-2f1cb9c34a",
    "thompson-calculus-made-easy-1914/eq-1896337166",
    "thompson-calculus-made-easy-1914/eq-dd26224cde",
    "thompson-calculus-made-easy-1914/eq-5788f5155d",
    "thompson-calculus-made-easy-1914/eq-4b49668077",
    "thompson-calculus-made-easy-1914/eq-4134466ca3",
    "thompson-calculus-made-easy-1914/eq-1fb045da24",
    "thompson-calculus-made-easy-1914/eq-e4baf7e24c",
    "thompson-calculus-made-easy-1914/eq-4b95faecad",
    "thompson-calculus-made-easy-1914/eq-35de85f576",
    "thompson-calculus-made-easy-1914/eq-36ea38e0d0",
    "thompson-calculus-made-easy-1914/eq-19814313f2",
    "thompson-calculus-made-easy-1914/eq-c3d3cb6f67",
    "thompson-calculus-made-easy-1914/eq-1422eebb5b",
    "thompson-calculus-made-easy-1914/eq-42fc8fbd29",
    "thompson-calculus-made-easy-1914/eq-6abe9e26a5",
    "thompson-calculus-made-easy-1914/eq-aefa094327",
    "thompson-calculus-made-easy-1914/eq-48ec2075ee",
    "thompson-calculus-made-easy-1914/eq-a2035197e1",
    "thompson-calculus-made-easy-1914/eq-14ef531dc7",
    "thompson-calculus-made-easy-1914/eq-85327723ec",
    "thompson-calculus-made-easy-1914/eq-57b95c695b",
    "thompson-calculus-made-easy-1914/eq-ec5dfb218c"
   ],
   "exercise_sets": []
  }
 ],
 "excerpts": [
  {
   "id": "thompson-calculus-made-easy-1914/x-200fc8d0ca",
   "chapter": "thompson-calculus-made-easy-1914/ch-preface-to-the-second-edition",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "vii",
   "location": "Preface to the Second Edition",
   "latex": "Advantage has also been taken to enlarge certain parts where experience showed that further explanations would be useful.",
   "markdown": "Advantage has also been taken to enlarge certain parts where experience showed that further explanations would be useful.",
   "why": "It shows the author revising explanations where real students stumbled, which is a useful model for a learner who finds a passage hard.",
   "use": [
    "lesson"
   ],
   "concepts": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-9ce804c0e5",
   "chapter": "thompson-calculus-made-easy-1914/ch-preface-to-the-second-edition",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "vii",
   "location": "Preface to the Second Edition",
   "latex": "\\First{The} surprising success of this work has led the author to add a considerable number of worked examples and exercises.",
   "markdown": "The surprising success of this work has led the author to add a considerable number of worked examples and exercises.",
   "why": "It shows learners that this book was popular enough to be revised, and that the second edition added worked examples and exercises.",
   "use": [
    "history",
    "website"
   ],
   "concepts": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-2b463e796d",
   "chapter": "thompson-calculus-made-easy-1914/ch-preface-to-the-second-edition",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "vii",
   "location": "Preface to the Second Edition",
   "latex": "The author acknowledges with gratitude many valuable suggestions and letters received from teachers, students, and---critics.",
   "markdown": "The author acknowledges with gratitude many valuable suggestions and letters received from teachers, students, and---critics.",
   "why": "It gives a light, human glimpse of the author's humour and of how readers' letters shaped the book.",
   "use": [
    "history",
    "website"
   ],
   "concepts": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f25e3d5340",
   "chapter": "thompson-calculus-made-easy-1914/ch-prologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "xi",
   "location": "Prologue",
   "latex": "Some calculus-tricks are quite easy. Some are enormously difficult.",
   "markdown": "Some calculus-tricks are quite easy. Some are enormously difficult.",
   "why": "It tells the learner at the start that calculus contains both easy and hard calculations, so the easy parts can be mastered first.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-2f58cbb0c8",
   "chapter": "thompson-calculus-made-easy-1914/ch-prologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "xi",
   "location": "Prologue",
   "latex": "The fools who write the textbooks of advanced mathematics---and they are mostly clever fools---seldom take the trouble to show you how easy the easy calculations are.",
   "markdown": "The fools who write the textbooks of advanced mathematics---and they are mostly clever fools---seldom take the trouble to show you how easy the easy calculations are.",
   "why": "It warns that much advanced writing hides how simple its basic calculations are, which is a useful caution for any reader.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-d71d640237",
   "chapter": "thompson-calculus-made-easy-1914/ch-prologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "xi",
   "location": "Prologue",
   "latex": "Master these thoroughly, and the rest will follow. What one fool can do, another can.",
   "markdown": "Master these thoroughly, and the rest will follow. What one fool can do, another can.",
   "why": "It gives the learner a clear strategy, which is to master the basics thoroughly before attempting the rest.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-458e43917b",
   "chapter": "thompson-calculus-made-easy-1914/ch-prologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "xi",
   "location": "Prologue",
   "latex": "\\First{Considering} how many fools can calculate, it is\nsurprising that it should be thought either a difficult\nor a tedious task for any other fool to learn how to\nmaster the same tricks.",
   "markdown": "Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.",
   "why": "It opens with a joke that tells the learner calculus is within reach of an ordinary person.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-70e1ded86f",
   "chapter": "thompson-calculus-made-easy-1914/ch-prologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "xi",
   "location": "Prologue",
   "latex": "Some calculus-tricks are quite easy. Some are\nenormously difficult. The fools who write the textbooks\nof advanced mathematics---and they are mostly\nclever fools---seldom take the trouble to show you how\neasy the easy calculations are.",
   "markdown": "Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics---and they are mostly clever fools---seldom take the trouble to show you how easy the easy calculations are.",
   "why": "It names a real teaching failure, that experts skip over the easy parts, and so reassures a learner who feels stuck.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/calculus",
    "concept/mathematics"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-bc213208ff",
   "chapter": "thompson-calculus-made-easy-1914/ch-prologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "xi",
   "location": "Prologue",
   "latex": "Being myself a remarkably stupid fellow, I have\nhad to unteach myself the difficulties, and now beg\nto present to my fellow fools the parts that are not\nhard. Master these thoroughly, and the rest will\nfollow. What one fool can do, another can.",
   "markdown": "Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.",
   "why": "It gives the book's promise and its encouragement in the author's own humble voice.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/calculus",
    "concept/mathematics"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-e8e575acc5",
   "chapter": "thompson-calculus-made-easy-1914/ch-i",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "1",
   "location": "To deliver you from the Preliminary Terrors",
   "latex": "(1) $d$ which merely means ``a little bit of.''",
   "markdown": "(1) $d$ which merely means “a little bit of.”",
   "why": "It gives the learner the plain-language meaning of the d symbol before any notation is used.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-b3750aba2f",
   "chapter": "thompson-calculus-made-easy-1914/ch-i",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "1",
   "location": "To deliver you from the Preliminary Terrors",
   "latex": "Thus $\\ds\\int dx$ means the sum of all the little bits of~$x$; or $\\ds\\int dt$ means the sum of all the little bits of~$t$.",
   "markdown": "Thus $\\ds\\int dx$ means the sum of all the little bits of $x$; or $\\ds\\int dt$ means the sum of all the little bits of $t$.",
   "why": "It pairs the integral sign with the differential so the learner reads an integral as adding up little bits.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5e1e99e2bb",
   "chapter": "thompson-calculus-made-easy-1914/ch-i",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "2",
   "location": "To deliver you from the Preliminary Terrors",
   "latex": "The word ``integral'' simply means ``the whole.''",
   "markdown": "The word “integral” simply means “the whole.”",
   "why": "It explains the name integral in common terms, so the word no longer seems mysterious.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-70dca9fa6f",
   "chapter": "thompson-calculus-made-easy-1914/ch-i",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "1",
   "location": "To deliver you from the Preliminary Terrors",
   "latex": "Thus $dx$ means a little bit of~$x$; or $du$ means a little bit of~$u$. Ordinary mathematicians think it more polite to say ``an element of,'' instead of ``a little bit of.'' Just as you please.",
   "markdown": "Thus $dx$ means a little bit of $x$; or $du$ means a little bit of $u$. Ordinary mathematicians think it more polite to say “an element of,” instead of “a little bit of.” Just as you please.",
   "why": "It gives the plain meaning of d and says the formal wording is optional.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-556b6caf34",
   "chapter": "thompson-calculus-made-easy-1914/ch-i",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "2",
   "location": "To deliver you from the Preliminary Terrors",
   "latex": "If you think of the duration of time for one hour, you may (if you like) think of it as cut up into $3600$ little bits called seconds. The whole of the $3600$ little bits added up together make one hour.",
   "markdown": "If you think of the duration of time for one hour, you may (if you like) think of it as cut up into $3600$ little bits called seconds. The whole of the $3600$ little bits added up together make one hour.",
   "why": "A familiar example of little bits adding up to a whole.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/differential",
    "concept/integral",
    "quantity/time",
    "unit/hour",
    "unit/second"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-eae80e9841",
   "chapter": "thompson-calculus-made-easy-1914/ch-i",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "1",
   "location": "To deliver you from the Preliminary Terrors",
   "latex": "\\First{The} preliminary terror, which chokes off most fifth-form boys from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning---in common-sense terms---of the two principal symbols that are used in calculating.",
   "markdown": "The preliminary terror, which chokes off most fifth-form boys from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning---in common-sense terms---of the two principal symbols that are used in calculating.",
   "why": "It names the fear of calculus and promises to remove it by explaining the two symbols in plain terms.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/differential",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-8a1600ab58",
   "chapter": "thompson-calculus-made-easy-1914/ch-i",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "2",
   "location": "To deliver you from the Preliminary Terrors",
   "latex": "Now any fool can see that if $x$~is considered as made up of a lot of little bits, each of which is called~$dx$, if you add them all up together you get the sum of all the~$dx$'s, (which is the same thing as the whole of~$x$). The word ``integral'' simply means ``the whole.''",
   "markdown": "Now any fool can see that if $x$ is considered as made up of a lot of little bits, each of which is called $dx$, if you add them all up together you get the sum of all the $dx$’s, (which is the same thing as the whole of $x$). The word “integral” simply means “the whole.”",
   "why": "It shows that the integral is the whole obtained by adding up the little bits.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/differential",
    "concept/integral",
    "concept/sum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f14492f21e",
   "chapter": "thompson-calculus-made-easy-1914/ch-i",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "2",
   "location": "To deliver you from the Preliminary Terrors",
   "latex": "When you see an expression that begins with this terrifying symbol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow. That's all.",
   "markdown": "When you see an expression that begins with this terrifying symbol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow. That’s all.",
   "why": "It tells the learner the integral sign is only an instruction to add up little bits, which takes the fear out of it.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/integral",
    "concept/operation",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-bf4acef86e",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "3",
   "location": "On Different Degrees of Smallness",
   "latex": "Obviously $1$~minute is a very small quantity of time compared with a whole week.",
   "markdown": "Obviously $1$ minute is a very small quantity of time compared with a whole week.",
   "why": "Shows through a familiar example that smallness is relative, which is the chapter's central point.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/infinitesimal"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-06a9da7d58",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "4",
   "location": "On Different Degrees of Smallness",
   "latex": "The mathematicians talk about the second order of ``magnitude''\n  (\\IE~greatness) when they really mean second order of \\emph{smallness}.\n  This is very confusing to beginners.",
   "markdown": "The mathematicians talk about the second order of “magnitude” (*i.e.* greatness) when they really mean second order of *smallness*. This is very confusing to beginners.",
   "why": "Warns learners about a common source of confusion in the wording used by mathematicians.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/order-of-smallness"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-341dc8a359",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "5",
   "location": "On Different Degrees of Smallness",
   "latex": "But, it must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a farthing becomes important if only it is multiplied by a few hundred.",
   "markdown": "But, it must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a farthing becomes important if only it is multiplied by a few hundred.",
   "why": "Cautions that smallness is relative and a small quantity multiplied by a large one may not be negligible.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/infinitesimal",
    "method/neglecting-higher-order-small-quantities"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-fe80269af9",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "5",
   "location": "On Different Degrees of Smallness",
   "latex": "Then we see that the smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become.",
   "markdown": "Then we see that the smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become.",
   "why": "States the key reason higher-order small quantities can be dropped.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/order-of-smallness",
    "method/neglecting-higher-order-small-quantities"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-d4ef91cf5f",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "3",
   "location": "On Different Degrees of Smallness",
   "latex": "Nowadays we call these small quantities of the second order of smallness ``seconds.'' But few people know \\emph{why} they are so called.",
   "markdown": "Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know *why* they are so called.",
   "why": "Shows that the familiar word 'second' comes from 'second order of smallness', which makes the idea of orders concrete.",
   "use": [
    "history",
    "website"
   ],
   "concepts": [
    "concept/order-of-smallness",
    "unit/second"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-e224d14e50",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "5",
   "location": "On Different Degrees of Smallness",
   "latex": "Now in the calculus we write $dx$ for a little bit of~$x$. These things such as~$dx$, and~$du$, and~$dy$, are called ``differentials,'' the differential of~$x$, or of~$u$, or of~$y$, as the case may be.",
   "markdown": "Now in the calculus we write $dx$ for a little bit of $x$. These things such as $dx$, and $du$, and $dy$, are called “differentials,” the differential of $x$, or of $u$, or of $y$, as the case may be.",
   "why": "Introduces differential notation as 'a little bit of' a quantity.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ef3b5f8fe7",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "6",
   "location": "On Different Degrees of Smallness",
   "latex": "Let us think of $x$ as a quantity that can grow by a small amount so as to become $x + dx$, where $dx$~is the small increment added by growth. The square of this is $x^2 + 2x · dx + (dx)^2$.",
   "markdown": "Let us think of $x$ as a quantity that can grow by a small amount so as to become $x + dx$, where $dx$ is the small increment added by growth. The square of this is $x^2 + 2x · dx + (dx)^2$.",
   "why": "Sets up the worked example that separates first-order and second-order terms.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "concept/increment",
    "concept/order-of-smallness",
    "concept/square",
    "method/neglecting-higher-order-small-quantities"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-09414216f6",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "7",
   "location": "On Different Degrees of Smallness",
   "latex": "Clearly $(dx)^2$ is negligible if only we consider the increment~$dx$ to be itself small enough.",
   "markdown": "Clearly $(dx)^2$ is negligible if only we consider the increment $dx$ to be itself small enough.",
   "why": "Gives the conclusion of the growing-square picture: the tiny corner square can be ignored.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/increment",
    "concept/square",
    "method/neglecting-higher-order-small-quantities"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-71a2d2ee71",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "8",
   "location": "On Different Degrees of Smallness",
   "latex": "An ox might worry about a flea of ordinary size---a small creature of the first order of smallness. But he would probably not trouble himself about a flea's flea; being of the second order of smallness, it would be negligible.",
   "markdown": "An ox might worry about a flea of ordinary size---a small creature of the first order of smallness. But he would probably not trouble himself about a flea’s flea; being of the second order of smallness, it would be negligible.",
   "why": "A humorous analogy that makes orders of smallness easy to remember.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/order-of-smallness",
    "person/jonathan-swift"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f00a19abb0",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "9",
   "location": "On Relative Growings",
   "latex": "Those which we regard as of fixed value, and call \\emph{constants}, we generally denote algebraically by letters from the beginning of the alphabet, such as $a$,~$b$, or~$c$; while those which we consider as capable of growing, or (as mathematicians say) of ``varying,'' we denote by letters from the end of the alphabet, such as $x$,~$y$,~$z$, $u$,~$v$,~$w$, or sometimes~$t$.",
   "markdown": "Those which we regard as of fixed value, and call *constants*, we generally denote algebraically by letters from the beginning of the alphabet, such as $a$, $b$, or $c$; while those which we consider as capable of growing, or (as mathematicians say) of “varying,” we denote by letters from the end of the alphabet, such as $x$, $y$, $z$, $u$, $v$, $w$, or sometimes $t$.",
   "why": "Gives the learner the basic split between fixed and varying quantities and the lettering habit that goes with it.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/constant",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-d87e13e5ff",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "12",
   "location": "On Relative Growings",
   "latex": "Suppose the ladder was so long that when the bottom end~$A$ was $19$~inches from the wall the top end~$B$ reached just $15$~feet from the ground. Now, if you were to pull the bottom end out $1$~inch more, how much would the top end come down?",
   "markdown": "Suppose the ladder was so long that when the bottom end $A$ was $19$ inches from the wall the top end $B$ reached just $15$ feet from the ground. Now, if you were to pull the bottom end out $1$ inch more, how much would the top end come down?",
   "why": "Poses a concrete ladder problem in which a positive step in x gives a negative step in y, and the answer is then worked out in numbers.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/increment",
    "concept/relation-between-variables",
    "theorem/pythagorean-theorem"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-7ccec4a881",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "15",
   "location": "On Relative Growings",
   "latex": "It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy.",
   "markdown": "It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy.",
   "why": "Eases a learner's fear of the term differential coefficient with the author's cheerful impatience for long names.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-1a4eb0dd0b",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "16",
   "location": "On Relative Growings",
   "latex": "You have now to learn to go hunting in a new way; the fox being now neither $x$ nor~$y$. Instead of this you have to hunt for this curious cub called~$\\dfrac{dy}{dx}$.",
   "markdown": "You have now to learn to go hunting in a new way; the fox being now neither $x$ nor $y$. Instead of this you have to hunt for this curious cub called $\\dfrac{dy}{dx}$.",
   "why": "Contrasts the school-algebra hunt for an unknown with the new hunt for a ratio, in a charming old image.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/derivative",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f2bbf3126b",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "9",
   "location": "On Relative Growings",
   "latex": "We classify all quantities into two classes: \\emph{constants} and \\emph{variables}. Those which we regard as of fixed value, and call \\emph{constants}, we generally denote algebraically by letters from the beginning of the alphabet, such as $a$,~$b$, or~$c$; while those which we consider as capable of growing, or (as mathematicians say) of ``varying,'' we denote by letters from the end of the alphabet, such as $x$,~$y$,~$z$, $u$,~$v$,~$w$, or sometimes~$t$.",
   "markdown": "We classify all quantities into two classes: *constants* and *variables*. Those which we regard as of fixed value, and call *constants*, we generally denote algebraically by letters from the beginning of the alphabet, such as $a$, $b$, or $c$; while those which we consider as capable of growing, or (as mathematicians say) of “varying,” we denote by letters from the end of the alphabet, such as $x$, $y$, $z$, $u$, $v$, $w$, or sometimes $t$.",
   "why": "It gives the learner the basic split between fixed letters and growing letters that the whole calculus rests on.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/constant",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-07e1886cc5",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "12",
   "location": "On Relative Growings",
   "latex": "So we see that making $dx$ an increase of $1$~inch has resulted in making $dy$ a decrease of $0.11$~inch.",
   "markdown": "So we see that making $dx$ an increase of $1$ inch has resulted in making $dy$ a decrease of $0.11$ inch.",
   "why": "It states in plain words what a pair of small changes in a ladder problem means, which a learner can check with numbers.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-2b02b9edc9",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "14",
   "location": "On Relative Growings",
   "latex": "For example $x^2 + 3 = 2y - 7$ is an implicit function in $x$~and~$y$; it may be written $y = \\dfrac{x^2 + 10}{2}$ (explicit function of~$x$) or $x = \\sqrt{2y - 10}$ (explicit function of~$y$).",
   "markdown": "For example $x^2 + 3 = 2y - 7$ is an implicit function in $x$ and $y$; it may be written $y = \\dfrac{x^2 + 10}{2}$ (explicit function of $x$) or $x = \\sqrt{2y - 10}$ (explicit function of $y$).",
   "why": "A single worked equation shows how the same relation can be read as an implicit or an explicit function.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/explicit-function",
    "concept/implicit-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-686644f489",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "12",
   "location": "On Relative Growings",
   "latex": "Now right through the differential calculus we are hunting, hunting, hunting for a curious thing, a mere ratio, namely, the proportion which $dy$~bears to~$dx$ when both of them are indefinitely small.",
   "markdown": "Now right through the differential calculus we are hunting, hunting, hunting for a curious thing, a mere ratio, namely, the proportion which $dy$ bears to $dx$ when both of them are indefinitely small.",
   "why": "States in one vivid sentence what the whole differential calculus is after.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/infinitesimal",
    "concept/ratio"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ed451fba05",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "13",
   "location": "On Relative Growings",
   "latex": "If, while $x$ is, as before, the distance of the foot of the ladder from the wall, $y$~is, instead of the height reached, the horizontal length of the wall, or the number of bricks in it, or the number of years since it was built, any change in~$x$ would naturally cause no change whatever in~$y$; in this case $\\dfrac{dy}{dx}$ has no meaning whatever, and it is not possible to find an expression for it.",
   "markdown": "If, while $x$ is, as before, the distance of the foot of the ladder from the wall, $y$ is, instead of the height reached, the horizontal length of the wall, or the number of bricks in it, or the number of years since it was built, any change in $x$ would naturally cause no change whatever in $y$; in this case $\\dfrac{dy}{dx}$ has no meaning whatever, and it is not possible to find an expression for it.",
   "why": "Warns that dy/dx only means something when y really depends on x, using a counter-example the learner can picture.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/function",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-846c195b12",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "17",
   "location": "On Relative Growings",
   "latex": "It will never do to fall into the schoolboy error of thinking that $dx$ means $d$~times~$x$, for $d$ is not a factor---it means ``an element of'' or ``a bit of'' whatever follows.",
   "markdown": "It will never do to fall into the schoolboy error of thinking that $dx$ means $d$ times $x$, for $d$ is not a factor---it means “an element of” or “a bit of” whatever follows.",
   "why": "Heads off the common mistake of reading dx as a product.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-dc72085496",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "18",
   "location": "Simplest Cases",
   "latex": "Now remember that the fundamental notion about the calculus is the idea of \\emph{growing}. Mathematicians call it \\emph{varying}.",
   "markdown": "Now remember that the fundamental notion about the calculus is the idea of *growing*. Mathematicians call it *varying*.",
   "why": "Gives the learner the central intuition of the calculus, growing, before any symbols are used.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/calculus",
    "concept/derivative",
    "concept/differential",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-aafdc7c3f5",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "19",
   "location": "Simplest Cases",
   "latex": "Then $(dx)^2$~will mean a little bit of a little bit of~$x$; that is, as explained above (\\Pageref{smallness}), it is a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms.",
   "markdown": "Then $(dx)^2$ will mean a little bit of a little bit of $x$; that is, as explained above (smallness), it is a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms.",
   "why": "Explains in plain words why the (dx)^2 term can be dropped in the working.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/order-of-smallness",
    "method/differentiating-from-first-principles"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-45feec2834",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "20",
   "location": "Simplest Cases",
   "latex": "But, you will say, we neglected a whole unit.",
   "markdown": "But, you will say, we neglected a whole unit.",
   "why": "Voices the learner's natural objection to discarding a term, which the next step answers by making dx smaller.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/order-of-smallness"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-777cd550fb",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "22",
   "location": "Simplest Cases",
   "latex": "Just look at these results: the operation of differentiating appears to have had the effect of diminishing the power of~$x$ by~$1$ (for example in the last case reducing $x^4$ to~$x^3$), and at the same time multiplying by a number (the same number in fact which originally appeared as the power).",
   "markdown": "Just look at these results: the operation of differentiating appears to have had the effect of diminishing the power of $x$ by $1$ (for example in the last case reducing $x^4$ to $x^3$), and at the same time multiplying by a number (the same number in fact which originally appeared as the power).",
   "why": "Shows how a general rule is guessed from a pattern in worked cases.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/exponent",
    "concept/power",
    "method/arranging-in-columns",
    "theorem/power-rule",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-236f2ba0fb",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Simplest Cases",
   "latex": "To differentiate~$x^n$, multiply by the power and reduce the power by one, so giving us~$nx^{n-1}$ as the result.",
   "markdown": "To differentiate $x^n$, multiply by the power and reduce the power by one, so giving us $nx^{n-1}$ as the result.",
   "why": "States the power rule as a short procedure a learner can follow.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/differentiation",
    "theorem/power-rule"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-739715ec41",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Simplest Cases",
   "latex": "You have now learned how to differentiate powers of~$x$. How easy it is!",
   "markdown": "You have now learned how to differentiate powers of $x$. How easy it is!",
   "why": "Ends the chapter with the book's encouraging tone.",
   "use": [
    "website"
   ],
   "concepts": [
    "method/differentiation",
    "theorem/power-rule",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6d13a57c77",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "21",
   "location": "Simplest Cases",
   "latex": "Now we know that we may neglect small quantities of the second and third orders; since, when $dy$~and~$dx$ are both made indefinitely small, $(dx)^2$~and~$(dx)^3$ will become indefinitely smaller by comparison.",
   "markdown": "Now we know that we may neglect small quantities of the second and third orders; since, when $dy$ and $dx$ are both made indefinitely small, $(dx)^2$ and $(dx)^3$ will become indefinitely smaller by comparison.",
   "why": "It gives the reason why second- and higher-order terms in dx can be dropped, which is the step learners most often question.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/order-of-smallness",
    "method/neglecting-higher-order-small-quantities"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-45273359b4",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "20",
   "location": "Simplest Cases",
   "latex": "But, you will say, we neglected a whole unit. Well, try again, making $dx$ a still smaller bit.",
   "markdown": "But, you will say, we neglected a whole unit. Well, try again, making $dx$ a still smaller bit.",
   "why": "It anticipates a learner's objection to the numerical check and shows how a smaller dx settles it.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "method/neglecting-higher-order-small-quantities"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-d06cb0e44a",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "27",
   "location": "Next Stage. What to do with Constants",
   "latex": "So if we take the letter~$a$, or~$b$, or~$c$ to represent any constant, it will simply disappear when we differentiate.",
   "markdown": "So if we take the letter $a$, or $b$, or $c$ to represent any constant, it will simply disappear when we differentiate.",
   "why": "It generalises the vanishing of a constant to any symbol standing for a constant.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant",
    "concept/constant-term"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-670d7d12ef",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "29",
   "location": "Next Stage. What to do with Constants",
   "latex": "If we had begun with $y = ax^n$, we should have had $\\dfrac{dy}{dx} = a×nx^{n-1}$.",
   "markdown": "If we had begun with $y = ax^n$, we should have had $\\dfrac{dy}{dx} = a×nx^{n-1}$.",
   "why": "It gives the general power rule with a constant coefficient in a single line a learner can test against examples.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-factor",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6791618ee4",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "29",
   "location": "Next Stage. What to do with Constants",
   "latex": "And, what is true about multiplication is equally true about \\emph{division}: for if, in the example above, we had taken as the constant~$\\frac{1}{7}$ instead of~$7$, we should have had the same~$\\frac{1}{7}$ come out in the result after differentiation.",
   "markdown": "And, what is true about multiplication is equally true about *division*: for if, in the example above, we had taken as the constant $\\frac{1}{7}$ instead of $7$, we should have had the same $\\frac{1}{7}$ come out in the result after differentiation.",
   "why": "It extends the constant-factor rule to division, so a learner sees that a fractional constant behaves the same way.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-factor"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-9b15e9e874",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "26",
   "location": "Next Stage. What to do with Constants",
   "latex": "We usually think of $x$ as a quantity that we can vary; and, regarding the variation of~$x$ as a sort of \\emph{cause}, we consider the resulting variation of~$y$ as an \\emph{effect}. In other words, we regard the value of~$y$ as depending on that of~$x$.",
   "markdown": "We usually think of $x$ as a quantity that we can vary; and, regarding the variation of $x$ as a sort of *cause*, we consider the resulting variation of $y$ as an *effect*. In other words, we regard the value of $y$ as depending on that of $x$.",
   "why": "Gives the learner a cause-and-effect picture of the independent and dependent variable.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ef0056bc24",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "27",
   "location": "Next Stage. What to do with Constants",
   "latex": "So the $5$ has quite disappeared. It added nothing to the growth of~$x$, and does not enter into the differential coefficient.",
   "markdown": "So the $5$ has quite disappeared. It added nothing to the growth of $x$, and does not enter into the differential coefficient.",
   "why": "Explains in plain words why an added constant drops out of the derivative.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/constant-term",
    "concept/derivative",
    "theorem/derivative-of-an-added-constant"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-c322b99b25",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "29",
   "location": "Next Stage. What to do with Constants",
   "latex": "So that any mere multiplication by a constant reappears as a mere multiplication when the thing is differentiated. And, what is true about multiplication is equally true about \\emph{division}: for if, in the example above, we had taken as the constant~$\\frac{1}{7}$ instead of~$7$, we should have had the same~$\\frac{1}{7}$ come out in the result after differentiation.",
   "markdown": "So that any mere multiplication by a constant reappears as a mere multiplication when the thing is differentiated. And, what is true about multiplication is equally true about *division*: for if, in the example above, we had taken as the constant $\\frac{1}{7}$ instead of $7$, we should have had the same $\\frac{1}{7}$ come out in the result after differentiation.",
   "why": "States the constant-multiple rule and extends it to division, in contrast with the added-constant case.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant",
    "concept/constant-factor",
    "theorem/derivative-of-a-constant-multiple"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ad5de5c50a",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "30",
   "location": "Next Stage. What to do with Constants",
   "latex": "As a rule an expression of this kind will need a little more knowledge than we have acquired so far; it is, however, always worth while to try whether the expression can be put in a simpler form.",
   "markdown": "As a rule an expression of this kind will need a little more knowledge than we have acquired so far; it is, however, always worth while to try whether the expression can be put in a simpler form.",
   "why": "Teaches the habit of trying to simplify an awkward expression before differentiating.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-b34480ce00",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "31",
   "location": "Next Stage. What to do with Constants",
   "latex": "If $r = 5.5$~in.\\ and $h=20$~in.\\ this becomes $690.8$. It means that a change of radius of $1$~inch will cause a change of volume of $690.8$~cub.~inch. This can be easily verified, for the volumes with $r = 5$~and $r = 6$ are $1570$~cub.~in.\\ and $2260.8$~cub.~in.\\ respectively, and $2260.8 - 1570 = 690.8$.",
   "markdown": "If $r = 5.5$ in. and $h=20$ in. this becomes $690.8$. It means that a change of radius of $1$ inch will cause a change of volume of $690.8$ cub. inch. This can be easily verified, for the volumes with $r = 5$ and $r = 6$ are $1570$ cub. in. and $2260.8$ cub. in. respectively, and $2260.8 - 1570 = 690.8$.",
   "why": "Shows how to interpret a derivative as a rate and check it against direct subtraction.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/rate-of-change",
    "theorem/volume-of-a-cylinder"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-dfaac2754d",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "32",
   "location": "Next Stage. What to do with Constants",
   "latex": "The sensitiveness is approximately doubled from $800°$~to $1000°$, and becomes three-quarters as great again up to~$1200°$.",
   "markdown": "The sensitiveness is approximately doubled from $800°$ to $1000°$, and becomes three-quarters as great again up to $1200°$.",
   "why": "Shows how to read a derivative's values in a physical setting, here the sensitivity of a pyrometer.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/rate-of-change",
    "instrument/pyrometer"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-724c67dad5",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "35",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "If you have any doubt whether this is right, try a more general case, working it by first principles. And this is the way.",
   "markdown": "If you have any doubt whether this is right, try a more general case, working it by first principles. And this is the way.",
   "why": "It encourages the learner to check a rule by deriving it from first principles rather than trusting memory.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/differentiating-from-first-principles",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ca635bc95f",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "36",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "This justifies the procedure. You differentiate each function separately and add the results.",
   "markdown": "This justifies the procedure. You differentiate each function separately and add the results.",
   "why": "States the sum rule plainly right after its first-principles proof.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/differentiation",
    "theorem/sum-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-89c1dff4ad",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "37",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "The result will certainly \\emph{not} be $2x × 4ax^3$; for it is easy to see that neither $c × ax^4$, nor $x^2 × b$, would have been taken into that product.",
   "markdown": "The result will certainly *not* be $2x × 4ax^3$; for it is easy to see that neither $c × ax^4$, nor $x^2 × b$, would have been taken into that product.",
   "why": "Warns against the tempting mistake of multiplying the separate derivatives.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/product",
    "theorem/product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-42695b5aab",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "38",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "\\emph{To differentiate the product of two functions, multiply each function by the differential coefficient of the other, and add together the two products so obtained.}",
   "markdown": "*To differentiate the product of two functions, multiply each function by the differential coefficient of the other, and add together the two products so obtained.*",
   "why": "The product rule in one clear sentence of instruction.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "theorem/product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-49893f9384",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "38",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "You should note that this process amounts to the following: Treat~$u$ as constant while you differentiate~$v$; then treat~$v$ as constant while you differentiate~$u$; and the whole differential coefficient~$\\dfrac{dy}{dx}$ will be the sum of these two treatments.",
   "markdown": "You should note that this process amounts to the following: Treat $u$ as constant while you differentiate $v$; then treat $v$ as constant while you differentiate $u$; and the whole differential coefficient $\\dfrac{dy}{dx}$ will be the sum of these two treatments.",
   "why": "Gives a memorable way to think about the product rule.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant",
    "concept/derivative",
    "theorem/product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-83e471f307",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "40",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "This gives us our instructions as to \\textit{how to differentiate a quotient \\emph{of two functions}. Multiply the divisor function by the differential coefficient of the dividend function; then multiply the dividend function by the differential coefficient of the divisor function; and subtract. Lastly divide by the square of the divisor function}.",
   "markdown": "This gives us our instructions as to *how to differentiate a quotient *of two functions*. Multiply the divisor function by the differential coefficient of the dividend function; then multiply the dividend function by the differential coefficient of the divisor function; and subtract. Lastly divide by the square of the divisor function*.",
   "why": "States the quotient rule as a step-by-step procedure in words.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/quotient",
    "theorem/quotient-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6feb96a5af",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "41",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "The working out of quotients is often tedious, but there is nothing difficult about it.",
   "markdown": "The working out of quotients is often tedious, but there is nothing difficult about it.",
   "why": "Reassures learners that the quotient rule is laborious but not hard.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/quotient",
    "theorem/quotient-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-bad55a502a",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "37",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "Now $du · dv$ is a small quantity of the second order of smallness, and therefore in the limit may be discarded, leaving",
   "markdown": "Now $du · dv$ is a small quantity of the second order of smallness, and therefore in the limit may be discarded, leaving",
   "why": "Shows why the du·dv term drops out of the product-rule derivation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/order-of-smallness",
    "theorem/product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-07d0fff3d7",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "39",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "In such a case it is no use to try to work out the division beforehand, because $x^2 + a$ will not divide into $bx^5 + c$, neither have they any common factor.",
   "markdown": "In such a case it is no use to try to work out the division beforehand, because $x^2 + a$ will not divide into $bx^5 + c$, neither have they any common factor.",
   "why": "Explains when the quotient rule is needed rather than simplifying first.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/quotient",
    "theorem/quotient-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-9296a72814",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "49",
   "location": "Successive Differentiation",
   "latex": "Begin with a concrete case.",
   "markdown": "Begin with a concrete case.",
   "why": "It tells the learner to start from a specific power before the general case, which is a sound teaching order.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-4aad470c43",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "49",
   "location": "Successive Differentiation",
   "latex": "This is called the ``derived function'' of~$x$.",
   "markdown": "This is called the “derived function” of $x$.",
   "why": "It names the derivative of a function, so the learner knows what the new function is called.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5684fabdfa",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "49",
   "location": "Successive Differentiation",
   "latex": "So the\nstatement $y=f(x)$ merely tells us that $y$ is a function\nof~$x$, it may be $x^2$ or $ax^n$, or $\\cos x$ or any other complicated\nfunction of~$x$.",
   "markdown": "So the statement $y=f(x)$ merely tells us that $y$ is a function of $x$, it may be $x^2$ or $ax^n$, or $\\cos x$ or any other complicated function of $x$.",
   "why": "It makes the general symbol f(x) concrete by listing several functions it could stand for.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-383421d8e3",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "49",
   "location": "Successive Differentiation",
   "latex": "This is to employ the general symbol~$f(x)$ for any function of~$x$. Here the symbol~$f(~)$ is read as ``function of,'' without saying what particular function is meant. So the statement $y=f(x)$ merely tells us that $y$ is a function of~$x$, it may be $x^2$ or $ax^n$, or $\\cos x$ or any other complicated function of~$x$.",
   "markdown": "This is to employ the general symbol $f(x)$ for any function of $x$. Here the symbol $f(~)$ is read as “function of,” without saying what particular function is meant. So the statement $y=f(x)$ merely tells us that $y$ is a function of $x$, it may be $x^2$ or $ax^n$, or $\\cos x$ or any other complicated function of $x$.",
   "why": "Explains plainly what f(x) means: a placeholder for any function, not one particular formula.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/function",
    "concept/function-notation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-83820ff740",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "49",
   "location": "Successive Differentiation",
   "latex": "The corresponding symbol for the differential coefficient is~$f'(x)$, which is simpler to write than $\\dfrac{dy}{dx}$. This is called the ``derived function'' of~$x$.",
   "markdown": "The corresponding symbol for the differential coefficient is $f'(x)$, which is simpler to write than $\\dfrac{dy}{dx}$. This is called the “derived function” of $x$.",
   "why": "Introduces the primed notation for the derivative and the older name 'derived function'.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/derivative",
    "concept/function-notation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-fd057f44dd",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "Suppose we differentiate over again, we shall get the ``second derived function'' or second differential coefficient, which is denoted by~$f''(x)$; and so on.",
   "markdown": "Suppose we differentiate over again, we shall get the “second derived function” or second differential coefficient, which is denoted by $f''(x)$; and so on.",
   "why": "Shows in one sentence that differentiating again gives the next derivative, with its notation.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-be021a4257",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "Similarly, we may write as the result of thrice differentiating, $\\dfrac{d^3y}{dx^3} = f'''(x)$.",
   "markdown": "Similarly, we may write as the result of thrice differentiating, $\\dfrac{d^3y}{dx^3} = f'''(x)$.",
   "why": "Links the d^3y/dx^3 notation to the primed notation so learners can read both.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f70c47ebb2",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "53",
   "location": "When Time Varies",
   "latex": "What do we mean by \\emph{rate}? In both these cases we are making a mental comparison of something that is happening, and the length of time that it takes to happen.",
   "markdown": "What do we mean by *rate*? In both these cases we are making a mental comparison of something that is happening, and the length of time that it takes to happen.",
   "why": "It gives learners a plain everyday meaning of rate as a comparison between what happens and the time it takes, before any calculus.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/rate-of-change"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5da336fc27",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "53",
   "location": "When Time Varies",
   "latex": "Ten yards is not the same as $600$~yards, nor is one second the same thing as one minute. What we mean by saying that the \\emph{rate} is the same, is this: that the proportion borne between distance passed over and time taken to pass over it, is the same in both cases.",
   "markdown": "Ten yards is not the same as $600$ yards, nor is one second the same thing as one minute. What we mean by saying that the *rate* is the same, is this: that the proportion borne between distance passed over and time taken to pass over it, is the same in both cases.",
   "why": "It shows that two different speeds can share one rate because the ratio of distance to time is what matters.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/rate-of-change",
    "quantity/distance",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-cde1e65486",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "58",
   "location": "When Time Varies",
   "latex": "He did not use the notation of the $dy$ and~$dx$, and~$dt$ (this was due to Leibnitz), but had instead a notation of his own.",
   "markdown": "He did not use the notation of the $dy$ and $dx$, and $dt$ (this was due to Leibnitz), but had instead a notation of his own.",
   "why": "It places the dot notation historically, so learners understand why two notations for the same rate exist.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/derivative",
    "concept/fluxional-notation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-3b1468536a",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "56",
   "location": "When Time Varies",
   "latex": "The force necessary to accelerate a mass is proportional to the mass, and it is also proportional to the acceleration which is being imparted.",
   "markdown": "The force necessary to accelerate a mass is proportional to the mass, and it is also proportional to the acceleration which is being imparted.",
   "why": "It gives the reasoning behind f = ma in plain words before the symbols are written.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "quantity/acceleration",
    "quantity/force",
    "quantity/mass"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-c21583f667",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "57",
   "location": "When Time Varies",
   "latex": "Again, if a force is employed to move something (against an equal and opposite counter-force), it does \\emph{work}; and the amount of work done is measured by the product of the force into the distance (in its own direction) through which its point of application moves forward.",
   "markdown": "Again, if a force is employed to move something (against an equal and opposite counter-force), it does *work*; and the amount of work done is measured by the product of the force into the distance (in its own direction) through which its point of application moves forward.",
   "why": "It defines work as force times displacement in the force's own direction, a definition learners can check by example.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/work",
    "quantity/distance",
    "quantity/force"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ea2803d89b",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "56",
   "location": "When Time Varies",
   "latex": "When a railway train has just begun to move, its velocity~$v$ is small; but it is rapidly gaining speed---it is being hurried up, or accelerated, by the effort of the engine. So its $\\dfrac{d^2y}{dt^2}$ is large. When it has got up its top speed it is no longer being accelerated, so that then $\\dfrac{d^2y}{dt^2}$ has fallen to zero.",
   "markdown": "When a railway train has just begun to move, its velocity $v$ is small; but it is rapidly gaining speed---it is being hurried up, or accelerated, by the effort of the engine. So its $\\dfrac{d^2y}{dt^2}$ is large. When it has got up its top speed it is no longer being accelerated, so that then $\\dfrac{d^2y}{dt^2}$ has fallen to zero.",
   "why": "Gives a physical picture of what the second derivative is doing as a train starts and then cruises.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/higher-order-derivative",
    "quantity/acceleration",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f147508200",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "59",
   "location": "When Time Varies",
   "latex": "But this notation does not tell us what is the independent variable with respect to which the differentiation has been effected. When we see $\\dfrac{dy}{dt}$ we know that $y$ is to be differentiated with respect to~$t$.",
   "markdown": "But this notation does not tell us what is the independent variable with respect to which the differentiation has been effected. When we see $\\dfrac{dy}{dt}$ we know that $y$ is to be differentiated with respect to $t$.",
   "why": "Shows the strength of Leibniz's notation over Newton's dots, which hide the variable of differentiation.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/derivative",
    "concept/fluxional-notation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-c95c798100",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "53",
   "location": "When Time Varies",
   "latex": "What we mean by saying that the \\emph{rate} is the same, is this: that the proportion borne between distance passed over and time taken to pass over it, is the same in both cases.",
   "markdown": "What we mean by saying that the *rate* is the same, is this: that the proportion borne between distance passed over and time taken to pass over it, is the same in both cases.",
   "why": "Explains why 10 yards per second and 600 yards per minute count as the same rate, even though neither the distances nor the times are equal.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/rate-of-change"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ba1b4087ab",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "53",
   "location": "When Time Varies",
   "latex": "It is said that Sandy had not been in London above five minutes when ``bang went saxpence.''",
   "markdown": "It is said that Sandy had not been in London above five minutes when “bang went saxpence.”",
   "why": "A humorous anecdote that makes the idea of spending money at a rate memorable.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/rate-of-change"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-303d843bf5",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "55",
   "location": "When Time Varies",
   "latex": "Now the speed was not actually constant all the way: at starting, and during the slowing up at the end of the journey, the speed was less. Probably at some part, when running downhill, the speed was over $60$~miles an hour. If, during any particular element of time~$dt$, the corresponding element of distance passed over was~$dy$, then at that part of the journey the speed was~$\\dfrac{dy}{dt}$.",
   "markdown": "Now the speed was not actually constant all the way: at starting, and during the slowing up at the end of the journey, the speed was less. Probably at some part, when running downhill, the speed was over $60$ miles an hour. If, during any particular element of time $dt$, the corresponding element of distance passed over was $dy$, then at that part of the journey the speed was $\\dfrac{dy}{dt}$.",
   "why": "Contrasts a journey's average speed with its speed at an instant, which leads into dy/dt.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/rate-of-change",
    "quantity/average-velocity",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-3cafa18767",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "57",
   "location": "When Time Varies",
   "latex": "That is to say, force may be expressed either as mass times acceleration, or as rate of change of momentum.",
   "markdown": "That is to say, force may be expressed either as mass times acceleration, or as rate of change of momentum.",
   "why": "States the two equivalent forms of force that the preceding derivation arrives at.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "quantity/acceleration",
    "quantity/force",
    "quantity/momentum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-57c709427f",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "58",
   "location": "When Time Varies",
   "latex": "In this last sentence the word \\emph{rate} is clearly not used in its time-sense, but in its meaning as ratio or proportion.",
   "markdown": "In this last sentence the word *rate* is clearly not used in its time-sense, but in its meaning as ratio or proportion.",
   "why": "Warns that rate does not always mean change per unit time, as in work done per unit of length.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/rate-of-change",
    "concept/work",
    "quantity/force"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6677eb02bb",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "60",
   "location": "When Time Varies",
   "latex": "(It is the same velocity as the velocity at the middle of the interval, $t = 5$; for, the acceleration being constant, the velocity has varied uniformly from zero when $t = 0$ to $4~\\text{ft./sec.}$ when $t = 10$.)",
   "markdown": "(It is the same velocity as the velocity at the middle of the interval, $t = 5$; for, the acceleration being constant, the velocity has varied uniformly from zero when $t = 0$ to $4~\\text{ft./sec.}$ when $t = 10$.)",
   "why": "Explains why average velocity equals the midpoint velocity when acceleration is constant, a point the later examples show failing when it is not.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "quantity/acceleration",
    "quantity/average-velocity",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-0da2a79318",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "67",
   "location": "Introducing a Useful Dodge",
   "latex": "Thus, the equation\\Pagelabel{dodge} \\[ y = (x^2+a^2)^{\\efrac{3}{2}} \\] is awkward to a beginner.",
   "markdown": "Thus, the equationdodge y = (x^2+a^2)^32 is awkward to a beginner.",
   "why": "It gives the concrete example that the rest of the chapter works through.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ec720cd927",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "67",
   "location": "Introducing a Useful Dodge",
   "latex": "\\First{Sometimes} one is stumped by finding that the expression\nto be differentiated is too complicated to\ntackle directly.",
   "markdown": "Sometimes one is stumped by finding that the expression to be differentiated is too complicated to tackle directly.",
   "why": "It names the problem the chapter solves: an expression that looks too tangled to differentiate in one step.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/differentiation",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-0fcb0bc29a",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "68",
   "location": "Introducing a Useful Dodge",
   "latex": "By and bye,\\DPnote{** TN: [sic], archaic spelling} when you have learned how to deal\nwith sines, and cosines, and exponentials, you will\nfind this dodge of increasing usefulness.",
   "markdown": "By and bye, when you have learned how to deal with sines, and cosines, and exponentials, you will find this dodge of increasing usefulness.",
   "why": "It tells the learner the dodge is a tool they will keep using later, in a charming old voice.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/cosine",
    "concept/sine",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-60240dfa8b",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "74",
   "location": "Introducing a Useful Dodge",
   "latex": "The process can be extended to three or more\ndifferential coefficients, so that $\\dfrac{dy}{dx} = \\dfrac{dy}{dz} × \\dfrac{dz}{dv} × \\dfrac{dv}{dx}$.",
   "markdown": "The process can be extended to three or more differential coefficients, so that $\\dfrac{dy}{dx} = \\dfrac{dy}{dz} × \\dfrac{dz}{dv} × \\dfrac{dv}{dx}$.",
   "why": "It shows the dodge can be chained through several intermediate variables.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f843e50b12",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "69",
   "location": "Introducing a Useful Dodge",
   "latex": "(We may also write $y = (1-x)^{\\efrac{1}{2}} (1+x)^{-\\efrac{1}{2}}$ and differentiate\nas a product.)",
   "markdown": "(We may also write $y = (1-x)^{\\efrac{1}{2}} (1+x)^{-\\efrac{1}{2}}$ and differentiate as a product.)",
   "why": "It reminds the learner that one function can be differentiated by more than one route.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/exponent",
    "method/differentiation",
    "theorem/product-rule",
    "theorem/quotient-rule"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-30ab00664e",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "68",
   "location": "Introducing a Useful Dodge",
   "latex": "(1) Differentiate $y = \\sqrt{a+x}$.\n\nLet $a+x = u$.",
   "markdown": "(1) Differentiate $y = \\sqrt{a+x}$. Let $a+x = u$.",
   "why": "It is the simplest worked case of the dodge: name the inner expression u, then proceed.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "theorem/chain-rule-for-differentiation",
    "theorem/power-rule"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-3c1a98925f",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "77",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "This tangent to the curve has evidently the same slope as~$QT$, so that $\\dfrac{dy}{dx}$ is the slope of the tangent to the curve at the point~$Q$ for which the value of~$\\dfrac{dy}{dx}$ is found.",
   "markdown": "This tangent to the curve has evidently the same slope as $QT$, so that $\\dfrac{dy}{dx}$ is the slope of the tangent to the curve at the point $Q$ for which the value of $\\dfrac{dy}{dx}$ is found.",
   "why": "It links the derivative directly to the tangent's slope, the central geometric idea of the chapter.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/slope-of-a-curve",
    "concept/tangent"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-fb108dc83a",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "79",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "For a horizontal line, or a horizontal place in a curve, $dy=0$, and therefore~$\\dfrac{dy}{dx}=0$.",
   "markdown": "For a horizontal line, or a horizontal place in a curve, $dy=0$, and therefore $\\dfrac{dy}{dx}=0$.",
   "why": "It tells a learner that a zero derivative means a horizontal tangent.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-3c9a0468ca",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "77",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "We have seen that the short expression ``the slope of a curve'' has no precise meaning, because a curve has so many slopes---in fact, every small portion of a curve has a different slope.",
   "markdown": "We have seen that the short expression “the slope of a curve” has no precise meaning, because a curve has so many slopes---in fact, every small portion of a curve has a different slope.",
   "why": "Warns learners that 'the slope of a curve' is ill-defined until a point is named.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-033f323dbd",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "77",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "``The slope of a curve \\emph{at a point}'' is, however, a perfectly defined thing; it is the slope of a very small portion of the curve situated just at that point; and we have seen that this is the same as ``the slope of the tangent to the curve at that point.''",
   "markdown": "“The slope of a curve *at a point*” is, however, a perfectly defined thing; it is the slope of a very small portion of the curve situated just at that point; and we have seen that this is the same as “the slope of the tangent to the curve at that point.”",
   "why": "Gives the precise meaning of slope at a point and ties it to the tangent.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/slope-of-a-curve",
    "concept/tangent"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5e63489376",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "78",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "Observe that $dx$ is a short step to the right, and $dy$ the corresponding short step upwards. These steps must be considered as short as possible---in fact indefinitely short,---though in diagrams we have to represent them by bits that are not infinitesimally small, otherwise they could not be seen.",
   "markdown": "Observe that $dx$ is a short step to the right, and $dy$ the corresponding short step upwards. These steps must be considered as short as possible---in fact indefinitely short,---though in diagrams we have to represent them by bits that are not infinitesimally small, otherwise they could not be seen.",
   "why": "Explains honestly why diagrams exaggerate the tiny steps dx and dy.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/increment"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ba68deb04f",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "78",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "If a curve is sloping up at~$45°$ at a particular point, as in \\Fig{8}, $dy$ and~$dx$ will be equal, and the value of $\\dfrac{dy}{dx} = 1$.",
   "markdown": "If a curve is sloping up at $45°$ at a particular point, as in [fig:8]Fig. 8, $dy$ and $dx$ will be equal, and the value of $\\dfrac{dy}{dx} = 1$.",
   "why": "Gives a concrete benchmark for reading slope values from a picture.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-701bb55c55",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "79",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "If a curve slopes \\emph{downward}, as in \\Fig{11}, $dy$~will be a step down, and must therefore be reckoned of negative value; hence $\\dfrac{dy}{dx}$~will have negative sign also.",
   "markdown": "If a curve slopes *downward*, as in [fig:11]Fig. 11, $dy$ will be a step down, and must therefore be reckoned of negative value; hence $\\dfrac{dy}{dx}$ will have negative sign also.",
   "why": "Shows why a descending curve has a negative slope.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-375950e9b3",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "81",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "The characteristic of a minimum is that $y$~must increase \\emph{on either side} of it.",
   "markdown": "The characteristic of a minimum is that $y$ must increase *on either side* of it.",
   "why": "Corrects the common idea that a minimum is simply the smallest value.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/minimum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-955ae548cb",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "88",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "The slope of the tangent is the slope of the curve at the point where they touch one another (see \\Pageref{slope}); that is, it is the $\\dfrac{dy}{dx}$ of the curve for that point.",
   "markdown": "The slope of the tangent is the slope of the curve at the point where they touch one another (see slope); that is, it is the $\\dfrac{dy}{dx}$ of the curve for that point.",
   "why": "Starts the worked method for finding a tangent: use dy/dx at the touching point.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/tangent",
    "theorem/equation-of-the-tangent"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-463b0b6006",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "90",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "In all exercises dealing with curves, students will find it extremely instructive to verify the deductions obtained by actually plotting the curves.",
   "markdown": "In all exercises dealing with curves, students will find it extremely instructive to verify the deductions obtained by actually plotting the curves.",
   "why": "Encourages checking calculations against a drawn curve.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "method/plotting-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-a0a4648296",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "97",
   "location": "Maxima and Minima",
   "latex": "So, writing $\\dfrac{dy}{dx} = 0$ does \\emph{not} mean that it always is $=0$; but you write it down \\emph{as a condition} in order to see how much $x$ will come out if $\\dfrac{dy}{dx}$ is to be zero.",
   "markdown": "So, writing $\\dfrac{dy}{dx} = 0$ does *not* mean that it always is $=0$; but you write it down *as a condition* in order to see how much $x$ will come out if $\\dfrac{dy}{dx}$ is to be zero.",
   "why": "It explains that the equation dy/dx = 0 is an equation of condition, true only at the particular point sought.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/equation",
    "concept/stationary-value",
    "method/equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f639b4580b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "98",
   "location": "Maxima and Minima",
   "latex": "It does not of itself discriminate; it finds for you the right value of~$x$ but leaves you to find out for yourselves whether the corresponding~$y$ is a maximum or a minimum.",
   "markdown": "It does not of itself discriminate; it finds for you the right value of $x$ but leaves you to find out for yourselves whether the corresponding $y$ is a maximum or a minimum.",
   "why": "It warns the learner that setting the derivative to zero finds the candidate point but does not say whether it is a maximum or a minimum.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/maximum",
    "concept/minimum",
    "method/equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6a2d721ab9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "108",
   "location": "Maxima and Minima",
   "latex": "It is necessary therefore always to check by taking one value on either side.",
   "markdown": "It is necessary therefore always to check by taking one value on either side.",
   "why": "It gives the learner the habit of checking a stationary point on both sides before calling it a maximum or minimum.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cusp",
    "method/testing-a-stationary-value-by-neighbouring-values"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-172de47127",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "99",
   "location": "Maxima and Minima",
   "latex": "Let the number to be cut into two parts be called~$n$. Then if $x$ is one part, the other will be~$n-x$, and the product will be $x(n-x)$ or~$nx-x^2$. So we write $y=nx-x^2$.",
   "markdown": "Let the number to be cut into two parts be called $n$. Then if $x$ is one part, the other will be $n-x$, and the product will be $x(n-x)$ or $nx-x^2$. So we write $y=nx-x^2$.",
   "why": "It shows how a word problem is turned into a function of one variable before differentiating.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/maximum",
    "method/stating-a-problem-as-an-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-e0d3323226",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "99",
   "location": "Maxima and Minima",
   "latex": "This is a very useful rule, and applies to any number of factors, so that if $m+n+p=$ a constant number, $m×n×p$ is a maximum when $m=n=p$.",
   "markdown": "This is a very useful rule, and applies to any number of factors, so that if $m+n+p=$ a constant number, $m×n×p$ is a maximum when $m=n=p$.",
   "why": "It states a general result the learner can apply: a fixed sum of factors has its largest product when the factors are equal.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/maximum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-b7e98d0a47",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "93",
   "location": "Maxima and Minima",
   "latex": "\\First{One} of the principal uses of the process of differentiating is to find out under what conditions the value of the thing differentiated becomes a maximum, or a minimum. This is often exceedingly important in engineering questions, where it is most desirable to know what conditions will make the cost of working a minimum, or will make the efficiency a maximum.",
   "markdown": "One of the principal uses of the process of differentiating is to find out under what conditions the value of the thing differentiated becomes a maximum, or a minimum. This is often exceedingly important in engineering questions, where it is most desirable to know what conditions will make the cost of working a minimum, or will make the efficiency a maximum.",
   "why": "States why maxima and minima matter in practice, giving learners a reason to care.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/maximum",
    "concept/minimum",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-8a83287832",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "95",
   "location": "Maxima and Minima",
   "latex": "Now it may sound like juggling to be assured that there is a way by which one can arrive straight at a maximum (or minimum) value without making a lot of preliminary trials or guesses.",
   "markdown": "Now it may sound like juggling to be assured that there is a way by which one can arrive straight at a maximum (or minimum) value without making a lot of preliminary trials or guesses.",
   "why": "Voices the learner's natural suspicion before the method is introduced.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/maximum",
    "concept/minimum",
    "method/differentiation",
    "method/equating-the-derivative-to-zero",
    "method/trial-and-error"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-d1dd9cce98",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "95",
   "location": "Maxima and Minima",
   "latex": "When there is put before you an equation, and you want to find that value of~$x$ that will make its~$y$ a minimum (or a maximum), \\emph{first differentiate it}, and having done so, write its $\\dfrac{dy}{dx}$ as \\emph{equal to zero}, and then solve for~$x$. Put this particular value of~$x$ into the original equation, and you will then get the required value of~$y$. This process is commonly called ``equating to zero.''",
   "markdown": "When there is put before you an equation, and you want to find that value of $x$ that will make its $y$ a minimum (or a maximum), *first differentiate it*, and having done so, write its $\\dfrac{dy}{dx}$ as *equal to zero*, and then solve for $x$. Put this particular value of $x$ into the original equation, and you will then get the required value of $y$. This process is commonly called “equating to zero.”",
   "why": "Gives the whole method in one passage, in order.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/maximum",
    "concept/minimum",
    "method/differentiation",
    "method/equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-73230e63b4",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "97",
   "location": "Maxima and Minima",
   "latex": "Ordinarily you are dealing with equations that are true in themselves, but, on occasions, of which the present are examples, you have to write down equations that are not necessarily true, but are only true if certain conditions are to be fulfilled; and you write them down in order, by solving them, to find the conditions which make them true.",
   "markdown": "Ordinarily you are dealing with equations that are true in themselves, but, on occasions, of which the present are examples, you have to write down equations that are not necessarily true, but are only true if certain conditions are to be fulfilled; and you write them down in order, by solving them, to find the conditions which make them true.",
   "why": "Explains why writing dy/dx = 0 is a condition we impose and not a claim that the slope is always zero.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/equation-of-condition",
    "method/equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-fd573b035c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "98",
   "location": "Maxima and Minima",
   "latex": "Quite so. It does not of itself discriminate; it finds for you the right value of~$x$ but leaves you to find out for yourselves whether the corresponding~$y$ is a maximum or a minimum. Of course, if you have plotted the curve, you know already which it will be.",
   "markdown": "Quite so. It does not of itself discriminate; it finds for you the right value of $x$ but leaves you to find out for yourselves whether the corresponding $y$ is a maximum or a minimum. Of course, if you have plotted the curve, you know already which it will be.",
   "why": "Warns that setting the derivative to zero finds the turning point but does not say which kind it is.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/maximum",
    "concept/minimum",
    "method/equating-the-derivative-to-zero",
    "method/second-derivative-test"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-90c6627e47",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "99",
   "location": "Maxima and Minima",
   "latex": "So now we \\emph{know} that whatever number $n$ may be, we must divide it into two equal parts if the product of the parts is to be a maximum; and the value of that maximum product will always be $ = \\tfrac{1}{4} n^2$.",
   "markdown": "So now we *know* that whatever number $n$ may be, we must divide it into two equal parts if the product of the parts is to be a maximum; and the value of that maximum product will always be $ = \\tfrac{1}{4} n^2$.",
   "why": "Shows the calculus giving a general result that the trial-and-error table only hinted at for 60.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/maximum",
    "method/equating-the-derivative-to-zero",
    "method/trial-and-error"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-7367fca914",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "107",
   "location": "Maxima and Minima",
   "latex": "On plotting the graph it will be found that the curve goes to the origin, as if there were a minimum there; but instead of continuing beyond, as it should do for a minimum, it retraces its steps (forming what is called a ``cusp''). There is no minimum, therefore, although the condition for a minimum is satisfied, namely $\\dfrac{dy}{dx} = 0$. It is necessary therefore always to check by taking one value on either side.",
   "markdown": "On plotting the graph it will be found that the curve goes to the origin, as if there were a minimum there; but instead of continuing beyond, as it should do for a minimum, it retraces its steps (forming what is called a “cusp”). There is no minimum, therefore, although the condition for a minimum is satisfied, namely $\\dfrac{dy}{dx} = 0$. It is necessary therefore always to check by taking one value on either side.",
   "why": "Warns that dy/dx = 0 can hold where there is no minimum, so the answer must be checked.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/cusp",
    "concept/minimum",
    "method/equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-1e595e4f58",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "112",
   "location": "Curvature of Curves",
   "latex": "Clearly it means the rate (per unit of length~$x$) at which the slope is changing---in brief, it is \\emph{a measure of the curvature of the slope}.",
   "markdown": "Clearly it means the rate (per unit of length $x$) at which the slope is changing---in brief, it is *a measure of the curvature of the slope*.",
   "why": "It gives the learner a physical picture of the second derivative as the rate at which the slope itself changes.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/curvature",
    "concept/higher-order-derivative",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-13c5ffbb80",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "113",
   "location": "Curvature of Curves",
   "latex": "If $\\dfrac{d^2y}{dx^2}$ comes out \\emph{positive}, then you know that the value of~$y$ which you got was a \\emph{minimum}; but if $\\dfrac{d^2y}{dx^2}$ comes out \\emph{negative}, then the value of~$y$ which you got must be a \\emph{maximum}. That's the rule.",
   "markdown": "If $\\dfrac{d^2y}{dx^2}$ comes out *positive*, then you know that the value of $y$ which you got was a *minimum*; but if $\\dfrac{d^2y}{dx^2}$ comes out *negative*, then the value of $y$ which you got must be a *maximum*. That’s the rule.",
   "why": "It states the second derivative test compactly so the learner can apply it to any stationary value.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/maximum",
    "concept/minimum",
    "method/second-derivative-test"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6c5cfce66a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "114",
   "location": "Curvature of Curves",
   "latex": "To the left of~$M$ the slope is downward, that is, negative, and is getting less negative.",
   "markdown": "To the left of $M$ the slope is downward, that is, negative, and is getting less negative.",
   "why": "It explains in plain terms why a positive second derivative goes with a minimum, by tracking the sign of the slope.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/minimum",
    "concept/slope-of-a-curve",
    "method/second-derivative-test"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-46e9e04aea",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "112",
   "location": "Curvature of Curves",
   "latex": "But what can $\\dfrac{d^2 y}{dx^2}$~mean in this case? Clearly it means the rate (per unit of length~$x$) at which the slope is changing---in brief, it is \\emph{a measure of the curvature of the slope}.",
   "markdown": "But what can $\\dfrac{d^2 y}{dx^2}$ mean in this case? Clearly it means the rate (per unit of length $x$) at which the slope is changing---in brief, it is *a measure of the curvature of the slope*.",
   "why": "Answers the learner's question of why anyone differentiates twice, by giving the second derivative a plain geometric meaning.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/curvature",
    "concept/higher-order-derivative",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-e9ca152537",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "113",
   "location": "Curvature of Curves",
   "latex": "It is now time to initiate you into another secret---how to tell whether the result that you get by ``equating to zero'' is a maximum or a minimum. The trick is this: After you have differentiated (so as to get the expression which you equate to zero), you then differentiate a second time, and look whether the result of the second differentiation is \\emph{positive} or \\emph{negative}. If $\\dfrac{d^2y}{dx^2}$ comes out \\emph{positive}, then you know that the value of~$y$ which you got was a \\emph{minimum}; but if $\\dfrac{d^2y}{dx^2}$ comes out \\emph{negative}, then the value of~$y$ which you got must be a \\emph{maximum}. That's the rule.",
   "markdown": "It is now time to initiate you into another secret---how to tell whether the result that you get by “equating to zero” is a maximum or a minimum. The trick is this: After you have differentiated (so as to get the expression which you equate to zero), you then differentiate a second time, and look whether the result of the second differentiation is *positive* or *negative*. If $\\dfrac{d^2y}{dx^2}$ comes out *positive*, then you know that the value of $y$ which you got was a *minimum*; but if $\\dfrac{d^2y}{dx^2}$ comes out *negative*, then the value of $y$ which you got must be a *maximum*. That’s the rule.",
   "why": "States the second-derivative test as a short, memorable procedure.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/maximum",
    "concept/minimum",
    "method/equating-the-derivative-to-zero",
    "method/second-derivative-test"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-b24488d912",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "114",
   "location": "Curvature of Curves",
   "latex": "Clearly the change of slope as the curve passes through~$M$ is such that $\\dfrac{d^2y}{dx^2}$~is \\emph{positive}, for its operation, as $x$~increases toward the right, is to convert a downward slope into an upward one.",
   "markdown": "Clearly the change of slope as the curve passes through $M$ is such that $\\dfrac{d^2y}{dx^2}$ is *positive*, for its operation, as $x$ increases toward the right, is to convert a downward slope into an upward one.",
   "why": "Explains why a minimum goes with a positive second derivative, using the picture of the slope turning from downward to upward.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/concave-curve",
    "concept/higher-order-derivative",
    "concept/minimum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-174085d72b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "114",
   "location": "Curvature of Curves",
   "latex": "In this case, as the curve passes through~$M$ from left to right, its upward slope is converted into a downward or negative slope, so that in this case the ``slope of the slope'' $\\dfrac{d^2y}{dx^2}$ is \\emph{negative}.",
   "markdown": "In this case, as the curve passes through $M$ from left to right, its upward slope is converted into a downward or negative slope, so that in this case the “slope of the slope” $\\dfrac{d^2y}{dx^2}$ is *negative*.",
   "why": "Gives the matching picture for a maximum, with the slope turning from upward to downward.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/convex-curve",
    "concept/higher-order-derivative",
    "concept/maximum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-e441f89ba6",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "116",
   "location": "Curvature of Curves",
   "latex": "The denominator is always positive, so it is sufficient to ascertain the sign of the numerator.",
   "markdown": "The denominator is always positive, so it is sufficient to ascertain the sign of the numerator.",
   "why": "Shows a practical shortcut: when testing the sign of a messy second derivative, only the sign of the numerator matters.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/second-derivative-test",
    "theorem/quotient-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-940b06766f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "116",
   "location": "Curvature of Curves",
   "latex": "The expense~$C$ of handling the products of a certain factory varies with the weekly output~$P$ according to the relation $C = aP + \\dfrac{b}{c+P} + d$, where $a$,~$b$, $c$,~$d$ are positive constants. For what output will the expense be least?",
   "markdown": "The expense $C$ of handling the products of a certain factory varies with the weekly output $P$ according to the relation $C = aP + \\dfrac{b}{c+P} + d$, where $a$, $b$, $c$, $d$ are positive constants. For what output will the expense be least?",
   "why": "Poses a real-world minimising problem that the chapter's method then solves.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/minimum",
    "method/equating-the-derivative-to-zero",
    "method/second-derivative-test"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-db0f6ece47",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "122",
   "location": "Other Useful Dodges",
   "latex": "If we perform many additions of two or more fractions the denominators of which contain only terms in~$x$, and no terms in $x^2$,~$x^3$, or any other powers of~$x$, we \\emph{always} find that \\emph{the denominator of the final resulting fraction is the product of the denominators} of the fractions which were added to form the result.",
   "markdown": "If we perform many additions of two or more fractions the denominators of which contain only terms in $x$, and no terms in $x^2$, $x^3$, or any other powers of $x$, we *always* find that *the denominator of the final resulting fraction is the product of the denominators* of the fractions which were added to form the result.",
   "why": "It states the rule that tells a learner how to recover the denominators of the partial fractions by factoring the final denominator.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/denominator",
    "method/factoring",
    "method/partial-fractions"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-da3f3e33af",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "124",
   "location": "Other Useful Dodges",
   "latex": "If we make $x=1$, we get $4 = (A × 0)+(B × 2)$, so that $B=2$; and if we make $x=-1$, we get $-2 = (A × -2) + (B × 0)$, so that $A=1$.",
   "markdown": "If we make $x=1$, we get $4 = (A × 0)+(B × 2)$, so that $B=2$; and if we make $x=-1$, we get $-2 = (A × -2) + (B × 0)$, so that $A=1$.",
   "why": "It shows a worked step in which substituting convenient values of x isolates each unknown numerator.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/partial-fractions",
    "method/substitution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-1f18d6359c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "126",
   "location": "Other Useful Dodges",
   "latex": "Since the given fraction and the fraction found by adding the partial fractions are equal, and have \\emph{identical} denominators, the numerators must also be identically the same. In such a case, and for such algebraical expressions as those with which we are dealing here, \\emph{the coefficients of the same powers of~$x$ are equal and of same sign}.",
   "markdown": "Since the given fraction and the fraction found by adding the partial fractions are equal, and have *identical* denominators, the numerators must also be identically the same. In such a case, and for such algebraical expressions as those with which we are dealing here, *the coefficients of the same powers of $x$ are equal and of same sign*.",
   "why": "It explains why equal polynomials give equations between their coefficients, which a learner needs before solving for the unknown numerators.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/equating-coefficients",
    "method/partial-fractions"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ecc010264a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "121",
   "location": "Other Useful Dodges",
   "latex": "If we could split the fraction into two or more simpler fractions such that their sum is equivalent to the original fraction, we could then proceed by differentiating each of these simpler expressions.",
   "markdown": "If we could split the fraction into two or more simpler fractions such that their sum is equivalent to the original fraction, we could then proceed by differentiating each of these simpler expressions.",
   "why": "States the motive for partial fractions: replace one hard differentiation by several easy ones.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/differentiation",
    "method/partial-fractions"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-a42394c528",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "122",
   "location": "Other Useful Dodges",
   "latex": "But it is important to bear in mind that all which follows applies only to what are called ``proper'' algebraic fractions, meaning fractions like the above, which have the numerator of \\emph{a lesser degree} than the denominator; that is, those in which the highest index of~$x$ is less in the numerator than in the denominator.",
   "markdown": "But it is important to bear in mind that all which follows applies only to what are called “proper” algebraic fractions, meaning fractions like the above, which have the numerator of *a lesser degree* than the denominator; that is, those in which the highest index of $x$ is less in the numerator than in the denominator.",
   "why": "Warns learners when the method applies and when to divide first.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/proper-algebraic-fraction",
    "method/partial-fractions"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-450325b719",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "124",
   "location": "Other Useful Dodges",
   "latex": "The equation must be true for all values of~$x$; therefore it must be true for such values of~$x$ as will cause $x-1$ and~$x+1$ to become zero, that is for $x=1$ and for $x=-1$ respectively.",
   "markdown": "The equation must be true for all values of $x$; therefore it must be true for such values of $x$ as will cause $x-1$ and $x+1$ to become zero, that is for $x=1$ and for $x=-1$ respectively.",
   "why": "Explains the key trick that gets round having two unknowns and only one equation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/unknown",
    "method/substituting-convenient-values-of-x"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-7fde6cf143",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "128",
   "location": "Other Useful Dodges",
   "latex": "We see that it is sufficient to allow for one numerical term in each numerator, and that we always get the ultimate partial fractions.",
   "markdown": "We see that it is sufficient to allow for one numerical term in each numerator, and that we always get the ultimate partial fractions.",
   "why": "Settles a common doubt about repeated factors after the book's own Ax+B attempt failed.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/numerator",
    "method/partial-fractions"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-33a89eb995",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "129",
   "location": "Other Useful Dodges",
   "latex": "It is useful to check the results obtained. The simplest way is to replace $x$ by a single value, say~$+1$, both in the given expression and in the partial fractions obtained.",
   "markdown": "It is useful to check the results obtained. The simplest way is to replace $x$ by a single value, say $+1$, both in the given expression and in the partial fractions obtained.",
   "why": "Gives learners a quick habit for catching algebra slips.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/checking-a-result-by-substitution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5312d8d97b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "132",
   "location": "Other Useful Dodges",
   "latex": "It follows that, being given a function, if it be easier to differentiate the inverse function, this may be done, and the reciprocal of the differential coefficient of the inverse function gives the differential coefficient of the given function itself.",
   "markdown": "It follows that, being given a function, if it be easier to differentiate the inverse function, this may be done, and the reciprocal of the differential coefficient of the inverse function gives the differential coefficient of the given function itself.",
   "why": "States the inverse-function dodge as a practical rule.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/inverse-function",
    "concept/reciprocal",
    "method/differentiating-an-inverse-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-9d39ce2ad1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "133",
   "location": "Other Useful Dodges",
   "latex": "You will surely realize from this chapter and the preceding, that in many respects the calculus is an \\emph{art} rather than a \\emph{science}: an art only to be acquired, as all other arts are, by practice.",
   "markdown": "You will surely realize from this chapter and the preceding, that in many respects the calculus is an *art* rather than a *science*: an art only to be acquired, as all other arts are, by practice.",
   "why": "Encourages learners to see skill in calculus as built by practice.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f56b9be03a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "135",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "It is easy to see that if the value of the yearly interest is $\\dfrac{1}{n}$~of the capital, he must go on hoarding for $n$~years in order to double his property.",
   "markdown": "It is easy to see that if the value of the yearly interest is $\\dfrac{1}{n}$ of the capital, he must go on hoarding for $n$ years in order to double his property.",
   "why": "It shows a learner, in the plainest case, how a fixed yearly increment builds up linearly, which sets up the contrast with compound growth.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/simple-interest"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-aa0c2a83de",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "145",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "Another reason why $\\epsilon$ is important is because it was made by Napier, the inventor of logarithms, the basis of his system.",
   "markdown": "Another reason why $\\epsilon$ is important is because it was made by Napier, the inventor of logarithms, the basis of his system.",
   "why": "It places Euler's number historically with John Napier and explains why it became the base of natural logarithms.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/euler-s-number",
    "concept/natural-logarithm",
    "person/john-napier"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-82aaddcf54",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "139",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "To this mysterious number $2.7182818$ etc., the mathematicians have assigned as a symbol the Greek letter~$\\epsilon$ (pronounced \\emph{epsilon}). All schoolboys know that the Greek letter~$\\pi$ (called \\emph{pi}) stands for $3.141592$ etc.; but how many of them know that \\emph{epsilon} means $2.71828$? Yet it is an even more important number than~$\\pi$!",
   "markdown": "To this mysterious number $2.7182818$ etc., the mathematicians have assigned as a symbol the Greek letter $\\epsilon$ (pronounced *epsilon*). All schoolboys know that the Greek letter $\\pi$ (called *pi*) stands for $3.141592$ etc.; but how many of them know that *epsilon* means $2.71828$? Yet it is an even more important number than $\\pi$!",
   "why": "Introduces epsilon with a playful comparison to pi, so learners feel its importance.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/euler-s-number",
    "concept/pi"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-47d869dc69",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "139",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "Suppose we were to let $1$ grow at simple interest till it became~$2$; then, if at the same nominal rate of interest, and for the same time, we were to let $1$ grow at true compound interest, instead of simple, it would grow to the value \\emph{epsilon}.",
   "markdown": "Suppose we were to let $1$ grow at simple interest till it became $2$; then, if at the same nominal rate of interest, and for the same time, we were to let $1$ grow at true compound interest, instead of simple, it would grow to the value *epsilon*.",
   "why": "Gives a concrete way to understand epsilon by comparing simple and compound growth.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/compound-interest",
    "concept/euler-s-number",
    "concept/simple-interest"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-cc3aa38f2e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "136",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "But this mode of reckoning compound interest once a year, is really not quite fair; for even during the first year the~£$100$ ought to have been growing. At the end of half a year it ought to have been at least~£$105$, and it certainly would have been fairer had the interest for the second half of the year been calculated on~£$105$.",
   "markdown": "But this mode of reckoning compound interest once a year, is really not quite fair; for even during the first year the £$100$ ought to have been growing. At the end of half a year it ought to have been at least £$105$, and it certainly would have been fairer had the interest for the second half of the year been calculated on £$105$.",
   "why": "Motivates compounding more and more often by an intuitive appeal to fairness, the first step toward the limit.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/compound-interest",
    "concept/euler-s-number"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-89ce79b44c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "139",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "This process of growing proportionately, at every instant, to the magnitude at that instant, some people call \\emph{a logarithmic rate} of growing. Unit logarithmic rate of growth is that rate which in unit time will cause $1$ to grow to $2.718281$.",
   "markdown": "This process of growing proportionately, at every instant, to the magnitude at that instant, some people call *a logarithmic rate* of growing. Unit logarithmic rate of growth is that rate which in unit time will cause $1$ to grow to $2.718281$.",
   "why": "Names the growth process in which size and rate of increase stay proportional.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/euler-s-number",
    "concept/logarithmic-rate-of-growth"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-b94f47f2f7",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "143",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "The great reason why $\\epsilon$ is regarded of importance is that $\\epsilon^x$ possesses a property, not possessed by any other function of~$x$, that \\emph{when you differentiate it its value remains unchanged}\\Pagelabel{unchanged}; or, in other words, its differential coefficient is the same as itself.",
   "markdown": "The great reason why $\\epsilon$ is regarded of importance is that $\\epsilon^x$ possesses a property, not possessed by any other function of $x$, that *when you differentiate it its value remains unchanged*unchanged; or, in other words, its differential coefficient is the same as itself.",
   "why": "States the single property that makes epsilon^x central to calculus.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/derivative",
    "concept/exponential-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-a19ac3ed18",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "148",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "Note that $x^{-1}$ is a result that we could never have got by the rule for differentiating powers.",
   "markdown": "Note that $x^{-1}$ is a result that we could never have got by the rule for differentiating powers.",
   "why": "Warns that the derivative of the logarithm falls outside the power rule, preparing for integration.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/natural-logarithm",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5ea5798ed9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "157",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "In fact $\\epsilon^{-at}$ serves as a \\emph{die-away factor} for all those phenomena in which the rate of decrease is proportional to the magnitude of that which is decreasing; or where, in our usual symbols, $\\dfrac{dy}{dt}$~is proportional at every moment to the value that~$y$ has at that moment.",
   "markdown": "In fact $\\epsilon^{-at}$ serves as a *die-away factor* for all those phenomena in which the rate of decrease is proportional to the magnitude of that which is decreasing; or where, in our usual symbols, $\\dfrac{dy}{dt}$ is proportional at every moment to the value that $y$ has at that moment.",
   "why": "Links one formula to cooling, leaking charge and fading oscillations.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/die-away-factor",
    "theorem/die-away-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6f7a7f37d8",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "145",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "For the benefit of those who have no tutor at hand it may be of use to state that $\\epsilon^x$ is read as ``\\emph{epsilon to the eksth power};'' or some people read it ``\\emph{exponential eks}.''",
   "markdown": "For the benefit of those who have no tutor at hand it may be of use to state that $\\epsilon^x$ is read as “*epsilon to the eksth power*;” or some people read it “*exponential eks*.”",
   "why": "A charming aside on how to say exponentials aloud, from a book written for self-teachers.",
   "use": [
    "history",
    "website"
   ],
   "concepts": [
    "concept/exponential-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-7ed6f0f566",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "165",
   "location": "How to deal with Sines and Cosines",
   "latex": "What we have to investigate is the value of $\\dfrac{d(\\sin \\theta)}{d \\theta}$; or, in other words, if the angle~$\\theta$ varies, we have to find the relation between the increment of the sine and the increment of the angle, both increments being indefinitely small in themselves.",
   "markdown": "What we have to investigate is the value of $\\dfrac{d(\\sin \\theta)}{d \\theta}$; or, in other words, if the angle $\\theta$ varies, we have to find the relation between the increment of the sine and the increment of the angle, both increments being indefinitely small in themselves.",
   "why": "It states plainly what the derivative of the sine means: how a small change in the sine follows from a small change in the angle.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-20f76f29c9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "166",
   "location": "How to deal with Sines and Cosines",
   "latex": "But if we regard $d \\theta$ as indefinitely small, then in the limit we may neglect~$\\frac{1}{2} d \\theta$ by comparison with~$\\theta$, and may also take $\\sin\\frac{1}{2} d \\theta$ as being the same as~$\\frac{1}{2} d \\theta$.",
   "markdown": "But if we regard $d \\theta$ as indefinitely small, then in the limit we may neglect $\\frac{1}{2} d \\theta$ by comparison with $\\theta$, and may also take $\\sin\\frac{1}{2} d \\theta$ as being the same as $\\frac{1}{2} d \\theta$.",
   "why": "It shows the approximation step a learner must justify before taking the limit, the place where small terms are dropped.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "concept/limit",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-d6e44d38f0",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "168",
   "location": "How to deal with Sines and Cosines",
   "latex": "Now $\\cos \\theta=\\sin\\left(\\dfrac{\\pi}{2}-\\theta\\right)$.",
   "markdown": "Now $\\cos \\theta=\\sin\\left(\\dfrac{\\pi}{2}-\\theta\\right)$.",
   "why": "It lets the learner obtain the derivative of the cosine from the derivative of the sine, instead of starting again.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cosine",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6515ac845d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "171",
   "location": "How to deal with Sines and Cosines",
   "latex": "Passing now from the inverse function to the original one, we get",
   "markdown": "Passing now from the inverse function to the original one, we get",
   "why": "It makes the reciprocal rule for an inverse function concrete through the arcsin example.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/inverse-function",
    "method/differentiating-an-inverse-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-c67a1ee9bc",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "171",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\emph{Sines and cosines are the only functions of which the second differential coefficient is equal \\emph{(and of opposite sign to)} the original function.}",
   "markdown": "*Sines and cosines are the only functions of which the second differential coefficient is equal *(and of opposite sign to)* the original function.*",
   "why": "It gives the striking property that a second differentiation returns the function with its sign reversed, a memorable point for a learner.",
   "use": [
    "website"
   ],
   "concepts": [
    "concept/cosine",
    "concept/higher-order-derivative",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ebc542f1d8",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "171",
   "location": "How to deal with Sines and Cosines",
   "latex": "So we have this curious result that we have found a function such that if we differentiate it twice over, we get the same thing from which we started, but with the sign changed from $+$~to~$-$.",
   "markdown": "So we have this curious result that we have found a function such that if we differentiate it twice over, we get the same thing from which we started, but with the sign changed from $+$ to $-$.",
   "why": "It is the chapter's most vivid turn of phrase and invites a reader to pause over why the sine behaves so.",
   "use": [
    "website"
   ],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6de2792c85",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "170",
   "location": "How to deal with Sines and Cosines",
   "latex": "If the \\emph{frequency}, or number of periods per second, be denoted by~$n$, then $n = \\dfrac{1}{T}$, and we may then write:",
   "markdown": "If the *frequency*, or number of periods per second, be denoted by $n$, then $n = \\dfrac{1}{T}$, and we may then write:",
   "why": "It ties the period of a repeating motion to its frequency, the link a learner needs before differentiating with respect to time.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/periodic-function",
    "quantity/angle",
    "quantity/frequency"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-9c0c0e7930",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "The first is obtained by supposing $y$ constant, the second is obtained by supposing $x$ constant; then",
   "markdown": "The first is obtained by supposing $y$ constant, the second is obtained by supposing $x$ constant; then",
   "why": "It shows the working rule for partial derivatives: hold one variable fixed and differentiate with respect to the other.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-9e4f83dcbe",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "Clearly $A$ is maximum when $P$ is maximum.",
   "markdown": "Clearly $A$ is maximum when $P$ is maximum.",
   "why": "It models the reduction of a geometry problem to maximising a single function, a step learners often skip.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/maximum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-624a1e39ea",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "The variation when both the radius and the height change is given by $dV = \\dfrac{2\\pi}{3} rh\\, dV + \\dfrac{\\pi}{3} r^2\\, dh$.",
   "markdown": "The variation when both the radius and the height change is given by $dV = \\dfrac{2\\pi}{3} rh\\, dV + \\dfrac{\\pi}{3} r^2\\, dh$.",
   "why": "FLAG, do not teach from this line as printed: the source writes dV on the right where dr is required, since V depends on r and h (the cone's partial derivatives in Example 3 are correct; this total differential has a misprinted differential, an erratum or transcription issue to be checked against the book's errata).",
   "use": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-4b8e657803",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "175",
   "location": "Partial Differentiation",
   "latex": "\\First{We} sometimes come across quantities that are functions of more than one independent variable.",
   "markdown": "We sometimes come across quantities that are functions of more than one independent variable.",
   "why": "Opens the chapter by naming the new situation: a quantity that depends on several variables at once.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/function-of-several-variables"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-3972a6c1da",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "175",
   "location": "Partial Differentiation",
   "latex": "The little letters here put as subscripts are to show which quantity has been taken as constant in the operation.",
   "markdown": "The little letters here put as subscripts are to show which quantity has been taken as constant in the operation.",
   "why": "Explains the subscript notation, which shows exactly which variable is being held still.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant",
    "concept/partial-derivative",
    "concept/partial-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-b8c4c68085",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "But, if you think of it, you will observe that the total variation of~$y$ depends on \\emph{both} these things at the same time.",
   "markdown": "But, if you think of it, you will observe that the total variation of $y$ depends on *both* these things at the same time.",
   "why": "Motivates the total differential: when both variables change, both partial changes must count.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-521e0a2ce1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "178",
   "location": "Partial Differentiation",
   "latex": "This differential equation is of immense importance in mathematical physics.",
   "markdown": "This differential equation is of immense importance in mathematical physics.",
   "why": "Tells learners that this abstract calculation leads to a result of real importance in physics.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/differential-equation",
    "concept/mathematical-physics"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6bff6eb201",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "175",
   "location": "Partial Differentiation",
   "latex": "Another way of indicating that the differentiation has been performed only \\emph{partially}, that is, has been performed only with respect to \\emph{one} of the independent variables, is to write the differential coefficients with Greek deltas, like~$\\partial$, instead of little~$d$.",
   "markdown": "Another way of indicating that the differentiation has been performed only *partially*, that is, has been performed only with respect to *one* of the independent variables, is to write the differential coefficients with Greek deltas, like $\\partial$, instead of little $d$.",
   "why": "Gives the plain definition of partial differentiation and the reason for the curly-delta symbol.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5df1d1f888",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "In the following example $F$~and~$f$ denote two arbitrary functions of any form whatsoever. For example, they may be sine-functions, or exponentials, or mere algebraic functions of the two independent variables, $t$~and~$x$.",
   "markdown": "In the following example $F$ and $f$ denote two arbitrary functions of any form whatsoever. For example, they may be sine-functions, or exponentials, or mere algebraic functions of the two independent variables, $t$ and $x$.",
   "why": "Sets up the wave-equation example by showing that the result holds for any functions F and f.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential-equation",
    "concept/function-of-several-variables"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-0686a9e413",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "Clearly $x=15$ gives minimum area; $x=10$ gives the maximum, for $\\dfrac{d^2 P}{dx^2} = 12x - 150$, which is $+30$ for $x=15$ and $-30$ for $x=10$.",
   "markdown": "Clearly $x=15$ gives minimum area; $x=10$ gives the maximum, for $\\dfrac{d^2 P}{dx^2} = 12x - 150$, which is $+30$ for $x=15$ and $-30$ for $x=10$.",
   "why": "Shows how the sign of the second derivative tells a maximum from a minimum in the string-triangle problem.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/maximum",
    "concept/minimum",
    "method/maxima-and-minima-of-a-function-of-two-variables",
    "method/second-derivative-test"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-810222b0cf",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Partial Differentiation",
   "latex": "The truck is a rectangular box open at the top. Let $x$ be the length and $y$ be the width; then the depth is~$\\dfrac{V}{xy}$. The surface area is $S=xy + \\dfrac{2V}{x} + \\dfrac{2V}{y}$.",
   "markdown": "The truck is a rectangular box open at the top. Let $x$ be the length and $y$ be the width; then the depth is $\\dfrac{V}{xy}$. The surface area is $S=xy + \\dfrac{2V}{x} + \\dfrac{2V}{y}$.",
   "why": "A concrete modelling step: turning a coal-truck problem into a function of two variables to minimise.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/minimum",
    "method/maxima-and-minima-of-a-function-of-two-variables"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-29fbd257d8",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "184",
   "location": "Integration",
   "latex": "The remainder needed will always be equal to the last term added.",
   "markdown": "The remainder needed will always be equal to the last term added.",
   "why": "It shows the learner, through the halving series, why an infinite sum can still reach an exact total.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-27c5707aef",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "184",
   "location": "Integration",
   "latex": "A microscope would not show even the $18^{\\text{th}}$~term! So the infinite number of operations is no such dreadful thing after all.",
   "markdown": "A microscope would not show even the $18^{\\text{th}}$ term! So the infinite number of operations is no such dreadful thing after all.",
   "why": "A vivid, memorable remark that makes the idea of infinitely many steps feel manageable.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-654cc5f7c1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "189",
   "location": "Integration",
   "latex": "Clearly, at any point~$P$ of the curve, the value of~$y$ will be the sum of all the little~$dy$'s from~$0$ up to that level, that is to say, $\\ds\\int dy = y$.",
   "markdown": "Clearly, at any point $P$ of the curve, the value of $y$ will be the sum of all the little $dy$’s from $0$ up to that level, that is to say, $\\ds\\int dy = y$.",
   "why": "It states the central idea that y is the integral of the little dy's, the step from differentials to a curve.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-039416112f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "189",
   "location": "Integration",
   "latex": "But, as in the previous case, this requires the addition of an undetermined constant~$C$, because we have not been told at what height above the origin the curve will begin, when $x = 0$.",
   "markdown": "But, as in the previous case, this requires the addition of an undetermined constant $C$, because we have not been told at what height above the origin the curve will begin, when $x = 0$.",
   "why": "It explains why every integral carries a constant, which a learner often forgets.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-of-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-7eb634817c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "182",
   "location": "Integration",
   "latex": "Any one can understand how the whole of anything can be conceived of as made up of a lot of little bits; and the smaller the bits the more of them there will be.",
   "markdown": "Any one can understand how the whole of anything can be conceived of as made up of a lot of little bits; and the smaller the bits the more of them there will be.",
   "why": "Gives the learner the core intuition of integration, a whole built from ever smaller pieces.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/infinitesimal",
    "concept/integral",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-fd919bb31c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "189",
   "location": "Integration",
   "latex": "But $x$~began by being~$0$, and increases to the particular value of~$x$ at the point~$P$, so that its average value from~$0$ to that point is~$\\frac{1}{2}x$. Hence $\\ds\\int \\tfrac{1}{5} x\\, dx = \\tfrac{1}{10} x^2$; or $y=\\frac{1}{10}x^2$.",
   "markdown": "But $x$ began by being $0$, and increases to the particular value of $x$ at the point $P$, so that its average value from $0$ to that point is $\\frac{1}{2}x$. Hence $\\ds\\int \\tfrac{1}{5} x\\, dx = \\tfrac{1}{10} x^2$; or $y=\\frac{1}{10}x^2$.",
   "why": "Shows a heuristic worked step for integrating a non-constant slope; it is an informal argument that learners should check by differentiating the result.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/average",
    "concept/integral",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-9062ffedfa",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "183",
   "location": "Integration",
   "latex": "The simple reason is that there are a vast number of cases in which one cannot calculate the bigness of the thing as a whole without reckoning up the sum of a lot of small parts. The process of ``\\emph{integrating}'' is to enable us to calculate totals that otherwise we should be unable to estimate directly.",
   "markdown": "The simple reason is that there are a vast number of cases in which one cannot calculate the bigness of the thing as a whole without reckoning up the sum of a lot of small parts. The process of “*integrating*” is to enable us to calculate totals that otherwise we should be unable to estimate directly.",
   "why": "Answers the learner's 'why bother?' by stating what integration is for.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-519dd5cae3",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "183",
   "location": "Integration",
   "latex": "If at any point of the operation we stop, there will still be a piece wanting to make up the whole $2$~inches; and the piece wanting will always be the same size as the last piece added.",
   "markdown": "If at any point of the operation we stop, there will still be a piece wanting to make up the whole $2$ inches; and the piece wanting will always be the same size as the last piece added.",
   "why": "Explains with a line of inches why the halving series approaches 2 but needs infinitely many steps to reach it.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/convergent-series",
    "concept/geometric-series"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f8bfe8ac67",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "184",
   "location": "Integration",
   "latex": "If we want to go so far that not even a Whitworth's measuring machine would detect it, we should merely have to go to about $20$~terms. A microscope would not show even the $18^{\\text{th}}$~term! So the infinite number of operations is no such dreadful thing after all.",
   "markdown": "If we want to go so far that not even a Whitworth’s measuring machine would detect it, we should merely have to go to about $20$ terms. A microscope would not show even the $18^{\\text{th}}$ term! So the infinite number of operations is no such dreadful thing after all.",
   "why": "A charming, reassuring touch that makes an infinite sum feel practical; it also records a period instrument, Whitworth's measuring machine.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/geometric-series",
    "concept/infinitesimal"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-e063224f9e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "185",
   "location": "Integration",
   "latex": "For we have seen that differentiating a curve means finding an expression for its slope (or for its slopes at different points). Can we perform the reverse process of reconstructing the whole curve if the slope (or slopes) are prescribed for us?",
   "markdown": "For we have seen that differentiating a curve means finding an expression for its slope (or for its slopes at different points). Can we perform the reverse process of reconstructing the whole curve if the slope (or slopes) are prescribed for us?",
   "why": "Frames integration as the undoing of differentiation, as a question the learner can pose.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/slope-of-a-curve",
    "method/differentiation",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-7ba65c603d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "187",
   "location": "Integration",
   "latex": "As the only information we have is as to the slope, we are without any instructions as to the particular height above~$O$; in fact the initial height is undetermined. The slope will be the same, whatever the initial height.",
   "markdown": "As the only information we have is as to the slope, we are without any instructions as to the particular height above $O$; in fact the initial height is undetermined. The slope will be the same, whatever the initial height.",
   "why": "Shows why an added constant is needed: the slope alone cannot fix where the curve sits.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ba160de5fd",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "193",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "So, therefore, when we reverse the process we must always remember to add on this undetermined constant, even if we do not yet know what its value will be.",
   "markdown": "So, therefore, when we reverse the process we must always remember to add on this undetermined constant, even if we do not yet know what its value will be.",
   "why": "It explains why every integral needs a constant C, the idea learners most often forget.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-4b76f48e9e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "199",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\NB---Here note this very remarkable fact, that we could not have integrated in the above case if we had not happened to know the corresponding differentiation.",
   "markdown": "*N.B.*---Here note this very remarkable fact, that we could not have integrated in the above case if we had not happened to know the corresponding differentiation.",
   "why": "It warns that integrals are found by recognising known derivatives, so memorised derivative results are needed to integrate.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/natural-logarithm",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-abee18b9b0",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "192",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "Clearly, in dealing with powers of~$x$, the rule for working backwards will be: Increase the power by~$1$, then divide by that increased power, and add the undetermined constant.",
   "markdown": "Clearly, in dealing with powers of $x$, the rule for working backwards will be: Increase the power by $1$, then divide by that increased power, and add the undetermined constant.",
   "why": "It states the power rule for integration in words a learner can apply to every power term.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/integration",
    "theorem/power-rule-for-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-00a4abfc54",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "197",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "Hence, when you work the other way and integrate, the constant reappears multiplied by~$x$.",
   "markdown": "Hence, when you work the other way and integrate, the constant reappears multiplied by $x$.",
   "why": "It shows why a constant term integrates to a multiple of x, which prevents a common slip.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-term",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-b7a2b507ad",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "193",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "Thus, if $\\dfrac{dy}{dx} = 4x^2$, the reverse process gives us $y = \\frac{4}{3}x^3$.",
   "markdown": "Thus, if $\\dfrac{dy}{dx} = 4x^2$, the reverse process gives us $y = \\frac{4}{3}x^3$.",
   "why": "A short worked example of a constant multiplier and the power rule together, ready to read aloud.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-factor",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-df408bcf42",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "191",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "But here comes in a curious point. We should get $\\dfrac{dy}{dx} = 4x^3$ if we had begun with \\emph{any} of the following:~$x^4$, or~$x^4 + a$, or~$x^4 + c$, or~$x^4$ with \\emph{any} added constant.",
   "markdown": "But here comes in a curious point. We should get $\\dfrac{dy}{dx} = 4x^3$ if we had begun with *any* of the following: $x^4$, or $x^4 + a$, or $x^4 + c$, or $x^4$ with *any* added constant.",
   "why": "Shows concretely why reversing differentiation leaves an unknown constant.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-4529bd4145",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "194",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "We haven't yet integrated: we have only written down instructions to integrate---if we can. Let us try. Plenty of other fools can do it---why not we also?",
   "markdown": "We haven’t yet integrated: we have only written down instructions to integrate---if we can. Let us try. Plenty of other fools can do it---why not we also?",
   "why": "Separates writing the integral sign from actually integrating, in an encouraging voice.",
   "use": [
    "website",
    "lesson"
   ],
   "concepts": [
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-250108942e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "194",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "But when we come to the right-hand side of the equation we must remember that what we have got to sum up together is not all the~$dx$'s, but all such terms as~$x^2\\, dx$; and this will \\emph{not} be the same as $x^2 \\ds\\int dx$, because $x^2$~is not a constant.",
   "markdown": "But when we come to the right-hand side of the equation we must remember that what we have got to sum up together is not all the $dx$’s, but all such terms as $x^2\\, dx$; and this will *not* be the same as $x^2 \\ds\\int dx$, because $x^2$ is not a constant.",
   "why": "Warns against pulling a non-constant factor outside the integral sign.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-factor",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-854570e8a6",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "196",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "If a stranger were set down in Trafalgar Square, and told to find his way to Euston Station, he might find the task hopeless. But if he had previously been personally conducted from Euston Station to Trafalgar Square, it would be comparatively easy to him to find his way back to Euston Station.",
   "markdown": "If a stranger were set down in Trafalgar Square, and told to find his way to Euston Station, he might find the task hopeless. But if he had previously been personally conducted from Euston Station to Trafalgar Square, it would be comparatively easy to him to find his way back to Euston Station.",
   "why": "A vivid analogy for why integration depends on having first done the differentiation.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/differentiation",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-68e963178a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "196",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "So, when we work backwards, integrating, the integration will be simply the sum of the two separate integrations.",
   "markdown": "So, when we work backwards, integrating, the integration will be simply the sum of the two separate integrations.",
   "why": "States the term-by-term rule and the reason it works.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "theorem/sum-rule-for-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-ba86b32b6d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "199",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "Then, of course, since we know that differentiating $\\log_\\epsilon x$ gives us~$x^{-1}$, we know that, by reversing the process, integrating $dy = x^{-1}\\, dx$ will give us $y = \\log_\\epsilon x$.",
   "markdown": "Then, of course, since we know that differentiating $\\log_\\epsilon x$ gives us $x^{-1}$, we know that, by reversing the process, integrating $dy = x^{-1}\\, dx$ will give us $y = \\log_\\epsilon x$.",
   "why": "Shows how the exceptional case x^{-1} is settled by reading a known derivative backwards.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/logarithm",
    "theorem/integral-of-1-x"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5d1c530ade",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "200",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "Indeed it should be frankly admitted that this is one of the curious features of the integral calculus:---that you can't integrate anything before the reverse process of differentiating something else has yielded that expression which you want to integrate.",
   "markdown": "Indeed it should be frankly admitted that this is one of the curious features of the integral calculus:---that you can’t integrate anything before the reverse process of differentiating something else has yielded that expression which you want to integrate.",
   "why": "A candid statement of the limits of integration as a reverse process.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/standard-forms-of-integration",
    "method/differentiation",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-9f921da6a6",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "You should make such a table for yourself, putting in it only the general functions which you have successfully differentiated and integrated. See to it that it grows steadily!",
   "markdown": "You should make such a table for yourself, putting in it only the general functions which you have successfully differentiated and integrated. See to it that it grows steadily!",
   "why": "Gives learners a practical habit: build their own table of integrals.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/standard-forms-of-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-e7983ea4f0",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "209",
   "location": "On Finding Areas by Integrating",
   "latex": "Here we have the clue as to what to do; the definite integral between the two limits is \\emph{the difference} between the integral worked out for the superior limit and the integral worked out for the lower limit.",
   "markdown": "Here we have the clue as to what to do; the definite integral between the two limits is *the difference* between the integral worked out for the superior limit and the integral worked out for the lower limit.",
   "why": "It states the key idea that a definite integral is a difference of two antiderivative values.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/definite-integral",
    "concept/limits-of-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-43c2a987d0",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "213",
   "location": "On Finding Areas by Integrating",
   "latex": "There are $18$~whole squares and four triangles, each of which has an area equal to $1\\frac{1}{2}$~squares; or, in total, $24$~squares. Hence $24$~is the numerical value of the integral of $\\dfrac{x}{3}\\, dx$ between the lower limit of $x = 0$ and the higher limit of $x = 12$.",
   "markdown": "There are $18$ whole squares and four triangles, each of which has an area equal to $1\\frac{1}{2}$ squares; or, in total, $24$ squares. Hence $24$ is the numerical value of the integral of $\\dfrac{x}{3}\\, dx$ between the lower limit of $x = 0$ and the higher limit of $x = 12$.",
   "why": "Counting squares on graph paper independently checks the integration result, so the learner can see why the method works.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/definite-integral",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-328e83d165",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "214",
   "location": "On Finding Areas by Integrating",
   "latex": "\\NB---Notice that in dealing with definite integrals the constant~$C$ always disappears by subtraction.",
   "markdown": "*N.B.*---Notice that in dealing with definite integrals the constant $C$ always disappears by subtraction.",
   "why": "It warns learners that the constant of integration need not be carried through when limits are used.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "concept/definite-integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-dadf41b10d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "208",
   "location": "On Finding Areas by Integrating",
   "latex": "That is all very well; but a little thought will show you that something more must be done.",
   "markdown": "That is all very well; but a little thought will show you that something more must be done.",
   "why": "This old turn of phrase introduces the need for limits and makes an easy hook for a lesson on why definite integrals are needed.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/limits-of-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f3abe7ab01",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "222",
   "location": "On Finding Areas by Integrating",
   "latex": "By ``quadratic mean'' is denoted the square root of the mean of the squares of all the values between the limits considered.",
   "markdown": "By “quadratic mean” is denoted the square root of the mean of the squares of all the values between the limits considered.",
   "why": "It defines the quadratic mean in words a learner can check against the formula that follows.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/quadratic-mean"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-16f1f2ccf1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "219",
   "location": "On Finding Areas by Integrating",
   "latex": "Instead of a strip of area, we consider a small triangle $OAB$, the angle at~$O$ being~$d\\theta$, and we find the sum of all the little triangles making up the required area.",
   "markdown": "Instead of a strip of area, we consider a small triangle $OAB$, the angle at $O$ being $d\\theta$, and we find the sum of all the little triangles making up the required area.",
   "why": "It explains why polar area uses thin triangles from the pole rather than vertical strips.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/polar-coordinates",
    "method/finding-an-area-in-polar-coordinates"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f7054e1888",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "207",
   "location": "On Finding Areas by Integrating",
   "latex": "The secret of solving this problem is to conceive the area as being divided up into a lot of narrow strips, each of them being of the width~$dx$. The smaller we take~$dx$, the more of them there will be between $x_1$ and~$x_2$. Now, the whole area is clearly equal to the sum of the areas of all such strips.",
   "markdown": "The secret of solving this problem is to conceive the area as being divided up into a lot of narrow strips, each of them being of the width $dx$. The smaller we take $dx$, the more of them there will be between $x_1$ and $x_2$. Now, the whole area is clearly equal to the sum of the areas of all such strips.",
   "why": "It states the central idea of finding area by integration: cut it into thin strips and add them up.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/area",
    "method/integration",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-7e0794e2d1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "209",
   "location": "On Finding Areas by Integrating",
   "latex": "If then we were to subtract the smaller area from the larger, we should have left as a remainder the area $PQNM$, which is what we want. Here we have the clue as to what to do; the definite integral between the two limits is \\emph{the difference} between the integral worked out for the superior limit and the integral worked out for the lower limit.",
   "markdown": "If then we were to subtract the smaller area from the larger, we should have left as a remainder the area $PQNM$, which is what we want. Here we have the clue as to what to do; the definite integral between the two limits is *the difference* between the integral worked out for the superior limit and the integral worked out for the lower limit.",
   "why": "It explains why a definite integral is a difference of two values of the general integral.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/definite-integral",
    "concept/limits-of-integration",
    "method/subtraction"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-1a9286e971",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "211",
   "location": "On Finding Areas by Integrating",
   "latex": "All integration between limits requires the difference between two values to be thus found. Also note that, in making the subtraction the added constant~$C$ has disappeared.",
   "markdown": "All integration between limits requires the difference between two values to be thus found. Also note that, in making the subtraction the added constant $C$ has disappeared.",
   "why": "It warns learners why the constant of integration drops out of a definite integral.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-term",
    "concept/definite-integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-8778bd96aa",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "213",
   "location": "On Finding Areas by Integrating",
   "latex": "Now reckon out the area beneath the curve \\emph{by counting the little squares} below the line, from $x = 0$ as far as $x = 12$ on the right. There are $18$~whole squares and four triangles, each of which has an area equal to $1\\frac{1}{2}$~squares; or, in total, $24$~squares.",
   "markdown": "Now reckon out the area beneath the curve *by counting the little squares* below the line, from $x = 0$ as far as $x = 12$ on the right. There are $18$ whole squares and four triangles, each of which has an area equal to $1\\frac{1}{2}$ squares; or, in total, $24$ squares.",
   "why": "It lets learners check the integral by hand, counting squares on graph paper, so the method is not taken on trust.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/area",
    "concept/definite-integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-833928bd2b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "216",
   "location": "On Finding Areas by Integrating",
   "latex": "Consider an elementary zone or annulus of the surface (\\Fig{59}), of breadth~$dr$, situated at a distance~$r$ from the centre. We may consider the entire surface as consisting of such narrow zones, and the whole area~$A$ will simply be the integral of all such elementary zones from centre to margin, that is, integrated from $r = 0$ to $r = R$.",
   "markdown": "Consider an elementary zone or annulus of the surface ([fig:59]Fig. 59), of breadth $dr$, situated at a distance $r$ from the centre. We may consider the entire surface as consisting of such narrow zones, and the whole area $A$ will simply be the integral of all such elementary zones from centre to margin, that is, integrated from $r = 0$ to $r = R$.",
   "why": "It shows how to build an integral from a well-chosen thin piece, by re-deriving the familiar area of a circle.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/annulus",
    "method/integration",
    "theorem/area-of-a-circle"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-a0aeb5110e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "217",
   "location": "On Finding Areas by Integrating",
   "latex": "To find the mean ordinate, we shall have to find the area of the piece~$OMN$, and then divide it by the length of the base~$ON$. But before we can find the area we must ascertain the length of the base, so as to know up to what limit we are to integrate.",
   "markdown": "To find the mean ordinate, we shall have to find the area of the piece $OMN$, and then divide it by the length of the base $ON$. But before we can find the area we must ascertain the length of the base, so as to know up to what limit we are to integrate.",
   "why": "It shows that the limits of integration come from where the curve meets the axis, and that an average height is area divided by base.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limits-of-integration",
    "concept/mean-ordinate",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-3719f3535c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "222",
   "location": "On Finding Areas by Integrating",
   "latex": "In certain branches of physics, particularly in the study of alternating electric currents, it is necessary to be able to calculate the \\emph{quadratic mean} of a variable quantity. By ``quadratic mean'' is denoted the square root of the mean of the squares of all the values between the limits considered. Other names for the quadratic mean of any quantity are its ``virtual'' value, or its ``\\textsc{r.m.s.}''\\ (meaning root-mean-square) value. The French term is \\textit{valeur efficace}.",
   "markdown": "In certain branches of physics, particularly in the study of alternating electric currents, it is necessary to be able to calculate the *quadratic mean* of a variable quantity. By “quadratic mean” is denoted the square root of the mean of the squares of all the values between the limits considered. Other names for the quadratic mean of any quantity are its “virtual” value, or its “r.m.s.” (meaning root-mean-square) value. The French term is *valeur efficace*.",
   "why": "It defines the quadratic mean in words, gives its other names, and links the calculus to alternating currents.",
   "use": [
    "lesson",
    "website",
    "history"
   ],
   "concepts": [
    "concept/quadratic-mean"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-118ce46b78",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "227",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "Write $u = w$, and for $\\sin w · dw$ write~$dx$. We shall then have $du = dw$, while $\\ds\\int \\sin w · dw = -\\cos w = x$.",
   "markdown": "Write $u = w$, and for $\\sin w · dw$ write $dx$. We shall then have $du = dw$, while $\\ds\\int \\sin w · dw = -\\cos w = x$.",
   "why": "It shows the choice of u and dx in a worked integration by parts step that a learner can follow line by line.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/integration-by-parts"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f2c13dfa15",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "There are whole treatises, such as Boole's \\textit{Differential Equations}, devoted to the subject of thus finding the ``solutions'' for different original forms.",
   "markdown": "There are whole treatises, such as Boole’s *Differential Equations*, devoted to the subject of thus finding the “solutions” for different original forms.",
   "why": "It is a historical remark pointing readers to the specialist literature on finding solutions of differential equations.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/differential-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-7077d235a4",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "226",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "A great part of the labour of integrating things consists in licking them into some shape that can be integrated.",
   "markdown": "A great part of the labour of integrating things consists in licking them into some shape that can be integrated.",
   "why": "Frames integration as preparing an expression, which motivates the rest of the chapter's tricks.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-28d110b7cb",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "226",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "It is useful in some cases that you can't tackle directly, for it shows that if in any case $\\ds\\int x\\, du$ can be found, then $\\ds\\int u\\, dx$ can also be found.",
   "markdown": "It is useful in some cases that you can’t tackle directly, for it shows that if in any case $\\ds\\int x\\, du$ can be found, then $\\ds\\int u\\, dx$ can also be found.",
   "why": "Explains in plain words what integration by parts is for: trading a hard integral for an easier one.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/integration-by-parts"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-a55e413174",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "231",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "A beginner is liable to overlook certain points that a practised hand would avoid; such as the use of factors that are equivalent to either zero or infinity, and the occurrence of indeterminate quantities such as $\\tfrac{0}{0}$. There is no golden rule that will meet every possible case. Nothing but practice and intelligent care will avail.",
   "markdown": "A beginner is liable to overlook certain points that a practised hand would avoid; such as the use of factors that are equivalent to either zero or infinity, and the occurrence of indeterminate quantities such as $\\tfrac{0}{0}$. There is no golden rule that will meet every possible case. Nothing but practice and intelligent care will avail.",
   "why": "An honest warning that integration has traps and no universal safeguard, only care and practice.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/pitfall",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5d3450d46b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "232",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "The solution often seems as different from the original expression as a butterfly does from the caterpillar that it was.",
   "markdown": "The solution often seems as different from the original expression as a butterfly does from the caterpillar that it was.",
   "why": "A vivid image for how a solution of a differential equation can look nothing like the equation itself.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/differential-equation",
    "concept/solution",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-1c637105c3",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "231",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "Generally it is much easier to state the appropriate differential equation than to solve it:---the real trouble begins then only when one wants to integrate, unless indeed the equation is seen to possess some standard form of which the integral is known, and then the triumph is easy.",
   "markdown": "Generally it is much easier to state the appropriate differential equation than to solve it:---the real trouble begins then only when one wants to integrate, unless indeed the equation is seen to possess some standard form of which the integral is known, and then the triumph is easy.",
   "why": "Shows why integration skill matters: writing a differential equation is easy, solving it is the hard part.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/differential-equation",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5fd9a74e0e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "231",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "By triumphs must be understood the successes with which the calculus has been applied to the solution of problems otherwise intractable.",
   "markdown": "By triumphs must be understood the successes with which the calculus has been applied to the solution of problems otherwise intractable.",
   "why": "Gives a short statement of what the calculus is for.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/differential-equation",
    "concept/solution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-9d8ee0ed36",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "230",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "Notice that the same integral can be expressed sometimes in more than one way (which are equivalent to one another).",
   "markdown": "Notice that the same integral can be expressed sometimes in more than one way (which are equivalent to one another).",
   "why": "Warns learners that two different-looking answers can both be correct.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/integration",
    "method/partial-fractions"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-269166d2fd",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "234",
   "location": "Finding some Solutions",
   "latex": "He who would attain that facility must work out examples, and more examples, and yet more examples, such as are found abundantly in all the regular treatises on the Calculus.",
   "markdown": "He who would attain that facility must work out examples, and more examples, and yet more examples, such as are found abundantly in all the regular treatises on the Calculus.",
   "why": "It is a vivid statement of why practice matters, suitable for a reader's introduction to the chapter.",
   "use": [
    "website"
   ],
   "concepts": [
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-f365c78b0a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "242",
   "location": "Finding some Solutions",
   "latex": "Now the test of the matter is this. If the expression is an exact differential, it must be true that",
   "markdown": "Now the test of the matter is this. If the expression is an exact differential, it must be true that",
   "why": "It states the cross-derivative test in the author's own words, the key check for whether an expression can be integrated directly.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/exact-differential",
    "method/testing-for-an-exact-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-a70480b0be",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "243",
   "location": "Finding some Solutions",
   "latex": "There is no one rule for discovering such an integrating factor; but experience will usually suggest one.",
   "markdown": "There is no one rule for discovering such an integrating factor; but experience will usually suggest one.",
   "why": "It warns the learner that finding an integrating factor is a matter of judgement, not a formula.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/integrating-factor"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-5d5e1cd3c5",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "245",
   "location": "Finding some Solutions",
   "latex": "The only function we know that has this property is the exponential function (see \\Pageref{unchanged}), and we may be certain therefore that the solution of the equation will be of that form.",
   "markdown": "The only function we know that has this property is the exponential function (see unchanged), and we may be certain therefore that the solution of the equation will be of that form.",
   "why": "It explains why the exponential is the natural trial solution when a function's second derivative is proportional to itself.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/exponential-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-402dccf65e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "234",
   "location": "Finding some Solutions",
   "latex": "The beginner, who now knows how easy most of those processes are in themselves, will here begin to realize that integration is \\emph{an art}. As in all arts, so in this, facility can be acquired only by diligent and regular practice.",
   "markdown": "The beginner, who now knows how easy most of those processes are in themselves, will here begin to realize that integration is *an art*. As in all arts, so in this, facility can be acquired only by diligent and regular practice.",
   "why": "Tells the learner honestly that skill in integration comes from repeated practice, not from a single rule.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-a54316e5a4",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "236",
   "location": "Finding some Solutions",
   "latex": "Now, as to the $C$, its meaning depends on the initial value of~$y$.",
   "markdown": "Now, as to the $C$, its meaning depends on the initial value of $y$.",
   "why": "Gives the arbitrary constant a physical meaning: the starting value of y.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "concept/initial-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-39232f14e6",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "241",
   "location": "Finding some Solutions",
   "latex": "This is indeed none other than the equation of an alternating electric current, where $g$ represents the amplitude of the electromotive force, $n$~the frequency, $a$~the resistance, $b$~the coefficient of self-induction of the circuit, and $\\phi$ is an angle of lag.",
   "markdown": "This is indeed none other than the equation of an alternating electric current, where $g$ represents the amplitude of the electromotive force, $n$ the frequency, $a$ the resistance, $b$ the coefficient of self-induction of the circuit, and $\\phi$ is an angle of lag.",
   "why": "Connects an abstract differential equation to an alternating-current circuit, naming each symbol's physical role.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/alternating-current",
    "concept/phase-lag",
    "quantity/coefficient-of-induction",
    "quantity/electromotive-force",
    "quantity/frequency",
    "quantity/resistance",
    "quantity/self-induction"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-209af684c9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "247",
   "location": "Finding some Solutions",
   "latex": "In this case the simplified equation represents the propagation of a wave (of any form) at a uniform speed along the $x$~direction.",
   "markdown": "In this case the simplified equation represents the propagation of a wave (of any form) at a uniform speed along the $x$ direction.",
   "why": "Shows that a bare differential equation can describe a wave of any shape travelling at steady speed.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/wave-equation",
    "concept/wave-propagation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-157b910651",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "235",
   "location": "Finding some Solutions",
   "latex": "Now the mere inspection of this relation tells us that we have got to do with a case in which $\\dfrac{dy}{dx}$ is proportional to~$y$. If we think of the curve which will represent $y$ as a function of~$x$, it will be such that its slope at any point will be proportional to the ordinate at that point, and will be a negative slope if $y$~is positive.",
   "markdown": "Now the mere inspection of this relation tells us that we have got to do with a case in which $\\dfrac{dy}{dx}$ is proportional to $y$. If we think of the curve which will represent $y$ as a function of $x$, it will be such that its slope at any point will be proportional to the ordinate at that point, and will be a negative slope if $y$ is positive.",
   "why": "Shows how to read an equation for the shape of its answer before doing any calculation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/slope-of-a-curve",
    "theorem/die-away-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-2cc9b13449",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "235",
   "location": "Finding some Solutions",
   "latex": "As both $y$~and~$dy$ occur in the equation and on opposite sides, we can do nothing until we get both $y$~and~$dy$ to one side, and $dx$~to the other. To do this, we must split our usually inseparable companions $dy$~and~$dx$ from one another.",
   "markdown": "As both $y$ and $dy$ occur in the equation and on opposite sides, we can do nothing until we get both $y$ and $dy$ to one side, and $dx$ to the other. To do this, we must split our usually inseparable companions $dy$ and $dx$ from one another.",
   "why": "Explains with some humour why and how the variables are separated before integrating.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/separating-the-variables"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-bb959d8a5c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "238",
   "location": "Finding some Solutions",
   "latex": "Now, as it stands, the left side is not integrable. But it can be made so by the artifice---and this is where skill and practice suggest a plan---of multiplying all the terms by $\\epsilon^{\\efrac{a}{b} t}$, giving us:",
   "markdown": "Now, as it stands, the left side is not integrable. But it can be made so by the artifice---and this is where skill and practice suggest a plan---of multiplying all the terms by $\\epsilon^{\\efrac{a}{b} t}$, giving us:",
   "why": "Shows the idea of multiplying by a well-chosen factor so that a non-integrable expression becomes integrable, and admits that this needs practised judgement.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/exact-differential",
    "concept/integrating-factor"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-59e443ef1e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "243",
   "location": "Finding some Solutions",
   "latex": "It is possible in such cases to discover, however, \\emph{an integrating factor}, that is to say, a factor such that if both are multiplied by this factor, the expression will become an exact differential. There is no one rule for discovering such an integrating factor; but experience will usually suggest one.",
   "markdown": "It is possible in such cases to discover, however, *an integrating factor*, that is to say, a factor such that if both are multiplied by this factor, the expression will become an exact differential. There is no one rule for discovering such an integrating factor; but experience will usually suggest one.",
   "why": "Defines the integrating factor and honestly says that finding one has no fixed rule.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/exact-differential",
    "concept/integrating-factor"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-10b4ac3681",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "248",
   "location": "Finding some Solutions",
   "latex": "You have now been personally conducted over the frontiers into the enchanted land.",
   "markdown": "You have now been personally conducted over the frontiers into the enchanted land.",
   "why": "A warm closing line that puts the whole book as a guided journey.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-4d9edf8892",
   "chapter": "thompson-calculus-made-easy-1914/ch-epilogue-and-apologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "250",
   "location": "Epilogue and Apologue",
   "latex": "You don't teach the rules of syntax to children until they have already become fluent in the \\emph{use} of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus.",
   "markdown": "You don’t teach the rules of syntax to children until they have already become fluent in the *use* of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus.",
   "why": "It argues that fluency in using a method should come before its rigorous proof, which is a useful stance for any beginner's course.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-78d83b4ce5",
   "chapter": "thompson-calculus-made-easy-1914/ch-epilogue-and-apologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "251",
   "location": "Epilogue and Apologue",
   "latex": "There are amongst young engineers a number on whose ears the adage that \\emph{what one fool can do, another can}, may fall with a familiar sound.",
   "markdown": "There are amongst young engineers a number on whose ears the adage that *what one fool can do, another can*, may fall with a familiar sound.",
   "why": "It is a lively, self-mocking turn of phrase that shows the author's tone and invites readers in.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-23d71fb8c0",
   "chapter": "thompson-calculus-made-easy-1914/ch-epilogue-and-apologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "250",
   "location": "Epilogue and Apologue",
   "latex": "You don't forbid the use of a watch to every person who does not know how to make one? You don't object to the musician playing on a violin that he has not himself constructed. You don't teach the rules of syntax to children until they have already become fluent in the \\emph{use} of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus.",
   "markdown": "You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the *use* of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus.",
   "why": "Through everyday analogies, it shows learners and teachers why using a method comes before proving it.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/calculus",
    "concept/mathematical-rigor"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-37862b1abe",
   "chapter": "thompson-calculus-made-easy-1914/ch-epilogue-and-apologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "250",
   "location": "Epilogue and Apologue",
   "latex": "Any subject can be made repulsive by presenting it bristling with difficulties.",
   "markdown": "Any subject can be made repulsive by presenting it bristling with difficulties.",
   "why": "It names a teaching mistake: piling difficulties on a beginner can put them off a subject.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-d66f32f981",
   "chapter": "thompson-calculus-made-easy-1914/ch-epilogue-and-apologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "250",
   "location": "Epilogue and Apologue",
   "latex": "The aim of this book is to enable beginners to learn its language, to acquire familiarity with its endearing simplicities, and to grasp its powerful methods of solving problems, without being compelled to toil through the intricate out-of-the-way (and mostly irrelevant) mathematical gymnastics so dear to the unpractical mathematician.",
   "markdown": "The aim of this book is to enable beginners to learn its language, to acquire familiarity with its endearing simplicities, and to grasp its powerful methods of solving problems, without being compelled to toil through the intricate out-of-the-way (and mostly irrelevant) mathematical gymnastics so dear to the unpractical mathematician.",
   "why": "It states the book's purpose, which is to build fluency and problem-solving power before formal subtlety.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/calculus",
    "concept/mathematical-rigor"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-a2a42fb24f",
   "chapter": "thompson-calculus-made-easy-1914/ch-epilogue-and-apologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "249",
   "location": "Epilogue and Apologue",
   "latex": "By showing you that \\emph{what one fool can do, other fools can do also}, it lets you see that these mathematical swells, who pride themselves on having mastered such an awfully difficult subject as the calculus, have no such great reason to be puffed up.",
   "markdown": "By showing you that *what one fool can do, other fools can do also*, it lets you see that these mathematical swells, who pride themselves on having mastered such an awfully difficult subject as the calculus, have no such great reason to be puffed up.",
   "why": "It encourages learners that calculus is within their reach and that its difficulty is partly a matter of reputation.",
   "use": [
    "website",
    "history"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-cb65ed32d1",
   "chapter": "thompson-calculus-made-easy-1914/ch-epilogue-and-apologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "249",
   "location": "Epilogue and Apologue",
   "latex": "First, it shows how ridiculously easy most of the operations of the calculus really are.",
   "markdown": "First, it shows how ridiculously easy most of the operations of the calculus really are.",
   "why": "It is a short, cheeky statement of the book's central promise.",
   "use": [
    "website"
   ],
   "concepts": [
    "concept/calculus"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-391923ee65",
   "chapter": "thompson-calculus-made-easy-1914/ch-epilogue-and-apologue",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "251",
   "location": "Epilogue and Apologue",
   "latex": "They are earnestly requested not to give the author away, nor to tell the mathematicians what a fool he really is.",
   "markdown": "They are earnestly requested not to give the author away, nor to tell the mathematicians what a fool he really is.",
   "why": "It is a charming closing joke that shows the author's humility and his warmth toward readers.",
   "use": [
    "website",
    "history"
   ],
   "concepts": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-624276693a",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\DStrut$nx^{n-1}$ & $x^n$ &$ \\dfrac{1}{n+1} x^{n+1} + C $\\\\",
   "markdown": "[-12pt]0pt32pt$nx^{n-1}$ & $x^n$ &$ \\dfrac{1}{n+1} x^{n+1} + C $",
   "why": "Shows the power rule in both directions in a single row, so a learner can see integration undoing differentiation.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "method/differentiation",
    "method/integration",
    "theorem/power-rule",
    "theorem/power-rule-for-differentiation",
    "theorem/power-rule-for-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-eb8f544e3b",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "$\\epsilon^x$ & $\\epsilon^x$ & $\\epsilon^x + C$ \\\\",
   "markdown": "$\\epsilon^x$ & $\\epsilon^x$ & $\\epsilon^x + C$",
   "why": "Shows the special property that ε^x is unchanged by both differentiation and integration, up to the constant.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/exponential-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-7805b97462",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\DStrut$u\\, \\dfrac{dv}{dx} + v\\, \\dfrac{du}{dx}$\n   & $uv$  & No general form known \\\\",
   "markdown": "[-12pt]0pt32pt$u\\, \\dfrac{dv}{dx} + v\\, \\dfrac{du}{dx}$ & $uv$ & No general form known",
   "why": "Warns that differentiating a product is easy but integrating one has no general rule.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "method/integration",
    "theorem/product-rule"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-342917ba59",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "$x^{-1}$     & $\\log_\\epsilon x$ & $ x(\\log_\\epsilon x - 1) + C$",
   "markdown": "$x^{-1}$ & $\\log_\\epsilon x$ & $ x(\\log_\\epsilon x - 1) + C$",
   "why": "It shows that the integral of a logarithm is not obvious and is found by reversing a product rule, which is a useful surprise for a learner.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/euler-s-number",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-6f701edc38",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "$\\sec^2 x$& $\\tan x$ & $-\\log_\\epsilon \\cos x + C $",
   "markdown": "$\\sec^2 x$& $\\tan x$ & $-\\log_\\epsilon \\cos x + C $",
   "why": "It pairs the tangent function with its integral, giving a learner a concrete trigonometric result to check by differentiating back.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/integral",
    "concept/natural-logarithm",
    "concept/tangent-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/x-3fa780aaf0",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "$ 2a·\\sin 2ax$ & $\\sin^2 ax$ & $\\dfrac{x}{2} - \\dfrac{\\sin 2ax}{4a} + C $ \\\\",
   "markdown": "$ 2a·\\sin 2ax$ & $\\sin^2 ax$ & $\\dfrac{x}{2} - \\dfrac{\\sin 2ax}{4a} + C $",
   "why": "FLAG, do not feature: the derivative entry reads 2a·sin 2ax, but d/dx sin^2 ax = a·sin 2ax, so this row disagrees with its own integral; the sin mx·sin nx row's integral column is also not the integral of its integrand, and both are recorded as findings, not corrected.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/derivative",
    "concept/integral"
   ]
  }
 ],
 "equations": [
  {
   "id": "thompson-calculus-made-easy-1914/eq-0fda498405",
   "chapter": "thompson-calculus-made-easy-1914/ch-ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "6",
   "location": "On Different Degrees of Smallness",
   "latex": "x^2 + 2x · dx + (dx)^2",
   "name": null,
   "statement": "The square of x grown by a small increment dx expands to x squared, plus twice x times dx, plus dx squared, where the last term is of the second order of smallness.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the quantity that grows by a small amount to become x + dx"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "the small increment added to x by growth (a differential, a little bit of x)"
    }
   ],
   "sympy": "Eq((x + dx)**2, x**2 + 2*x*dx + dx**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/infinitesimal",
    "concept/order-of-smallness",
    "theorem/square-of-a-sum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-475cdd1488",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "11",
   "location": "On Relative Growings",
   "latex": "\\frac{dy}{dx} = \\frac{1}{1.73}",
   "name": null,
   "statement": "For a right triangle whose other side slopes at a fixed 30°, the ratio of a small increase in height to the matching small increase in base is 1/1.73.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small increment of the height y of the triangle"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "small increment of the base x of the triangle"
    }
   ],
   "sympy": "Eq(dy/dx, 1/1.73)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/tangent-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7fe626655f",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "12",
   "location": "On Relative Growings",
   "latex": "\\frac{dy}{dx} = - \\frac{0.11}{1}",
   "name": null,
   "statement": "In the ladder example, moving the foot of the ladder 1 inch further from the wall lowers the top by 0.11 inch, so dy/dx is -0.11 per inch.",
   "kind": "result",
   "symbols": [
    {
     "unit": "inch",
     "symbol": "dy",
     "meaning": "change in height of the ladder top on the wall"
    },
    {
     "unit": "inch",
     "symbol": "dx",
     "meaning": "change in distance of the ladder foot from the wall"
    }
   ],
   "sympy": "Eq(dy/dx, -0.11/1)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/rate-of-change"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-04404005bc",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "13",
   "location": "On Relative Growings",
   "latex": "x^2 + y^2 = l^2",
   "name": "Pythagorean theorem",
   "statement": "The foot distance x, the height y and the fixed ladder length l of a ladder against a wall satisfy the Pythagorean relation.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal distance of the bottom of the ladder from the wall"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height the ladder reaches up the wall"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "length of the ladder"
    }
   ],
   "sympy": "Eq(x**2 + y**2, l**2)",
   "physics": false,
   "states": [
    "theorem/pythagorean-theorem"
   ],
   "concepts": [
    "concept/function",
    "concept/implicit-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-87c041607f",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "14",
   "location": "On Relative Growings",
   "latex": "y &= x \\tan 30°",
   "name": null,
   "statement": "The height of the triangle written explicitly as a function of its base, for a 30° angle.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "base of the right-angled triangle"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height of the right-angled triangle"
    }
   ],
   "sympy": "Eq(y, x*tan(30*pi/180))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/explicit-function",
    "concept/function",
    "concept/tangent-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5884d32bdf",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "14",
   "location": "On Relative Growings",
   "latex": "y &= \\sqrt{ l^2 - x^2}",
   "name": null,
   "statement": "The height of the ladder on the wall written explicitly as a function of the foot distance from the wall.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal distance of the bottom of the ladder from the wall"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height the ladder reaches up the wall"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "length of the ladder"
    }
   ],
   "sympy": "Eq(y, sqrt(l**2 - x**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/explicit-function",
    "concept/function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5663b53b18",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "14",
   "location": "On Relative Growings",
   "latex": "x^2 + 3 = 2y - 7",
   "name": null,
   "statement": "An example of an implicit function relating x and y, which can be rewritten either way as an explicit function.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "an independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "a variable depending on x"
    }
   ],
   "sympy": "Eq(x**2 + 3, 2*y - 7)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/implicit-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-54f6bbf103",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "14",
   "location": "On Relative Growings",
   "latex": "y = \\dfrac{x^2 + 10}{2}",
   "name": null,
   "statement": "The example implicit relation x^2 + 3 = 2y - 7 written as y explicitly in terms of x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "an independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the dependent variable"
    }
   ],
   "sympy": "Eq(y, (x**2 + 10)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/explicit-function",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-04c4f2ed99",
   "chapter": "thompson-calculus-made-easy-1914/ch-iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "15",
   "location": "On Relative Growings",
   "latex": "u = x^2 \\sin \\theta",
   "name": null,
   "statement": "An example function in which u depends on x and theta, so x and theta are the independent variables and u the dependent variable.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "the dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "an independent variable"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "an independent variable (angle)"
    }
   ],
   "sympy": "Eq(u, x**2*sin(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-fd1a2eb2a8",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "19",
   "location": "Simplest Cases",
   "latex": "\\frac{dy}{dx} = 2x",
   "name": null,
   "statement": "Differentiating y = x^2 with respect to x gives the ratio of the growing of y to the growing of x equal to 2x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function x^2 (the dependent quantity)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(dydx, 2*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/exponent",
    "concept/power",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-51925fb234",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "21",
   "location": "Simplest Cases",
   "latex": "\\frac{dy}{dx} = 3x^2",
   "name": null,
   "statement": "Differentiating y = x^3 with respect to x gives dy/dx equal to 3x^2.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function x^3"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(dydx, 3*x**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/exponent",
    "concept/power",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4890a57bfb",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "21",
   "location": "Simplest Cases",
   "latex": "\\frac{dy}{dx} = 4x^3",
   "name": null,
   "statement": "Differentiating y = x^4 with respect to x gives dy/dx equal to 4x^3.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function x^4"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(dydx, 4*x**3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/exponent",
    "concept/power",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4ac9f44ca4",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "23",
   "location": "Simplest Cases",
   "latex": "\\frac{dy}{dx} = 5x^4",
   "name": null,
   "statement": "Differentiating y = x^5 with respect to x gives dy/dx equal to 5x^4.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function x^5"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(dydx, 5*x**4)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/exponent",
    "concept/power",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-03195012b7",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "23",
   "location": "Simplest Cases",
   "latex": "\\frac{dy}{dx} = nx^{(n-1)}",
   "name": "power rule",
   "statement": "For y = x^n, the differential coefficient of y with respect to x is n times x raised to the power n minus one; the chapter states this as the general rule, holding for whole positive n and checked for negative and fractional n.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the power x^n"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the power (exponent) to which x is raised"
    },
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(dydx, n*x**(n - 1))",
   "physics": false,
   "states": [
    "theorem/power-rule"
   ],
   "concepts": [
    "concept/derivative",
    "concept/exponent",
    "concept/power",
    "method/differentiation",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b535c5eecd",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "24",
   "location": "Simplest Cases",
   "latex": "\\frac{dy}{dx} = -2x^{-3}",
   "name": null,
   "statement": "Differentiating y = x^(-2) with respect to x gives dy/dx equal to -2x^(-3), in agreement with the general power rule.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function x^(-2)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(dydx, -2*x**(-3))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/exponent",
    "concept/negative-number",
    "concept/power",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-57301c9ad4",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "24",
   "location": "Simplest Cases",
   "latex": "\\dfrac{dy}{dx} = \\dfrac{1}{2} x^{-\\efrac{1}{2}}",
   "name": null,
   "statement": "Differentiating y = x^(1/2) with respect to x gives dy/dx equal to one half times x^(-1/2), agreeing with the general power rule for a fractional power.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function x^(1/2)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(dydx, Rational(1, 2)*x**Rational(-1, 2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-fraction",
    "concept/derivative",
    "concept/exponent",
    "concept/power",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8fbafa85db",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "27",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\frac{dy}{dx} = 3x^2.",
   "name": null,
   "statement": "The derivative of x^3 + 5 with respect to x is 3x^2, so the added constant 5 drops out.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, equal to x^3 + 5 in this example"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 3*x**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-term",
    "concept/derivative",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8b547a23d3",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "27",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\frac{dy}{dx} = 14x.",
   "name": null,
   "statement": "The derivative of 7x^2 with respect to x is 14x, so the constant factor 7 is carried through.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, equal to 7x^2 in this example"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 14*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-factor",
    "concept/derivative",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dd661a4283",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "29",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\frac{dy}{dx} = a × 2x.",
   "name": null,
   "statement": "For y = ax^2, the derivative is a times 2x, so a constant multiplier reappears unchanged in the derivative.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant multiplier"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, equal to ax^2"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 2*a*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/constant-factor",
    "concept/derivative",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-37c50c3c87",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "29",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\dfrac{dy}{dx} = a×nx^{n-1}",
   "name": "power rule for differentiation",
   "statement": "For y = ax^n, the derivative is a n x^(n-1): a constant multiplier passes through and the power drops by one.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant multiplier"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant exponent"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, equal to ax^n"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), a*n*x**(n-1))",
   "physics": false,
   "states": [
    "theorem/power-rule-for-differentiation"
   ],
   "concepts": [
    "concept/constant",
    "concept/constant-factor",
    "concept/derivative",
    "concept/exponent",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5f39ab91bd",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "30",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\frac{dy}{dx} = \\frac{5}{7} x^4.",
   "name": null,
   "statement": "The derivative of x^5/7 - 3/5 is 5x^4/7; the constant 3/5 vanishes and the 1/7 stays as a factor.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, equal to x^5/7 - 3/5"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 5*x**4/7)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-factor",
    "concept/constant-term",
    "concept/derivative",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b46eaa9665",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "30",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\frac{dy}{dx} = \\frac{a}{2\\sqrt{x}}.",
   "name": null,
   "statement": "The derivative of a times the square root of x, minus the constant term (1/2)sqrt(a), is a divided by 2 sqrt(x).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, equal to a sqrt(x) - (1/2) sqrt(a)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), a/(2*sqrt(x)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-factor",
    "concept/constant-term",
    "concept/derivative",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-cbd702d5be",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "31",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\frac{dy}{dx} = \\sqrt{\\frac{a+b}{a-b}}.",
   "name": null,
   "statement": "After squaring and simplifying the given relation, y is proportional to x and the derivative of y with respect to x is the square root of (a+b)/(a-b).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable defined implicitly by the given relation"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), sqrt((a+b)/(a-b)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/derivative",
    "concept/implicit-function",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ca747ce947",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "31",
   "location": "Next Stage. What to do with Constants",
   "latex": "V = \\pi r^2 h",
   "name": null,
   "statement": "The volume of a cylinder equals pi times the square of its radius times its height.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "cubic inch",
     "symbol": "V",
     "meaning": "volume of the cylinder"
    },
    {
     "unit": "inch",
     "symbol": "r",
     "meaning": "radius of the cylinder"
    },
    {
     "unit": "inch",
     "symbol": "h",
     "meaning": "height of the cylinder"
    }
   ],
   "sympy": "Eq(V, pi*r**2*h)",
   "physics": false,
   "states": [],
   "concepts": [
    "quantity/height",
    "quantity/pi",
    "quantity/radius",
    "theorem/volume-of-a-cylinder"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dab651fb79",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "31",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\frac{dV}{dr} = 2 \\pi r h.",
   "name": null,
   "statement": "The rate of change of the cylinder's volume with its radius is 2 pi r h.",
   "kind": "result",
   "symbols": [
    {
     "unit": "cubic inch",
     "symbol": "V",
     "meaning": "volume of the cylinder"
    },
    {
     "unit": "inch",
     "symbol": "r",
     "meaning": "radius of the cylinder"
    },
    {
     "unit": "inch",
     "symbol": "h",
     "meaning": "height of the cylinder"
    }
   ],
   "sympy": "Eq(Derivative(V, r), 2*pi*r*h)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/rate-of-change",
    "quantity/height",
    "quantity/pi",
    "quantity/radius",
    "theorem/volume-of-a-cylinder"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4d74c495a7",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "32",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\dfrac{dV}{dr} = 2\\pi r^2 = 400",
   "name": null,
   "statement": "When r = h, the rate of change of volume with radius equals 2 pi r^2, and this is set equal to 400 cubic inches per inch.",
   "kind": "result",
   "symbols": [
    {
     "unit": "cubic inch",
     "symbol": "V",
     "meaning": "volume of the cylinder"
    },
    {
     "unit": "inch",
     "symbol": "r",
     "meaning": "radius of the cylinder, equal to h here"
    }
   ],
   "sympy": "Eq(2*pi*r**2, 400)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/rate-of-change",
    "quantity/height",
    "quantity/pi",
    "quantity/radius",
    "theorem/volume-of-a-cylinder"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-174b881fa3",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "32",
   "location": "Next Stage. What to do with Constants",
   "latex": "r = h = \\sqrt{\\dfrac{400}{2\\pi}} = 7.98~\\text{in}.",
   "name": null,
   "statement": "Solving 2 pi r^2 = 400 with r = h gives a radius and height of about 7.98 inches.",
   "kind": "result",
   "symbols": [
    {
     "unit": "inch",
     "symbol": "r",
     "meaning": "radius of the cylinder"
    },
    {
     "unit": "inch",
     "symbol": "h",
     "meaning": "height of the cylinder"
    }
   ],
   "sympy": "Eq(r, sqrt(400/(2*pi)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "quantity/height",
    "quantity/pi",
    "quantity/radius"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f5088d8771",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "32",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\dfrac{\\theta}{\\theta_1} = \\left(\\dfrac{t}{t_1}\\right)^4",
   "name": null,
   "statement": "The pyrometer reading is proportional to the fourth power of the Centigrade temperature, relative to a reading taken at a known temperature.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "reading of the Féry radiation pyrometer"
    },
    {
     "unit": null,
     "symbol": "θ_1",
     "meaning": "reading corresponding to the known temperature t_1"
    },
    {
     "unit": "degree Centigrade",
     "symbol": "t",
     "meaning": "Centigrade temperature of the observed body"
    },
    {
     "unit": "degree Centigrade",
     "symbol": "t_1",
     "meaning": "known temperature of the observed body"
    }
   ],
   "sympy": "Eq(theta/theta_1, (t/t_1)**4)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/function",
    "concept/power",
    "concept/temperature"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-33a36cf519",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "32",
   "location": "Next Stage. What to do with Constants",
   "latex": "\\dfrac{d\\theta}{dt} = \\dfrac{100t^3}{1000^4} = \\dfrac{t^3}{10,000,000,000}.",
   "name": null,
   "statement": "With the reading fixed at 25 at 1000 degrees C, the sensitiveness d(theta)/dt of the pyrometer equals t^3 divided by 10^10.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "reading of the Féry radiation pyrometer"
    },
    {
     "unit": "degree Centigrade",
     "symbol": "t",
     "meaning": "Centigrade temperature of the observed body"
    }
   ],
   "sympy": "Eq(Derivative(theta, t), t**3/10000000000)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/power",
    "concept/rate-of-change",
    "concept/temperature"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d5043f0e8c",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "36",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "\\dfrac{dy}{dx} &= \\dfrac{du}{dx} + \\dfrac{dv}{dx}",
   "name": null,
   "statement": "The derivative of a sum of two functions of x is the sum of their derivatives.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the sum y = u + v, a function of x"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "one function of x"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "another function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), Derivative(u, x) + Derivative(v, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/sum",
    "theorem/sum-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a2339bcac3",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "36",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "\\frac{dy}{dx} &= \\frac{du}{dx} + \\frac{dv}{dx} + \\frac{dw}{dx}",
   "name": null,
   "statement": "The derivative of a sum of three functions of x is the sum of their three derivatives.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the sum y = u + v + w, a function of x"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "first function of x"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "second function of x"
    },
    {
     "unit": null,
     "symbol": "w",
     "meaning": "third function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), Derivative(u, x) + Derivative(v, x) + Derivative(w, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/sum",
    "theorem/sum-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3380a420b0",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "36",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "\\frac{dy}{dx} &= \\frac{du}{dx} - \\frac{dv}{dx}",
   "name": null,
   "statement": "The derivative of a difference of two functions of x is the difference of their derivatives.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the difference y = u - v, a function of x"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "first function of x"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "second function of x, subtracted"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), Derivative(u, x) - Derivative(v, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/difference",
    "theorem/sum-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e61bb29e2d",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "38",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "\\dfrac{dy}{dx} = u\\, \\dfrac{dv}{dx} + v\\, \\dfrac{du}{dx}",
   "name": null,
   "statement": "The derivative of a product of two functions is each function times the derivative of the other, added together.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the product y = u × v, a function of x"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "one function of x"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "another function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), u*Derivative(v, x) + v*Derivative(u, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/product",
    "theorem/product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-57dd510b08",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "40",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "\\dfrac{dy}{dx} &= \\dfrac{v\\, \\dfrac{du}{dx} - u\\, \\dfrac{dv}{dx}}{v^2}",
   "name": null,
   "statement": "The derivative of a quotient of two functions is the divisor times the derivative of the dividend, minus the dividend times the derivative of the divisor, all over the square of the divisor.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the quotient y = u / v, a function of x"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "dividend function of x"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "divisor function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), (v*Derivative(u, x) - u*Derivative(v, x))/v**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/quotient",
    "theorem/quotient-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f487d8da66",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "45",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "V = \\dfrac{H}{3} (A + a + \\sqrt{Aa} )",
   "name": null,
   "statement": "The volume of a frustum of a pyramid equals one third of its height times the sum of the two base areas and the square root of their product.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume of the frustum of a pyramid"
    },
    {
     "unit": null,
     "symbol": "H",
     "meaning": "height of the frustum"
    },
    {
     "unit": null,
     "symbol": "A",
     "meaning": "area of one base"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "area of the other base"
    }
   ],
   "sympy": "Eq(V, H/3*(A + a + sqrt(A*a)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/frustum-of-a-pyramid",
    "quantity/volume",
    "theorem/volume-of-a-frustum-of-a-pyramid"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0f28110e6a",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "45",
   "location": "Sums, Differences, Products and Quotients",
   "latex": "P = \\left( \\dfrac{40 + t}{140} \\right)^5",
   "name": "Dulong's formula for saturated steam",
   "statement": "Dulong's empirical relation gives the absolute pressure of saturated steam as the fifth power of (40 + t) divided by 140, valid for t above 80 degrees.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "atmosphere",
     "symbol": "P",
     "meaning": "absolute pressure of saturated steam"
    },
    {
     "unit": "degree Centigrade",
     "symbol": "t",
     "meaning": "temperature of the steam"
    }
   ],
   "sympy": "Eq(P, ((40 + t)/140)**5)",
   "physics": true,
   "states": [
    "theorem/dulong-s-formula-for-saturated-steam"
   ],
   "concepts": [
    "concept/pressure",
    "concept/temperature"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0239b587f4",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "49",
   "location": "Successive Differentiation",
   "latex": "y = x^5",
   "name": null,
   "statement": "The concrete function y = x^5 is differentiated repeatedly as a first example of successive differentiation.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, x**5)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2464303fa5",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "y = f(x) = x^n",
   "name": null,
   "statement": "A general power function y = f(x) = x^n is introduced to generalise the repeated differentiation.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "f(x)",
     "meaning": "general function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "exponent"
    }
   ],
   "sympy": "Eq(y, Function('f')(x), x**n)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/function",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b9ff4f629a",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "f'(x) &= nx^{n-1}",
   "name": null,
   "statement": "The first derivative of x^n is n x^(n-1).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "f(x)",
     "meaning": "general function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "exponent"
    }
   ],
   "sympy": "Eq(Derivative(f(x), x), n*x**(n - 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "method/differentiation",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ea644f753a",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "f''(x) &= n(n-1)x^{n-2}",
   "name": null,
   "statement": "The second derivative of x^n is n(n-1) x^(n-2).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "f(x)",
     "meaning": "general function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "exponent"
    }
   ],
   "sympy": "Eq(Derivative(f(x), (x, 2)), n*(n - 1)*x**(n - 2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e6f0a20bb7",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "f'''(x) &= n(n-1)(n-2)x^{n-3}",
   "name": null,
   "statement": "The third derivative of x^n is n(n-1)(n-2) x^(n-3).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "f(x)",
     "meaning": "general function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "exponent"
    }
   ],
   "sympy": "Eq(Derivative(f(x), (x, 3)), n*(n - 1)*(n - 2)*x**(n - 3))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2495612627",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "f''''(x) &= n(n-1)(n-2)(n-3)x^{n-4}",
   "name": null,
   "statement": "The fourth derivative of x^n is n(n-1)(n-2)(n-3) x^(n-4).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "f(x)",
     "meaning": "general function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "exponent"
    }
   ],
   "sympy": "Eq(Derivative(f(x), (x, 4)), n*(n - 1)*(n - 2)*(n - 3)*x**(n - 4))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-95964be744",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "49",
   "location": "Successive Differentiation",
   "latex": "y &= f(x)",
   "name": null,
   "statement": "The original function is written y = f(x), so y is a function of x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "f(x)",
     "meaning": "general function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, Function('f')(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e7c0e41e3a",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "\\frac{dy}{dx} = f'(x)",
   "name": null,
   "statement": "The differential coefficient of y with respect to x is denoted f'(x), the derived function.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "f'(x)",
     "meaning": "derived function of x"
    }
   ],
   "sympy": "Eq(Derivative(y, x), Derivative(f(x), x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2238ea562a",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "\\frac{d\\left(\\dfrac{dy}{dx}\\right)}{dx} &= f''(x)",
   "name": null,
   "statement": "Differentiating dy/dx again gives the second derived function f''(x).",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "f''(x)",
     "meaning": "second derived function of x"
    }
   ],
   "sympy": "Eq(Derivative(Derivative(y, x), x), Derivative(f(x), (x, 2)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a6c46d0e4b",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "50",
   "location": "Successive Differentiation",
   "latex": "\\dfrac{d^3y}{dx^3} = f'''(x)",
   "name": null,
   "statement": "Thrice differentiating y with respect to x is written d^3y/dx^3 and equals f'''(x).",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "f'''(x)",
     "meaning": "third derived function of x"
    }
   ],
   "sympy": "Eq(Derivative(y, (x, 3)), Derivative(f(x), (x, 3)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4c6e72f8ea",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "y = f(x) = 7x^4 + 3.5x^3 - \\frac{1}{2}x^2 + x - 2",
   "name": null,
   "statement": "A specific polynomial y = f(x) is chosen as an example for successive differentiation.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "f(x)",
     "meaning": "general function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, Function('f')(x), 7*x**4 + Rational(7, 2)*x**3 - Rational(1, 2)*x**2 + x - 2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e45a286b18",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "\\frac{dy}{dx}     &= f'(x) = 28x^3 + 10.5x^2 - x + 1",
   "name": null,
   "statement": "The first derivative of the example polynomial is 28x^3 + 10.5x^2 - x + 1.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "f'(x)",
     "meaning": "derived function of x"
    }
   ],
   "sympy": "Eq(Derivative(y, x), Derivative(7*x**4 + Rational(7, 2)*x**3 - Rational(1, 2)*x**2 + x - 2, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/polynomial",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-119a5c8009",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "\\frac{d^2y}{dx^2} &= f''(x) = 84x^2 + 21x - 1",
   "name": null,
   "statement": "The second derivative of the example polynomial is 84x^2 + 21x - 1.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "f''(x)",
     "meaning": "second derived function of x"
    }
   ],
   "sympy": "Eq(Derivative(y, (x, 2)), 84*x**2 + 21*x - 1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3db345c570",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "\\frac{d^3y}{dx^3} &= f'''(x) = 168x + 21",
   "name": null,
   "statement": "The third derivative of the example polynomial is 168x + 21.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "f'''(x)",
     "meaning": "third derived function of x"
    }
   ],
   "sympy": "Eq(Derivative(y, (x, 3)), 168*x + 21)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-cfe5ebfaff",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "\\frac{d^4y}{dx^4} &= f''''(x) = 168",
   "name": null,
   "statement": "The fourth derivative of the example polynomial is the constant 168.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "f''''(x)",
     "meaning": "fourth derived function of x"
    }
   ],
   "sympy": "Eq(Derivative(y, (x, 4)), 168)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/higher-order-derivative",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1a2449be3c",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "\\frac{d^5y}{dx^5} &= f'''''(x) = 0",
   "name": null,
   "statement": "The fifth derivative of the example polynomial is zero.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "f'''''(x)",
     "meaning": "fifth derived function of x"
    }
   ],
   "sympy": "Eq(Derivative(y, (x, 5)), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ce6bc7cdd3",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "y = \\phi(x) = 3x(x^2 - 4)",
   "name": null,
   "statement": "A second example function y = phi(x) = 3x(x^2 - 4) is given for successive differentiation.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "phi(x)",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, 3*x*(x**2 - 4))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-22d6a162a7",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "\\phi'(x)    &= \\frac{dy}{dx} = 3\\bigl[x × 2x + (x^2 - 4) × 1\\bigr] = 3(3x^2 - 4)",
   "name": null,
   "statement": "The first derivative of 3x(x^2 - 4) is 3(3x^2 - 4), obtained by the product rule.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "phi'(x)",
     "meaning": "derived function of phi(x)"
    }
   ],
   "sympy": "Eq(Derivative(3*x*(x**2 - 4), x), 3*(3*x**2 - 4))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/polynomial",
    "theorem/product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b9c02bf07e",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "\\phi''(x)   &= \\frac{d^2y}{dx^2} = 3 × 6x = 18x",
   "name": null,
   "statement": "The second derivative of the second example function is 18x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "phi''(x)",
     "meaning": "second derived function of phi(x)"
    }
   ],
   "sympy": "Eq(Derivative(3*x*(x**2 - 4), (x, 2)), 18*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d3190b81fa",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "\\phi'''(x)  &= \\frac{d^3y}{dx^3} = 18",
   "name": null,
   "statement": "The third derivative of the second example function is the constant 18.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "phi'''(x)",
     "meaning": "third derived function of phi(x)"
    }
   ],
   "sympy": "Eq(Derivative(3*x*(x**2 - 4), (x, 3)), 18)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/higher-order-derivative",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b5c3308c2a",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Successive Differentiation",
   "latex": "\\phi''''(x) &= \\frac{d^4y}{dx^4} = 0",
   "name": null,
   "statement": "The fourth derivative of the second example function is zero.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "phi''''(x)",
     "meaning": "fourth derived function of phi(x)"
    }
   ],
   "sympy": "Eq(Derivative(3*x*(x**2 - 4), (x, 4)), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2b97326852",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "55",
   "location": "When Time Varies",
   "latex": "v = \\dfrac{dy}{dt}",
   "name": null,
   "statement": "Velocity is the rate at which distance changes with time: the differential coefficient of distance with respect to time.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "v",
     "meaning": "velocity"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "distance passed over"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(v, Derivative(y, t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/rate-of-change",
    "quantity/distance",
    "quantity/time",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-030c1614f7",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "55",
   "location": "When Time Varies",
   "latex": "a = \\dfrac{dv}{dt}",
   "name": null,
   "statement": "Acceleration is the rate at which velocity changes with time.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "acceleration at an instant"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "velocity"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(a, Derivative(v, t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/rate-of-change",
    "quantity/acceleration",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-469d8dbfa7",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "56",
   "location": "When Time Varies",
   "latex": "a = \\frac{d\\left( \\dfrac{dy}{dt} \\right)}{dt}",
   "name": null,
   "statement": "Acceleration is the second differential coefficient of distance with respect to time.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "acceleration"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "distance passed over"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(a, Derivative(y, (t, 2)))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative",
    "quantity/acceleration",
    "quantity/distance",
    "quantity/time"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1392020674",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "56",
   "location": "When Time Varies",
   "latex": "a = \\dfrac{d^2y}{dt^2}",
   "name": null,
   "statement": "The acceleration equals the second derivative of distance with respect to time.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "acceleration"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "distance passed over"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(a, Derivative(y, (t, 2)))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "quantity/acceleration",
    "quantity/distance",
    "quantity/time"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-11d6cc09c6",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "57",
   "location": "When Time Varies",
   "latex": "f = m \\frac{dv}{dt}",
   "name": null,
   "statement": "The force needed to accelerate a mass is the mass times the rate of change of velocity, that is, mass times acceleration.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "f",
     "meaning": "force"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "mass"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "velocity"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(f, m*Derivative(v, t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/rate-of-change",
    "quantity/acceleration",
    "quantity/force",
    "quantity/mass",
    "quantity/momentum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7f03c75e44",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "57",
   "location": "When Time Varies",
   "latex": "w = f × y",
   "name": null,
   "statement": "For a constant force, the work done equals the force times the distance moved in its own direction.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "w",
     "meaning": "work done"
    },
    {
     "unit": null,
     "symbol": "f",
     "meaning": "constant force"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "length through which the force moves"
    }
   ],
   "sympy": "Eq(w, f*y)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/work",
    "quantity/distance",
    "quantity/force"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a3d741432a",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "59",
   "location": "When Time Varies",
   "latex": "v = \\dot{x}",
   "name": null,
   "statement": "In fluxional notation, velocity is the dot over the distance symbol, meaning the derivative of distance with respect to time.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "v",
     "meaning": "velocity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "distance"
    }
   ],
   "sympy": "Eq(v, Derivative(x, t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/fluxional-notation",
    "quantity/distance",
    "quantity/time",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-00d5f13566",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "59",
   "location": "When Time Varies",
   "latex": "a = \\dot{v} = \\ddot{x}",
   "name": null,
   "statement": "In fluxional notation, acceleration is the dot over velocity, equal to the double dot over distance, with time as the independent variable.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "acceleration"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "velocity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "distance"
    }
   ],
   "sympy": "Eq(a, Derivative(v, t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/fluxional-notation",
    "concept/higher-order-derivative",
    "quantity/acceleration",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9a49794374",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "59",
   "location": "When Time Varies",
   "latex": "f = m\\dot{v} = m\\ddot{x}",
   "name": null,
   "statement": "Force equals mass times the rate of change of velocity, which is mass times the second time-derivative of distance.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "f",
     "meaning": "force"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "mass"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "velocity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "distance"
    }
   ],
   "sympy": "Eq(f, m*Derivative(v, t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/fluxional-notation",
    "concept/rate-of-change",
    "quantity/acceleration",
    "quantity/force",
    "quantity/mass"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d6838a5b6c",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "59",
   "location": "When Time Varies",
   "latex": "w = x × m \\ddot{x}",
   "name": null,
   "statement": "In fluxional notation the work is written as distance times mass times the second time-derivative of distance.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "w",
     "meaning": "work"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "distance"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "mass"
    }
   ],
   "sympy": "Eq(w, x*m*Derivative(x, (t, 2)))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/fluxional-notation",
    "concept/higher-order-derivative",
    "concept/work",
    "quantity/distance",
    "quantity/mass"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-de7b3de8c1",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "67",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dy}{du} = \\frac{3}{2} u^{\\efrac{1}{2}}",
   "name": null,
   "statement": "Differentiating y = u^(3/2) with respect to u gives three halves times u to the power one half.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function being differentiated, y = (x^2+a^2)^(3/2) in the dodge"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "the substituted expression x^2 + a^2"
    }
   ],
   "sympy": "Eq(dydu, Rational(3,2)*u**Rational(1,2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-84ea8aa82d",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "67",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{du}{dx} = 2x",
   "name": null,
   "statement": "The derivative of u = x^2 + a^2 with respect to x is 2x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "the substituted expression x^2 + a^2"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dudx, 2*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-20028de417",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "68",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dy}{dx} = \\frac{dy}{du}×\\frac{du}{dx}",
   "name": "Chain rule",
   "statement": "The derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function being differentiated"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "the substituted expression standing for a function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, dydu*dudx)",
   "physics": false,
   "states": [
    "theorem/chain-rule"
   ],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-bbb20245af",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "68",
   "location": "Introducing a Useful Dodge",
   "latex": "&= 3x(x^2 + a^2)^{\\efrac{1}{2}}",
   "name": null,
   "statement": "The derivative of (x^2 + a^2)^(3/2) with respect to x is 3x times (x^2 + a^2) to the power one half.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    }
   ],
   "sympy": "Eq(dydx, 3*x*(x**2+a**2)**Rational(1,2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-88d96171cf",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "68",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dy}{dx} &= \\frac{dy}{du} × \\frac{du}{dx} = \\frac{1}{2\\sqrt{a+x}}",
   "name": null,
   "statement": "The derivative of y = sqrt(a+x) with respect to x is 1 over twice the square root of a+x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function sqrt(a+x)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, 1/(2*sqrt(a+x)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-950a17643c",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "68",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dy}{dx} &= \\frac{dy}{du}×\\frac{du}{dx} = - \\frac{x}{\\sqrt{(a+x^2)^3}}",
   "name": null,
   "statement": "The derivative of y = 1/sqrt(a+x^2) with respect to x is minus x over the square root of (a+x^2) cubed.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function 1/sqrt(a+x^2)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, -x/sqrt((a+x**2)**3))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-691de85314",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "69",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dy}{dx} &= \\frac{dy}{du} × \\frac{du}{dx} = -\\frac{3x^2}{2\\sqrt{(x^3 - a^2)^3}}",
   "name": null,
   "statement": "The derivative of y = 1/sqrt(x^3 - a^2) with respect to x is minus 3x^2 over twice the square root of (x^3 - a^2) cubed.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function 1/sqrt(x^3 - a^2)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, -3*x**2/(2*sqrt((x**3-a**2)**3)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3a83076c90",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "70",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dy}{dx} &= - \\frac{1}{(1+x)\\sqrt{1-x^2}}",
   "name": null,
   "statement": "The derivative of y = sqrt((1-x)/(1+x)) with respect to x is minus 1 over (1+x) times the square root of (1-x^2).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function sqrt((1-x)/(1+x))"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, -1/((1+x)*sqrt(1-x**2)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/product-rule-for-differentiation",
    "theorem/quotient-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ae836b5c89",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "70",
   "location": "Introducing a Useful Dodge",
   "latex": "= \\frac{\\sqrt{x}(3+x^2)}{2\\sqrt{(1+x^2)^3}}",
   "name": null,
   "statement": "The derivative of y = sqrt(x^3/(1+x^2)) with respect to x is sqrt(x) times (3+x^2), over twice the square root of (1+x^2) cubed.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function sqrt(x^3/(1+x^2))"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, sqrt(x)*(3+x**2)/(2*sqrt((1+x**2)**3)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation",
    "theorem/product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0654b7a640",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "71",
   "location": "Introducing a Useful Dodge",
   "latex": "&= 3\\left(x+\\sqrt{x^2+x+a}\\right)^2 \\left(1 +\\frac{2x+1}{2\\sqrt{x^2+x+a}}\\right)",
   "name": null,
   "statement": "The derivative of y = (x + sqrt(x^2+x+a))^3 with respect to x is three times the square of (x + sqrt(x^2+x+a)), times (1 + (2x+1) over twice sqrt(x^2+x+a)).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function (x + sqrt(x^2+x+a))^3"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, 3*(x+sqrt(x**2+x+a))**2*(1+(2*x+1)/(2*sqrt(x**2+x+a))))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1d601b014e",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "73",
   "location": "Introducing a Useful Dodge",
   "latex": "= \\frac{x(3a-4x)}{2b\\sqrt{(a-x)x}}",
   "name": null,
   "statement": "The first derivative of y = (x/b) sqrt((a-x)x) with respect to x is x(3a-4x) over 2b times the square root of (a-x)x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function (x/b) sqrt((a-x)x)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, x*(3*a-4*x)/(2*b*sqrt((a-x)*x)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation",
    "theorem/product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-995e34b769",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "73",
   "location": "Introducing a Useful Dodge",
   "latex": "&= \\frac{3a^2-12ax+8x^2}{4b(a-x)\\sqrt{(a-x)x}}",
   "name": null,
   "statement": "The second derivative of y = (x/b) sqrt((a-x)x) with respect to x is (3a^2 - 12ax + 8x^2) over 4b(a-x) times the square root of (a-x)x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function (x/b) sqrt((a-x)x)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(d2ydx2, (3*a**2-12*a*x+8*x**2)/(4*b*(a-x)*sqrt((a-x)*x)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-45dbca0671",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "72",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{d(y^n)}{d(y^5)} = \\frac{ny^{n-1}}{5y^{5-1}} = \\frac{n}{5} y^{n-5}",
   "name": null,
   "statement": "The derivative of y^n with respect to y^5 equals n/5 times y to the power n minus 5.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a constant exponent"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the variable being raised to powers"
    }
   ],
   "sympy": "Eq(dyn_dy5, n*y**(n-5)/5)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3914dd9c18",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "74",
   "location": "Introducing a Useful Dodge",
   "latex": "\\dfrac{dy}{dx} = \\dfrac{dy}{dz} × \\dfrac{dz}{dv} × \\dfrac{dv}{dx}",
   "name": "Chain rule",
   "statement": "The chain rule extends to any number of intermediate quantities, so dy/dx is the product of successive derivatives along the chain of variables.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the dependent quantity at the end of the chain"
    },
    {
     "unit": null,
     "symbol": "z",
     "meaning": "an intermediate quantity"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "an intermediate quantity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, dydz*dzdv*dvdx)",
   "physics": false,
   "states": [
    "theorem/chain-rule"
   ],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7d41bcaf3c",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "74",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dv}{dx} = \\frac{7x(5x-6)}{3\\sqrt[3]{(x-1)^4}}",
   "name": null,
   "statement": "The derivative of v = 7x^2 / cube root of (x-1) with respect to x is 7x(5x-6) over 3 times the cube root of (x-1)^4.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "v",
     "meaning": "the function 7x^2/cube root(x-1)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dvdx, 7*x*(5*x-6)/(3*(x-1)**Rational(4,3)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/power-rule-for-differentiation",
    "theorem/quotient-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b4e5b7cb08",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "74",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dx}{dt} = 3t^2 + \\tfrac{1}{2}",
   "name": null,
   "statement": "The derivative of x = t^3 + t/2 with respect to t is 3t^2 + 1/2.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the function t^3 + t/2"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "an intermediate variable"
    }
   ],
   "sympy": "Eq(dxdt, 3*t**2+Rational(1,2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-fae65530ab",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "74",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dt}{d\\theta} = -\\frac{1}{10\\sqrt{\\theta^3}}",
   "name": null,
   "statement": "The derivative of t = 1/(5 sqrt(theta)) with respect to theta is minus 1 over 10 times the square root of theta cubed.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "t",
     "meaning": "the function 1/(5 sqrt(theta))"
    },
    {
     "unit": null,
     "symbol": "theta",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dtdtheta, -1/(10*sqrt(theta**3)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5d8083c40b",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "74",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{dv}{d\\theta} = -\\frac{7x(5x-6)(3t^2+\\frac{1}{2})} {30\\sqrt[3]{(x-1)^4} \\sqrt{\\theta^3}}",
   "name": null,
   "statement": "The derivative of v with respect to theta, by the chain of intermediate variables, is minus 7x(5x-6)(3t^2 + 1/2) over 30 times the cube root of (x-1)^4 times the square root of theta cubed.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "v",
     "meaning": "the function 7x^2/cube root(x-1)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "intermediate variable t^3 + t/2"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "intermediate variable 1/(5 sqrt(theta))"
    },
    {
     "unit": null,
     "symbol": "theta",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dvdtheta, -7*x*(5*x-6)*(3*t**2+Rational(1,2))/(30*(x-1)**Rational(4,3)*sqrt(theta**3)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3233f32bc3",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "74",
   "location": "Introducing a Useful Dodge",
   "latex": "= -\\frac{28}{3x^5\\sqrt{9x^8+7}}",
   "name": null,
   "statement": "For y = sqrt(1+v), v = 7/z^2, z = 3x^4, the book's chain of derivatives gives this value, which it labels dy/dx although the question asks for dv/dx.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function sqrt(1+v)"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "the function 7/z^2"
    },
    {
     "unit": null,
     "symbol": "z",
     "meaning": "the function 3x^4"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    }
   ],
   "sympy": "Eq(dydx, -28/(3*x**5*sqrt(9*x**8+7)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a614f9d5d0",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "75",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{d\\phi}{d\\omega} = \\frac{1}{\\sqrt{2}\\omega^2}",
   "name": null,
   "statement": "The derivative of phi = sqrt(3) - 1/(omega sqrt(2)) with respect to omega is 1 over sqrt(2) times omega squared.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "the function sqrt(3) - 1/(omega sqrt(2))"
    },
    {
     "unit": null,
     "symbol": "omega",
     "meaning": "the intermediate variable sqrt((1-theta)/(1+theta))"
    }
   ],
   "sympy": "Eq(dphidomega, 1/(sqrt(2)*omega**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2f9b1c3b73",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "75",
   "location": "Introducing a Useful Dodge",
   "latex": "\\frac{d\\omega}{d\\theta} = -\\frac{1}{(1+\\theta)\\sqrt{1-\\theta^2}}",
   "name": null,
   "statement": "The derivative of omega with respect to theta is minus 1 over (1+theta) times the square root of (1-theta^2).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "omega",
     "meaning": "the intermediate variable sqrt(1-theta^2)/(1+theta)"
    },
    {
     "unit": null,
     "symbol": "theta",
     "meaning": "the intermediate variable 3a^2 x / sqrt(x^3)"
    }
   ],
   "sympy": "Eq(domegadtheta, -1/((1+theta)*sqrt(1-theta**2)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "theorem/quotient-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1a76990960",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "83",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "y=x+b",
   "name": null,
   "statement": "A straight line of slope 1 that crosses the y-axis at height b and rises at 45 degrees.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa, the horizontal coordinate of a point on the curve"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate, the vertical coordinate of a point on the curve"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant height at which the curve crosses the y-axis"
    }
   ],
   "sympy": "Eq(y, x + b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/function",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7768d9fff6",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "78",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "\\dfrac{dy}{dx} = 1",
   "name": null,
   "statement": "Differentiating y = x + b gives a slope of 1 at every point, so the line has constant slope.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "short step upward along the curve"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "short step to the right along the curve"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9a63a2d72f",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "84",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "y = ax+b",
   "name": null,
   "statement": "A straight line with constant slope a that crosses the y-axis at height b.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant slope of the line, the tangent of its angle of slope"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant height at which the line crosses the y-axis"
    }
   ],
   "sympy": "Eq(y, a*x + b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/function",
    "concept/linear-equation",
    "concept/slope-of-a-curve"
   ],
   "pages": [
    "84",
    "185"
   ],
   "chapters": [
    "thompson-calculus-made-easy-1914/ch-x",
    "thompson-calculus-made-easy-1914/ch-xvii"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ab730b882d",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "84",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "\\dfrac{dy}{dx} = a",
   "name": null,
   "statement": "Differentiating y = ax + b gives a constant slope equal to a.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant slope of the line"
    }
   ],
   "sympy": "Eq(Derivative(y, x), a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/derivative",
    "concept/slope-of-a-curve"
   ],
   "pages": [
    "84",
    "186"
   ],
   "chapters": [
    "thompson-calculus-made-easy-1914/ch-x",
    "thompson-calculus-made-easy-1914/ch-xvii"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-fa903cb6b7",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "85",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "y = ax^2 + b",
   "name": null,
   "statement": "A parabola with its vertex at height b on the y-axis, whose steepness changes with x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient of x squared"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant height at which the curve meets the y-axis"
    }
   ],
   "sympy": "Eq(y, a*x**2 + b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/curve",
    "concept/function",
    "concept/power"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dd9cc42873",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "85",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "\\frac{dy}{dx} = 2ax",
   "name": null,
   "statement": "Differentiating y = ax^2 + b gives a slope that is proportional to x, so the steepness increases with x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient of x squared"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 2*a*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/slope-of-a-curve",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7cccfd51b2",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "79",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "\\dfrac{dy}{dx}=0",
   "name": null,
   "statement": "At a horizontal place on a curve the slope dy/dx is zero.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/slope-of-a-curve",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8dc090ccf1",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "79",
   "location": "Geometrical Meaning of Differentiation",
   "latex": "\\dfrac{dy}{dx}= 0",
   "name": null,
   "statement": "At the x where y is a minimum or a maximum, the slope dy/dx is zero.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa of the curve"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/maximum",
    "concept/minimum",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-97d11fd666",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "93",
   "location": "Maxima and Minima",
   "latex": "y = x^2 - 4x + 7.",
   "name": null,
   "statement": "The example function whose curve has a minimum, used to introduce the method.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, x**2 - 4*x + 7)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/minimum",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-33a88a2752",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "94",
   "location": "Maxima and Minima",
   "latex": "y = 3x - x^2",
   "name": null,
   "statement": "The second example function, whose curve has a maximum between x = 1 and x = 2.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, 3*x - x**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/maximum",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ee70a11391",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "96",
   "location": "Maxima and Minima",
   "latex": "\\dfrac{dy}{dx} = 2x - 4",
   "name": null,
   "statement": "The derivative of y = x^2 - 4x + 7 with respect to x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 2*x - 4)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8502897ee8",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "96",
   "location": "Maxima and Minima",
   "latex": "2x - 4 = 0",
   "name": null,
   "statement": "The condition obtained by equating the derivative to zero, whose solution gives the stationary value of x.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(2*x - 4, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/method-equating-the-derivative-to-zero",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5ac14c66fc",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "97",
   "location": "Maxima and Minima",
   "latex": "\\frac{dy}{dx} = 0",
   "name": null,
   "statement": "The condition that the curve is neither rising nor falling, written as an equation of condition to be solved for x.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "differential coefficient of y with respect to x"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/equation",
    "concept/method-equating-the-derivative-to-zero",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0b5060d776",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "98",
   "location": "Maxima and Minima",
   "latex": "y = 4x + \\frac{1}{x}.",
   "name": null,
   "statement": "An example function used to show that equating the derivative to zero does not say whether the stationary value is a maximum or a minimum.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, 4*x + 1/x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/maximum",
    "concept/minimum",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4bf18f4852",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "99",
   "location": "Maxima and Minima",
   "latex": "y = nx - x^2",
   "name": null,
   "statement": "The product y of the two parts when a number n is cut into parts x and n - x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "product of the two parts"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the number cut into two parts"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "one of the two parts"
    }
   ],
   "sympy": "Eq(y, n*x - x**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/maximum",
    "concept/product",
    "method/stating-a-problem-as-an-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5ecac923e2",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "99",
   "location": "Maxima and Minima",
   "latex": "\\dfrac{n}{2} = x",
   "name": null,
   "statement": "The value of one part for which the product of the two parts is a maximum, namely half the number.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the number cut into two parts"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "one of the two parts"
    }
   ],
   "sympy": "Eq(x, n/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/maximum",
    "concept/method-equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f06c657a66",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "101",
   "location": "Maxima and Minima",
   "latex": "x^2 - 4x +3 = 0",
   "name": null,
   "statement": "The quadratic obtained by equating dy/dx to zero for y = (1/3)x^3 - 2x^2 + 3x + 1, whose two roots give the maximum and minimum.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(x**2 - 4*x + 3, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/quadratic-equation",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ae47a5c558",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "101",
   "location": "Maxima and Minima",
   "latex": "y =\\tfrac{1}{3} x^3 - 2x^2 + 3x + 1.",
   "name": null,
   "statement": "The cubic function in the further examples, which has both a maximum and a minimum.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, Rational(1,3)*x**3 - 2*x**2 + 3*x + 1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/maximum",
    "concept/minimum",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ae1cabf995",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "102",
   "location": "Maxima and Minima",
   "latex": "(y-b)^2 + (x-a)^2 = r^2.",
   "name": null,
   "statement": "The equation of a circle of radius r with centre at (a, b).",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal coordinate of a point on the circle"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "vertical coordinate of a point on the circle"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "x-coordinate of the centre"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "y-coordinate of the centre"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq((y - b)**2 + (x - a)**2, r**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/equation",
    "quantity/radius"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2e6a185a09",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "102",
   "location": "Maxima and Minima",
   "latex": "y = \\sqrt{r^2-(x-a)^2} + b.",
   "name": null,
   "statement": "The upper-branch form of the circle's equation, solved for y.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "vertical coordinate on the circle"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal coordinate on the circle"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "x-coordinate of the centre"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "y-coordinate of the centre"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(y, sqrt(r**2 - (x - a)**2) + b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/function",
    "quantity/radius"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a28f2dc210",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "103",
   "location": "Maxima and Minima",
   "latex": "\\frac{a-x}{\\sqrt{r^2-(x-a)^2}} = 0.",
   "name": null,
   "statement": "The condition for the circle's height y to be a maximum or minimum, which holds only at x = a.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal coordinate on the circle"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "x-coordinate of the centre"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq((a - x)/sqrt(r**2 - (x - a)**2), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/method-equating-the-derivative-to-zero",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9f9aa5bb6b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "104",
   "location": "Maxima and Minima",
   "latex": "3ax^2 + b = 0",
   "name": null,
   "statement": "The condition from equating dy/dx to zero for y = ax^3 + bx + c; it gives no real x when a and b have the same sign, so y has no maximum or minimum.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(3*a*x**2 + b, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/equation",
    "concept/method-equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1d793625ae",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "104",
   "location": "Maxima and Minima",
   "latex": "\\text{the other side} = \\sqrt{(\\text{diagonal})^2 - x^2}",
   "name": null,
   "statement": "The side of a rectangle inscribed in a circle, found from its diagonal, which is a diameter of the circle.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "one side of the rectangle"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/diagonal",
    "concept/rectangle",
    "concept/root"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-717a946266",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "104",
   "location": "Maxima and Minima",
   "latex": "S = x\\sqrt{4R^2 - x^2}",
   "name": null,
   "statement": "The area of a rectangle inscribed in a circle of radius R, with one side x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "S",
     "meaning": "area of the rectangle"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "one side of the rectangle"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(S, x*sqrt(4*R**2 - x**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/rectangle",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b65b6703b3",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "105",
   "location": "Maxima and Minima",
   "latex": "4R^2 - 2x^2 = 0",
   "name": null,
   "statement": "The condition from equating dS/dx to zero for the inscribed rectangle.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "one side of the rectangle"
    }
   ],
   "sympy": "Eq(4*R**2 - 2*x**2, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/method-equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-62732c6249",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "105",
   "location": "Maxima and Minima",
   "latex": "x = R\\sqrt{2}",
   "name": null,
   "statement": "The side of the maximum-area inscribed rectangle, which makes it a square.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "one side of the rectangle"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(x, R*sqrt(2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/maximum",
    "concept/rectangle",
    "concept/root"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-500aed3e51",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "105",
   "location": "Maxima and Minima",
   "latex": "H = \\sqrt{l^2 - R^2}",
   "name": null,
   "statement": "The height of a cone in terms of its slant length l and base radius R.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "H",
     "meaning": "height of the cone"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "sloping side (slant length) of the cone"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the base"
    }
   ],
   "sympy": "Eq(H, sqrt(l**2 - R**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cone",
    "concept/root",
    "quantity/height"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-607332e8f8",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "105",
   "location": "Maxima and Minima",
   "latex": "V = \\pi R^2 × \\dfrac{H}{3} = \\pi R^2 × \\dfrac{\\sqrt{l^2 - R^2}}{3}",
   "name": null,
   "statement": "The volume of a cone of base radius R and height H, with H expressed through l and R.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume of the cone"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the base"
    },
    {
     "unit": null,
     "symbol": "H",
     "meaning": "height of the cone"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "sloping side of the cone"
    }
   ],
   "sympy": "Eq(V, pi*R**2*H/3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cone",
    "quantity/radius",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-26b00246e4",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "105",
   "location": "Maxima and Minima",
   "latex": "2\\pi R(l^2 - R^2) - \\pi R^2 = 0",
   "name": null,
   "statement": "The condition printed in the cone problem after dV/dR is set to zero. FLAG: this does not follow from the line before it, which has numerator 2\\pi R(l^2 - R^2) - \\pi R^3; the book's printed R^2 appears to be a typo for R^3, and the stated result R = l\\sqrt{2/3} follows only from the R^3 version.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the base"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "sloping side of the cone"
    }
   ],
   "sympy": "Eq(2*pi*R*(l**2 - R**2) - pi*R**2, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cone",
    "concept/equation",
    "concept/method-equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4216dad603",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "105",
   "location": "Maxima and Minima",
   "latex": "R = l\\sqrt{\\tfrac{2}{3}}",
   "name": null,
   "statement": "The base radius of the cone of maximum volume for a given slant length l.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the base"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "sloping side of the cone"
    }
   ],
   "sympy": "Eq(R, l*sqrt(2/3))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cone",
    "concept/maximum",
    "concept/root"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-806df7957f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "106",
   "location": "Maxima and Minima",
   "latex": "y = \\dfrac{x}{4-x} + \\dfrac{4-x}{x}",
   "name": null,
   "statement": "The function of the third worked example, which has a single minimum at x = 2.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, x/(4 - x) + (4 - x)/x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/minimum",
    "concept/rational-expression"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c11c740a30",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "106",
   "location": "Maxima and Minima",
   "latex": "\\dfrac{4}{(4-x)^2} - \\dfrac{4}{x^2} = 0",
   "name": null,
   "statement": "The condition from equating dy/dx to zero for the third example, which gives x = 2.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(4/(4 - x)**2 - 4/x**2, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/method-equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3f1951f80c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "106",
   "location": "Maxima and Minima",
   "latex": "y = \\sqrt{1+x} + \\sqrt{1-x}",
   "name": null,
   "statement": "The function of the fourth worked example, which has a maximum at x = 0.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, sqrt(1 + x) + sqrt(1 - x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/maximum",
    "concept/root"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-15deb18216",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "106",
   "location": "Maxima and Minima",
   "latex": "\\dfrac{dy}{dx} = \\dfrac{1}{2\\sqrt{1+x}} - \\dfrac{1}{2\\sqrt{1-x}} = 0",
   "name": null,
   "statement": "The derivative of the fourth example set equal to zero for maximum or minimum.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(1/(2*sqrt(1 + x)) - 1/(2*sqrt(1 - x)), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/equation",
    "concept/method-equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-800ec7f740",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "106",
   "location": "Maxima and Minima",
   "latex": "\\sqrt{1+x} = \\sqrt{1-x}",
   "name": null,
   "statement": "The condition whose only solution is x = 0 in the fourth example.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(sqrt(1 + x), sqrt(1 - x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/root"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4ebf478238",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "107",
   "location": "Maxima and Minima",
   "latex": "y = \\dfrac{x^2-5}{2x-4}",
   "name": null,
   "statement": "The function of the fifth worked example, which has neither a maximum nor a minimum.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value of the function"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, (x**2 - 5)/(2*x - 4))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/rational-expression"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dbb06339ed",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "107",
   "location": "Maxima and Minima",
   "latex": "x^2 - 4x + 5 = 0",
   "name": null,
   "statement": "The quadratic obtained from dy/dx = 0 in the fifth example; its roots are 2 ± i, not the 5/2 ± i printed in the next line.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(x**2 - 4*x + 5, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/imaginary-root",
    "concept/quadratic-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-402756a770",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "107",
   "location": "Maxima and Minima",
   "latex": "x = \\tfrac{5}{2} ± \\sqrt{-1}",
   "name": null,
   "statement": "FLAG: the book's printed roots of x^2 - 4x + 5 = 0 are wrong; completing the square gives x = 2 ± i. The conclusion that there are no real roots, hence no maximum or minimum, is unaffected.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/imaginary-root",
    "concept/quadratic-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-77179615e9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "107",
   "location": "Maxima and Minima",
   "latex": "(y-x^2)^2 = x^5",
   "name": null,
   "statement": "The implicit curve of the sixth example, which splits into two branches y = x^2 ± x^(5/2).",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    }
   ],
   "sympy": "Eq((y - x**2)**2, x**5)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/implicit-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-6d5b7a3bcb",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "107",
   "location": "Maxima and Minima",
   "latex": "y = x^2 ± x^{\\efrac{5}{2}}",
   "name": null,
   "statement": "The sixth example's curve written as two explicit functions of x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/explicit-function",
    "concept/function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-02ea1873db",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "107",
   "location": "Maxima and Minima",
   "latex": "\\dfrac{dy}{dx} = 2x ± \\tfrac{5}{2} x^{\\efrac{3}{2}} = 0",
   "name": null,
   "statement": "The derivative of the two branches set equal to zero for a maximum or minimum.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/method-equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2f86321752",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "107",
   "location": "Maxima and Minima",
   "latex": "2 ± \\tfrac{5}{2} x^{\\efrac{1}{2}} = 0",
   "name": null,
   "statement": "The condition on the non-zero stationary point of the sixth example.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/equation",
    "concept/root"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3616554166",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "107",
   "location": "Maxima and Minima",
   "latex": "x = \\tfrac{16}{25}",
   "name": null,
   "statement": "The second stationary value of x for the sixth example, obtained from the non-zero condition.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(x, Rational(16, 25))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-725d6b5fbb",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "109",
   "location": "Maxima and Minima",
   "latex": "S = 2(\\pi r^2)+ 2 \\pi r × 2r = 6 \\pi r^2",
   "name": null,
   "statement": "The total surface area of a cylinder whose height is twice the radius r of its base.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "S",
     "meaning": "total area of the cylinder"
    },
    {
     "unit": "foot",
     "symbol": "r",
     "meaning": "radius of the base"
    }
   ],
   "sympy": "Eq(S, 2*(pi*r**2) + 2*pi*r*2*r)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cylinder",
    "quantity/radius",
    "quantity/total-area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5656f64a41",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "109",
   "location": "Maxima and Minima",
   "latex": "V = \\pi r^2 × 2r=2 \\pi r^3",
   "name": null,
   "statement": "The volume of a cylinder whose height is twice the radius r of its base.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume of the cylinder"
    },
    {
     "unit": "foot",
     "symbol": "r",
     "meaning": "radius of the base"
    }
   ],
   "sympy": "Eq(V, 2*pi*r**3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cylinder",
    "quantity/radius",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-93ef987bb2",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "109",
   "location": "Maxima and Minima",
   "latex": "\\frac{dS}{dr} = 12\\pi r",
   "name": null,
   "statement": "The rate of change of the surface area with the base radius r.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "S",
     "meaning": "total area of the cylinder"
    },
    {
     "unit": "foot",
     "symbol": "r",
     "meaning": "radius of the base"
    }
   ],
   "sympy": "Eq(Derivative(S, r), 12*pi*r)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/rate-of-change",
    "quantity/total-area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-876bb3bc96",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "109",
   "location": "Maxima and Minima",
   "latex": "\\frac{dV}{dr}=6 \\pi r^2",
   "name": null,
   "statement": "The rate of change of the volume with the base radius r.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume of the cylinder"
    },
    {
     "unit": "foot",
     "symbol": "r",
     "meaning": "radius of the base"
    }
   ],
   "sympy": "Eq(Derivative(V, r), 6*pi*r**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/rate-of-change",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5025b30a18",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "116",
   "location": "Curvature of Curves",
   "latex": "C = aP + \\dfrac{b}{c+P} + d",
   "name": null,
   "statement": "The expense C of handling the factory's products is modelled as a function of the weekly output P, plus a term b/(c+P) and a constant d.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "C",
     "meaning": "expense of handling the products"
    },
    {
     "unit": null,
     "symbol": "P",
     "meaning": "weekly output"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "positive constant"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "positive constant"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "positive constant"
    },
    {
     "unit": null,
     "symbol": "d",
     "meaning": "positive constant"
    }
   ],
   "sympy": "Eq(C, a*P + b/(c + P) + d)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f4179b8d63",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "116",
   "location": "Curvature of Curves",
   "latex": "\\dfrac{dC}{dP} = a - \\frac{b}{(c+P)^2} = 0",
   "name": null,
   "statement": "Setting the derivative of the expense with respect to the output to zero gives the candidate outputs for a maximum or minimum.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "C",
     "meaning": "expense of handling the products"
    },
    {
     "unit": null,
     "symbol": "P",
     "meaning": "weekly output"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "positive constant"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "positive constant"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "positive constant"
    }
   ],
   "sympy": "Eq(a - b/(c + P)**2, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/stationary-value",
    "method/equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-6206efcc63",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "116",
   "location": "Curvature of Curves",
   "latex": "P = ±\\sqrt{\\dfrac{b}{a}} - c",
   "name": null,
   "statement": "The stationary output is P equal to the square root of b/a minus c; the book keeps only the positive sign, since output cannot be negative.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "P",
     "meaning": "weekly output"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "positive constant"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "positive constant"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "positive constant"
    }
   ],
   "sympy": "Eq(P, sqrt(b/a) - c)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/root",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dd77c51d00",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "117",
   "location": "Curvature of Curves",
   "latex": "C = N\\left(\\frac{C_l}{t} + \\frac{EPC_e}{1000}\\right)",
   "name": null,
   "statement": "The total cost per hour of lighting a building with N lamps is the renewal cost per lamp plus the energy cost per lamp.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "C",
     "meaning": "total cost per hour of lighting the building"
    },
    {
     "unit": null,
     "symbol": "N",
     "meaning": "number of lamps"
    },
    {
     "unit": "pence",
     "symbol": "C_l",
     "meaning": "cost of renewal per hour of use"
    },
    {
     "unit": "hours",
     "symbol": "t",
     "meaning": "average life of each lamp"
    },
    {
     "unit": "watts per candle-power",
     "symbol": "E",
     "meaning": "commercial efficiency"
    },
    {
     "unit": null,
     "symbol": "P",
     "meaning": "candle power of each lamp"
    },
    {
     "unit": null,
     "symbol": "C_e",
     "meaning": "cost of energy per 1000 watts per hour"
    }
   ],
   "sympy": "Eq(C, N*(C_l/t + E*P*C_e/1000))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/function-of-several-variables"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4fa343c1bf",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "117",
   "location": "Curvature of Curves",
   "latex": "t = mE^n",
   "name": null,
   "statement": "The average life of a lamp is approximately a constant power of its commercial efficiency.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": "hours",
     "symbol": "t",
     "meaning": "average life of each lamp"
    },
    {
     "unit": "watts per candle-power",
     "symbol": "E",
     "meaning": "commercial efficiency"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "constant depending on the kind of lamp"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant depending on the kind of lamp"
    }
   ],
   "sympy": "Eq(t, m*E**n)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/exponent",
    "concept/power"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f5f5920db7",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "117",
   "location": "Curvature of Curves",
   "latex": "\\frac{PC_e}{1000} - \\frac{nC_l}{m} E^{-(n+1)} = 0",
   "name": null,
   "statement": "Setting the derivative of the total cost with respect to the commercial efficiency to zero gives the candidate efficiency for a maximum or minimum.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "C",
     "meaning": "total cost per hour of lighting"
    },
    {
     "unit": "watts per candle-power",
     "symbol": "E",
     "meaning": "commercial efficiency"
    },
    {
     "unit": null,
     "symbol": "P",
     "meaning": "candle power of each lamp"
    },
    {
     "unit": null,
     "symbol": "C_e",
     "meaning": "cost of energy per 1000 watts per hour"
    },
    {
     "unit": "pence",
     "symbol": "C_l",
     "meaning": "cost of renewal per hour of use"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "constant depending on the kind of lamp"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant depending on the kind of lamp"
    }
   ],
   "sympy": "Eq(P*C_e/1000 - n*C_l/m*E**(-(n+1)), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/stationary-value",
    "method/equating-the-derivative-to-zero"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-6d8c567678",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "117",
   "location": "Curvature of Curves",
   "latex": "E = \\sqrt[n+1]{\\frac{1000 × nC_l}{mPC_e}}",
   "name": null,
   "statement": "The commercial efficiency that makes the total cost of lighting least is the (n+1)th root of 1000 n C_l divided by m P C_e.",
   "kind": "result",
   "symbols": [
    {
     "unit": "watts per candle-power",
     "symbol": "E",
     "meaning": "commercial efficiency for least total cost"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant depending on the kind of lamp"
    },
    {
     "unit": "pence",
     "symbol": "C_l",
     "meaning": "cost of renewal per hour of use"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "constant depending on the kind of lamp"
    },
    {
     "unit": null,
     "symbol": "P",
     "meaning": "candle power of each lamp"
    },
    {
     "unit": null,
     "symbol": "C_e",
     "meaning": "cost of energy per 1000 watts per hour"
    }
   ],
   "sympy": "Eq(E, (1000*n*C_l/(m*P*C_e))**(1/(n+1)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/minimum",
    "concept/root",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4e08db8e31",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "118",
   "location": "Curvature of Curves",
   "latex": "\\frac{d^2C}{dE^2} = (n + 1) \\frac{nC_l}{m} E^{-(n+2)}",
   "name": null,
   "statement": "The second derivative of the total cost with respect to efficiency is positive for positive E, so the stationary value found is a minimum.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "C",
     "meaning": "total cost per hour of lighting"
    },
    {
     "unit": "watts per candle-power",
     "symbol": "E",
     "meaning": "commercial efficiency"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant depending on the kind of lamp"
    },
    {
     "unit": "pence",
     "symbol": "C_l",
     "meaning": "cost of renewal per hour of use"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "constant depending on the kind of lamp"
    }
   ],
   "sympy": "Eq(Derivative(C, (E, 2)), (n + 1)*n*C_l/m*E**(-(n + 2)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/minimum",
    "method/second-derivative-test"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-66c2a74851",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "125",
   "location": "Other Useful Dodges",
   "latex": "\\frac{3}{x+1} - \\frac{1}{x-1} + \\frac{2}{x+3}",
   "name": null,
   "statement": "The fraction (4x^2+2x-14)/(x^3+3x^2-x-3) splits into three partial fractions with denominators x+1, x-1 and x+3.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq((4*x**2 + 2*x - 14)/(x**3 + 3*x**2 - x - 3), 3/(x+1) - 1/(x-1) + 2/(x+3))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-fraction",
    "concept/method-partial-fractions",
    "concept/proper-algebraic-fraction"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-40daf6d98c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "126",
   "location": "Other Useful Dodges",
   "latex": "\\frac{x-1}{x^2+1} - \\frac{2}{x+1}",
   "name": null,
   "statement": "The fraction (-x^2-3)/((x^2+1)(x+1)) splits into (x-1)/(x^2+1) minus 2/(x+1).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq((-x**2 - 3)/((x**2 + 1)*(x + 1)), (x - 1)/(x**2 + 1) - 2/(x + 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-fraction",
    "concept/method-partial-fractions",
    "concept/proper-algebraic-fraction"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-fdfbc1d57f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "128",
   "location": "Other Useful Dodges",
   "latex": "\\frac{2}{x+1} - \\frac{2}{(x+1)^2} + \\frac{1}{x-2}",
   "name": null,
   "statement": "The fraction (3x^2-2x+1)/((x+1)^2(x-2)) splits into partial fractions with denominators x+1, (x+1)^2 and x-2.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq((3*x**2 - 2*x + 1)/((x+1)**2*(x-2)), 2/(x+1) - 2/(x+1)**2 + 1/(x-2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-fraction",
    "concept/method-partial-fractions",
    "concept/proper-algebraic-fraction"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e4ac49f6b8",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "129",
   "location": "Other Useful Dodges",
   "latex": "\\frac{(8x - 5)}{(2x^2 - 1)^2} + \\frac{8(x - 1)}{2x^2 - 1} - \\frac{4}{x + 1}",
   "name": null,
   "statement": "The fraction (3x-1)/((2x^2-1)^2(x+1)) splits into (8x-5)/(2x^2-1)^2, 8(x-1)/(2x^2-1) and -4/(x+1).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq((3*x - 1)/((2*x**2 - 1)**2*(x + 1)), (8*x - 5)/(2*x**2 - 1)**2 + 8*(x - 1)/(2*x**2 - 1) - 4/(x + 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-fraction",
    "concept/method-partial-fractions",
    "concept/proper-algebraic-fraction"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7ae3912f20",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "129",
   "location": "Other Useful Dodges",
   "latex": "\\frac{4}{(x + 1)^2} - \\frac{3}{(x + 1)^3}",
   "name": null,
   "statement": "The fraction (4x+1)/(x+1)^3 splits into 4/(x+1)^2 minus 3/(x+1)^3, found by substituting z = x+1.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq((4*x + 1)/(x + 1)**3, 4/(x + 1)**2 - 3/(x + 1)**3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-fraction",
    "concept/method-partial-fractions",
    "concept/proper-algebraic-fraction"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2d8b0c74c7",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "127",
   "location": "Other Useful Dodges",
   "latex": "\\frac{3x^2 - 2x + 1}{(x+1)^2(x-2)} = \\frac{x-1}{(x+1)^2} + \\frac{1}{x+1} + \\frac{1}{x-2}",
   "name": null,
   "statement": "The fraction (3x^2-2x+1)/((x+1)^2(x-2)) equals the three-term sum with numerator x-1 over (x+1)^2, which is not the form that the method of unknown numerators yields.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq((3*x**2 - 2*x + 1)/((x + 1)**2*(x - 2)), (x - 1)/(x + 1)**2 + 1/(x + 1) + 1/(x - 2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-fraction",
    "concept/method-partial-fractions",
    "concept/proper-algebraic-fraction"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b71f9ccfa3",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "130",
   "location": "Other Useful Dodges",
   "latex": "\\frac{dy}{dx} = -\\frac{3}{(3x-1)^2} + \\frac{4}{(2x+3)^2}",
   "name": null,
   "statement": "The derivative of y = (5-4x)/(6x^2+7x-3), obtained by differentiating its partial fractions 1/(3x-1) - 2/(2x+3).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, the function (5-4x)/(6x^2+7x-3)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), -3/(3*x - 1)**2 + 4/(2*x + 3)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/method-differentiation",
    "concept/method-partial-fractions"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b0bf03b642",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "132",
   "location": "Other Useful Dodges",
   "latex": "\\frac{dy}{dx} = -\\frac{3}{2x^2\\sqrt{\\dfrac{3}{x} -1}}",
   "name": null,
   "statement": "The derivative of y = sqrt(3/x - 1), found by differentiating the inverse function x = 3/(1+y^2).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, the function sqrt(3/x - 1)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), -3/(2*x**2*sqrt(3/x - 1)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/inverse-function",
    "concept/method-differentiating-an-inverse-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dca1b357ea",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "133",
   "location": "Other Useful Dodges",
   "latex": "\\dfrac{dy}{dx} = -\\dfrac{1}{3\\sqrt{(\\theta +5)^4}}",
   "name": null,
   "statement": "The derivative of y = 1/cuberoot(theta+5) with respect to x, obtained from the inverse function theta = y^(-3) - 5.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, the function 1/cuberoot(theta+5)"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "theta",
     "meaning": "the variable in the function y = 1/cuberoot(theta+5)"
    }
   ],
   "sympy": "Eq(Derivative(y, x), -1/(3*sqrt((theta + 5)**4)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/inverse-function",
    "concept/method-differentiating-an-inverse-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3a66c391c0",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Other Useful Dodges",
   "latex": "\\frac{dy}{dx} × \\frac{dx}{dy} = 1",
   "name": null,
   "statement": "The derivative of y with respect to x times the derivative of x with respect to y equals 1, illustrated for y = 3x and y = 4x^2.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x)*Derivative(x, y), 1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/inverse-function",
    "concept/reciprocal"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-73d1546250",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Other Useful Dodges",
   "latex": "\\frac{dy}{dx} = \\frac{1}{\\ \\dfrac{dx}{dy}\\ }",
   "name": null,
   "statement": "For any function that has an inverse form, the derivative of the function is the reciprocal of the derivative of its inverse function.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 1/Derivative(x, y))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/inverse-function",
    "concept/method-differentiating-an-inverse-function",
    "concept/reciprocal"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7052ac9d5a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "135",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y + n\\dfrac{y}{n} = 2y",
   "name": null,
   "statement": "At simple interest, a capital y that earns a yearly interest of y/n for n years has doubled.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "original capital"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "number of years"
    }
   ],
   "sympy": "Eq(y + n*(y/n), 2*y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/simple-interest",
    "quantity/amount",
    "quantity/principal"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-aaa5492e0c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "136",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y_n = y_0\\left(1 + \\frac{1}{n}\\right)^n",
   "name": null,
   "statement": "Compound interest added n times over the period multiplies the original capital by (1 + 1/n) at each operation, giving the capital after n operations.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y_0",
     "meaning": "original capital"
    },
    {
     "unit": null,
     "symbol": "y_n",
     "meaning": "value of the capital at the end of the nth operation"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "number of compounding operations"
    }
   ],
   "sympy": "Eq(y_n, y_0*(1 + 1/n)**n)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/compound-interest",
    "concept/euler-s-number"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2243c3cab6",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "148",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\frac{d(\\log_\\epsilon x)}{dx} = x^{-1}",
   "name": null,
   "statement": "The differential coefficient of the natural logarithm of x with respect to x is 1/x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(log(x), x), x**(-1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/natural-logarithm",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a07b8ba3b1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "141",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\epsilon = 1 + 1 + \\dfrac{1}{2!} + \\dfrac{1}{3!} + \\dfrac{1}{4!} + \\text{etc}.\\ldots",
   "name": null,
   "statement": "Epsilon, the limit of (1 + 1/n) to the n as n grows, equals the sum of the series 1 + 1 + 1/2! + 1/3! + ...",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "epsilon",
     "meaning": "Euler's number, the limiting value of (1 + 1/n)^n, about 2.71828"
    }
   ],
   "sympy": "Eq(epsilon, Sum(1/factorial(k), (k, 0, oo)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximation",
    "concept/convergent-series",
    "concept/euler-s-number",
    "concept/infinite-sequence",
    "theorem/exponential-series"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8dcff52f3f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "147",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y = \\log_\\epsilon x",
   "name": null,
   "statement": "y is the natural (Naperian) logarithm of x, the power to which epsilon must be raised to give x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "natural logarithm of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, log(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/euler-s-number",
    "concept/logarithm",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-904d7931c1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "146",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\log_\\epsilon a + \\log_\\epsilon b = \\log_\\epsilon ab",
   "name": null,
   "statement": "The natural logarithm of a product is the sum of the natural logarithms of the factors.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "positive number"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "positive number"
    }
   ],
   "sympy": "Eq(log(a) + log(b), log(a*b))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/natural-logarithm",
    "theorem/logarithm-of-a-product"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-cdfa4fe8ed",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "155",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y = b\\epsilon^{ax}",
   "name": null,
   "statement": "A quantity growing in geometrical progression with a constant ratio per unit of x follows the exponential curve with initial height b.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "initial height of y (value at x = 0)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant equal to the natural logarithm of the ratio p"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, b*exp(a*x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponential-function",
    "concept/geometrical-progression",
    "concept/logarithmic-rate-of-growth"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-31a51320f7",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "155",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\log_\\epsilon \\frac{y}{b}=ax",
   "name": null,
   "statement": "The natural logarithm of y divided by its initial value b is proportional to x, with constant a.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "value at x"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "initial value of y"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant, the natural logarithm of p"
    }
   ],
   "sympy": "Eq(log(y/b), a*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/logarithmic-rate-of-growth",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-352f515c18",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "156",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y=b\\epsilon^{-ax}",
   "name": null,
   "statement": "A quantity that dies away exponentially: y starts at b and decays with constant a as x increases.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "initial value of y"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant of decrement"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable (time in the physical uses)"
    }
   ],
   "sympy": "Eq(y, b*exp(-a*x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-decrement",
    "concept/die-away-factor",
    "concept/exponential-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8e257b8aaa",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "156",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\theta_t=\\theta_0 \\epsilon^{-at}",
   "name": "Newton's law of cooling",
   "statement": "The excess of temperature of a hot body over its surroundings falls exponentially with time at a constant rate a.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\theta_0",
     "meaning": "original excess of temperature of a hot body over its surroundings"
    },
    {
     "unit": null,
     "symbol": "\\theta_t",
     "meaning": "excess of temperature at time t"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant of decrement, depending on exposed surface, conductivity and emissivity"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(theta_t, theta_0*exp(-a*t))",
   "physics": true,
   "states": [
    "law/newton-s-law-of-cooling"
   ],
   "concepts": [
    "concept/die-away-factor",
    "concept/heat",
    "concept/rate-of-change",
    "concept/temperature"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-30be9bfc0e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "157",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "Q_t=Q_0 \\epsilon^{-at}",
   "name": null,
   "statement": "The charge of an electrified body leaking away through a resistance decays exponentially with time.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "Q_0",
     "meaning": "original charge of the body"
    },
    {
     "unit": null,
     "symbol": "Q_t",
     "meaning": "charge remaining at time t"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant of decrement, depending on capacity and leakage resistance"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(Q_t, Q_0*exp(-a*t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/die-away-factor",
    "quantity/capacity",
    "quantity/electric-charge",
    "quantity/resistance"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-cf999a08ae",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "160",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "C = \\dfrac{E}{R}\\left\\{1 - \\epsilon^{-\\efrac{Rt}{L}}\\right\\}",
   "name": null,
   "statement": "The strength of an electric current in a conductor rises toward E/R as the exponential term dies away, with time constant L/R.",
   "kind": "result",
   "symbols": [
    {
     "unit": "ampere",
     "symbol": "C",
     "meaning": "strength of the electric current at time t"
    },
    {
     "unit": "volt",
     "symbol": "E",
     "meaning": "electromotive force producing the current"
    },
    {
     "unit": "ohm",
     "symbol": "R",
     "meaning": "resistance of the conductor"
    },
    {
     "unit": null,
     "symbol": "L",
     "meaning": "not defined in this passage; the time constant is L/R"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time after the electromotive force is applied"
    }
   ],
   "sympy": "Eq(C, E/R*(1 - epsilon**(-R*t/L)))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/die-away-factor",
    "concept/electric-current",
    "quantity/electromotive-force",
    "quantity/resistance",
    "quantity/time-constant"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-be5eb2b5eb",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "161",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "I = I_0\\epsilon^{-Kl}",
   "name": null,
   "statement": "The intensity of a light beam decreases exponentially with the thickness of the transparent medium it passes through.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "I",
     "meaning": "intensity of the beam after thickness l"
    },
    {
     "unit": null,
     "symbol": "I_0",
     "meaning": "initial intensity of the beam"
    },
    {
     "unit": "per centimetre",
     "symbol": "K",
     "meaning": "constant of absorption"
    },
    {
     "unit": "centimetre",
     "symbol": "l",
     "meaning": "thickness of the medium"
    }
   ],
   "sympy": "Eq(I, I_0*epsilon**(-K*l))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/absorption-of-light",
    "concept/constant-of-absorption",
    "concept/die-away-factor",
    "concept/exponential-function",
    "concept/intensity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7260ac1080",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "135",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y + n\\dfrac{y}{n} = 2y.",
   "name": null,
   "statement": "At simple interest, if the yearly interest is y/n, then after n years the hoarded property is the original capital y plus n times y/n, which equals 2y, so it has doubled.",
   "kind": "result",
   "symbols": [
    {
     "unit": "pound",
     "symbol": "y",
     "meaning": "original capital"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "number of years of hoarding"
    }
   ],
   "sympy": "Eq(y + n*y/n, 2*y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/simple-interest",
    "quantity/amount"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c15187bd6d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "136",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y_n = y_0\\left(1 + \\frac{1}{n}\\right)^n.",
   "name": null,
   "statement": "Compound interest added n times, each time by the fraction 1/n of the capital, multiplies the original capital by (1 + 1/n) raised to the power n.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "pound",
     "symbol": "y_n",
     "meaning": "value of the capital at the end of the nth operation"
    },
    {
     "unit": "pound",
     "symbol": "y_0",
     "meaning": "original capital"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "number of operations"
    }
   ],
   "sympy": "Eq(y_n, y_0*(1 + 1/n)**n)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/compound-interest",
    "concept/power"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-592273159a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "137",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y_n = £100 \\left( 1 + \\tfrac{1}{100} \\right)^{100};",
   "name": null,
   "statement": "Compounding 1 per cent for each of 100 tenth-year periods over ten years grows the £100 capital to about £270 9s 7½d.",
   "kind": "result",
   "symbols": [
    {
     "unit": "pound",
     "symbol": "y_n",
     "meaning": "value of the capital at the end of ten years"
    }
   ],
   "sympy": "Eq(y_n, 100*(1 + Rational(1, 100))**100)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/compound-interest"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3fafd9025a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "137",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y_n = £100 \\left( 1 + \\tfrac{1}{1000} \\right)^{1000};",
   "name": null,
   "statement": "Compounding one-tenth of a per cent for each of 1000 periods over ten years grows the £100 capital to about £271 13s 10d.",
   "kind": "result",
   "symbols": [
    {
     "unit": "pound",
     "symbol": "y_n",
     "meaning": "value of the capital at the end of ten years"
    }
   ],
   "sympy": "Eq(y_n, 100*(1 + Rational(1, 1000))**1000)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/compound-interest"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-cf563fbd9f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "137",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y_n = £100 \\left( 1 + \\tfrac{1}{10,000} \\right)^{10,000};",
   "name": null,
   "statement": "Compounding in 10,000 periods of one-thousandth of a year over ten years grows the £100 capital to about £271 16s 3½d.",
   "kind": "result",
   "symbols": [
    {
     "unit": "pound",
     "symbol": "y_n",
     "meaning": "value of the capital at the end of ten years"
    }
   ],
   "sympy": "Eq(y_n, 100*(1 + Rational(1, 10000))**10000)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/compound-interest"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-de3b335fe5",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "143",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\epsilon^x = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\text{etc.}\\dots",
   "name": null,
   "statement": "Epsilon to the power x is the infinite series 1 + x + x^2/2! + x^3/3! + ..., called the exponential series.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "epsilon",
     "meaning": "Euler's number"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable exponent"
    }
   ],
   "sympy": "Eq(epsilon**x, Sum(x**k/factorial(k), (k, 0, oo)))",
   "physics": false,
   "states": [
    "theorem/exponential-series"
   ],
   "concepts": [
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/power-series",
    "theorem/exponential-series"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c3958c1cc4",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "145",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\epsilon^x = 1 + \\dfrac{x}{1} + \\dfrac{x^2}{1·2} + \\dfrac{x^3}{1· 2· 3} + \\dfrac{x^4}{1· 2· 3· 4} + \\text{etc}.",
   "name": null,
   "statement": "The series 1 + x + x^2/2! + x^3/3! + ... is shown to equal epsilon to the power x, since it is unchanged by differentiation and equals epsilon when x is 1.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "epsilon",
     "meaning": "Euler's number"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable exponent"
    }
   ],
   "sympy": "Eq(epsilon**x, Sum(x**k/factorial(k), (k, 0, oo)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/euler-s-number",
    "concept/exponential-function",
    "theorem/exponential-series"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-760a737dd7",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "147",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y = \\log_\\epsilon x.",
   "name": null,
   "statement": "Y is defined as the natural logarithm, to base epsilon, of x, so that x equals epsilon to the power y.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "natural logarithm of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    }
   ],
   "sympy": "Eq(y, log(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/logarithm",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dd5d2e0991",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "148",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\frac{d(\\log_\\epsilon x)}{dx} = x^{-1}.",
   "name": null,
   "statement": "The differential coefficient of the natural logarithm of x with respect to x is 1/x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "variable"
    }
   ],
   "sympy": "Eq(Derivative(log(x), x), x**(-1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a7ed6dda75",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "146",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\log_\\epsilon a + \\log_\\epsilon b = \\log_\\epsilon ab.",
   "name": null,
   "statement": "The sum of natural logarithms of a and b is the natural logarithm of their product.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "positive number"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "positive number"
    }
   ],
   "sympy": "Eq(log(a) + log(b), log(a*b))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/logarithm",
    "concept/natural-logarithm",
    "concept/product"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-648e8f4ccf",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "146",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "n × \\log_\\epsilon a = \\log_\\epsilon a^n.",
   "name": null,
   "statement": "n times the natural logarithm of a equals the natural logarithm of a to the power n.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "exponent"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "positive number"
    }
   ],
   "sympy": "Eq(n*log(a), log(a**n))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/natural-logarithm",
    "concept/power"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3c037289f5",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "156",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "p=\\epsilon^{-a}",
   "name": null,
   "statement": "A proper fraction p, less than one, can be written as epsilon to the power minus a, where a is minus the natural logarithm of p.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "p",
     "meaning": "ratio between successive ordinates"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "minus the natural logarithm of p"
    }
   ],
   "sympy": "Eq(p, epsilon**(-a))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/euler-s-number",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-6ca4e37ce5",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "156",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y=bp^x;",
   "name": null,
   "statement": "A curve whose successive ordinates are in geometrical progression: y equals b times p to the power x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "initial height of y, the value at x = 0"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "ratio of each ordinate to the preceding one"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, b*p**x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/function",
    "concept/geometrical-progression"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a0a5823e09",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "155",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y = b\\epsilon^{ax}.",
   "name": null,
   "statement": "The logarithmic curve, the geometric-progression curve written with epsilon as base, has y equal to b times epsilon to the power a x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "initial height of y"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "natural logarithm of p"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, b*epsilon**(a*x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/logarithmic-rate-of-growth"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-526bf8e68a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "155",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\log_\\epsilon \\frac{y}{b}=ax,",
   "name": null,
   "statement": "The natural logarithm of y divided by b is a times x, so the logarithm of the ordinate is a straight line in x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "initial height of y"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "natural logarithm of p"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(log(y/b), a*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/linear-relation",
    "concept/natural-logarithm",
    "concept/relation-between-variables"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b9c6d80f8e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "156",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "y=b\\epsilon^{-ax}.",
   "name": null,
   "statement": "The die-away curve: y equals b times epsilon to the power minus a x, a quantity that decreases exponentially with x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "quantity that is gradually dying away"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "initial value of y"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant of decrement"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable, often time"
    }
   ],
   "sympy": "Eq(y, b*epsilon**(-a*x))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/constant-of-decrement",
    "concept/die-away-factor",
    "concept/exponential-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-66efe184f8",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "156",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "\\theta_t=\\theta_0 \\epsilon^{-at};",
   "name": "Newton's law of cooling",
   "statement": "The excess of temperature of a cooling hot body above its surroundings falls exponentially with time, as theta_0 times epsilon to the power minus a t.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "theta_t",
     "meaning": "excess of temperature at the end of time t"
    },
    {
     "unit": null,
     "symbol": "theta_0",
     "meaning": "original excess of temperature over the surroundings"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant of decrement, depending on surface exposed, conductivity and emissivity"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(theta_t, theta_0*epsilon**(-a*t))",
   "physics": true,
   "states": [
    "law/newton-s-law-of-cooling"
   ],
   "concepts": [
    "concept/constant-of-decrement",
    "concept/exponential-function",
    "concept/temperature"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e9b04ad5f3",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "157",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "Q_t=Q_0 \\epsilon^{-at},",
   "name": null,
   "statement": "The charge of an electrified body leaking away decays exponentially with time, as Q_0 times epsilon to the power minus a t.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "Q_t",
     "meaning": "charge at time t"
    },
    {
     "unit": null,
     "symbol": "Q_0",
     "meaning": "original charge"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant of decrement, depending on capacity and resistance of the leakage path"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(Q_t, Q_0*epsilon**(-a*t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/constant-of-decrement",
    "concept/die-away-factor",
    "concept/exponential-function",
    "quantity/electric-charge"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-80a36b9cff",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "162",
   "location": "On true Compound Interest and the Law of Organic Growth",
   "latex": "Q = Q_0 \\epsilon^{-\\lambda t}",
   "name": null,
   "statement": "The quantity of a radio-active substance not yet transformed falls exponentially with time, at a constant rate lambda.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "Q",
     "meaning": "quantity of the substance not yet transformed at time t"
    },
    {
     "unit": null,
     "symbol": "Q_0",
     "meaning": "initial quantity of the substance"
    },
    {
     "unit": null,
     "symbol": "lambda",
     "meaning": "constant of the substance"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time elapsed since transformation began"
    }
   ],
   "sympy": "Eq(Q, Q_0*epsilon**(-lambda_*t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/constant-of-decrement",
    "concept/die-away-factor",
    "concept/exponential-function",
    "concept/radioactive-decay",
    "concept/rate-of-change"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f60c095eab",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "165",
   "location": "How to deal with Sines and Cosines",
   "latex": "y= \\sin \\theta",
   "name": null,
   "statement": "The height y is defined as the sine of the angle theta.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height of the sine on the unit circle, the value of sin θ"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    }
   ],
   "sympy": "Eq(y, sin(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/function",
    "concept/sine",
    "quantity/angle"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-01e611e566",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "166",
   "location": "How to deal with Sines and Cosines",
   "latex": "dy = \\sin(\\theta + d \\theta)- \\sin \\theta",
   "name": null,
   "statement": "The increment dy of the sine equals the sine of the increased angle minus the sine of the original angle.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small increment of the sine y"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    },
    {
     "unit": null,
     "symbol": "dθ",
     "meaning": "small increment (element) of the angle θ"
    }
   ],
   "sympy": "Eq(dy, sin(theta + dtheta) - sin(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/increment",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-85259a47d9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "166",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\sin M - \\sin N = 2 \\cos\\frac{M+N}{2}·\\sin\\frac{M-N}{2}",
   "name": null,
   "statement": "The difference of two sines equals twice the cosine of half the sum times the sine of half the difference of the angles.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "M",
     "meaning": "first of two different angles"
    },
    {
     "unit": null,
     "symbol": "N",
     "meaning": "second of two different angles"
    }
   ],
   "sympy": "Eq(sin(M) - sin(N), 2*cos((M+N)/2)*sin((M-N)/2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/sine",
    "theorem/difference-of-two-sines",
    "theorem/trigonometric-identity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d1abf927ed",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "166",
   "location": "How to deal with Sines and Cosines",
   "latex": "dy = \\cos \\theta · d \\theta",
   "name": null,
   "statement": "In the limit of an indefinitely small angle, the increment of the sine is cosine theta times the increment of the angle.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small increment of the sine y"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    },
    {
     "unit": null,
     "symbol": "dθ",
     "meaning": "small increment (element) of the angle θ"
    }
   ],
   "sympy": "Eq(dy, cos(theta)*dtheta)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/derivative",
    "concept/differential",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1cac35fd85",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "168",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\cos \\theta=\\sin\\left(\\dfrac{\\pi}{2}-\\theta\\right)",
   "name": null,
   "statement": "Cosine of an angle equals sine of its complement, pi/2 minus the angle.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    }
   ],
   "sympy": "Eq(cos(theta), sin(pi/2 - theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/complementary-angles",
    "concept/cosine",
    "concept/sine",
    "theorem/trigonometric-identity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-81b74fcbf1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "168",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\frac{dy}{d\\theta} = -\\sin \\theta",
   "name": null,
   "statement": "The derivative of cosine theta with respect to theta is minus sine theta.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the cosine of θ"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    }
   ],
   "sympy": "Eq(Derivative(y, theta), -sin(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/derivative",
    "concept/sine",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-14856c66bc",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "169",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\frac{dy}{d\\theta} = \\sec^2 \\theta",
   "name": null,
   "statement": "The derivative of tangent theta with respect to theta is secant squared theta.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the tangent of θ"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    }
   ],
   "sympy": "Eq(Derivative(y, theta), sec(theta)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/secant",
    "concept/tangent-function",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8758ad57f0",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "172",
   "location": "How to deal with Sines and Cosines",
   "latex": "-(1+\\cot^2 \\theta) = -\\cosec^2 \\theta",
   "name": null,
   "statement": "Minus one plus cotangent squared equals minus cosecant squared, the step by which the derivative of cotangent is reduced.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    }
   ],
   "sympy": "Eq(-(1 + cot(theta)**2), -csc(theta)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosecant",
    "concept/cotangent",
    "concept/derivative",
    "theorem/trigonometric-identity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c2be6d43f6",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "172",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\frac{1}{\\sin\\theta} × \\cos\\theta = \\cot\\theta",
   "name": null,
   "statement": "Cosine divided by sine equals cotangent, which gives the derivative of log of sine in the example.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    }
   ],
   "sympy": "Eq(1/sin(theta)*cos(theta), cot(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/cotangent",
    "concept/logarithm",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d4e35ee42f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "169",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\theta = 2\\pi\\frac{t}{T}",
   "name": null,
   "statement": "The angle moved through in time t, for a motion with period T, is two pi times t over T in radians.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "θ",
     "meaning": "angle moved through in time t"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": null,
     "symbol": "T",
     "meaning": "time of one complete period"
    }
   ],
   "sympy": "Eq(theta, 2*pi*t/T)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "concept/periodic-function",
    "quantity/angle",
    "quantity/time"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7c62bbffbf",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "169",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\theta = 360\\frac{t}{T}",
   "name": null,
   "statement": "The angle moved through in time t, for a motion with period T, is 360 times t over T in degrees.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "θ",
     "meaning": "angle moved through in time t"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": null,
     "symbol": "T",
     "meaning": "time of one complete period"
    }
   ],
   "sympy": "Eq(theta, 360*t/T)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/periodic-function",
    "quantity/angle",
    "quantity/time",
    "unit/degree-of-angle"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-81081b9e3f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "170",
   "location": "How to deal with Sines and Cosines",
   "latex": "n = \\dfrac{1}{T}",
   "name": null,
   "statement": "The frequency n, the number of periods per second, is the reciprocal of the period T.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency, number of periods per second"
    },
    {
     "unit": null,
     "symbol": "T",
     "meaning": "time of one complete period"
    }
   ],
   "sympy": "Eq(n, 1/T)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/periodic-function",
    "quantity/frequency",
    "quantity/time"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-056fe6f33d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "170",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\theta=2\\pi nt.",
   "name": null,
   "statement": "The angle equals two pi times the frequency times the time.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "θ",
     "meaning": "angle moved through in time t"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency, number of periods per second"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(theta, 2*pi*n*t)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/periodic-function",
    "quantity/angle",
    "quantity/frequency"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-352d01f370",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "170",
   "location": "How to deal with Sines and Cosines",
   "latex": "y = \\sin 2\\pi nt.",
   "name": null,
   "statement": "A sine varying with time as sine of two pi n t, a simple periodic motion.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height of the sine at time t"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency, number of periods per second"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(y, sin(2*pi*n*t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/periodic-function",
    "concept/sine",
    "quantity/frequency"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-191d4fb27e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "170",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\frac{dy}{dt} = \\frac{dy}{d\\theta} · \\frac{d\\theta}{dt}",
   "name": null,
   "statement": "The rate of change of y with time is the rate with respect to the angle times the rate of the angle with time.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height of the sine"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "angle"
    }
   ],
   "sympy": "Eq(Derivative(y, t), Derivative(y, theta)*Derivative(theta, t))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "quantity/time",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ff6779bcde",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "170",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\frac{d(\\cos 2\\pi nt)}{dt} = -2\\pi n · \\sin 2\\pi nt",
   "name": null,
   "statement": "The derivative of cosine of two pi n t with respect to time is minus two pi n times sine of two pi n t.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency, number of periods per second"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(Derivative(cos(2*pi*n*t), t), -2*pi*n*sin(2*pi*n*t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/derivative",
    "concept/sine",
    "quantity/frequency",
    "quantity/time"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-31b38e8267",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "170",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\frac{d^2(\\DPtypo{\\cos \\theta}{\\sin \\theta})}{d\\theta^2} = -\\sin \\theta",
   "name": null,
   "statement": "The second derivative of sine theta with respect to theta is minus sine theta, so sine is a function whose second derivative is its own negative. The source's DPtypo markup shows cos theta struck for sin theta; the corrected reading (sin theta) is what the text's argument uses, and this is a typesetting note to flag rather than a silent fix.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    }
   ],
   "sympy": "Eq(Derivative(sin(theta), (theta, 2)), -sin(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/higher-order-derivative",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c8b8556bed",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "171",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\frac{d^2(\\cos\\theta)}{d\\theta^2} = -\\cos\\theta",
   "name": null,
   "statement": "The second derivative of cosine theta with respect to theta is minus cosine theta.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    }
   ],
   "sympy": "Eq(Derivative(cos(theta), (theta, 2)), -cos(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/derivative",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-61cca4f3ed",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "172",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\frac{dy}{dx}=\\cos(x+a)",
   "name": null,
   "statement": "The derivative of sine of x plus a with respect to x is cosine of x plus a.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "sine of x plus a"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant added to x"
    }
   ],
   "sympy": "Eq(Derivative(y, x), cos(x + a))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/derivative",
    "concept/sine",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0a0eb7c28f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "172",
   "location": "How to deal with Sines and Cosines",
   "latex": "\\frac{dy}{d\\theta}=3 \\sec^2 3\\theta",
   "name": null,
   "statement": "The derivative of tangent of three theta with respect to theta is three times secant squared of three theta.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "tangent of 3θ"
    },
    {
     "unit": null,
     "symbol": "θ",
     "meaning": "variable angle"
    }
   ],
   "sympy": "Eq(Derivative(y, theta), 3*sec(3*theta)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/secant",
    "concept/tangent-function",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0cc50021bc",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "175",
   "location": "Partial Differentiation",
   "latex": "y &= u×v",
   "name": null,
   "statement": "The simplest concrete case of two variables: y is the product of u and v.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, u*v)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function-of-several-variables",
    "concept/variable",
    "method/multiplication"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-075c515c62",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "175",
   "location": "Partial Differentiation",
   "latex": "y &= f(u, v)",
   "name": null,
   "statement": "y depends on two independent variables u and v through a function f.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "f",
     "meaning": "function of u and v"
    }
   ],
   "sympy": "Eq(y, f(u, v))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function-notation",
    "concept/function-of-several-variables"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3108f28f11",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "175",
   "location": "Partial Differentiation",
   "latex": "dy_v &= v\\, du",
   "name": null,
   "statement": "With v held constant, the partial differential of y = uv with respect to u is v du.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy_v",
     "meaning": "partial differential of y with v held constant"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "independent variable held constant"
    },
    {
     "unit": null,
     "symbol": "du",
     "meaning": "small change in u"
    }
   ],
   "sympy": "Eq(dy_v, v*du)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/partial-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-37ba606fc7",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "175",
   "location": "Partial Differentiation",
   "latex": "dy_u &= u\\, dv",
   "name": null,
   "statement": "With u held constant, the partial differential of y = uv with respect to v is u dv.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy_u",
     "meaning": "partial differential of y with u held constant"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "independent variable held constant"
    },
    {
     "unit": null,
     "symbol": "dv",
     "meaning": "small change in v"
    }
   ],
   "sympy": "Eq(dy_u, u*dv)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/partial-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-39bc7f55e9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "\\frac{\\partial y}{\\partial u} &= v",
   "name": null,
   "statement": "The partial derivative of y = uv with respect to u, treating v as constant, is v.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "independent variable held constant"
    }
   ],
   "sympy": "Eq(Derivative(y, u), v)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-746f2d64f2",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "\\frac{\\partial y}{\\partial v} &= u",
   "name": null,
   "statement": "The partial derivative of y = uv with respect to v, treating u as constant, is u.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "u",
     "meaning": "independent variable held constant"
    }
   ],
   "sympy": "Eq(Derivative(y, v), u)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-6fef4e3d05",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "dy_v &= \\frac{\\partial y}{\\partial u}\\, du",
   "name": null,
   "statement": "The partial differential with respect to u, written with the partial derivative.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy_v",
     "meaning": "partial differential of y with v held constant"
    },
    {
     "unit": null,
     "symbol": "du",
     "meaning": "small change in u"
    }
   ],
   "sympy": "Eq(dy_v, Derivative(y, u)*du)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/partial-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1339629dbb",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "dy_u &= \\frac{\\partial y}{\\partial v}\\, dv",
   "name": null,
   "statement": "The partial differential with respect to v, written with the partial derivative.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy_u",
     "meaning": "partial differential of y with u held constant"
    },
    {
     "unit": null,
     "symbol": "dv",
     "meaning": "small change in v"
    }
   ],
   "sympy": "Eq(dy_u, Derivative(y, v)*dv)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/partial-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dd75bd9097",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "dy = \\frac{\\partial y}{\\partial u}\\, du + \\dfrac{\\partial y}{\\partial v}\\, dv",
   "name": "total differential",
   "statement": "When both u and v vary, the total change in y is the sum of its two partial differentials.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "total differential of y"
    },
    {
     "unit": null,
     "symbol": "du",
     "meaning": "small change in u"
    },
    {
     "unit": null,
     "symbol": "dv",
     "meaning": "small change in v"
    }
   ],
   "sympy": "Eq(dy, Derivative(y, u)*du + Derivative(y, v)*dv)",
   "physics": false,
   "states": [
    "concept/total-differential"
   ],
   "concepts": [
    "concept/partial-derivative",
    "concept/partial-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-18c7d9b8d9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "dy = \\left(\\dfrac{dy}{du}\\right)\\, du + \\left(\\dfrac{dy}{dv}\\right)\\, dv",
   "name": null,
   "statement": "The same total differential written in the alternative notation some books use.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "total differential of y"
    },
    {
     "unit": null,
     "symbol": "du",
     "meaning": "small change in u"
    },
    {
     "unit": null,
     "symbol": "dv",
     "meaning": "small change in v"
    }
   ],
   "sympy": "Eq(dy, (dy/du)*du + (dy/dv)*dv)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7e6337cc3a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "w = 2ax^2 + 3bxy + 4cy^3",
   "name": null,
   "statement": "Example 1's expression, a polynomial in x and y with constant coefficients a, b, c.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "w",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "constant coefficient"
    }
   ],
   "sympy": "Eq(w, 2*a*x**2 + 3*b*x*y + 4*c*y**3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/function-of-several-variables",
    "concept/polynomial"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-03b870d573",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "\\frac{\\partial w}{\\partial x} &= 4ax + 3by",
   "name": null,
   "statement": "Partial derivative of w with respect to x, with y held constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "w",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "independent variable held constant"
    }
   ],
   "sympy": "Eq(Derivative(w, x), 4*a*x + 3*b*y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c54307323e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "176",
   "location": "Partial Differentiation",
   "latex": "\\frac{\\partial w}{\\partial y} &= 3bx + 12cy^2",
   "name": null,
   "statement": "Partial derivative of w with respect to y, with x held constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "w",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable held constant"
    }
   ],
   "sympy": "Eq(Derivative(w, y), 3*b*x + 12*c*y**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b62da49d8a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "dw = (4ax+3by)\\, dx + (3bx+12cy^2)\\, dy",
   "name": null,
   "statement": "The total differential of w assembled from its two partial derivatives.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dw",
     "meaning": "total differential of w"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "small change in x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small change in y"
    }
   ],
   "sympy": "Eq(dw, (4*a*x + 3*b*y)*dx + (3*b*x + 12*c*y**2)*dy)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9f1d1909d1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "z = x^y",
   "name": null,
   "statement": "Example 2's function: z is x raised to the power y.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(z, x**y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/function-of-several-variables"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e1976ae675",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "\\dfrac{\\partial z}{\\partial x} &= yx^{y-1}",
   "name": null,
   "statement": "Partial derivative of z = x^y with respect to x, with y held constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "independent variable held constant"
    }
   ],
   "sympy": "Eq(Derivative(z, x), y*x**(y - 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "theorem/power-rule"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e8f74d6dc7",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "\\dfrac{\\partial z}{\\partial y} &= x^y × \\log_\\epsilon x",
   "name": null,
   "statement": "Partial derivative of z = x^y with respect to y, with x held constant; the result involves the natural logarithm of x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "z",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable held constant"
    },
    {
     "unit": null,
     "symbol": "epsilon",
     "meaning": "base of the natural logarithm"
    }
   ],
   "sympy": "Eq(Derivative(z, y), x**y*log(x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/euler-s-number",
    "concept/natural-logarithm",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-bb308f801a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "dz = yx^{y-1}\\, dx + x^y \\log_\\epsilon x \\, dy",
   "name": null,
   "statement": "The total differential of z = x^y.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dz",
     "meaning": "total differential of z"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "small change in x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small change in y"
    }
   ],
   "sympy": "Eq(dz, y*x**(y - 1)*dx + x**y*log(x)*dy)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/natural-logarithm",
    "concept/total-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-79ec7d2af9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "V=\\frac{1}{3} \\pi r^2 h",
   "name": null,
   "statement": "The volume of a cone of base radius r and height h.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume of the cone"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the base"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "height"
    }
   ],
   "sympy": "Eq(V, pi*r**2*h/3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cone",
    "quantity/height",
    "quantity/pi",
    "quantity/radius",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-6ea96ed433",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "\\frac{\\partial V}{\\partial r} &= \\dfrac{2\\pi}{3} rh",
   "name": null,
   "statement": "Rate of change of the cone's volume with radius, with height held constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume of the cone"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the base"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "height held constant"
    }
   ],
   "sympy": "Eq(Derivative(V, r), 2*pi*r*h/3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b6f255250c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "\\frac{\\partial V}{\\partial h} &= \\dfrac{\\pi}{3} r^2",
   "name": null,
   "statement": "Rate of change of the cone's volume with height, with radius held constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume of the cone"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "height"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the base held constant"
    }
   ],
   "sympy": "Eq(Derivative(V, h), pi*r**2/3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-53ccb9dfbe",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "177",
   "location": "Partial Differentiation",
   "latex": "dV = \\dfrac{2\\pi}{3} rh\\, dV + \\dfrac{\\pi}{3} r^2\\, dh",
   "name": null,
   "statement": "As printed, the total differential of the cone's volume; the first term reads dV where dr is evidently intended (erratum flag: the correct term is (2π/3)rh dr).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dV",
     "meaning": "change in volume"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the base"
    },
    {
     "unit": null,
     "symbol": "h",
     "meaning": "height"
    },
    {
     "unit": null,
     "symbol": "dh",
     "meaning": "small change in height"
    }
   ],
   "sympy": "Eq(dV, 2*pi*r*h*dV/3 + pi*r**2*dh/3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/total-differential",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b0aeeaeb3b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "178",
   "location": "Partial Differentiation",
   "latex": "y &= F(x+at) + f(x-at)",
   "name": null,
   "statement": "The general form built from two arbitrary functions F and f of x + at and x - at.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "F",
     "meaning": "arbitrary function"
    },
    {
     "unit": null,
     "symbol": "f",
     "meaning": "arbitrary function"
    }
   ],
   "sympy": "Eq(y, F(x + a*t) + f(x - a*t))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function-notation",
    "concept/function-of-several-variables",
    "concept/wave-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-917cb1e041",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "178",
   "location": "Partial Differentiation",
   "latex": "\\frac{\\partial^2 y}{\\partial x^2} &= F''(w) + f''(v)",
   "name": null,
   "statement": "Second partial derivative of y with respect to x, in terms of the second derivatives of F and f.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "w",
     "meaning": "x + at"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "x - at"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9ed2ae84b9",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "178",
   "location": "Partial Differentiation",
   "latex": "\\frac{\\partial^2 y}{\\partial t^2} &= F''(w)a^2 + f''(v)a^2",
   "name": null,
   "statement": "Second partial derivative of y with respect to t, in terms of the second derivatives of F and f.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent quantity"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "w",
     "meaning": "x + at"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "x - at"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-783a6856db",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "178",
   "location": "Partial Differentiation",
   "latex": "\\frac{\\partial^2 y}{\\partial t^2} &= a^2\\, \\frac{\\partial^2 y}{\\partial x^2}",
   "name": null,
   "statement": "The second partial derivative in t equals a² times the second partial derivative in x; the book calls this differential equation of immense importance in mathematical physics.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "displacement-type dependent quantity"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant speed"
    }
   ],
   "sympy": "Eq(Derivative(y, (t, 2)), a**2*Derivative(y, (x, 2)))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/partial-derivative",
    "concept/wave-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d86c1470a1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "A = \\sqrt{s(s-x)(s-y)(s-30+x+y)}",
   "name": "Heron's formula",
   "statement": "The area of the triangle from its semi-perimeter s and its three sides.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "A",
     "meaning": "area of the triangle"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "half perimeter"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length of one portion of the string"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "length of another portion of the string"
    }
   ],
   "sympy": "Eq(A, sqrt(s*(s - x)*(s - y)*(s - 30 + x + y)))",
   "physics": false,
   "states": [
    "theorem/heron-s-formula"
   ],
   "concepts": [
    "concept/theorem-heron-s-formula",
    "quantity/area",
    "quantity/semi-perimeter"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d481f6a022",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "A = \\sqrt{15P}",
   "name": null,
   "statement": "With s = 15 the area reduces to the square root of 15P.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "A",
     "meaning": "area of the triangle"
    },
    {
     "unit": null,
     "symbol": "P",
     "meaning": "product of the side differences"
    }
   ],
   "sympy": "Eq(A, sqrt(15*P))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/theorem-heron-s-formula",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-fa4e0ffcef",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "P &= (15-x)(15-y)(x+y-15)",
   "name": null,
   "statement": "P written as the product of the three side-differences when s = 15.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "P",
     "meaning": "product of the side differences"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length of one portion of the string"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "length of another portion of the string"
    }
   ],
   "sympy": "Eq(P, (15 - x)*(15 - y)*(x + y - 15))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function-of-several-variables",
    "concept/maximum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d1e3dd27fc",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "dP = \\dfrac{\\partial P}{\\partial x}\\, dx + \\dfrac{\\partial P}{\\partial y}\\, dy",
   "name": null,
   "statement": "Total differential of P as a function of x and y.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "dP",
     "meaning": "total differential of P"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "small change in x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small change in y"
    }
   ],
   "sympy": "Eq(dP, Derivative(P, x)*dx + Derivative(P, y)*dy)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7e987cde51",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "\\dfrac{\\partial P}{\\partial x} = 0 \\quad\\text{and}\\quad \\dfrac{\\partial P}{\\partial y} = 0",
   "name": null,
   "statement": "For a maximum, both partial derivatives of P must vanish simultaneously.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "P",
     "meaning": "product of the side differences"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length of one portion of the string"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "length of another portion of the string"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/method-equating-the-derivative-to-zero",
    "concept/partial-derivative",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4f74e70f1f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "2xy - 30x + y^2 - 45y + 450 &= 0",
   "name": null,
   "statement": "The first condition for a stationary point of P, written out.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length of one portion of the string"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "length of another portion of the string"
    }
   ],
   "sympy": "Eq(2*x*y - 30*x + y**2 - 45*y + 450, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-085a4861bd",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "2xy - 30y + x^2 - 45x + 450 &= 0",
   "name": null,
   "statement": "The second condition for a stationary point of P, written out.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length of one portion of the string"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "length of another portion of the string"
    }
   ],
   "sympy": "Eq(2*x*y - 30*y + x**2 - 45*x + 450, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/stationary-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2c669761f2",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "P = (15-x)^2 (2x-15) = 2x^3 - 75x^2 + 900x - 3375",
   "name": null,
   "statement": "With x = y, P reduces to the cubic 2x³ - 75x² + 900x - 3375.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "P",
     "meaning": "product of the side differences"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length of one portion of the string, with y = x"
    }
   ],
   "sympy": "Eq(P, 2*x**3 - 75*x**2 + 900*x - 3375)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/polynomial",
    "method/substitution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-801bf7fc83",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "6x^2 - 150x + 900 = 0",
   "name": null,
   "statement": "Setting dP/dx to zero gives the stationary condition for the one-variable cubic.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length of one portion of the string, with y = x"
    }
   ],
   "sympy": "Eq(6*x**2 - 150*x + 900, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/method-equating-the-derivative-to-zero",
    "concept/quadratic-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8fac84a73c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "179",
   "location": "Partial Differentiation",
   "latex": "\\dfrac{d^2 P}{dx^2} = 12x - 150",
   "name": null,
   "statement": "Second derivative of the cubic, used to tell maximum from minimum.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "P",
     "meaning": "product of the side differences, with y = x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length of one portion of the string"
    }
   ],
   "sympy": "Eq(Derivative(P, (x, 2)), 12*x - 150)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/maximum",
    "concept/method-second-derivative-test",
    "concept/minimum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5cc9a8b86e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Partial Differentiation",
   "latex": "S=xy + \\dfrac{2V}{x} + \\dfrac{2V}{y}",
   "name": null,
   "statement": "Surface area of an open rectangular truck of given volume V with length x and width y, depth V/(xy).",
   "kind": "formula",
   "symbols": [
    {
     "unit": "square feet",
     "symbol": "S",
     "meaning": "area of sides and floor"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "width"
    },
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume"
    }
   ],
   "sympy": "Eq(S, x*y + 2*V/x + 2*V/y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function-of-several-variables",
    "quantity/area",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-051f9f812b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Partial Differentiation",
   "latex": "dS = \\frac{\\partial S}{\\partial x}\\, dx + \\frac{\\partial S}{\\partial y}\\, dy",
   "name": null,
   "statement": "Total differential of the surface area S as a function of x and y.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "dS",
     "meaning": "total differential of S"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "small change in x"
    },
    {
     "unit": null,
     "symbol": "dy",
     "meaning": "small change in y"
    }
   ],
   "sympy": "Eq(dS, Derivative(S, x)*dx + Derivative(S, y)*dy)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/partial-derivative",
    "concept/total-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8dad9dc515",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Partial Differentiation",
   "latex": "y - \\frac{2V}{x^2} = 0",
   "name": null,
   "statement": "Stationary condition for S with respect to x, for minimum.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "width"
    },
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume"
    }
   ],
   "sympy": "Eq(y - 2*V/x**2, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/method-equating-the-derivative-to-zero",
    "concept/minimum",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e07086df09",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Partial Differentiation",
   "latex": "x - \\frac{2V}{y^2} = 0",
   "name": null,
   "statement": "Stationary condition for S with respect to y, for minimum.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "width"
    },
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume"
    }
   ],
   "sympy": "Eq(x - 2*V/y**2, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/method-equating-the-derivative-to-zero",
    "concept/minimum",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b6db8f54bf",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Partial Differentiation",
   "latex": "S = x^2 + \\dfrac{4V}{x}",
   "name": null,
   "statement": "With x = y, the surface area becomes a function of x alone.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "S",
     "meaning": "area of sides and floor"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length, equal to width"
    },
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume"
    }
   ],
   "sympy": "Eq(S, x**2 + 4*V/x)",
   "physics": false,
   "states": [],
   "concepts": [
    "method/substitution",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f9c00b6b7a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Partial Differentiation",
   "latex": "\\dfrac{dS}{dx}= 2x - \\dfrac{4V}{x^2} =0",
   "name": null,
   "statement": "Setting dS/dx to zero for a minimum.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "S",
     "meaning": "area of sides and floor"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length, equal to width"
    },
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume"
    }
   ],
   "sympy": "Eq(2*x - 4*V/x**2, 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/method-equating-the-derivative-to-zero",
    "concept/minimum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d6eadd8bd5",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Partial Differentiation",
   "latex": "x = \\sqrt[3]{2V}",
   "name": null,
   "statement": "The length that minimises the surface area for a given volume V: the cube root of 2V.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "length"
    },
    {
     "unit": null,
     "symbol": "V",
     "meaning": "volume"
    }
   ],
   "sympy": "Eq(x, cbrt(2*V))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/minimum",
    "concept/root"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8f31be2d78",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "182",
   "location": "Integration",
   "latex": "\\ds\\int dy = y",
   "name": null,
   "statement": "The integral sign summed over the small pieces dy gives the total y, so integrating dy recovers y.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the total height built up from the small pieces dy"
    }
   ],
   "sympy": "Eq(Integral(1, y), y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/integral",
    "concept/sum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b8f451fe18",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "182",
   "location": "Integration",
   "latex": "\\ds\\int dx = x",
   "name": null,
   "statement": "The integral of the small pieces dx gives the total x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the total of the small pieces dx"
    }
   ],
   "sympy": "Eq(Integral(1, x), x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/integral",
    "concept/sum"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8195fe4b70",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "187",
   "location": "Integration",
   "latex": "\\dfrac{y}{x} = a",
   "name": null,
   "statement": "If y and x are the totals of all the dy's and dx's of a constant-slope line, their ratio equals the slope a.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "total of all the dy's"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "total of all the dx's"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant slope"
    }
   ],
   "sympy": "Eq(y/x, a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/integral",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-aef0c5588d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "187",
   "location": "Integration",
   "latex": "y = ax + C.",
   "name": null,
   "statement": "Reconstructing a line from its constant slope gives y = ax + C, where the undetermined constant C is the height at x = 0.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height of the line above the origin"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant slope"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal coordinate"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "undetermined constant; the height above the origin where the curve begins, i.e. y when x = 0"
    }
   ],
   "sympy": "Eq(y, a*x + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/linear-equation",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a9755c7adf",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "187",
   "location": "Integration",
   "latex": "\\frac{dy}{dx} = ax.",
   "name": null,
   "statement": "The slope of the curve increases in proportion to x.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "slope of the curve at a point"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant of proportionality"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal coordinate"
    }
   ],
   "sympy": "Eq(Derivative(y, x), a*x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/derivative",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9a206fc1ba",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "187",
   "location": "Integration",
   "latex": "\\frac{dy}{dx} = \\tfrac{1}{5} x",
   "name": null,
   "statement": "A concrete case of the increasing slope, with a = 1/5, so the slope equals x/5.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "dy/dx",
     "meaning": "slope of the curve at a point"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal coordinate"
    }
   ],
   "sympy": "Eq(Derivative(y, x), x/5)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/slope-of-a-curve"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-de1c65827e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "189",
   "location": "Integration",
   "latex": "\\ds\\int \\tfrac{1}{5} x\\, dx = \\tfrac{1}{10} x^2",
   "name": "power rule for integration",
   "statement": "The sum of the pieces (1/5)x dx equals (1/10)x^2, found by using the average value of x, which is x/2, from 0 to x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal coordinate"
    }
   ],
   "sympy": "Eq(Integral(x/5, x), x**2/10)",
   "physics": false,
   "states": [
    "theorem/power-rule-for-integration"
   ],
   "concepts": [
    "concept/integral",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d13e1cec80",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "189",
   "location": "Integration",
   "latex": "y=\\frac{1}{10}x^2",
   "name": null,
   "statement": "The height y of the curve with slope x/5 is x^2/10, up to an added constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height of the curve above the origin"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal coordinate"
    }
   ],
   "sympy": "Eq(y, x**2/10)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/curve",
    "concept/integral",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5ba5d309bf",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "190",
   "location": "Integration",
   "latex": "y = \\tfrac{1}{10}x^2 + C.",
   "name": null,
   "statement": "The equation of the curve with slope x/5 includes the undetermined constant C, because the starting height above the origin was not given.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height of the curve above the origin"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "horizontal coordinate"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "undetermined constant; the height where the curve begins when x = 0"
    }
   ],
   "sympy": "Eq(y, x**2/10 + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/curve",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1d81cdfd2e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "192",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "y = \\frac{1}{n + 1} x^{n+1} + C",
   "name": "power rule for integration",
   "statement": "Working backwards from dy/dx = x^n gives y equal to x raised to n+1, divided by n+1, plus an undetermined constant.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the whole quantity to be found, a function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the power of x"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration, undetermined until ascertained otherwise"
    }
   ],
   "sympy": "Eq(y, x**(n + 1)/(n + 1) + C)",
   "physics": false,
   "states": [
    "theorem/power-rule-for-integration"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/power"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-890c8e50c1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "199",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int x^n\\, dx = \\dfrac{1}{n+1} x^{n+1}",
   "name": "power rule for integration",
   "statement": "The integral of x to the power n with respect to x is x to the power n+1 divided by n+1.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the power of x"
    }
   ],
   "sympy": "Eq(Integral(x**n, x), x**(n + 1)/(n + 1))",
   "physics": false,
   "states": [
    "theorem/power-rule-for-integration"
   ],
   "concepts": [
    "concept/integral",
    "concept/power"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-781cc487d5",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "192",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\frac{dy}{dx} = anx^{n-1}",
   "name": null,
   "statement": "Differentiating y = a x^n gives a times n times x to the power n-1, the constant factor a carried through.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant multiplier"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the power of x"
    }
   ],
   "sympy": "Eq(Derivative(y, x), a*n*x**(n - 1))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-factor",
    "concept/derivative",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c2d6e017d5",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "197",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "y = \\frac{1}{n+1} x^{n+1} + bx + C",
   "name": null,
   "statement": "Integrating dy/dx = x^n + b gives x to the power n+1 over n+1, plus b times x, plus a constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the whole quantity to be found, a function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "the power of x"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant term in dy/dx"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(y, x**(n + 1)/(n + 1) + b*x + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/constant-term",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1a024152aa",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int x^{-1}\\, dx = \\log_\\epsilon x + C",
   "name": null,
   "statement": "The integral of 1/x is the natural logarithm of x plus a constant; this is the exceptional case the power rule does not cover.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of the natural logarithm (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(x**(-1), x), log(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d3f3ce7879",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int \\frac{1}{x+a}\\, dx = \\log_\\epsilon (x+a) + C",
   "name": null,
   "statement": "The integral of 1/(x+a) with respect to x is the natural logarithm of x+a plus a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant added to x"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of the natural logarithm (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(1/(x + a), x), log(x + a) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0b52d71969",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int \\epsilon^x\\, dx = \\epsilon ^x + C",
   "name": null,
   "statement": "The exponential function e^x is its own integral, up to a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of the natural logarithm (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(exp(x), x), exp(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-728aebef93",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int \\epsilon^{-x}\\, dx = -\\epsilon^{-x} + C",
   "name": null,
   "statement": "The integral of e^(-x) with respect to x is minus e^(-x), plus a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of the natural logarithm (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(exp(-x), x), -exp(-x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b62c304af1",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int \\sin x\\, dx = -\\cos x + C",
   "name": null,
   "statement": "The integral of sin x is minus cos x, plus a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle, independent variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sin(x), x), -cos(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-49f1746e9f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int \\cos x\\, dx = \\sin x + C",
   "name": null,
   "statement": "The integral of cos x is sin x, plus a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle, independent variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(cos(x), x), sin(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8b6821bc89",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int\\log_\\epsilon x\\, dx = x(\\log_\\epsilon x - 1) + C",
   "name": null,
   "statement": "The integral of the natural logarithm of x is x times (the natural logarithm of x minus 1), plus a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of the natural logarithm (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(log(x), x), x*(log(x) - 1) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-13dd054127",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int\\log_{10} x\\, dx = 0.4343x (\\log_\\epsilon x - 1) + C",
   "name": null,
   "statement": "The integral of the common logarithm of x is about 0.4343 times x times (the natural logarithm of x minus 1), plus a constant; 0.4343 is the book's rounded value of 1/ln 10.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of the natural logarithm (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(log(x, 10), x), 0.4343*x*(log(x) - 1) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/constant-of-integration",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-84cb0d8fc5",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int a^x\\, dx = \\dfrac{a^x}{\\log_\\epsilon a} + C",
   "name": null,
   "statement": "The integral of a to the power x is a to the power x divided by the natural logarithm of a, plus a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "base of the exponential"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of the natural logarithm (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(a**x, x), a**x/log(a) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/exponential-function",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3d8c55f5e0",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int\\cos ax\\, dx = \\frac{1}{a} \\sin ax + C",
   "name": null,
   "statement": "The integral of cos(ax) with respect to x is sin(ax) divided by a, plus a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant multiplier of x inside the angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(cos(a*x), x), sin(a*x)/a + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b6cf0444ce",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int\\sin ax\\, dx = -\\frac{1}{a} \\cos ax + C",
   "name": null,
   "statement": "The integral of sin(ax) with respect to x is minus cos(ax) divided by a, plus a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant multiplier of x inside the angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sin(a*x), x), -cos(a*x)/a + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b69529f8de",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\cos 2\\theta = \\cos^2\\theta - \\sin^2\\theta",
   "name": null,
   "statement": "The cosine of twice an angle equals the square of its cosine minus the square of its sine.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "angle"
    }
   ],
   "sympy": "Eq(cos(2*theta), cos(theta)**2 - sin(theta)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/cosine",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-792f35c5da",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "199",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int x^n\\, dx = \\dfrac{1}{n+1} x^{n+1}.",
   "name": "power rule for integration",
   "statement": "Integrating x to the power n gives x to the power n+1 divided by n+1 (the constant of integration is added separately).",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "power (any number except -1)"
    }
   ],
   "sympy": "Eq(Integral(x**n, x), x**(n + 1)/(n + 1))",
   "physics": false,
   "states": [
    "theorem/power-rule-for-integration"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/power",
    "concept/variable",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-13d074fc0b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "199",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "y = a \\log_\\epsilon x + C.",
   "name": null,
   "statement": "The integral of a^... type constant-multiplied x^-1 dx is a times the natural logarithm of x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the integrated quantity"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "given constant multiplier"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number (log base)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(y, a*log(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-factor",
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/integral",
    "concept/logarithm",
    "concept/variable",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-86a9dc0637",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int x^{-1}\\, dx      &&= \\log_\\epsilon x + C.",
   "name": null,
   "statement": "The integral of 1/x with respect to x is the natural logarithm of x plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number (log base)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(x**(-1), x), log(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/integral",
    "concept/logarithm",
    "concept/variable",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-deff2dd9c7",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int \\epsilon^x\\, dx    &&= \\epsilon ^x + C.",
   "name": null,
   "statement": "The integral of Euler's number to the power x is itself, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(exp(x), x), exp(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/integral",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0afbffc3e0",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int \\sin x\\, dx        &&= -\\cos x + C.",
   "name": null,
   "statement": "The integral of sin x with respect to x is minus cos x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sin(x), x), -cos(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1297fb846b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int \\cos x\\, dx        &&= \\sin x + C.",
   "name": null,
   "statement": "The integral of cos x with respect to x is sin x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(cos(x), x), sin(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a38cbd1334",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "201",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int\\log_\\epsilon x\\, dx &&= x(\\log_\\epsilon x - 1) + C",
   "name": null,
   "statement": "The integral of the natural logarithm of x is x times (log x minus 1), plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number (log base)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(log(x), x), x*(log(x) - 1) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/integral",
    "concept/logarithm",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9328efa062",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int a^x\\, dx        &&= \\dfrac{a^x}{\\log_\\epsilon a} + C.",
   "name": null,
   "statement": "The integral of a to the power x is a to the power x divided by the natural logarithm of a, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "base of the exponential"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number (log base)"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(a**x, x), a**x/log(a) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/exponential-function",
    "concept/integral",
    "concept/logarithm",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e14783d448",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int\\cos ax\\, dx     &&= \\frac{1}{a} \\sin ax + C",
   "name": null,
   "statement": "The integral of cos(ax) with respect to x is sin(ax) divided by a, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant multiplier of the angle"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(cos(a*x), x), sin(a*x)/a + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-factor",
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-60143ebd06",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int\\sin ax\\, dx     &&= -\\frac{1}{a} \\cos ax + C.",
   "name": null,
   "statement": "The integral of sin(ax) with respect to x is minus cos(ax) divided by a, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant multiplier of the angle"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sin(a*x), x), -cos(a*x)/a + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-factor",
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0d8ec6a754",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "202",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\int\\log_{10} x\\, dx &&= 0.4343x (\\log_\\epsilon x - 1) + C.",
   "name": null,
   "statement": "The integral of the base-10 logarithm of x is about 0.4343 times x times (natural log of x minus 1), plus the constant of integration.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(log(x, 10), x), 0.4343*x*(log(x) - 1) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/integral",
    "concept/logarithm",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4d0f15a3c6",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "204",
   "location": "Integrating as the Reverse of Differentiating",
   "latex": "\\text{volume} = \\iiint f(x,y,z) · dx · dy · dz.",
   "name": null,
   "statement": "The volume of a solid is the triple integral of the function f over the small cubes dx dy dz filling the solid.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "cubic feet",
     "symbol": "volume",
     "meaning": "volume of the solid"
    },
    {
     "unit": null,
     "symbol": "f",
     "meaning": "function describing the figure of the solid"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "first coordinate"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "second coordinate"
    },
    {
     "unit": null,
     "symbol": "z",
     "meaning": "third coordinate"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/element-of-a-cylinder",
    "concept/multiple-integral",
    "concept/three-dimensional-figure",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-68f78bd090",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "207",
   "location": "On Finding Areas by Integrating",
   "latex": "dS = y · dx",
   "name": null,
   "statement": "The area of one narrow vertical strip under the curve is its height times its width dx.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "dS",
     "meaning": "area of one narrow strip (a bit of the whole area S)"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height of the curve above the strip"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "width of the strip"
    }
   ],
   "sympy": "Eq(dS, y*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/integral",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ee687e7d43",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "210",
   "location": "On Finding Areas by Integrating",
   "latex": "\\int^{x=x_2}_{x=x_1} y\\, dx = y_2 - y_1",
   "name": null,
   "statement": "A definite integral between limits equals the difference between the integrated value at the superior limit and at the inferior limit.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x_1",
     "meaning": "inferior limit of integration"
    },
    {
     "unit": null,
     "symbol": "x_2",
     "meaning": "superior limit of integration"
    },
    {
     "unit": null,
     "symbol": "y_1",
     "meaning": "integrated value of y dx corresponding to x_1"
    },
    {
     "unit": null,
     "symbol": "y_2",
     "meaning": "integrated value of y dx corresponding to x_2"
    }
   ],
   "sympy": "Eq(Integral(y, (x, x1, x2)), y2 - y1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/integral",
    "concept/limits-of-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4fae0c49d3",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "210",
   "location": "On Finding Areas by Integrating",
   "latex": "\\text{area~$S$} = b(x_2 - x_1) + \\frac{a}{3}(x_2^3 - x_1^3)",
   "name": null,
   "statement": "The area under the curve y = b + ax^2 between x_1 and x_2 is b(x_2 - x_1) + (a/3)(x_2^3 - x_1^3).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "S",
     "meaning": "area under the curve between the limits x_1 and x_2"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient in the curve y = b + ax^2"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant term in the curve y = b + ax^2"
    },
    {
     "unit": null,
     "symbol": "x_1",
     "meaning": "inferior limit"
    },
    {
     "unit": null,
     "symbol": "x_2",
     "meaning": "superior limit"
    }
   ],
   "sympy": "Eq(S, b*(x2 - x1) + a/3*(x2**3 - x1**3))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dc790294fa",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "218",
   "location": "On Finding Areas by Integrating",
   "latex": "\\text{mean~$y$} = \\frac{1}{x_1} \\int^{x=x_1}_{x=0} y · dx",
   "name": null,
   "statement": "The mean ordinate of a curve from x = 0 to x = x_1 is the area under the curve divided by the length of the base.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "x_1",
     "meaning": "upper end of the base range"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/mean-ordinate",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d78ee45a1a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "217",
   "location": "On Finding Areas by Integrating",
   "latex": "dA = 2 \\pi r\\, dr",
   "name": null,
   "statement": "The area of a narrow circular zone at radius r with width dr is its circumference 2 pi r times dr.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "dA",
     "meaning": "area of the narrow zone"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "distance from the centre"
    },
    {
     "unit": null,
     "symbol": "dr",
     "meaning": "breadth of the zone"
    }
   ],
   "sympy": "Eq(dA, 2*pi*r*dr)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/differential",
    "quantity/area",
    "quantity/circumference"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-79193fc0d2",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "217",
   "location": "On Finding Areas by Integrating",
   "latex": "A &= \\pi R^2",
   "name": null,
   "statement": "The area of a circle of radius R is pi times R squared, found by integrating the zones from the centre to the margin.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "A",
     "meaning": "area of the circle"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(A, pi*R**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "quantity/area",
    "quantity/pi",
    "theorem/area-of-a-circle"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8c8d53b6db",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "220",
   "location": "On Finding Areas by Integrating",
   "latex": "\\tfrac{1}{2} \\int^{\\theta=\\theta_2}_{\\theta=\\theta_1} r^2\\, d\\theta",
   "name": null,
   "statement": "The area between a polar curve and two rays at angles theta_1 and theta_2 is one half the integral of r squared d theta.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "r",
     "meaning": "distance of a point of the boundary from the pole"
    },
    {
     "unit": null,
     "symbol": "theta",
     "meaning": "angle which r makes with the positive horizontal direction"
    },
    {
     "unit": null,
     "symbol": "theta_1",
     "meaning": "lower angle limit"
    },
    {
     "unit": null,
     "symbol": "theta_2",
     "meaning": "upper angle limit"
    }
   ],
   "sympy": "Eq(Area, Integral(r**2/2, (theta, theta1, theta2)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/polar-coordinates",
    "method/finding-an-area-in-polar-coordinates",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ebb7a56089",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "219",
   "location": "On Finding Areas by Integrating",
   "latex": "2\\pi y\\, dx",
   "name": null,
   "statement": "The area of a narrow belt of a surface of revolution is its circumference 2 pi y times its width dx.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve, equal to the radius of revolution"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "width of the belt"
    }
   ],
   "sympy": "Eq(dS_belt, 2*pi*y*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/area-of-a-surface-of-revolution",
    "concept/differential",
    "quantity/circumference"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-62fc6c1855",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "215",
   "location": "On Finding Areas by Integrating",
   "latex": "&= b(1-\\epsilon^{-a})",
   "name": null,
   "statement": "The area under the die-away curve y = b e^(-x) from x = 0 to x = a is b(1 - e^(-a)).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant factor in the curve y = b epsilon^(-x)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "upper limit of x"
    },
    {
     "unit": null,
     "symbol": "epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    }
   ],
   "sympy": "Eq(Area, b*(1 - exp(-a)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/euler-s-number",
    "concept/exponential-function",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7cf8c6a05e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "207",
   "location": "On Finding Areas by Integrating",
   "latex": "\\text{area of $1$~strip} = dS = y · dx.",
   "name": null,
   "statement": "The area of a narrow vertical strip under the curve is its average height y times its width dx, written as the small area element dS.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dS",
     "meaning": "area of one narrow strip"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "average height of the strip, taken as the height of the curve"
    },
    {
     "unit": null,
     "symbol": "dx",
     "meaning": "width of the strip"
    }
   ],
   "sympy": "Eq(dS, y*dx)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/infinitesimal",
    "concept/integral",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-73d243f09c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "207",
   "location": "On Finding Areas by Integrating",
   "latex": "\\text{total area~$S$} = \\int dS = \\int y\\, dx.",
   "name": null,
   "statement": "The total area S is the sum (integral) of all the strip areas, so S equals the integral of y dx.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "S",
     "meaning": "total area under the curve (surface)"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "height of the curve"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa"
    }
   ],
   "sympy": "Eq(S, Integral(y, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/differential",
    "concept/integral",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-04869f075a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "210",
   "location": "On Finding Areas by Integrating",
   "latex": "\\int^{x=x_2}_{x=x_1} y\\, dx = y_2 - y_1,",
   "name": null,
   "statement": "The definite integral between limits equals the difference between the integrated values at the superior and inferior limits.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x_1",
     "meaning": "inferior limit of x (OM)"
    },
    {
     "unit": null,
     "symbol": "x_2",
     "meaning": "superior limit of x (ON)"
    },
    {
     "unit": null,
     "symbol": "y_1",
     "meaning": "integrated value of y dx corresponding to x_1"
    },
    {
     "unit": null,
     "symbol": "y_2",
     "meaning": "integrated value of y dx corresponding to x_2"
    }
   ],
   "sympy": "Eq(Integral(y, (x, x_1, x_2)), y_2 - y_1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/difference",
    "concept/integral",
    "concept/limits-of-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-63e8456a71",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "210",
   "location": "On Finding Areas by Integrating",
   "latex": "\\text{area~$S$} = b(x_2 - x_1) + \\frac{a}{3}(x_2^3 - x_1^3).",
   "name": null,
   "statement": "The area under the curve y = b + a x^2 between x_1 and x_2 is b(x_2 - x_1) + (a/3)(x_2^3 - x_1^3).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "S",
     "meaning": "area under the curve between the limits"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant term of the curve y = b + ax^2"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "coefficient of x^2 in the curve"
    },
    {
     "unit": null,
     "symbol": "x_1",
     "meaning": "inferior limit (OM)"
    },
    {
     "unit": null,
     "symbol": "x_2",
     "meaning": "superior limit (ON)"
    }
   ],
   "sympy": "Eq(S, b*(x_2 - x_1) + a/3*(x_2**3 - x_1**3))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-term",
    "concept/definite-integral",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8702c5480b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "218",
   "location": "On Finding Areas by Integrating",
   "latex": "\\text{mean~$y$} = \\frac{1}{x_1} \\int^{x=x_1}_{x=0} y · dx.",
   "name": "mean ordinate",
   "statement": "The mean ordinate of a curve from x = 0 to x = x_1 is the integral of y dx over that range divided by the length x_1.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the curve"
    },
    {
     "unit": null,
     "symbol": "x_1",
     "meaning": "upper limit of the range, also the length of the base"
    }
   ],
   "sympy": "Eq(mean_y, Integral(y, (x, 0, x_1))/x_1)",
   "physics": false,
   "states": [
    "concept/mean-ordinate"
   ],
   "concepts": [
    "concept/arithmetical-mean",
    "concept/definite-integral",
    "concept/limits-of-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ca965ed37d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "217",
   "location": "On Finding Areas by Integrating",
   "latex": "dA = 2 \\pi r\\, dr.",
   "name": null,
   "statement": "The area of a narrow circular zone of breadth dr at distance r from the centre is its length 2πr times its breadth.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "dA",
     "meaning": "area of the elementary zone"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "distance from the centre"
    },
    {
     "unit": null,
     "symbol": "dr",
     "meaning": "breadth of the zone"
    }
   ],
   "sympy": "Eq(dA, 2*pi*r*dr)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/differential",
    "quantity/area",
    "quantity/circumference"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2461951643",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "217",
   "location": "On Finding Areas by Integrating",
   "latex": "A &= \\pi R^2.",
   "name": null,
   "statement": "The area of a circle of radius R equals pi times R squared.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "A",
     "meaning": "area of the circle"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(A, pi*R**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "quantity/area",
    "quantity/radius"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a3b75dc585",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "214",
   "location": "On Finding Areas by Integrating",
   "latex": "\\pi(r_2^2 - r_1^2)",
   "name": null,
   "statement": "The area of a plane ring is the outer circle's area minus the inner circle's area, pi times (r_2 squared minus r_1 squared).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "r_2",
     "meaning": "outer radius of the ring"
    },
    {
     "unit": null,
     "symbol": "r_1",
     "meaning": "inner radius of the ring"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/annulus",
    "concept/circle",
    "quantity/area",
    "quantity/radius"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e010fe9d8c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "215",
   "location": "On Finding Areas by Integrating",
   "latex": "pv^n = c",
   "name": null,
   "statement": "The adiabatic curve of a perfect gas: pressure times volume raised to the index n stays constant.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "p",
     "meaning": "pressure"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "volume"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "index of the adiabatic curve, given in the book as 1.42"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "constant"
    }
   ],
   "sympy": "Eq(p*v**n, c)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/adiabatic-process",
    "concept/constant",
    "concept/perfect-gas",
    "concept/pressure",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ce98597298",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "215",
   "location": "On Finding Areas by Integrating",
   "latex": "y &= b\\epsilon^{-x}.",
   "name": "die-away curve",
   "statement": "The die-away curve is given by y equal to b times epsilon to the power minus x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "ordinate of the die-away curve"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant multiplier"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "abscissa"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number (base of natural logarithms)"
    }
   ],
   "sympy": "Eq(y, b*exp(-x))",
   "physics": false,
   "states": [
    "theorem/die-away-curve"
   ],
   "concepts": [
    "concept/curve",
    "concept/euler-s-number",
    "concept/exponential-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-88fdf5281e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "215",
   "location": "On Finding Areas by Integrating",
   "latex": "&= b(1-\\epsilon^{-a}).",
   "name": null,
   "statement": "The area under the die-away curve from x = 0 to x = a equals b times (1 minus epsilon to the minus a).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant multiplier of the die-away curve"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "upper limit of x"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number"
    }
   ],
   "sympy": "Eq(Area, b*(1 - exp(-a)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/definite-integral",
    "concept/exponential-function",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a7416eabd3",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "222",
   "location": "On Finding Areas by Integrating",
   "latex": "\\sqrt[2] {\\frac{1}{l} \\int^l_0 y^2\\, dx}.",
   "name": "quadratic mean",
   "statement": "The quadratic mean of a function y over x from 0 to l is the square root of the mean of y squared over that range.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the function under consideration"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "upper limit of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Q, sqrt(Integral(y**2, (x, 0, l))/l))",
   "physics": false,
   "states": [
    "concept/quadratic-mean"
   ],
   "concepts": [
    "concept/arithmetical-mean",
    "concept/integral",
    "concept/root"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-68f039c7ec",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "223",
   "location": "On Finding Areas by Integrating",
   "latex": "\\text{quadratic mean} = \\frac{1}{\\sqrt 3}\\, al.",
   "name": null,
   "statement": "For y = a x from 0 to l, the quadratic mean is a l divided by the square root of 3.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "coefficient in y = ax"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "upper limit of x"
    }
   ],
   "sympy": "Eq(Q, a*l/sqrt(3))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/quadratic-mean"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e26bc583ff",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "223",
   "location": "On Finding Areas by Integrating",
   "latex": "\\dfrac{2}{\\sqrt 3}=1.155",
   "name": "form-factor",
   "statement": "The form-factor, the ratio of quadratic to arithmetical mean for y = ax, equals 2 over the square root of 3, about 1.155.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "2/√3",
     "meaning": "form-factor for the function y = ax"
    }
   ],
   "sympy": "Eq(2/sqrt(3), 1.155)",
   "physics": false,
   "states": [
    "concept/form-factor"
   ],
   "concepts": [
    "concept/arithmetical-mean",
    "concept/common-ratio",
    "concept/quadratic-mean"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a401ac6fd4",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "223",
   "location": "On Finding Areas by Integrating",
   "latex": "\\text{quadratic mean} = \\sqrt[2]{\\dfrac{l^{2a}}{2a+1}}.",
   "name": null,
   "statement": "For y = x^a from 0 to l, the quadratic mean is the square root of l to the power 2a over (2a+1).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "exponent in y = x^a"
    },
    {
     "unit": null,
     "symbol": "l",
     "meaning": "upper limit of x"
    }
   ],
   "sympy": "Eq(Q, sqrt(l**(2*a)/(2*a+1)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/power",
    "concept/quadratic-mean"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e50a14f883",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "220",
   "location": "On Finding Areas by Integrating",
   "latex": "\\tfrac{1}{2} \\int^{\\theta=\\theta_2}_{\\theta=\\theta_1} r^2\\, d\\theta.",
   "name": null,
   "statement": "The area between a polar curve and two radii at angles theta_1 and theta_2 is one half the integral of r squared d theta.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "r",
     "meaning": "distance of a point of the curve from the pole"
    },
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "angle the radius makes with the initial line OX"
    },
    {
     "unit": "radian",
     "symbol": "\\theta_1",
     "meaning": "lower angle limit"
    },
    {
     "unit": "radian",
     "symbol": "\\theta_2",
     "meaning": "upper angle limit"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/integral",
    "concept/plane-angle",
    "concept/polar-coordinates",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-cb13b3f4e7",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "220",
   "location": "On Finding Areas by Integrating",
   "latex": "r=a(1+\\cos \\theta)",
   "name": null,
   "statement": "The polar equation of Pascal's snail gives r as a times one plus the cosine of theta.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "r",
     "meaning": "distance from the pole"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant scale of the curve"
    },
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "angle with the initial line"
    }
   ],
   "sympy": "Eq(r, a*(1+cos(theta)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/curve",
    "concept/polar-coordinates"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9549b92db4",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "220",
   "location": "On Finding Areas by Integrating",
   "latex": "&= \\frac{a^2(3\\pi+8)}{8}.",
   "name": null,
   "statement": "The area of the first quadrant of Pascal's snail equals a squared times (3 pi + 8) over 8.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant in the polar equation r = a(1+cos θ)"
    }
   ],
   "sympy": "Eq(Area, a**2*(3*pi+8)/8)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/polar-coordinates",
    "concept/quadrant",
    "quantity/area"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b5e5f54d1a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "221",
   "location": "On Finding Areas by Integrating",
   "latex": "y^2 = r^2 - x^2.",
   "name": null,
   "statement": "For a sphere of radius r, a slice at abscissa x has the squared ordinate y squared equal to r squared minus x squared.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "radius of a circular slice of the sphere at abscissa x"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the sphere"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "distance of the slice from the centre"
    }
   ],
   "sympy": "Eq(y**2, r**2 - x**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cartesian-coordinates",
    "concept/circle",
    "concept/sphere",
    "concept/square"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f36105247b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "226",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "\\int u\\, dx = ux - \\int x\\, du + C.",
   "name": "integration by parts formula",
   "statement": "The integral of u dx equals ux minus the integral of x du, plus a constant, so an integral that is hard to find directly can be traded for one that may be easier. In this book x denotes the antiderivative of the second factor, as in the chapter's examples.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "the first factor, a function of the variable of integration"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the antiderivative of the second factor dx (the book's v), so that du is the differential of u"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [
    "theorem/integration-by-parts-formula"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "concept/differential",
    "concept/method-integration-by-parts",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4d9549004f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "226",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "d(ux) = u\\, dx + x\\, du,",
   "name": null,
   "statement": "The differential of the product ux equals u dx plus x du, which is the product rule written in differentials and is the basis of integration by parts.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "a function of the variable x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable (the book's second factor in the product ux)"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/method-integration-by-parts",
    "concept/theorem-product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c8faa15c3e",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "229",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "\\int \\sqrt{1-x^2}\\, dx = \\frac{x \\sqrt{1-x^2}}{2} + \\tfrac{1}{2} \\arcsin x +C.",
   "name": null,
   "statement": "The integral of the square root of 1 minus x squared equals half of x times that square root plus half of arcsin x, plus a constant, obtained from the chapter's integration-by-parts and dodge argument.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable of integration"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sqrt(1 - x**2), x), x*sqrt(1 - x**2)/2 + asin(x)/2 + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/inverse-function",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d4aa9340cf",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "232",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "\\frac{1}{a^2-x^2} = \\frac{1}{2a(a+x)} + \\frac{1}{2a(a-x)},",
   "name": null,
   "statement": "The function 1 over (a squared minus x squared) splits into the sum of two simpler fractions, 1 over 2a(a+x) and 1 over 2a(a-x).",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(1/(a**2 - x**2), 1/(2*a*(a + x)) + 1/(2*a*(a - x)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-fraction",
    "concept/method-partial-fractions"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f44292469d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "232",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "\\dfrac{dy}{dx} = \\dfrac{1}{a^2-x^2}",
   "name": null,
   "statement": "The derivative of y with respect to x equals 1 over (a squared minus x squared); this is a differential equation whose solution is found by integration.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the dependent variable"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    }
   ],
   "sympy": "Eq(Derivative(y, x), 1/(a**2 - x**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d6838ccf1d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "232",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "y = \\dfrac{1}{2a} \\log_\\epsilon \\dfrac{a+x}{a-x} + C?",
   "name": null,
   "statement": "The solution of the differential equation dy/dx = 1/(a squared minus x squared) is y equal to 1 over 2a times the natural logarithm of (a+x)/(a-x), plus a constant.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "the dependent variable, the solution of the differential equation"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the independent variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number, the base of the logarithm"
    }
   ],
   "sympy": "Eq(y, log((a + x)/(a - x))/(2*a) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/differential-equation",
    "concept/euler-s-number",
    "concept/natural-logarithm",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9d17621a84",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "230",
   "location": "Dodges, Pitfalls, and Triumphs",
   "latex": "u=\\dfrac{1}{a} \\arctan \\dfrac{u}{a}",
   "name": null,
   "statement": "As printed, this says u equals (1/a) arctan(u/a); the book means that differentiating (1/a) arctan(u/a) gives du/(u squared + a squared), so the equation is the integral result and is not a literal identity in u. Flagged for the book's loose notation.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "the variable of integration"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "a constant"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/inverse-function",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7d76322ac5",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "234",
   "location": "Finding some Solutions",
   "latex": "ay + b \\frac{dy}{dx} = 0",
   "name": null,
   "statement": "The differential equation of Example 1, in which dy/dx is proportional to y.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(a*y + b*Derivative(y, x), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/derivative",
    "concept/differential-equation",
    "concept/function",
    "concept/variable"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-84169f93af",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "236",
   "location": "Finding some Solutions",
   "latex": "y = C \\epsilon^{-\\efrac{a}{b} x}",
   "name": null,
   "statement": "The solution of Example 1: y equals C times the exponential of minus (a/b) x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of x"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, C*exp(-a*x/b))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/differential-equation",
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/natural-logarithm",
    "concept/solution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0ba207cf50",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "238",
   "location": "Finding some Solutions",
   "latex": "ay+b\\frac{dy}{dt} = g · \\sin 2\\pi nt",
   "name": null,
   "statement": "The differential equation of Example 3, a forced equation with a sinusoidal right-hand side, describing an alternating electric circuit.",
   "kind": "law",
   "symbols": [
    {
     "unit": "ohm",
     "symbol": "a",
     "meaning": "resistance of the circuit"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "coefficient of self-induction of the circuit"
    },
    {
     "unit": null,
     "symbol": "g",
     "meaning": "amplitude of the electromotive force (as the book reads it later)"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, a function of t (current in the book's reading)"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(a*y + b*Derivative(y, t), g*sin(2*pi*n*t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/differential-equation",
    "concept/electric-circuit",
    "concept/electric-current",
    "concept/sine",
    "quantity/coefficient-of-induction",
    "quantity/electromotive-force",
    "quantity/frequency",
    "quantity/resistance"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e1d1db5dde",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "241",
   "location": "Finding some Solutions",
   "latex": "y = g \\left\\{\\frac{ a · \\sin 2 \\pi n t - 2 \\pi n b · \\cos 2 \\pi nt}{ a^2 + 4 \\pi^2  n^2 b^2}\\right\\}",
   "name": null,
   "statement": "The steady-state solution of Example 3 before its amplitude and phase are simplified.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, a function of t"
    },
    {
     "unit": null,
     "symbol": "g",
     "meaning": "amplitude of the electromotive force"
    },
    {
     "unit": "ohm",
     "symbol": "a",
     "meaning": "resistance"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "coefficient of self-induction"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(y, g*(a*sin(2*pi*n*t) - 2*pi*n*b*cos(2*pi*n*t))/(a**2 + 4*pi**2*n**2*b**2))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/differential-equation",
    "concept/electric-current",
    "concept/sine",
    "quantity/frequency",
    "quantity/resistance"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-858d3636f3",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "239",
   "location": "Finding some Solutions",
   "latex": "\\ds\\int u dv = uv - \\int v du",
   "name": "integration by parts",
   "statement": "The general formula for integrating a product by parts.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "a function of the variable chosen to be differentiated"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "a function whose differential is dv, integrated to give v"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [
    "method/integration-by-parts"
   ],
   "concepts": [
    "concept/differential",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d7c4ddcc25",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "242",
   "location": "Finding some Solutions",
   "latex": "M\\, dx + N\\, dy = 0",
   "name": null,
   "statement": "The general first-order differential equation in differential form, to be tested for exactness.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "M",
     "meaning": "function of x and y multiplying dx"
    },
    {
     "unit": null,
     "symbol": "N",
     "meaning": "function of x and y multiplying dy"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential",
    "concept/differential-equation",
    "concept/exact-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9979996f39",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "244",
   "location": "Finding some Solutions",
   "latex": "2 \\frac{d^2y}{dt^2}\\, \\frac{dy}{dt} = \\frac{d \\left(\\dfrac{dy}{dt}\\right)^2}{dt}",
   "name": null,
   "statement": "Multiplying by 2 dy/dt turns the left side into the exact derivative of (dy/dt) squared.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, a function of t"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "independent variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/exact-differential",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-39e7722f6c",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "236",
   "location": "Finding some Solutions",
   "latex": "y = y_0 \\epsilon^{-\\efrac{a}{b} x}",
   "name": null,
   "statement": "Example 1's solution rewritten with the constant C identified as the starting value y_0 of y at x = 0.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y_0",
     "meaning": "value of y at starting (x = 0)"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of x"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, y0*exp(-a*x/b))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/differential-equation",
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/initial-value"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-fab79c6135",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "236",
   "location": "Finding some Solutions",
   "latex": "ay + b \\frac{dy}{dx} = g",
   "name": null,
   "statement": "The differential equation of Example 2, with a constant term g on the right.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "g",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(a*y + b*Derivative(y, x), g)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/derivative",
    "concept/differential-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-82126f0eaa",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "237",
   "location": "Finding some Solutions",
   "latex": "y = \\frac{g}{a} + C\\epsilon^{-\\efrac{a}{b}x}",
   "name": null,
   "statement": "The general solution of Example 2: y equals g/a plus C times the decaying exponential.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of x"
    },
    {
     "unit": null,
     "symbol": "g",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, g/a + C*exp(-a*x/b))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/differential-equation",
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/solution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8b42feb43b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "238",
   "location": "Finding some Solutions",
   "latex": "y = \\frac{g}{a} (1-\\epsilon^{-\\efrac{a}{b} x})",
   "name": null,
   "statement": "The solution of Example 2 satisfying y = 0 when x = 0.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of x"
    },
    {
     "unit": null,
     "symbol": "g",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, g/a*(1 - exp(-a*x/b)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential-equation",
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/initial-value",
    "concept/solution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-854385e821",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "238",
   "location": "Finding some Solutions",
   "latex": "y_{\\text{max.}} = \\dfrac{g}{a}",
   "name": null,
   "statement": "The maximum value that y approaches as x grows indefinitely, in Example 2.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y_{max.}",
     "meaning": "maximum value of y"
    },
    {
     "unit": null,
     "symbol": "g",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    }
   ],
   "sympy": "Eq(y_max, g/a)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/maximum",
    "concept/value-of-a-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0b93ffafcf",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "238",
   "location": "Finding some Solutions",
   "latex": "y = y_{\\text{max.}}(1-\\epsilon^{-\\efrac{a}{b} x})",
   "name": null,
   "statement": "Example 2's solution written in terms of its maximum value y_max; y grows toward that maximum as x increases.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of x"
    },
    {
     "unit": null,
     "symbol": "y_{max.}",
     "meaning": "maximum value of y"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, y_max*(1 - exp(-a*x/b)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/differential-equation",
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/maximum",
    "concept/solution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7a9b9d76b2",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "241",
   "location": "Finding some Solutions",
   "latex": "\\tan \\phi = \\dfrac{2 \\pi n b}{ a}",
   "name": null,
   "statement": "Definition of the angle of lag phi in the alternating-current solution of Example 3.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "angle of lag"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "coefficient of self-induction of the circuit"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "resistance"
    }
   ],
   "sympy": "Eq(tan(phi), 2*pi*n*b/a)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/electric-circuit",
    "concept/electric-induction",
    "concept/plane-angle",
    "concept/tangent-function",
    "quantity/angle",
    "quantity/coefficient-of-induction",
    "quantity/frequency",
    "quantity/resistance"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-794581565a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "241",
   "location": "Finding some Solutions",
   "latex": "\\sin \\phi = \\frac{2 \\pi nb}{\\sqrt{a^2 + 4 \\pi^2 n^2 b^2}}",
   "name": null,
   "statement": "The sine of the lag angle phi, expressed in the circuit constants.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "angle of lag"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "coefficient of self-induction of the circuit"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "resistance"
    }
   ],
   "sympy": "Eq(sin(phi), 2*pi*n*b/sqrt(a**2 + 4*pi**2*n**2*b**2))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/electric-circuit",
    "concept/plane-angle",
    "concept/sine",
    "quantity/angle",
    "quantity/frequency"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b454b9e594",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "241",
   "location": "Finding some Solutions",
   "latex": "\\cos \\phi = \\frac{a}{\\sqrt{a^2 + 4 \\pi^2 n^2 b^2}}",
   "name": null,
   "statement": "The cosine of the lag angle phi, expressed in the circuit constants.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "angle of lag"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "resistance"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "coefficient of self-induction of the circuit"
    }
   ],
   "sympy": "Eq(cos(phi), a/sqrt(a**2 + 4*pi**2*n**2*b**2))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/electric-circuit",
    "concept/plane-angle",
    "quantity/angle",
    "quantity/frequency"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d92c6b1133",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "241",
   "location": "Finding some Solutions",
   "latex": "y = g \\frac{\\sin(2 \\pi nt - \\phi)}{\\sqrt{a^2 + 4 \\pi^2 n^2 b^2}}",
   "name": null,
   "statement": "The steady solution of Example 3: the alternating current y is a sine wave lagging the electromotive force by the angle phi.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "current in the circuit, function of t"
    },
    {
     "unit": null,
     "symbol": "g",
     "meaning": "amplitude of the electromotive force"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "frequency"
    },
    {
     "unit": null,
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": null,
     "symbol": "phi",
     "meaning": "angle of lag"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "resistance"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "coefficient of self-induction of the circuit"
    }
   ],
   "sympy": "Eq(y, g*sin(2*pi*n*t - phi)/sqrt(a**2 + 4*pi**2*n**2*b**2))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/alternating-current",
    "concept/electric-circuit",
    "concept/electric-current",
    "concept/electric-induction",
    "concept/sine",
    "quantity/angle",
    "quantity/coefficient-of-induction",
    "quantity/electromotive-force",
    "quantity/frequency",
    "quantity/resistance",
    "quantity/time"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-bec276802f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "239",
   "location": "Finding some Solutions",
   "latex": "\\int u dv = uv - \\int v du",
   "name": "integration by parts",
   "statement": "The rule of integration by parts, turning the integral of u dv into uv minus the integral of v du.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "function of the variable"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "function of the variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [
    "method/integration-by-parts"
   ],
   "concepts": [
    "concept/differential",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-cba570ff5a",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "242",
   "location": "Finding some Solutions",
   "latex": "\\frac{dM}{dy} = \\frac{dN}{dx}",
   "name": null,
   "statement": "The test for an exact differential: M dx + N dy is exact only if dM/dy equals dN/dx.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "M",
     "meaning": "coefficient of dx in M dx + N dy"
    },
    {
     "unit": null,
     "symbol": "N",
     "meaning": "coefficient of dy in M dx + N dy"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    }
   ],
   "sympy": "Eq(Derivative(M, y), Derivative(N, x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exact-differential",
    "concept/partial-derivative",
    "method/testing-for-an-exact-differential"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-d0800304b6",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "242",
   "location": "Finding some Solutions",
   "latex": "\\frac{\\partial U}{\\partial x} = M",
   "name": null,
   "statement": "If M dx + N dy is exact, M is the partial derivative of a common function U with respect to x.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "U",
     "meaning": "common function from which M and N are formed"
    },
    {
     "unit": null,
     "symbol": "M",
     "meaning": "coefficient of dx"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(Derivative(U, x), M)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exact-differential",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-29b73a251f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "242",
   "location": "Finding some Solutions",
   "latex": "\\frac{\\partial U}{\\partial y} = N",
   "name": null,
   "statement": "If M dx + N dy is exact, N is the partial derivative of the common function U with respect to y.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "U",
     "meaning": "common function from which M and N are formed"
    },
    {
     "unit": null,
     "symbol": "N",
     "meaning": "coefficient of dy"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    }
   ],
   "sympy": "Eq(Derivative(U, y), N)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exact-differential",
    "concept/partial-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a064e51410",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "243",
   "location": "Finding some Solutions",
   "latex": "w = 2x^3y",
   "name": null,
   "statement": "The function w whose differential is the exact expression 6x^2y dx + 2x^3 dy in Example 4.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "w",
     "meaning": "function whose differential is exact"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    }
   ],
   "sympy": "Eq(w, 2*x**3*y)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exact-differential",
    "concept/function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a49f4cfb28",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "243",
   "location": "Finding some Solutions",
   "latex": "U = x^2 + 2x^3y + C",
   "name": null,
   "statement": "The integrated common function U of the exact differential in Example 4, with constant C.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "U",
     "meaning": "common function of the exact differential"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(U, x**2 + 2*x**3*y + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/exact-differential",
    "concept/integral",
    "concept/integrating-factor",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e0c86d815f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "244",
   "location": "Finding some Solutions",
   "latex": "\\left(\\frac{dy}{dt}\\right)^2 + n^2 (y^2-C^2) = 0",
   "name": null,
   "statement": "The first integral of Example 5's equation, obtained by multiplying by 2 dy/dt and integrating. Erratum flagged, not corrected: the chapter then writes dy/dt = -n sqrt(y^2 - C^2), but this relation gives dy/dt = -n sqrt(C^2 - y^2), so the radicand sign in the chapter's next line does not match.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of t"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Derivative(y, t)**2 + n**2*(y**2 - C**2), 0)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/derivative",
    "concept/differential-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c403af7a35",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "244",
   "location": "Finding some Solutions",
   "latex": "\\frac{1}{\\sqrt{C^2 - y^2}} = \\frac{d (\\arcsin \\dfrac{y}{C})}{dy}",
   "name": null,
   "statement": "The derivative of arcsin(y/C) with respect to y equals 1 over the square root of C squared minus y squared.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant"
    }
   ],
   "sympy": "Eq(1/sqrt(C**2 - y**2), Derivative(asin(y/C), y))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/inverse-circular-function",
    "concept/square"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e69472fe2b",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "245",
   "location": "Finding some Solutions",
   "latex": "y = A \\sin nt + B \\cos nt",
   "name": null,
   "statement": "The solution of Example 5 written as a sum of sine and cosine terms with constants A and B.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of t"
    },
    {
     "unit": null,
     "symbol": "A",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "B",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(y, A*sin(n*t) + B*cos(n*t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/differential-equation",
    "concept/sine",
    "concept/solution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3df345ca31",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "244",
   "location": "Finding some Solutions",
   "latex": "y = C \\sin (nt + C_1)",
   "name": null,
   "statement": "Equivalent form of Example 5's solution with amplitude C and constant angle C_1.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of t"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "amplitude"
    },
    {
     "unit": null,
     "symbol": "C_1",
     "meaning": "constant angle from integration"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(y, C*sin(n*t + C_1))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/plane-angle",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-17748598d4",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "245",
   "location": "Finding some Solutions",
   "latex": "\\frac{dw}{dy} = \\frac{1}{\\sqrt{ y^2 + c^2}}",
   "name": null,
   "statement": "The derivative of w = log(y + sqrt(y^2 + c^2)) with respect to y, used in Example 6.",
   "kind": "identity",
   "symbols": [
    {
     "unit": null,
     "symbol": "w",
     "meaning": "logarithm of y + sqrt(y^2 + c^2)"
    },
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "constant"
    }
   ],
   "sympy": "Eq(Derivative(w, y), 1/sqrt(y**2 + c**2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/logarithm",
    "concept/square"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ce738cf544",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "245",
   "location": "Finding some Solutions",
   "latex": "y + \\sqrt{y^2 + c^2} = C \\epsilon^{nx}",
   "name": null,
   "statement": "Result (1) of Example 6: integrating gives y + sqrt(y^2 + c^2) as C times the growing exponential.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of x"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y + sqrt(y**2 + c**2), C*exp(n*x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/exponential-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1cf4c92cec",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "245",
   "location": "Finding some Solutions",
   "latex": "-y + \\sqrt{y^2 + c^2} = \\dfrac{c^2}{C} \\epsilon^{-nx}",
   "name": null,
   "statement": "Result (2) of Example 6, obtained from (1) by multiplying through by the conjugate expression.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of x"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(-y + sqrt(y**2 + c**2), c**2/C*exp(-n*x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/exponential-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-092c804380",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "246",
   "location": "Finding some Solutions",
   "latex": "y = \\frac{1}{2} C \\epsilon^{nx} - \\frac{1}{2}\\, \\frac{c^2}{C} \\epsilon^{-nx}",
   "name": null,
   "statement": "Half the difference of results (1) and (2), giving y as a combination of growing and decaying exponentials.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of x"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, C*exp(n*x)/2 - c**2/(2*C)*exp(-n*x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/solution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-44a0772ea2",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "246",
   "location": "Finding some Solutions",
   "latex": "y = A \\epsilon^{nx} + B \\epsilon^{-nx}",
   "name": null,
   "statement": "The solution of Example 6 as a sum of a growing and a dying exponential with constants A and B.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of x"
    },
    {
     "unit": null,
     "symbol": "A",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "B",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "independent variable"
    }
   ],
   "sympy": "Eq(y, A*exp(n*x) + B*exp(-n*x))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/solution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-bda730c0db",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "246",
   "location": "Finding some Solutions",
   "latex": "b \\frac{d^2y}{dt^2} + a \\frac{dy}{dt} + gy = 0",
   "name": null,
   "statement": "The damped second-order differential equation of Example 7, combining the forms of Examples 1 and 6.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of t"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "g",
     "meaning": "constant coefficient"
    }
   ],
   "sympy": "Eq(b*Derivative(y, (t, 2)) + a*Derivative(y, t) + g*y, 0)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/derivative",
    "concept/differential-equation",
    "concept/higher-order-derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-fcc5d9aabb",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "246",
   "location": "Finding some Solutions",
   "latex": "m = \\frac{a}{2b}",
   "name": null,
   "statement": "Definition of m used in the solution of Example 7.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "m",
     "meaning": "constant from the coefficients a and b"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    }
   ],
   "sympy": "Eq(m, a/(2*b))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-0cdd5fd386",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "246",
   "location": "Finding some Solutions",
   "latex": "n = \\sqrt{\\frac{a^2}{4b^2}} - \\frac{g}{b}",
   "name": null,
   "statement": "Definition of n in Example 7 as printed. Erratum flagged, not corrected: the square root covers only a^2/(4b^2), so the printed expression differs from the usual sqrt(a^2/(4b^2) - g/b).",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant from the coefficients a, b and g"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "constant coefficient"
    },
    {
     "unit": null,
     "symbol": "g",
     "meaning": "constant coefficient"
    }
   ],
   "sympy": "Eq(n, sqrt(a**2/(4*b**2)) - g/b)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/root-of-an-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1b70be9b9d",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "246",
   "location": "Finding some Solutions",
   "latex": "y = (\\epsilon^{-mt})(A \\epsilon^{nt} + B \\epsilon^{-nt})",
   "name": null,
   "statement": "The solution of Example 7 for the damped equation, with m and n as defined there.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "dependent variable, function of t"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "constant a/(2b)"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant defined in Example 7"
    },
    {
     "unit": null,
     "symbol": "A",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "B",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "base of natural logarithms (Euler's number)"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(y, exp(-m*t)*(A*exp(n*t) + B*exp(-n*t)))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/exponential-function",
    "concept/solution",
    "quantity/time"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-f472af2c45",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "247",
   "location": "Finding some Solutions",
   "latex": "\\frac{d^2y}{dt^2} = a^2 \\frac{d^2y}{dx^2}",
   "name": null,
   "statement": "The wave equation of Example 8: the second derivative in time equals a^2 times the second derivative in space.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "displacement of the curve at position x and time t"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position along the x direction"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "velocity of propagation"
    }
   ],
   "sympy": "Eq(Derivative(y, (t, 2)), a**2*Derivative(y, (x, 2)))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/partial-derivative",
    "concept/wave-equation",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7387993e71",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "247",
   "location": "Finding some Solutions",
   "latex": "y = F(x+at) + f(x-at)",
   "name": null,
   "statement": "The general solution of the wave equation in Example 8, with F and f arbitrary functions.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "displacement of the curve at position x and time t"
    },
    {
     "unit": null,
     "symbol": "F",
     "meaning": "arbitrary function"
    },
    {
     "unit": null,
     "symbol": "f",
     "meaning": "arbitrary function"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position along the x direction"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "velocity of propagation"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    }
   ],
   "sympy": "Eq(y, F(x + a*t) + f(x - a*t))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/function",
    "concept/wave-equation",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-6376bae709",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "248",
   "location": "Finding some Solutions",
   "latex": "a = \\sqrt{\\frac{k}{m}}",
   "name": null,
   "statement": "The velocity of propagation of a wave for the more general equation m d^2y/dt^2 = k d^2y/dx^2.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "velocity of propagation"
    },
    {
     "unit": null,
     "symbol": "k",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "constant"
    }
   ],
   "sympy": "Eq(a, sqrt(k/m))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/root",
    "concept/uniform-motion",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-baedea5c2f",
   "chapter": "thompson-calculus-made-easy-1914/ch-xxi",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "248",
   "location": "Finding some Solutions",
   "latex": "m \\frac{d^2y}{dt^2} = k\\, \\frac{d^2y}{dx^2}",
   "name": null,
   "statement": "The more general wave equation of Example 8, whose propagation velocity is sqrt(k/m).",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "y",
     "meaning": "displacement of the curve at position x and time t"
    },
    {
     "unit": "second",
     "symbol": "t",
     "meaning": "time"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "position along the x direction"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "k",
     "meaning": "constant"
    }
   ],
   "sympy": "Eq(m*Derivative(y, (t, 2)), k*Derivative(y, (x, 2)))",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/higher-order-derivative",
    "concept/partial-derivative",
    "concept/wave-equation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a46e74cc91",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\frac{1}{2} x^2 + C",
   "name": null,
   "statement": "The integral of x with respect to x is one half x squared plus the constant of integration (row with dy/dx = 1, y = x).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(x, x), x**2/2 + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "theorem/power-rule-for-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-cd7d495dd0",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "ax + C",
   "name": null,
   "statement": "The integral of the constant a with respect to x is a x plus the constant of integration (row with y = a).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(a, x), a*x + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant",
    "concept/constant-of-integration",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-716bfee8b4",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\frac{1}{2} ax^2 + C",
   "name": null,
   "statement": "The integral of a x with respect to x is one half a x squared plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(a*x, x), a*x**2/2 + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "theorem/power-rule-for-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-3d360f534f",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\frac{1}{3} x^3 + C",
   "name": null,
   "statement": "The integral of x squared with respect to x is one third x cubed plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(x**2, x), x**3/3 + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "theorem/power-rule-for-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-b745cc87f3",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\dfrac{1}{n+1} x^{n+1} + C",
   "name": null,
   "statement": "The integral of x to the power n is x to the power n+1 divided by n+1, plus the constant of integration.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "exponent"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(x**n, x), x**(n+1)/(n+1) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/exponent",
    "concept/integral",
    "theorem/power-rule-for-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-180be9d000",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\log_\\epsilon x + C",
   "name": null,
   "statement": "The integral of 1/x with respect to x is the natural logarithm of x plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number, base of natural logarithm"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(x**(-1), x), log(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e1186b8e51",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\int u\\, dx ± \\int v\\, dx ± \\int w\\, dx",
   "name": "sum rule for integration",
   "statement": "The integral of a sum or difference of functions is the sum or difference of their integrals.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "w",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [
    "theorem/sum-rule-for-integration"
   ],
   "concepts": [
    "concept/function",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a87f82215e",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "u\\, \\dfrac{dv}{dx} + v\\, \\dfrac{du}{dx}",
   "name": "product rule for differentiation",
   "statement": "The derivative of the product uv is u times dv/dx plus v times du/dx (the table gives no general integral of this row).",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(Derivative(u*v, x), u*Derivative(v, x) + v*Derivative(u, x))",
   "physics": false,
   "states": [
    "theorem/product-rule-for-differentiation"
   ],
   "concepts": [
    "concept/derivative"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-13d61cb689",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\dfrac{v\\, \\dfrac{du}{dx} - u\\, \\dfrac{dv}{dx}}{v^2}",
   "name": "quotient rule for differentiation",
   "statement": "The derivative of the quotient u/v is (v du/dx minus u dv/dx) divided by v squared.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "v",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    }
   ],
   "sympy": "Eq(Derivative(u/v, x), (v*Derivative(u, x) - u*Derivative(v, x))/v**2)",
   "physics": false,
   "states": [
    "theorem/quotient-rule-for-differentiation"
   ],
   "concepts": [
    "concept/derivative",
    "concept/quotient"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e35c85441c",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "ux - \\int x\\, du + C",
   "name": "integration by parts",
   "statement": "The integral of u with respect to x equals u x minus the integral of x du, plus the constant of integration.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "u",
     "meaning": "function of x"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [
    "method/integration-by-parts"
   ],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-9d7bc865d0",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\epsilon^x + C",
   "name": null,
   "statement": "The integral of the exponential function epsilon to the x is itself plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\epsilon",
     "meaning": "Euler's number"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(exp(x), x), exp(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/euler-s-number",
    "concept/exponential-function",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-57d855071e",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "x(\\log_\\epsilon x - 1) + C",
   "name": null,
   "statement": "The integral of the natural logarithm of x with respect to x is x times (log x minus 1), plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(log(x), x), x*(log(x) - 1) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-8815df15f1",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "0.4343x (\\log_\\epsilon x - 1) + C",
   "name": null,
   "statement": "The integral of the common logarithm of x with respect to x is 0.4343 x times (natural log of x minus 1), plus the constant of integration; 0.4343 is the modulus 1/log to base epsilon of 10.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(log(x)/log(10), x), 0.4343*x*(log(x) - 1) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/constant-of-integration",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-7403babbc9",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\dfrac{a^x}{\\log_\\epsilon a} + C",
   "name": null,
   "statement": "The integral of a to the power x with respect to x is a^x divided by the natural logarithm of a, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "positive base"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(a**x*log(a), x), a**x/log(a) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/exponential-function",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-48ff44bf6a",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "-\\cos x + C",
   "name": null,
   "statement": "The integral of sin x with respect to x is minus cos x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sin(x), x), -cos(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-2f1cb9c34a",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "\\sin x + C",
   "name": null,
   "statement": "The integral of cos x with respect to x is sin x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(cos(x), x), sin(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1896337166",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "-\\log_\\epsilon \\cos x + C",
   "name": null,
   "statement": "The integral of tan x with respect to x is minus the natural logarithm of cos x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(tan(x), x), -log(cos(x)) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/natural-logarithm",
    "concept/tangent-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-dd26224cde",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "x · \\arcsin x + \\sqrt{1 - x^2} + C",
   "name": null,
   "statement": "The integral of arcsin x with respect to x is x arcsin x plus the square root of one minus x squared, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(asin(x), x), x*asin(x) + sqrt(1 - x**2) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/constant-of-integration",
    "concept/integral",
    "concept/inverse-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-5788f5155d",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "x · \\arccos x - \\sqrt{1 - x^2} + C",
   "name": null,
   "statement": "The integral of arccos x with respect to x is x arccos x minus the square root of one minus x squared, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(acos(x), x), x*acos(x) - sqrt(1 - x**2) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/constant-of-integration",
    "concept/integral",
    "concept/inverse-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4b49668077",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "252",
   "location": "Table of Standard Forms",
   "latex": "x · \\arctan x - \\frac{1}{2} \\log_\\epsilon (1 + x^2) + C",
   "name": null,
   "statement": "The integral of arctan x with respect to x is x arctan x minus one half the natural logarithm of one plus x squared, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(atan(x), x), x*atan(x) - log(1 + x**2)/2 + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/inverse-function",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4134466ca3",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\cosh x + C",
   "name": null,
   "statement": "The integral of sinh x with respect to x is cosh x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sinh(x), x), cosh(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/hyperbolic-function",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1fb045da24",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\sinh x + C",
   "name": null,
   "statement": "The integral of cosh x with respect to x is sinh x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(cosh(x), x), sinh(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/hyperbolic-function",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-e4baf7e24c",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\log_\\epsilon \\cosh x + C",
   "name": null,
   "statement": "The integral of tanh x with respect to x is the natural logarithm of cosh x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(tanh(x), x), log(cosh(x)) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/hyperbolic-function",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-4b95faecad",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\log_\\epsilon (x+a) + C",
   "name": null,
   "statement": "The integral of 1/(x+a) with respect to x is the natural logarithm of x+a, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(1/(x + a), x), log(x + a) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-35de85f576",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\log_\\epsilon (x + \\sqrt{a^2 + x^2}) + C",
   "name": null,
   "statement": "The integral of 1 over the square root of a squared plus x squared is the natural logarithm of x plus that square root, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(1/sqrt(a**2 + x**2), x), log(x + sqrt(a**2 + x**2)) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-36ea38e0d0",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\dfrac{x}{\\sqrt{a^2 + x^2}} + C",
   "name": null,
   "statement": "The integral of a squared over (a squared plus x squared) to the three halves is x over the square root of a squared plus x squared, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(a**2/(a**2 + x**2)**Rational(3,2), x), x/sqrt(a**2 + x**2) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-19814313f2",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "-\\dfrac{1}{a} \\cos ax + C",
   "name": null,
   "statement": "The integral of sin ax with respect to x is minus cos ax divided by a, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sin(a*x), x), -cos(a*x)/a + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-c3d3cb6f67",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\dfrac{1}{a} \\sin ax + C",
   "name": null,
   "statement": "The integral of cos ax with respect to x is sin ax divided by a, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(cos(a*x), x), sin(a*x)/a + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-1422eebb5b",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "-\\dfrac{1}{a} \\log_\\epsilon \\cos ax + C",
   "name": null,
   "statement": "The integral of tan ax with respect to x is minus one over a times the natural logarithm of cos ax, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(tan(a*x), x), -log(cos(a*x))/a + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/natural-logarithm",
    "concept/tangent-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-42fc8fbd29",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\dfrac{x}{2} - \\dfrac{\\sin 2x}{4} + C",
   "name": null,
   "statement": "The integral of sin squared x with respect to x is x over 2 minus sin 2x over 4, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sin(x)**2, x), x/2 - sin(2*x)/4 + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/constant-of-integration",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-6abe9e26a5",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\dfrac{x}{2} + \\dfrac{\\sin 2x}{4} + C",
   "name": null,
   "statement": "The integral of cos squared x with respect to x is x over 2 plus sin 2x over 4, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(cos(x)**2, x), x/2 + sin(2*x)/4 + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-aefa094327",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "-\\frac{\\cos x}{n} \\sin^{n-1} x + \\frac{n-1}{n} \\int \\sin^{n-2} x\\, dx + C",
   "name": null,
   "statement": "Reduction formula: the integral of sin to the power n of x is minus cos x sin to the power n-1 over n, plus (n-1)/n times the integral of sin to the power n-2, plus the constant of integration.",
   "kind": "rule",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "exponent"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/integral",
    "concept/sine",
    "method/formulae-of-reduction"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-48ec2075ee",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\log_\\epsilon \\tan \\dfrac{x}{2} + C",
   "name": null,
   "statement": "The integral of csc x with respect to x is the natural logarithm of tan(x/2), plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(1/sin(x), x), log(tan(x/2)) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosecant",
    "concept/integral",
    "concept/natural-logarithm"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-a2035197e1",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "-\\cotan x + C",
   "name": null,
   "statement": "The integral of csc squared x with respect to x is minus the cotangent of x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(1/sin(x)**2, x), -cot(x) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/cosecant",
    "concept/cotangent",
    "concept/integral"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-14ef531dc7",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\log_\\epsilon \\tan x + C",
   "name": null,
   "statement": "The integral of 1 over sin x cos x with respect to x is the natural logarithm of tan x, plus the constant of integration.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "x",
     "meaning": "angle"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(1/(sin(x)*cos(x)), x), log(tan(x)) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/constant-of-integration",
    "concept/integral",
    "concept/natural-logarithm",
    "concept/tangent-function"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-85327723ec",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\frac{1}{2} \\cos(m - n)x - \\frac{1}{2} \\cos(m + n)x + C",
   "name": null,
   "statement": "FLAG (book/extraction discrepancy): the table gives this as the integral of sin mx sin nx. Differentiating the stated result does not return sin mx sin nx (for m = n it gives an x-independent expression), and the correct integral is sin((m-n)x)/(2(m-n)) minus sin((m+n)x)/(2(m+n)). Recorded as printed, not corrected.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "m",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/integral",
    "concept/product-of-trigonometric-functions",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-57b95c695b",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\dfrac{x}{2} - \\dfrac{\\sin 2ax}{4a} + C",
   "name": null,
   "statement": "The integral of sin squared ax with respect to x is x over 2 minus sin 2ax over 4a, plus the constant of integration. FLAG: the same row prints the derivative of sin squared ax as 2a sin 2ax; differentiation gives a sin 2ax, so the derivative column disagrees with this integral by a factor of 2. Recorded as printed.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(sin(a*x)**2, x), x/2 - sin(2*a*x)/(4*a) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/constant-of-integration",
    "concept/integral",
    "concept/sine"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/eq-ec5dfb218c",
   "chapter": "thompson-calculus-made-easy-1914/ch-table-of-standard-forms",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "253",
   "location": "Table of Standard Forms",
   "latex": "\\dfrac{x}{2} + \\dfrac{\\sin 2ax}{4a} + C",
   "name": null,
   "statement": "The integral of cos squared ax with respect to x is x over 2 plus sin 2ax over 4a, plus the constant of integration. FLAG: the same row prints the derivative of cos squared ax as minus 2a sin 2ax; differentiation gives minus a sin 2ax, a discrepancy of a factor of 2 in the derivative column. Recorded as printed.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "constant"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the variable"
    },
    {
     "unit": null,
     "symbol": "C",
     "meaning": "constant of integration"
    }
   ],
   "sympy": "Eq(Integral(cos(a*x)**2, x), x/2 + sin(2*a*x)/(4*a) + C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/constant-of-integration",
    "concept/cosine",
    "concept/integral"
   ]
  }
 ],
 "exercise_sets": [
  {
   "id": "thompson-calculus-made-easy-1914/ex-i",
   "set": "I",
   "page": "25",
   "chapter": "thompson-calculus-made-easy-1914/ch-iv",
   "practices": [
    "concept/exponent",
    "concept/negative-number",
    "concept/power",
    "concept/root",
    "method/differentiation",
    "theorem/power-rule",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v",
   "set": "V",
   "page": "64",
   "chapter": "thompson-calculus-made-easy-1914/ch-viii",
   "practices": [
    "concept/derivative",
    "concept/higher-order-derivative",
    "concept/rate-of-change",
    "method/differentiation",
    "quantity/acceleration",
    "quantity/angular-acceleration",
    "quantity/angular-velocity",
    "quantity/velocity"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x",
   "set": "X",
   "page": "118",
   "chapter": "thompson-calculus-made-easy-1914/ch-xii",
   "practices": [
    "concept/function",
    "concept/higher-order-derivative",
    "concept/maximum",
    "concept/minimum",
    "concept/stationary-value",
    "method/differentiation",
    "method/equating-the-derivative-to-zero",
    "method/second-derivative-test",
    "theorem/quotient-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii",
   "set": "II",
   "page": "33",
   "chapter": "thompson-calculus-made-easy-1914/ch-v",
   "practices": [
    "concept/constant",
    "concept/constant-factor",
    "concept/constant-term",
    "concept/derivative",
    "concept/diameter",
    "concept/pressure",
    "concept/rate-of-change",
    "concept/specific-gravity",
    "method/differentiating-from-first-principles",
    "method/differentiation",
    "quantity/circumference",
    "quantity/frequency",
    "quantity/lateral-area",
    "theorem/area-of-a-circle",
    "theorem/area-of-a-sphere",
    "theorem/derivative-of-a-constant-multiple",
    "theorem/derivative-of-an-added-constant",
    "theorem/power-rule-for-derivatives",
    "theorem/power-rule-for-differentiation",
    "theorem/volume-of-a-cone",
    "theorem/volume-of-a-sphere"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iv",
   "set": "IV",
   "page": "51",
   "chapter": "thompson-calculus-made-easy-1914/ch-vii",
   "practices": [
    "concept/derivative",
    "concept/higher-order-derivative",
    "method/differentiation",
    "theorem/power-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix",
   "set": "IX",
   "page": "109",
   "chapter": "thompson-calculus-made-easy-1914/ch-xi",
   "practices": [
    "concept/battery",
    "concept/cone",
    "concept/cylinder",
    "concept/cylindrical-surface",
    "concept/derivative",
    "concept/electric-current",
    "concept/maximum",
    "concept/minimum",
    "concept/rate-of-change",
    "concept/rectangle",
    "concept/sphere",
    "concept/square",
    "concept/triangle",
    "method/differentiation",
    "method/equating-the-derivative-to-zero",
    "method/plotting-a-curve",
    "quantity/area",
    "quantity/lateral-area",
    "quantity/total-area",
    "quantity/volume"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi",
   "set": "VI",
   "page": "73",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "practices": [
    "concept/derivative",
    "concept/exponent",
    "method/differentiation",
    "method/substitution",
    "theorem/chain-rule-for-differentiation",
    "theorem/power-rule",
    "theorem/power-rule-for-differentiation",
    "theorem/quotient-rule"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi",
   "set": "XI",
   "page": "130",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiii",
   "practices": [
    "concept/algebraic-fraction",
    "concept/degree",
    "concept/denominator",
    "concept/factor",
    "concept/numerator",
    "concept/proper-algebraic-fraction",
    "method/division",
    "method/equating-coefficients",
    "method/factoring",
    "method/partial-fractions",
    "method/substituting-convenient-values-of-x"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv",
   "set": "XV",
   "page": "180",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvi",
   "practices": [
    "concept/maximum",
    "concept/minimum",
    "concept/partial-derivative",
    "concept/partial-differential",
    "concept/total-differential",
    "method/differentiation",
    "method/equating-the-derivative-to-zero",
    "method/maxima-and-minima-of-a-function-of-two-variables",
    "method/second-derivative-test"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii",
   "set": "III",
   "page": "46",
   "chapter": "thompson-calculus-made-easy-1914/ch-vi",
   "practices": [
    "concept/constant",
    "concept/derivative",
    "concept/exponent",
    "concept/function",
    "concept/polynomial",
    "concept/product",
    "concept/quotient",
    "concept/variable",
    "method/differentiation",
    "theorem/difference-rule-for-differentiation",
    "theorem/power-rule-for-differentiation",
    "theorem/product-rule-for-differentiation",
    "theorem/quotient-rule-for-differentiation",
    "theorem/sum-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vii",
   "set": "VII",
   "page": "75",
   "chapter": "thompson-calculus-made-easy-1914/ch-ix",
   "practices": [
    "concept/derivative",
    "method/differentiation",
    "method/substitution",
    "theorem/chain-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii",
   "set": "XII",
   "page": "153",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "practices": [
    "concept/derivative",
    "concept/exponential-function",
    "concept/logarithm",
    "concept/maximum",
    "concept/minimum",
    "concept/natural-logarithm",
    "method/differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv",
   "set": "XIV",
   "page": "173",
   "chapter": "thompson-calculus-made-easy-1914/ch-xv",
   "practices": [
    "concept/circular-function",
    "concept/cosine",
    "concept/derivative",
    "concept/exponential-function",
    "concept/inverse-function",
    "concept/maximum",
    "concept/minimum",
    "concept/natural-logarithm",
    "concept/secant",
    "concept/sine",
    "concept/tangent-function",
    "method/differentiating-an-inverse-function",
    "theorem/chain-rule-for-differentiation",
    "theorem/product-rule-for-differentiation"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix",
   "set": "XIX",
   "page": "233",
   "chapter": "thompson-calculus-made-easy-1914/ch-xx",
   "practices": [
    "concept/exponential-function",
    "concept/inverse-circular-function",
    "concept/logarithm",
    "concept/natural-logarithm",
    "method/integration",
    "method/integration-by-parts",
    "method/partial-fractions",
    "method/substitution"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvi",
   "set": "XVI",
   "page": "190",
   "chapter": "thompson-calculus-made-easy-1914/ch-xvii",
   "practices": [
    "concept/constant-of-integration",
    "concept/convergent-series",
    "concept/cosine",
    "concept/derivative",
    "concept/geometric-series",
    "concept/geometrical-progression",
    "concept/infinite-sequence",
    "concept/integral",
    "concept/logarithm",
    "concept/natural-logarithm",
    "method/integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii",
   "set": "VIII",
   "page": "91",
   "chapter": "thompson-calculus-made-easy-1914/ch-x",
   "practices": [
    "concept/abscissa",
    "concept/angle-between-two-curves",
    "concept/derivative",
    "concept/intersection-of-two-curves",
    "concept/minimum",
    "concept/point-of-tangency",
    "concept/simultaneous-equations",
    "concept/slope-of-a-curve",
    "concept/tangent",
    "concept/tangent-function",
    "method/differentiation",
    "method/plotting-a-curve",
    "theorem/equation-of-the-tangent"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii",
   "set": "XIII",
   "page": "162",
   "chapter": "thompson-calculus-made-easy-1914/ch-xiv",
   "practices": [
    "concept/constant-of-decrement",
    "concept/die-away-factor",
    "concept/exponential-function",
    "concept/maximum",
    "concept/minimum",
    "concept/natural-logarithm",
    "law/newton-s-law-of-cooling",
    "method/differentiation",
    "quantity/time-constant"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii",
   "set": "XVII",
   "page": "205",
   "chapter": "thompson-calculus-made-easy-1914/ch-xviii",
   "practices": [
    "concept/constant-factor",
    "concept/constant-of-integration",
    "concept/constant-term",
    "concept/cosine",
    "concept/exponential-function",
    "concept/natural-logarithm",
    "concept/sine",
    "method/integration",
    "theorem/constant-multiple-rule-for-integration",
    "theorem/integral-of-1-x",
    "theorem/power-rule-for-integration",
    "theorem/sum-rule-for-integration"
   ]
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii",
   "set": "XVIII",
   "page": "224",
   "chapter": "thompson-calculus-made-easy-1914/ch-xix",
   "practices": [
    "concept/area",
    "concept/area-of-a-surface-of-revolution",
    "concept/arithmetical-mean",
    "concept/definite-integral",
    "concept/exponential-function",
    "concept/form-factor",
    "concept/logarithm",
    "concept/mean-ordinate",
    "concept/parabola",
    "concept/polar-coordinates",
    "concept/quadratic-mean",
    "concept/sine",
    "method/area-of-a-surface-of-revolution",
    "method/finding-an-area-in-polar-coordinates",
    "method/finding-volumes-by-integrating",
    "method/integration",
    "quantity/area",
    "quantity/volume",
    "theorem/area-of-a-parabolic-segment",
    "theorem/volume-of-a-cone"
   ]
  }
 ],
 "problems": [
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/1",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 1",
   "problem_latex": "$y = x^{13}$",
   "markdown": "$y = x^{13}$",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = 13x^{12}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = 13x^{12}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**13",
     "answer_expr": "13*x**12"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: x**13"
   ],
   "shape": [
    "differentiate: x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/10",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 10,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 10",
   "problem_latex": "$y = \\sqrt[n]{\\dfrac{1}{x^m}}$",
   "markdown": "$y = \\sqrt[n]{\\dfrac{1}{x^m}}$",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = -\\dfrac{m}{n} x^{-\\efrac{m+n}{n}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = -\\dfrac{m}{n} x^{-\\efrac{m+n}{n}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(1/x**m)**(1/n)",
     "answer_expr": "-(m/n)*x**(-(m+n)/n)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (x**(-a))**(1/b)"
   ],
   "shape": [
    "differentiate: (x**(-a))**(1/b)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/2",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 2",
   "problem_latex": "$y = x^{-\\efrac{3}{2}}$",
   "markdown": "$y = x^{-\\efrac{3}{2}}$",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = - \\dfrac{3}{2} x^{-\\efrac{5}{2}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = - \\dfrac{3}{2} x^{-\\efrac{5}{2}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**(-Rational(3,2))",
     "answer_expr": "-Rational(3,2)*x**(-Rational(5,2))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: x**(-3/2)"
   ],
   "shape": [
    "differentiate: x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/3",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 3",
   "problem_latex": "$y = x^{2a}$",
   "markdown": "$y = x^{2a}$",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = 2ax^{(2a-1)}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = 2ax^{(2a-1)}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**(2*a)",
     "answer_expr": "2*a*x**(2*a-1)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: x**(2*a)"
   ],
   "shape": [
    "differentiate: x**(N*a)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/4",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 4",
   "problem_latex": "$u = t^{2.4}$",
   "markdown": "$u = t^{2.4}$",
   "answer_latex": [
    "$\\dfrac{du}{dt} = 2.4t^{1.4}$."
   ],
   "answer_markdown": [
    "$\\dfrac{du}{dt} = 2.4t^{1.4}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "t**2.4",
     "answer_expr": "2.4*t**1.4"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: x**(12/5)"
   ],
   "shape": [
    "differentiate: x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/5",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 5",
   "problem_latex": "$z = \\sqrt[3]{u}$",
   "markdown": "$z = \\sqrt[3]{u}$",
   "answer_latex": [
    "$\\dfrac{dz}{du} = \\dfrac{1}{3} u^{-\\efrac{2}{3}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dz}{du} = \\dfrac{1}{3} u^{-\\efrac{2}{3}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "u**(Rational(1,3))",
     "answer_expr": "Rational(1,3)*u**(-Rational(2,3))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: x**(1/3)"
   ],
   "shape": [
    "differentiate: x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/6",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 6",
   "problem_latex": "$y = \\sqrt[3]{x^{-5}}$",
   "markdown": "$y = \\sqrt[3]{x^{-5}}$",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = -\\dfrac{5}{3}x^{\\DPtypo{-\\efrac{8}{5}}{-\\efrac{8}{3}}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = -\\dfrac{5}{3}x^{\\DPtypo{-\\efrac{8}{5}}{-\\efrac{8}{3}}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-ERRATUM",
     "judge_why": "the corrected answer holds; the printed one is the book's misprint",
     "problem_expr": "(x**(-5))**(Rational(1,3))",
     "answer_expr": "-Rational(5,3)*x**(-Rational(8,3))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ERRATUM"
    ]
   },
   "form": [
    "differentiate: (x**(-5))**(1/3)"
   ],
   "shape": [
    "differentiate: (x**N)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/7",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 7",
   "problem_latex": "$u = \\sqrt[5]{\\dfrac{1}{x^8}}$",
   "markdown": "$u = \\sqrt[5]{\\dfrac{1}{x^8}}$",
   "answer_latex": [
    "$\\dfrac{du}{dx} = -\\dfrac{8}{5}x^{-\\efrac{13}{5}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{du}{dx} = -\\dfrac{8}{5}x^{-\\efrac{13}{5}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(1/x**8)**(Rational(1,5))",
     "answer_expr": "-Rational(8,5)*x**(-Rational(13,5))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: Abs(x)**(-8/5)"
   ],
   "shape": [
    "differentiate: Abs(x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/8",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 8",
   "problem_latex": "$y = 2x^a$\\DPtypo{.}{}",
   "markdown": "$y = 2x^a$.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = 2ax^{a-1}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = 2ax^{a-1}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "2*x**a",
     "answer_expr": "2*a*x**(a-1)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 2*x**a"
   ],
   "shape": [
    "differentiate: N*x**a"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-i/9",
   "set": "thompson-calculus-made-easy-1914/ex-i",
   "number": 9,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "25",
   "location": "Exercise I, problem 9",
   "problem_latex": "$y = \\sqrt[q]{x^3}$",
   "markdown": "$y = \\sqrt[q]{x^3}$",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = \\dfrac{3}{q} x^{\\efrac{3-q}{q}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = \\dfrac{3}{q} x^{\\efrac{3-q}{q}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x**3)**(1/q)",
     "answer_expr": "(3/q)*x**((3-q)/q)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (x**3)**(1/a)"
   ],
   "shape": [
    "differentiate: (x**N)**(1/a)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/1",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 1",
   "problem_latex": "$y = ax^3 + 6$.",
   "markdown": "$y = ax^3 + 6$.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = 3ax^2$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = 3ax^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a*x**3 + 6",
     "answer_expr": "3*a*x**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a*x**3 + 6"
   ],
   "shape": [
    "differentiate: N + a*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/10",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 10,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "34",
   "location": "Exercise II, problem 10",
   "problem_latex": "The greatest external pressure~$P$ which a tube\ncan support without collapsing is given by\n\\[\n  P = \\left(\\dfrac{2E}{1-\\sigma^2}\\right) \\dfrac{t^3}{D^3},\n\\]\nwhere $E$~and~$\\sigma$ are constants, $t$~is the thickness of the\ntube and~$D$ is its diameter. (This formula assumes\nthat $4t$~is small compared to~$D$.)\n\nCompare the rate at which $P$~varies for a small\nchange of thickness and for a small change of diameter\ntaking place separately.",
   "markdown": "The greatest external pressure $P$ which a tube can support without collapsing is given by P = (2E1-^2) t^3D^3, where $E$ and $\\sigma$ are constants, $t$ is the thickness of the tube and $D$ is its diameter. (This formula assumes that $4t$ is small compared to $D$.) Compare the rate at which $P$ varies for a small change of thickness and for a small change of diameter taking place separately.",
   "answer_latex": [
    "$\\dfrac{\\text{Rate of change of~$P$ when $t$~varies}}\n            {\\text{Rate of change of~$P$ when $D$~varies}}\n  = - \\dfrac{D}{t}$."
   ],
   "answer_markdown": [
    "$\\dfrac{\\text{Rate of change of~$P$ when $t$~varies}} {\\text{Rate of change of~$P$ when $D$~varies}} = - \\dfrac{D}{t}$."
   ],
   "checks": [
    {
     "task": "partial_ratio",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(2*E/(1 - sigma**2))*t**3/D**3",
     "answer_expr": "-D/t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "partial_ratio: 2*a*x**3/(y**3*(-b**2 + 1))"
   ],
   "shape": [
    "partial_ratio: N*a*x**N*y**N/(-b**N + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "core.ratio.partials"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/11a",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 11,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "34",
   "location": "Exercise II, problem 11a",
   "problem_latex": "Find, from first principles, the rate at which\nthe following vary with respect to a change in\nradius:\n\\begin{SubProbs}\n\\item[(\\textit{a})] the circumference of a circle of radius~$r$;\n\n\\item[(\\textit{b})] the area of a circle of radius~$r$;\n\n\\item[(\\textit{c})] the lateral area of a cone of slant dimension~$l$;\n\n\\item[(\\textit{d})] the volume of a cone of radius~$r$ and height~$h$;\n\n\\item[(\\textit{e})] the area of a sphere of radius~$r$;\n\n\\item[(\\textit{f})] the volume of a sphere of radius~$r$.\n\\end{SubProbs}",
   "markdown": "Find, from first principles, the rate at which the following vary with respect to a change in radius: SubProbs [(*a*)] the circumference of a circle of radius $r$; [(*b*)] the area of a circle of radius $r$; [(*c*)] the lateral area of a cone of slant dimension $l$; [(*d*)] the volume of a cone of radius $r$ and height $h$; [(*e*)] the area of a sphere of radius $r$; [(*f*)] the volume of a sphere of radius $r$. SubProbs",
   "answer_latex": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "answer_markdown": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "2*pi*r",
     "answer_expr": "2*pi"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: 2*pi*x"
   ],
   "shape": [
    "differentiate: pi*N*x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.limit",
    "core.const"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/11b",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 11,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "34",
   "location": "Exercise II, problem 11b",
   "problem_latex": "Find, from first principles, the rate at which\nthe following vary with respect to a change in\nradius:\n\\begin{SubProbs}\n\\item[(\\textit{a})] the circumference of a circle of radius~$r$;\n\n\\item[(\\textit{b})] the area of a circle of radius~$r$;\n\n\\item[(\\textit{c})] the lateral area of a cone of slant dimension~$l$;\n\n\\item[(\\textit{d})] the volume of a cone of radius~$r$ and height~$h$;\n\n\\item[(\\textit{e})] the area of a sphere of radius~$r$;\n\n\\item[(\\textit{f})] the volume of a sphere of radius~$r$.\n\\end{SubProbs}",
   "markdown": "Find, from first principles, the rate at which the following vary with respect to a change in radius: SubProbs [(*a*)] the circumference of a circle of radius $r$; [(*b*)] the area of a circle of radius $r$; [(*c*)] the lateral area of a cone of slant dimension $l$; [(*d*)] the volume of a cone of radius $r$ and height $h$; [(*e*)] the area of a sphere of radius $r$; [(*f*)] the volume of a sphere of radius $r$. SubProbs",
   "answer_latex": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "answer_markdown": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "pi*r**2",
     "answer_expr": "2*pi*r"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: pi*x**2"
   ],
   "shape": [
    "differentiate: pi*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.limit",
    "core.const"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/11c",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 11,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "34",
   "location": "Exercise II, problem 11c",
   "problem_latex": "Find, from first principles, the rate at which\nthe following vary with respect to a change in\nradius:\n\\begin{SubProbs}\n\\item[(\\textit{a})] the circumference of a circle of radius~$r$;\n\n\\item[(\\textit{b})] the area of a circle of radius~$r$;\n\n\\item[(\\textit{c})] the lateral area of a cone of slant dimension~$l$;\n\n\\item[(\\textit{d})] the volume of a cone of radius~$r$ and height~$h$;\n\n\\item[(\\textit{e})] the area of a sphere of radius~$r$;\n\n\\item[(\\textit{f})] the volume of a sphere of radius~$r$.\n\\end{SubProbs}",
   "markdown": "Find, from first principles, the rate at which the following vary with respect to a change in radius: SubProbs [(*a*)] the circumference of a circle of radius $r$; [(*b*)] the area of a circle of radius $r$; [(*c*)] the lateral area of a cone of slant dimension $l$; [(*d*)] the volume of a cone of radius $r$ and height $h$; [(*e*)] the area of a sphere of radius $r$; [(*f*)] the volume of a sphere of radius $r$. SubProbs",
   "answer_latex": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "answer_markdown": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "pi*r*l",
     "answer_expr": "pi*l"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: pi*a*x"
   ],
   "shape": [
    "differentiate: pi*a*x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.limit",
    "core.const"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/11d",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 11,
   "part": "d",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "34",
   "location": "Exercise II, problem 11d",
   "problem_latex": "Find, from first principles, the rate at which\nthe following vary with respect to a change in\nradius:\n\\begin{SubProbs}\n\\item[(\\textit{a})] the circumference of a circle of radius~$r$;\n\n\\item[(\\textit{b})] the area of a circle of radius~$r$;\n\n\\item[(\\textit{c})] the lateral area of a cone of slant dimension~$l$;\n\n\\item[(\\textit{d})] the volume of a cone of radius~$r$ and height~$h$;\n\n\\item[(\\textit{e})] the area of a sphere of radius~$r$;\n\n\\item[(\\textit{f})] the volume of a sphere of radius~$r$.\n\\end{SubProbs}",
   "markdown": "Find, from first principles, the rate at which the following vary with respect to a change in radius: SubProbs [(*a*)] the circumference of a circle of radius $r$; [(*b*)] the area of a circle of radius $r$; [(*c*)] the lateral area of a cone of slant dimension $l$; [(*d*)] the volume of a cone of radius $r$ and height $h$; [(*e*)] the area of a sphere of radius $r$; [(*f*)] the volume of a sphere of radius $r$. SubProbs",
   "answer_latex": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "answer_markdown": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "pi*r**2*h/3",
     "answer_expr": "Rational(2,3)*pi*r*h"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: pi*a*x**2/3"
   ],
   "shape": [
    "differentiate: pi*N*a*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.limit",
    "core.const"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/11e",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 11,
   "part": "e",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "34",
   "location": "Exercise II, problem 11e",
   "problem_latex": "Find, from first principles, the rate at which\nthe following vary with respect to a change in\nradius:\n\\begin{SubProbs}\n\\item[(\\textit{a})] the circumference of a circle of radius~$r$;\n\n\\item[(\\textit{b})] the area of a circle of radius~$r$;\n\n\\item[(\\textit{c})] the lateral area of a cone of slant dimension~$l$;\n\n\\item[(\\textit{d})] the volume of a cone of radius~$r$ and height~$h$;\n\n\\item[(\\textit{e})] the area of a sphere of radius~$r$;\n\n\\item[(\\textit{f})] the volume of a sphere of radius~$r$.\n\\end{SubProbs}",
   "markdown": "Find, from first principles, the rate at which the following vary with respect to a change in radius: SubProbs [(*a*)] the circumference of a circle of radius $r$; [(*b*)] the area of a circle of radius $r$; [(*c*)] the lateral area of a cone of slant dimension $l$; [(*d*)] the volume of a cone of radius $r$ and height $h$; [(*e*)] the area of a sphere of radius $r$; [(*f*)] the volume of a sphere of radius $r$. SubProbs",
   "answer_latex": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "answer_markdown": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "4*pi*r**2",
     "answer_expr": "8*pi*r"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: 4*pi*x**2"
   ],
   "shape": [
    "differentiate: pi*N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.limit",
    "core.const"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/11f",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 11,
   "part": "f",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "34",
   "location": "Exercise II, problem 11f",
   "problem_latex": "Find, from first principles, the rate at which\nthe following vary with respect to a change in\nradius:\n\\begin{SubProbs}\n\\item[(\\textit{a})] the circumference of a circle of radius~$r$;\n\n\\item[(\\textit{b})] the area of a circle of radius~$r$;\n\n\\item[(\\textit{c})] the lateral area of a cone of slant dimension~$l$;\n\n\\item[(\\textit{d})] the volume of a cone of radius~$r$ and height~$h$;\n\n\\item[(\\textit{e})] the area of a sphere of radius~$r$;\n\n\\item[(\\textit{f})] the volume of a sphere of radius~$r$.\n\\end{SubProbs}",
   "markdown": "Find, from first principles, the rate at which the following vary with respect to a change in radius: SubProbs [(*a*)] the circumference of a circle of radius $r$; [(*b*)] the area of a circle of radius $r$; [(*c*)] the lateral area of a cone of slant dimension $l$; [(*d*)] the volume of a cone of radius $r$ and height $h$; [(*e*)] the area of a sphere of radius $r$; [(*f*)] the volume of a sphere of radius $r$. SubProbs",
   "answer_latex": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "answer_markdown": [
    "$2\\pi$, $2\\pi r$, $\\pi l$, $\\frac{2}{3}\\pi rh$, $8\\pi r$, $4\\pi r^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "4*pi*r**3/3",
     "answer_expr": "4*pi*r**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: 4*pi*x**3/3"
   ],
   "shape": [
    "differentiate: pi*N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.limit",
    "core.const"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/12",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 12,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "34",
   "location": "Exercise II, problem 12",
   "problem_latex": "The length~$L$ of an iron rod at the temperature~$T$\nbeing given by $L = l_t\\bigl[1 + 0.000012(T-t)\\bigr]$, where~$l_t$\nis the length at the temperature~$t$, find the rate of\nvariation of the diameter~$D$ of an iron tyre suitable\nfor being shrunk on a wheel, when the temperature~$T$\nvaries.",
   "markdown": "The length $L$ of an iron rod at the temperature $T$ being given by $L = l_t\\bigl[1 + 0.000012(T-t)\\bigr]$, where $l_t$ is the length at the temperature $t$, find the rate of variation of the diameter $D$ of an iron tyre suitable for being shrunk on a wheel, when the temperature $T$ varies.",
   "answer_latex": [
    "(12)~$\\dfrac{dD}{dT} = \\dfrac{0.000012l_t}{\\pi}$."
   ],
   "answer_markdown": [
    "(12) $\\dfrac{dD}{dT} = \\dfrac{0.000012l_t}{\\pi}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "l_t*(1 + 0.000012*(T - t))/pi",
     "answer_expr": "0.000012*l_t/pi"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: a*(-3*b/250000 + 3*x/250000 + 1)/pi"
   ],
   "shape": [
    "differentiate: a*(N*b + N*x + 1)/pi"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.const"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/2",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 2",
   "problem_latex": "$y = 13x^{\\efrac{3}{2}} - c$.",
   "markdown": "$y = 13x^{\\efrac{3}{2}} - c$.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = 13 × \\frac{3}{2}x^{\\efrac{1}{2}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = 13 × \\frac{3}{2}x^{\\efrac{1}{2}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "13*x**Rational(3,2) - c",
     "answer_expr": "13*Rational(3,2)*x**Rational(1,2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: -a + 13*x**(3/2)"
   ],
   "shape": [
    "differentiate: N*x**N - a"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/3",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 3",
   "problem_latex": "$y = 12x^{\\efrac{1}{2}} + c^{\\efrac{1}{2}}$.",
   "markdown": "$y = 12x^{\\efrac{1}{2}} + c^{\\efrac{1}{2}}$.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = 6x^{-\\efrac{1}{2}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = 6x^{-\\efrac{1}{2}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "12*x**Rational(1,2) + c**Rational(1,2)",
     "answer_expr": "6*x**Rational(-1,2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(a) + 12*sqrt(x)"
   ],
   "shape": [
    "differentiate: N*x**N + a**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/4",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 4",
   "problem_latex": "$y = c^{\\efrac{1}{2}} x^{\\efrac{1}{2}}$.",
   "markdown": "$y = c^{\\efrac{1}{2}} x^{\\efrac{1}{2}}$.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = \\dfrac{1}{2}c^{\\efrac{1}{2}} x^{-\\efrac{1}{2}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = \\dfrac{1}{2}c^{\\efrac{1}{2}} x^{-\\efrac{1}{2}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "c**Rational(1,2)*x**Rational(1,2)",
     "answer_expr": "Rational(1,2)*c**Rational(1,2)*x**Rational(-1,2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(a)*sqrt(x)"
   ],
   "shape": [
    "differentiate: a**N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/5",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 5",
   "problem_latex": "$u = \\dfrac{az^n - 1}{c}$.",
   "markdown": "$u = \\dfrac{az^n - 1}{c}$.",
   "answer_latex": [
    "$\\dfrac{du}{dz} = \\dfrac{an}{c} z^{n-1}$."
   ],
   "answer_markdown": [
    "$\\dfrac{du}{dz} = \\dfrac{an}{c} z^{n-1}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(a*z**n - 1)/c",
     "answer_expr": "a*n/c*z**(n-1)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (a*x**c - 1)/b"
   ],
   "shape": [
    "differentiate: (a*x**c - 1)/b"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/6",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 6",
   "problem_latex": "$y = 1.18t^2 + 22.4$.",
   "markdown": "$y = 1.18t^2 + 22.4$.",
   "answer_latex": [
    "$\\dfrac{dy}{dt} = 2.36t$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dt} = 2.36t$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1.18*t**2 + 22.4",
     "answer_expr": "2.36*t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 59*x**2/50 + 112/5"
   ],
   "shape": [
    "differentiate: N*x**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/7",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 7",
   "problem_latex": "If $l_t$~and $l_0$ be the lengths of a rod of iron at\nthe temperatures $t°$\\;C.~and $0°$\\;C. respectively, then\n$l_t = l_0(1 + 0.000012t)$. Find the change of length of the\nrod per degree Centigrade.",
   "markdown": "If $l_t$ and $l_0$ be the lengths of a rod of iron at the temperatures $t°$C. and $0°$C. respectively, then $l_t = l_0(1 + 0.000012t)$. Find the change of length of the rod per degree Centigrade.",
   "answer_latex": [
    "$\\dfrac{dl_t}{dt} = 0.000012×l_0$."
   ],
   "answer_markdown": [
    "$\\dfrac{dl_t}{dt} = 0.000012×l_0$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "l_0*(1 + 0.000012*t)",
     "answer_expr": "0.000012*l_0"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a*(3*x/250000 + 1)"
   ],
   "shape": [
    "differentiate: a*(N*x + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/8",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 8",
   "problem_latex": "It has been found that if $c$~be the candle power\nof an incandescent electric lamp, and~$V$ be the voltage,\n$c = aV^b$, where $a$~and $b$ are constants.\n\nFind the rate of change of the candle power with\nthe voltage, and calculate the change of candle power\nper volt at $80$, $100$ and~$120$ volts in the case of a lamp\nfor which $a = 0.5×10^{-10}$~and $b=6$.",
   "markdown": "It has been found that if $c$ be the candle power of an incandescent electric lamp, and $V$ be the voltage, $c = aV^b$, where $a$ and $b$ are constants. Find the rate of change of the candle power with the voltage, and calculate the change of candle power per volt at $80$, $100$ and $120$ volts in the case of a lamp for which $a = 0.5×10^{-10}$ and $b=6$.",
   "answer_latex": [
    "$\\dfrac{dC}{dV} = abV^{b-1}$, $0.98$, $3.00$ and $7.47$~candle power per volt respectively."
   ],
   "answer_markdown": [
    "$\\dfrac{dC}{dV} = abV^{b-1}$, $0.98$, $3.00$ and $7.47$ candle power per volt respectively."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a*V**b",
     "answer_expr": "a*b*V**(b-1)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.98304, printed 0.98 (half-unit 0.005; correctly rounded at the printed digits: 0.98)",
     "problem_expr": "a*V**b",
     "answer_expr": "a*b*V**(b-1)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 3.0, printed 3.00 (half-unit 0.005; correctly rounded at the printed digits: 3.0)",
     "problem_expr": "a*V**b",
     "answer_expr": "a*b*V**(b-1)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS-LOOSE",
     "judge_why": "computed 7.46496, printed 7.47 (half-unit 0.005; correctly rounded at the printed digits: 7.46): one unit off in the last printed place",
     "problem_expr": "a*V**b",
     "answer_expr": "a*b*V**(b-1)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "PASS",
     "PASS-LOOSE"
    ]
   },
   "form": [
    "differentiate: a*x**b",
    "evaluate: a*x**b at V=80, a=0.5e-10, b=6",
    "evaluate: a*x**b at V=100, a=0.5e-10, b=6",
    "evaluate: a*x**b at V=120, a=0.5e-10, b=6"
   ],
   "shape": [
    "differentiate: a*x**b",
    "evaluate: a*x**b"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/91",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 9,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 91",
   "problem_latex": "The frequency~$n$ of vibration of a string of\ndiameter~$D$, length~$L$ and specific gravity~$\\sigma$, stretched\nwith a force~$T$, is given by\n\\[\nn = \\dfrac{1}{DL} \\sqrt{\\dfrac{gT}{\\pi\\sigma}}.\n\\]\n\nFind the rate of change of the frequency when $D$,~$L$,\n$\\sigma$ and~$T$ are varied singly.",
   "markdown": "The frequency $n$ of vibration of a string of diameter $D$, length $L$ and specific gravity $\\sigma$, stretched with a force $T$, is given by n = 1DL gT. Find the rate of change of the frequency when $D$, $L$, $\\sigma$ and $T$ are varied singly.",
   "answer_latex": [
    "$\\begin{aligned}[t]\n  \\dfrac{dn}{dD} &= -\\dfrac{1}{LD^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, &\n  \\dfrac{dn}{dL} &= -\\dfrac{1}{DL^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, \\\\\n%\n  \\dfrac{dn}{d \\sigma}\n  &= -\\dfrac{1}{2DL} \\sqrt{\\dfrac{gT}{\\pi \\sigma^3}}, &\n  \\dfrac{dn}{dT} &=  \\dfrac{1}{2DL} \\sqrt{\\dfrac{g}{\\pi \\sigma T}}.\n\\end{aligned}$"
   ],
   "answer_markdown": [
    "$\\begin{aligned}[t] \\dfrac{dn}{dD} &= -\\dfrac{1}{LD^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, & \\dfrac{dn}{dL} &= -\\dfrac{1}{DL^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, \\\\ % \\dfrac{dn}{d \\sigma} &= -\\dfrac{1}{2DL} \\sqrt{\\dfrac{gT}{\\pi \\sigma^3}}, & \\dfrac{dn}{dT} &= \\dfrac{1}{2DL} \\sqrt{\\dfrac{g}{\\pi \\sigma T}}. \\end{aligned}$"
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/(D*L)*sqrt(g*T/(pi*sigma))",
     "answer_expr": "-1/(L*D**2)*sqrt(g*T/(pi*sigma))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(b*c/d)/(sqrt(pi)*a*x)"
   ],
   "shape": [
    "differentiate: pi**N*(b*c/d)**N/(a*x)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-ii/92"
   ],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/92",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 9,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 92",
   "problem_latex": "The frequency~$n$ of vibration of a string of\ndiameter~$D$, length~$L$ and specific gravity~$\\sigma$, stretched\nwith a force~$T$, is given by\n\\[\nn = \\dfrac{1}{DL} \\sqrt{\\dfrac{gT}{\\pi\\sigma}}.\n\\]\n\nFind the rate of change of the frequency when $D$,~$L$,\n$\\sigma$ and~$T$ are varied singly.",
   "markdown": "The frequency $n$ of vibration of a string of diameter $D$, length $L$ and specific gravity $\\sigma$, stretched with a force $T$, is given by n = 1DL gT. Find the rate of change of the frequency when $D$, $L$, $\\sigma$ and $T$ are varied singly.",
   "answer_latex": [
    "$\\begin{aligned}[t]\n  \\dfrac{dn}{dD} &= -\\dfrac{1}{LD^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, &\n  \\dfrac{dn}{dL} &= -\\dfrac{1}{DL^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, \\\\\n%\n  \\dfrac{dn}{d \\sigma}\n  &= -\\dfrac{1}{2DL} \\sqrt{\\dfrac{gT}{\\pi \\sigma^3}}, &\n  \\dfrac{dn}{dT} &=  \\dfrac{1}{2DL} \\sqrt{\\dfrac{g}{\\pi \\sigma T}}.\n\\end{aligned}$"
   ],
   "answer_markdown": [
    "$\\begin{aligned}[t] \\dfrac{dn}{dD} &= -\\dfrac{1}{LD^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, & \\dfrac{dn}{dL} &= -\\dfrac{1}{DL^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, \\\\ % \\dfrac{dn}{d \\sigma} &= -\\dfrac{1}{2DL} \\sqrt{\\dfrac{gT}{\\pi \\sigma^3}}, & \\dfrac{dn}{dT} &= \\dfrac{1}{2DL} \\sqrt{\\dfrac{g}{\\pi \\sigma T}}. \\end{aligned}$"
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/(D*L)*sqrt(g*T/(pi*sigma))",
     "answer_expr": "-1/(D*L**2)*sqrt(g*T/(pi*sigma))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(b*c/d)/(sqrt(pi)*a*x)"
   ],
   "shape": [
    "differentiate: pi**N*(b*c/d)**N/(a*x)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-ii/91"
   ],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/93",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 9,
   "part": "3",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 93",
   "problem_latex": "The frequency~$n$ of vibration of a string of\ndiameter~$D$, length~$L$ and specific gravity~$\\sigma$, stretched\nwith a force~$T$, is given by\n\\[\nn = \\dfrac{1}{DL} \\sqrt{\\dfrac{gT}{\\pi\\sigma}}.\n\\]\n\nFind the rate of change of the frequency when $D$,~$L$,\n$\\sigma$ and~$T$ are varied singly.",
   "markdown": "The frequency $n$ of vibration of a string of diameter $D$, length $L$ and specific gravity $\\sigma$, stretched with a force $T$, is given by n = 1DL gT. Find the rate of change of the frequency when $D$, $L$, $\\sigma$ and $T$ are varied singly.",
   "answer_latex": [
    "$\\begin{aligned}[t]\n  \\dfrac{dn}{dD} &= -\\dfrac{1}{LD^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, &\n  \\dfrac{dn}{dL} &= -\\dfrac{1}{DL^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, \\\\\n%\n  \\dfrac{dn}{d \\sigma}\n  &= -\\dfrac{1}{2DL} \\sqrt{\\dfrac{gT}{\\pi \\sigma^3}}, &\n  \\dfrac{dn}{dT} &=  \\dfrac{1}{2DL} \\sqrt{\\dfrac{g}{\\pi \\sigma T}}.\n\\end{aligned}$"
   ],
   "answer_markdown": [
    "$\\begin{aligned}[t] \\dfrac{dn}{dD} &= -\\dfrac{1}{LD^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, & \\dfrac{dn}{dL} &= -\\dfrac{1}{DL^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, \\\\ % \\dfrac{dn}{d \\sigma} &= -\\dfrac{1}{2DL} \\sqrt{\\dfrac{gT}{\\pi \\sigma^3}}, & \\dfrac{dn}{dT} &= \\dfrac{1}{2DL} \\sqrt{\\dfrac{g}{\\pi \\sigma T}}. \\end{aligned}$"
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/(D*L)*sqrt(g*T/(pi*sigma))",
     "answer_expr": "-1/(2*D*L)*sqrt(g*T/(pi*sigma**3))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(c*d/x)/(sqrt(pi)*a*b)"
   ],
   "shape": [
    "differentiate: pi**N*(c*d/x)**N/(a*b)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ii/94",
   "set": "thompson-calculus-made-easy-1914/ex-ii",
   "number": 9,
   "part": "4",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "33",
   "location": "Exercise II, problem 94",
   "problem_latex": "The frequency~$n$ of vibration of a string of\ndiameter~$D$, length~$L$ and specific gravity~$\\sigma$, stretched\nwith a force~$T$, is given by\n\\[\nn = \\dfrac{1}{DL} \\sqrt{\\dfrac{gT}{\\pi\\sigma}}.\n\\]\n\nFind the rate of change of the frequency when $D$,~$L$,\n$\\sigma$ and~$T$ are varied singly.",
   "markdown": "The frequency $n$ of vibration of a string of diameter $D$, length $L$ and specific gravity $\\sigma$, stretched with a force $T$, is given by n = 1DL gT. Find the rate of change of the frequency when $D$, $L$, $\\sigma$ and $T$ are varied singly.",
   "answer_latex": [
    "$\\begin{aligned}[t]\n  \\dfrac{dn}{dD} &= -\\dfrac{1}{LD^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, &\n  \\dfrac{dn}{dL} &= -\\dfrac{1}{DL^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, \\\\\n%\n  \\dfrac{dn}{d \\sigma}\n  &= -\\dfrac{1}{2DL} \\sqrt{\\dfrac{gT}{\\pi \\sigma^3}}, &\n  \\dfrac{dn}{dT} &=  \\dfrac{1}{2DL} \\sqrt{\\dfrac{g}{\\pi \\sigma T}}.\n\\end{aligned}$"
   ],
   "answer_markdown": [
    "$\\begin{aligned}[t] \\dfrac{dn}{dD} &= -\\dfrac{1}{LD^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, & \\dfrac{dn}{dL} &= -\\dfrac{1}{DL^2} \\sqrt{\\dfrac{gT}{\\pi \\sigma}}, \\\\ % \\dfrac{dn}{d \\sigma} &= -\\dfrac{1}{2DL} \\sqrt{\\dfrac{gT}{\\pi \\sigma^3}}, & \\dfrac{dn}{dT} &= \\dfrac{1}{2DL} \\sqrt{\\dfrac{g}{\\pi \\sigma T}}. \\end{aligned}$"
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/(D*L)*sqrt(g*T/(pi*sigma))",
     "answer_expr": "1/(2*D*L)*sqrt(g/(pi*sigma*T))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(c*x/d)/(sqrt(pi)*a*b)"
   ],
   "shape": [
    "differentiate: pi**N*(c*x/d)**N/(a*b)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/10",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 10,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "47",
   "location": "Exercise III, problem 10",
   "problem_latex": "$y = \\dfrac{x^n + a}{x^{-n} + b}$.",
   "markdown": "$y = \\dfrac{x^n + a}{x^{-n} + b}$.",
   "answer_latex": [
    "$\\dfrac{anx^{-n-1} + bnx^{n-1} + 2nx^{-1}}{(x^{-n} + b)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{anx^{-n-1} + bnx^{n-1} + 2nx^{-1}}{(x^{-n} + b)^2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x**n + a)/(x**(-n) + b)",
     "answer_expr": "(a*n*x**(-n - 1) + b*n*x**(n - 1) + 2*n*x**(-1))/(x**(-n) + b)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (a + x**c)/(b + x**(-c))"
   ],
   "shape": [
    "differentiate: (a + x**c)/(b + x**(-c))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/11",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 11,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "47",
   "location": "Exercise III, problem 11",
   "problem_latex": "The temperature~$t$ of the filament of an incandescent\nelectric lamp is connected to the current\npassing through the lamp by the relation\n\\[\nC = a + bt + ct^2.\n\\]\n\nFind an expression giving the variation of the\ncurrent corresponding to a variation of temperature.",
   "markdown": "The temperature $t$ of the filament of an incandescent electric lamp is connected to the current passing through the lamp by the relation C = a + bt + ct^2. Find an expression giving the variation of the current corresponding to a variation of temperature.",
   "answer_latex": [
    "$b + 2ct$."
   ],
   "answer_markdown": [
    "$b + 2ct$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a + b*t + c*t**2",
     "answer_expr": "b + 2*c*t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a + b*x + c*x**2"
   ],
   "shape": [
    "differentiate: a + b*x + c*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/121",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 12,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "47",
   "location": "Exercise III, problem 121",
   "problem_latex": "The following formulae have been proposed to\nexpress the relation between the electric resistance $R$\nof a wire at the temperature $t°$\\;C., and the resistance\n$R_0$ of that same wire at $0°$ Centigrade, $a$, $b$, $c$ being\nconstants.\n\\begin{align*}\nR &= R_0(1 + at + bt^2). \\\\\nR &= R_0(1 + at + b\\sqrt{t}). \\\\\nR &= R_0(1 + at + bt^2)^{-1}.\n\\end{align*}\n\nFind the rate of variation of the resistance with\nregard to temperature as given by each of these\nformulae.",
   "markdown": "The following formulae have been proposed to express the relation between the electric resistance $R$ of a wire at the temperature $t°$C., and the resistance $R_0$ of that same wire at $0°$ Centigrade, $a$, $b$, $c$ being constants. align* R &= R_0(1 + at + bt^2). R &= R_0(1 + at + bt). R &= R_0(1 + at + bt^2)^-1. align* Find the rate of variation of the resistance with regard to temperature as given by each of these formulae.",
   "answer_latex": [
    "$R_0(a + 2bt)$,\\quad $R_0 \\left(a + \\dfrac{b}{2\\sqrt{t}}\\right)$,\\quad\n  $-\\dfrac{R_0(a + 2bt)}{(1 + at + bt^2)^2}$\\quad or\\quad $\\dfrac{R^2 (a + 2bt)}{R_0}$."
   ],
   "answer_markdown": [
    "$R_0(a + 2bt)$, $R_0 \\left(a + \\dfrac{b}{2\\sqrt{t}}\\right)$, $-\\dfrac{R_0(a + 2bt)}{(1 + at + bt^2)^2}$ or $\\dfrac{R^2 (a + 2bt)}{R_0}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "R_0*(1 + a*t + b*t**2)",
     "answer_expr": "R_0*(a + 2*b*t)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a*(b*x + c*x**2 + 1)"
   ],
   "shape": [
    "differentiate: a*(b*x + c*x**N + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/122",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 12,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "47",
   "location": "Exercise III, problem 122",
   "problem_latex": "The following formulae have been proposed to\nexpress the relation between the electric resistance $R$\nof a wire at the temperature $t°$\\;C., and the resistance\n$R_0$ of that same wire at $0°$ Centigrade, $a$, $b$, $c$ being\nconstants.\n\\begin{align*}\nR &= R_0(1 + at + bt^2). \\\\\nR &= R_0(1 + at + b\\sqrt{t}). \\\\\nR &= R_0(1 + at + bt^2)^{-1}.\n\\end{align*}\n\nFind the rate of variation of the resistance with\nregard to temperature as given by each of these\nformulae.",
   "markdown": "The following formulae have been proposed to express the relation between the electric resistance $R$ of a wire at the temperature $t°$C., and the resistance $R_0$ of that same wire at $0°$ Centigrade, $a$, $b$, $c$ being constants. align* R &= R_0(1 + at + bt^2). R &= R_0(1 + at + bt). R &= R_0(1 + at + bt^2)^-1. align* Find the rate of variation of the resistance with regard to temperature as given by each of these formulae.",
   "answer_latex": [
    "$R_0(a + 2bt)$,\\quad $R_0 \\left(a + \\dfrac{b}{2\\sqrt{t}}\\right)$,\\quad\n  $-\\dfrac{R_0(a + 2bt)}{(1 + at + bt^2)^2}$\\quad or\\quad $\\dfrac{R^2 (a + 2bt)}{R_0}$."
   ],
   "answer_markdown": [
    "$R_0(a + 2bt)$, $R_0 \\left(a + \\dfrac{b}{2\\sqrt{t}}\\right)$, $-\\dfrac{R_0(a + 2bt)}{(1 + at + bt^2)^2}$ or $\\dfrac{R^2 (a + 2bt)}{R_0}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "R_0*(1 + a*t + b*sqrt(t))",
     "answer_expr": "R_0*(a + b/(2*sqrt(t)))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a*(b*x + c*sqrt(x) + 1)"
   ],
   "shape": [
    "differentiate: a*(b*x + c*x**N + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/123",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 12,
   "part": "3",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "47",
   "location": "Exercise III, problem 123",
   "problem_latex": "The following formulae have been proposed to\nexpress the relation between the electric resistance $R$\nof a wire at the temperature $t°$\\;C., and the resistance\n$R_0$ of that same wire at $0°$ Centigrade, $a$, $b$, $c$ being\nconstants.\n\\begin{align*}\nR &= R_0(1 + at + bt^2). \\\\\nR &= R_0(1 + at + b\\sqrt{t}). \\\\\nR &= R_0(1 + at + bt^2)^{-1}.\n\\end{align*}\n\nFind the rate of variation of the resistance with\nregard to temperature as given by each of these\nformulae.",
   "markdown": "The following formulae have been proposed to express the relation between the electric resistance $R$ of a wire at the temperature $t°$C., and the resistance $R_0$ of that same wire at $0°$ Centigrade, $a$, $b$, $c$ being constants. align* R &= R_0(1 + at + bt^2). R &= R_0(1 + at + bt). R &= R_0(1 + at + bt^2)^-1. align* Find the rate of variation of the resistance with regard to temperature as given by each of these formulae.",
   "answer_latex": [
    "$R_0(a + 2bt)$,\\quad $R_0 \\left(a + \\dfrac{b}{2\\sqrt{t}}\\right)$,\\quad\n  $-\\dfrac{R_0(a + 2bt)}{(1 + at + bt^2)^2}$\\quad or\\quad $\\dfrac{R^2 (a + 2bt)}{R_0}$."
   ],
   "answer_markdown": [
    "$R_0(a + 2bt)$, $R_0 \\left(a + \\dfrac{b}{2\\sqrt{t}}\\right)$, $-\\dfrac{R_0(a + 2bt)}{(1 + at + bt^2)^2}$ or $\\dfrac{R^2 (a + 2bt)}{R_0}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-ALT-ERRATUM",
     "judge_why": "the main answer holds; a printed alternative fails under its stated relation (an unmarked misprint)",
     "problem_expr": "R_0*(1 + a*t + b*t**2)**(-1)",
     "answer_expr": "-R_0*(a + 2*b*t)/(1 + a*t + b*t**2)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ALT-ERRATUM"
    ]
   },
   "form": [
    "differentiate: a/(b*x + c*x**2 + 1)"
   ],
   "shape": [
    "differentiate: a/(b*x + c*x**N + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/13",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 13,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "47",
   "location": "Exercise III, problem 13",
   "problem_latex": "The electromotive-force~$E$ of a certain type of\nstandard cell has been found to vary with the temperature~$t$\naccording to the relation\n\\[\nE = 1.4340 \\bigl[1 - 0.000814(t-15)\n                 + 0.000007(t-15)^2\\bigr] \\text{ volts}.\n\\]\n\nFind the change of electromotive-force per degree,\nat $15°$, $20°$ and~$25°$.",
   "markdown": "The electromotive-force $E$ of a certain type of standard cell has been found to vary with the temperature $t$ according to the relation E = 1.4340 [1 - 0.000814(t-15) + 0.000007(t-15)^2] volts. Find the change of electromotive-force per degree, at $15°$, $20°$ and $25°$.",
   "answer_latex": [
    "$1.4340(0.000014t - \\DPtypo{0.000828}{0.001024})$,\\quad $-0.00117$,\\quad $-0.00107$,\\quad $-0.00097$."
   ],
   "answer_markdown": [
    "$1.4340(0.000014t - \\DPtypo{0.000828}{0.001024})$, $-0.00117$, $-0.00107$, $-0.00097$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-ERRATUM",
     "judge_why": "the corrected answer holds; the printed one is the book's misprint",
     "problem_expr": "1.4340*(1 - 0.000814*(t - 15) + 0.000007*(t - 15)**2)",
     "answer_expr": "1.4340*(0.000014*t - 0.001024)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -0.001167276, printed -0.00117 (half-unit 5.0e-6; correctly rounded at the printed digits: -0.00117)",
     "problem_expr": "1.4340*(1 - 0.000814*(t - 15) + 0.000007*(t - 15)**2)",
     "answer_expr": "1.4340*(0.000014*t - 0.001024)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -0.001066896, printed -0.00107 (half-unit 5.0e-6; correctly rounded at the printed digits: -0.00107)",
     "problem_expr": "1.4340*(1 - 0.000814*(t - 15) + 0.000007*(t - 15)**2)",
     "answer_expr": "1.4340*(0.000014*t - 0.001024)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -0.000966516, printed -0.00097 (half-unit 5.0e-6; correctly rounded at the printed digits: -0.00097)",
     "problem_expr": "1.4340*(1 - 0.000814*(t - 15) + 0.000007*(t - 15)**2)",
     "answer_expr": "1.4340*(0.000014*t - 0.001024)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ERRATUM",
     "PASS",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: -291819*x/250000000 + 5019*(x - 15)**2/500000000 + 72575457/50000000",
    "evaluate: -291819*x/250000000 + 5019*(x - 15)**2/500000000 + 72575457/50000000 at t=15",
    "evaluate: -291819*x/250000000 + 5019*(x - 15)**2/500000000 + 72575457/50000000 at t=20",
    "evaluate: -291819*x/250000000 + 5019*(x - 15)**2/500000000 + 72575457/50000000 at t=25"
   ],
   "shape": [
    "differentiate: N*x + N*(N + x)**N + N",
    "evaluate: N*x + N*(N + x)**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/14a",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 14,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "48",
   "location": "Exercise III, problem 14a",
   "problem_latex": "The electromotive-force necessary to maintain\nan electric arc of length~$l$ with a current of intensity~$i$\nhas been found by Mrs.~Ayrton to be\n\\[\nE = a + bl + \\frac{c + kl}{i},\n\\]\nwhere $a$, $b$, $c$, $k$ are constants.\n\nFind an expression for the variation of the electromotive\\DPtypo{ }{-}force\n(\\textit{a})~with regard to the length of the arc;\n(\\textit{b})~with regard to the strength of the current.",
   "markdown": "The electromotive-force necessary to maintain an electric arc of length $l$ with a current of intensity $i$ has been found by Mrs. Ayrton to be E = a + bl + c + kli, where $a$, $b$, $c$, $k$ are constants. Find an expression for the variation of the electromotiveforce (*a*) with regard to the length of the arc; (*b*) with regard to the strength of the current.",
   "answer_latex": [
    "$\\dfrac{dE}{dl} = b + \\dfrac{k}{i}$,\\quad $\\dfrac{dE}{di} = -\\dfrac{c + kl}{i^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dE}{dl} = b + \\dfrac{k}{i}$, $\\dfrac{dE}{di} = -\\dfrac{c + kl}{i^2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a + b*l + (c + k*l)/i",
     "answer_expr": "b + k/i"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a + b*x + (c + e*x)/d"
   ],
   "shape": [
    "differentiate: a + b*x + (c + e*x)/d"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/14b",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 14,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "48",
   "location": "Exercise III, problem 14b",
   "problem_latex": "The electromotive-force necessary to maintain\nan electric arc of length~$l$ with a current of intensity~$i$\nhas been found by Mrs.~Ayrton to be\n\\[\nE = a + bl + \\frac{c + kl}{i},\n\\]\nwhere $a$, $b$, $c$, $k$ are constants.\n\nFind an expression for the variation of the electromotive\\DPtypo{ }{-}force\n(\\textit{a})~with regard to the length of the arc;\n(\\textit{b})~with regard to the strength of the current.",
   "markdown": "The electromotive-force necessary to maintain an electric arc of length $l$ with a current of intensity $i$ has been found by Mrs. Ayrton to be E = a + bl + c + kli, where $a$, $b$, $c$, $k$ are constants. Find an expression for the variation of the electromotiveforce (*a*) with regard to the length of the arc; (*b*) with regard to the strength of the current.",
   "answer_latex": [
    "$\\dfrac{dE}{dl} = b + \\dfrac{k}{i}$,\\quad $\\dfrac{dE}{di} = -\\dfrac{c + kl}{i^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dE}{dl} = b + \\dfrac{k}{i}$, $\\dfrac{dE}{di} = -\\dfrac{c + kl}{i^2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a + b*l + (c + k*l)/i",
     "answer_expr": "-(c + k*l)/i**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a + b*e + (c + d*e)/x"
   ],
   "shape": [
    "differentiate: a + b*e + (c + d*e)/x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/1a",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 1,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "46",
   "location": "Exercise III, problem 1a",
   "problem_latex": "Differentiate\\Pagelabel{examples2}\n\\begin{SubProbs}\n\\item[(\\textit{a})] $u = 1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3} + \\dotsb$.\n\n\\item[(\\textit{b})] $y = ax^2 + bx + c$. \\hfil (\\textit{c})~$y = (x + a)^2$.\n\n\\item[(\\textit{d})] $y = (x + a)^3$.\n\\end{SubProbs}",
   "markdown": "Differentiateexamples2 SubProbs [(*a*)] $u = 1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3} + \\dotsb$. [(*b*)] $y = ax^2 + bx + c$. (*c*) $y = (x + a)^2$. [(*d*)] $y = (x + a)^3$. SubProbs",
   "answer_latex": [
    "(\\textit{a}) $1 + x + \\dfrac{x^2}{2} + \\dfrac{x^3}{6} + \\dfrac{x^4}{24} + \\ldots$ \\qquad\n    (\\textit{b}) $2ax + b$. \\qquad (\\textit{c}) $2x + 2a$.\n\n    (\\textit{d}) $3x^2 + 6ax + 3a^2$."
   ],
   "answer_markdown": [
    "(*a*) $1 + x + \\dfrac{x^2}{2} + \\dfrac{x^3}{6} + \\dfrac{x^4}{24} + \\ldots$ (*b*) $2ax + b$. (*c*) $2x + 2a$. (*d*) $3x^2 + 6ax + 3a^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "exp(x)",
     "answer_expr": "exp(x)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: exp(x)"
   ],
   "shape": [
    "differentiate: exp(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.series"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/1b",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 1,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "46",
   "location": "Exercise III, problem 1b",
   "problem_latex": "Differentiate\\Pagelabel{examples2}\n\\begin{SubProbs}\n\\item[(\\textit{a})] $u = 1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3} + \\dotsb$.\n\n\\item[(\\textit{b})] $y = ax^2 + bx + c$. \\hfil (\\textit{c})~$y = (x + a)^2$.\n\n\\item[(\\textit{d})] $y = (x + a)^3$.\n\\end{SubProbs}",
   "markdown": "Differentiateexamples2 SubProbs [(*a*)] $u = 1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3} + \\dotsb$. [(*b*)] $y = ax^2 + bx + c$. (*c*) $y = (x + a)^2$. [(*d*)] $y = (x + a)^3$. SubProbs",
   "answer_latex": [
    "(\\textit{a}) $1 + x + \\dfrac{x^2}{2} + \\dfrac{x^3}{6} + \\dfrac{x^4}{24} + \\ldots$ \\qquad\n    (\\textit{b}) $2ax + b$. \\qquad (\\textit{c}) $2x + 2a$.\n\n    (\\textit{d}) $3x^2 + 6ax + 3a^2$."
   ],
   "answer_markdown": [
    "(*a*) $1 + x + \\dfrac{x^2}{2} + \\dfrac{x^3}{6} + \\dfrac{x^4}{24} + \\ldots$ (*b*) $2ax + b$. (*c*) $2x + 2a$. (*d*) $3x^2 + 6ax + 3a^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a*x**2 + b*x + c",
     "answer_expr": "2*a*x + b"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a*x**2 + b*x + c"
   ],
   "shape": [
    "differentiate: a*x**N + b*x + c"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/1c",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 1,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "46",
   "location": "Exercise III, problem 1c",
   "problem_latex": "Differentiate\\Pagelabel{examples2}\n\\begin{SubProbs}\n\\item[(\\textit{a})] $u = 1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3} + \\dotsb$.\n\n\\item[(\\textit{b})] $y = ax^2 + bx + c$. \\hfil (\\textit{c})~$y = (x + a)^2$.\n\n\\item[(\\textit{d})] $y = (x + a)^3$.\n\\end{SubProbs}",
   "markdown": "Differentiateexamples2 SubProbs [(*a*)] $u = 1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3} + \\dotsb$. [(*b*)] $y = ax^2 + bx + c$. (*c*) $y = (x + a)^2$. [(*d*)] $y = (x + a)^3$. SubProbs",
   "answer_latex": [
    "(\\textit{a}) $1 + x + \\dfrac{x^2}{2} + \\dfrac{x^3}{6} + \\dfrac{x^4}{24} + \\ldots$ \\qquad\n    (\\textit{b}) $2ax + b$. \\qquad (\\textit{c}) $2x + 2a$.\n\n    (\\textit{d}) $3x^2 + 6ax + 3a^2$."
   ],
   "answer_markdown": [
    "(*a*) $1 + x + \\dfrac{x^2}{2} + \\dfrac{x^3}{6} + \\dfrac{x^4}{24} + \\ldots$ (*b*) $2ax + b$. (*c*) $2x + 2a$. (*d*) $3x^2 + 6ax + 3a^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x + a)**2",
     "answer_expr": "2*x + 2*a"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (a + x)**2"
   ],
   "shape": [
    "differentiate: (a + x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.expand"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/1d",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 1,
   "part": "d",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "46",
   "location": "Exercise III, problem 1d",
   "problem_latex": "Differentiate\\Pagelabel{examples2}\n\\begin{SubProbs}\n\\item[(\\textit{a})] $u = 1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3} + \\dotsb$.\n\n\\item[(\\textit{b})] $y = ax^2 + bx + c$. \\hfil (\\textit{c})~$y = (x + a)^2$.\n\n\\item[(\\textit{d})] $y = (x + a)^3$.\n\\end{SubProbs}",
   "markdown": "Differentiateexamples2 SubProbs [(*a*)] $u = 1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3} + \\dotsb$. [(*b*)] $y = ax^2 + bx + c$. (*c*) $y = (x + a)^2$. [(*d*)] $y = (x + a)^3$. SubProbs",
   "answer_latex": [
    "(\\textit{a}) $1 + x + \\dfrac{x^2}{2} + \\dfrac{x^3}{6} + \\dfrac{x^4}{24} + \\ldots$ \\qquad\n    (\\textit{b}) $2ax + b$. \\qquad (\\textit{c}) $2x + 2a$.\n\n    (\\textit{d}) $3x^2 + 6ax + 3a^2$."
   ],
   "answer_markdown": [
    "(*a*) $1 + x + \\dfrac{x^2}{2} + \\dfrac{x^3}{6} + \\dfrac{x^4}{24} + \\ldots$ (*b*) $2ax + b$. (*c*) $2x + 2a$. (*d*) $3x^2 + 6ax + 3a^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x + a)**3",
     "answer_expr": "3*x**2 + 6*a*x + 3*a**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (a + x)**3"
   ],
   "shape": [
    "differentiate: (a + x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.expand"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/2",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "46",
   "location": "Exercise III, problem 2",
   "problem_latex": "If $w = at - \\frac{1}{2}bt^2$, find $\\dfrac{dw}{dt}$.",
   "markdown": "If $w = at - \\frac{1}{2}bt^2$, find $\\dfrac{dw}{dt}$.",
   "answer_latex": [
    "$\\dfrac{dw}{dt} = a - bt$."
   ],
   "answer_markdown": [
    "$\\dfrac{dw}{dt} = a - bt$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a*t - Rational(1,2)*b*t**2",
     "answer_expr": "a - b*t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a*x - b*x**2/2"
   ],
   "shape": [
    "differentiate: N*b*x**N + a*x"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-v/3a"
   ],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/3",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "46",
   "location": "Exercise III, problem 3",
   "problem_latex": "Find the differential coefficient of\n\\[\ny = (x + \\sqrt{-1}) × (x - \\sqrt{-1}).\n\\]",
   "markdown": "Find the differential coefficient of y = (x + -1) × (x - -1).",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = 2x$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = 2x$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x + sqrt(-1))*(x - sqrt(-1))",
     "answer_expr": "2*x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (x - I)*(x + I)"
   ],
   "shape": [
    "differentiate: (x - I)*(x + I)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.expand",
    "core.complex"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/4",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "46",
   "location": "Exercise III, problem 4",
   "problem_latex": "Differentiate\n\\[\ny = (197x - 34x^2) × (7 + 22x - 83x^3).\n\\]",
   "markdown": "Differentiate y = (197x - 34x^2) × (7 + 22x - 83x^3).",
   "answer_latex": [
    "$14110x^4 - 65404x^3 - 2244x^2 + 8192x + 1379$."
   ],
   "answer_markdown": [
    "$14110x^4 - 65404x^3 - 2244x^2 + 8192x + 1379$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(197*x - 34*x**2)*(7 + 22*x - 83*x**3)",
     "answer_expr": "14110*x**4 - 65404*x**3 - 2244*x**2 + 8192*x + 1379"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (-34*x**2 + 197*x)*(-83*x**3 + 22*x + 7)"
   ],
   "shape": [
    "differentiate: (N*x + N*x**N)*(N*x + N*x**N + N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.expand"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/5",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "46",
   "location": "Exercise III, problem 5",
   "problem_latex": "If $x = (y + 3) × (y + 5)$, find $\\dfrac{dx}{dy}$.",
   "markdown": "If $x = (y + 3) × (y + 5)$, find $\\dfrac{dx}{dy}$.",
   "answer_latex": [
    "$\\dfrac{dx}{dy} = 2y + 8$."
   ],
   "answer_markdown": [
    "$\\dfrac{dx}{dy} = 2y + 8$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(y + 3)*(y + 5)",
     "answer_expr": "2*y + 8"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (x + 3)*(x + 5)"
   ],
   "shape": [
    "differentiate: (N + x)**2"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.expand"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/6",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "46",
   "location": "Exercise III, problem 6",
   "problem_latex": "Differentiate $y = 1.3709x × (112.6 + 45.202x^2)$.",
   "markdown": "Differentiate $y = 1.3709x × (112.6 + 45.202x^2)$.",
   "answer_latex": [
    "$185.9022654x^2 + 154.36334$."
   ],
   "answer_markdown": [
    "$185.9022654x^2 + 154.36334$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1.3709*x*(112.6 + 45.202*x**2)",
     "answer_expr": "185.9022654*x**2 + 154.36334"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 13709*x*(22601*x**2/500 + 563/5)/10000"
   ],
   "shape": [
    "differentiate: N*x*(N*x**N + N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.expand",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/7",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "47",
   "location": "Exercise III, problem 7",
   "problem_latex": "$y = \\dfrac{2x + 3}{3x + 2}$.",
   "markdown": "$y = \\dfrac{2x + 3}{3x + 2}$.",
   "answer_latex": [
    "$\\dfrac{-5}{(3x + 2)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{-5}{(3x + 2)^2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(2*x + 3)/(3*x + 2)",
     "answer_expr": "-5/(3*x + 2)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (2*x + 3)/(3*x + 2)"
   ],
   "shape": [
    "differentiate: 1"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/8",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "47",
   "location": "Exercise III, problem 8",
   "problem_latex": "$y = \\dfrac{1 + x + 2x^2 + 3x^3}{1 + x + 2x^2}$.",
   "markdown": "$y = \\dfrac{1 + x + 2x^2 + 3x^3}{1 + x + 2x^2}$.",
   "answer_latex": [
    "$\\dfrac{6x^4 + 6x^3 + 9x^2}{(1 + x + 2x^2)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{6x^4 + 6x^3 + 9x^2}{(1 + x + 2x^2)^2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(1 + x + 2*x**2 + 3*x**3)/(1 + x + 2*x**2)",
     "answer_expr": "(6*x**4 + 6*x**3 + 9*x**2)/(1 + x + 2*x**2)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (3*x**3 + 2*x**2 + x + 1)/(2*x**2 + x + 1)"
   ],
   "shape": [
    "differentiate: (2*N*x**N + x + 1)/(N*x**N + x + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iii/9",
   "set": "thompson-calculus-made-easy-1914/ex-iii",
   "number": 9,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "47",
   "location": "Exercise III, problem 9",
   "problem_latex": "$y = \\dfrac{ax + b}{cx + d}$.",
   "markdown": "$y = \\dfrac{ax + b}{cx + d}$.",
   "answer_latex": [
    "$\\dfrac{ad - bc}{(cx + d)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{ad - bc}{(cx + d)^2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(a*x + b)/(c*x + d)",
     "answer_expr": "(a*d - b*c)/(c*x + d)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (a*x + b)/(c*x + d)"
   ],
   "shape": [
    "differentiate: (a*x + b)/(c*x + d)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iv/1a",
   "set": "thompson-calculus-made-easy-1914/ex-iv",
   "number": 1,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Exercise IV, problem 1a",
   "problem_latex": "$y = 17x + 12x^2$.",
   "markdown": "$y = 17x + 12x^2$.",
   "answer_latex": [
    "$17 + 24x$;\\quad $24$."
   ],
   "answer_markdown": [
    "$17 + 24x$; $24$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "17*x + 12*x**2",
     "answer_expr": "17 + 24*x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 12*x**2 + 17*x"
   ],
   "shape": [
    "differentiate: N*x + N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iv/1b",
   "set": "thompson-calculus-made-easy-1914/ex-iv",
   "number": 1,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Exercise IV, problem 1b",
   "problem_latex": "$y = 17x + 12x^2$.",
   "markdown": "$y = 17x + 12x^2$.",
   "answer_latex": [
    "$17 + 24x$;\\quad $24$."
   ],
   "answer_markdown": [
    "$17 + 24x$; $24$."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "17*x + 12*x**2",
     "answer_expr": "24"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate2: 12*x**2 + 17*x"
   ],
   "shape": [
    "differentiate2: N*x + N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iv/2a",
   "set": "thompson-calculus-made-easy-1914/ex-iv",
   "number": 2,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Exercise IV, problem 2a",
   "problem_latex": "$y = \\dfrac{x^2 + a}{x + a}$.",
   "markdown": "$y = \\dfrac{x^2 + a}{x + a}$.",
   "answer_latex": [
    "$\\dfrac{x^2 + 2ax - a}{(x + a)^2}$;\\quad $\\dfrac{2a(a + 1)}{(x + a)^3}$."
   ],
   "answer_markdown": [
    "$\\dfrac{x^2 + 2ax - a}{(x + a)^2}$; $\\dfrac{2a(a + 1)}{(x + a)^3}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x**2 + a)/(x + a)",
     "answer_expr": "(x**2 + 2*a*x - a)/(x + a)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (a + x**2)/(a + x)"
   ],
   "shape": [
    "differentiate: (a + x**N)/(a + x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iv/2b",
   "set": "thompson-calculus-made-easy-1914/ex-iv",
   "number": 2,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Exercise IV, problem 2b",
   "problem_latex": "$y = \\dfrac{x^2 + a}{x + a}$.",
   "markdown": "$y = \\dfrac{x^2 + a}{x + a}$.",
   "answer_latex": [
    "$\\dfrac{x^2 + 2ax - a}{(x + a)^2}$;\\quad $\\dfrac{2a(a + 1)}{(x + a)^3}$."
   ],
   "answer_markdown": [
    "$\\dfrac{x^2 + 2ax - a}{(x + a)^2}$; $\\dfrac{2a(a + 1)}{(x + a)^3}$."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x**2 + a)/(x + a)",
     "answer_expr": "2*a*(a + 1)/(x + a)**3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate2: (a + x**2)/(a + x)"
   ],
   "shape": [
    "differentiate2: (a + x**N)/(a + x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iv/3a",
   "set": "thompson-calculus-made-easy-1914/ex-iv",
   "number": 3,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Exercise IV, problem 3a",
   "problem_latex": "$y = 1 + \\dfrac{x}{1} + \\dfrac{x^2}{1×2} + \\dfrac{x^3}{1×2×3} + \\dfrac{x^4}{1×2×3×4}$.",
   "markdown": "$y = 1 + \\dfrac{x}{1} + \\dfrac{x^2}{1×2} + \\dfrac{x^3}{1×2×3} + \\dfrac{x^4}{1×2×3×4}$.",
   "answer_latex": [
    "$1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3}$"
   ],
   "answer_markdown": [
    "$1 + x + \\dfrac{x^2}{1 × 2} + \\dfrac{x^3}{1 × 2 × 3}$"
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1 + x/1 + x**2/(1*2) + x**3/(1*2*3) + x**4/(1*2*3*4)",
     "answer_expr": "1 + x + x**2/(1*2) + x**3/(1*2*3)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: x**4/24 + x**3/6 + x**2/2 + x + 1"
   ],
   "shape": [
    "differentiate: 3*N*x**N + x + 1"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iv/3b",
   "set": "thompson-calculus-made-easy-1914/ex-iv",
   "number": 3,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Exercise IV, problem 3b",
   "problem_latex": "$y = 1 + \\dfrac{x}{1} + \\dfrac{x^2}{1×2} + \\dfrac{x^3}{1×2×3} + \\dfrac{x^4}{1×2×3×4}$.",
   "markdown": "$y = 1 + \\dfrac{x}{1} + \\dfrac{x^2}{1×2} + \\dfrac{x^3}{1×2×3} + \\dfrac{x^4}{1×2×3×4}$.",
   "answer_latex": [
    "$1 + x + \\dfrac{x^2}{1 × 2}$."
   ],
   "answer_markdown": [
    "$1 + x + \\dfrac{x^2}{1 × 2}$."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1 + x/1 + x**2/(1*2) + x**3/(1*2*3) + x**4/(1*2*3*4)",
     "answer_expr": "1 + x + x**2/(1*2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate2: x**4/24 + x**3/6 + x**2/2 + x + 1"
   ],
   "shape": [
    "differentiate2: 3*N*x**N + x + 1"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-iv/4",
   "set": "thompson-calculus-made-easy-1914/ex-iv",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "51",
   "location": "Exercise IV, problem 4",
   "problem_latex": "Find the 2nd and~3rd derived functions in\nthe Exercises~III. (\\Pageref{examples2}), No.~1 to No.~7, and in the\nExamples given (\\Pageref{examples3}), No.~1 to No.~7.",
   "markdown": "Find the 2nd and 3rd derived functions in the Exercises III. (examples2), No. 1 to No. 7, and in the Examples given (examples3), No. 1 to No. 7.",
   "answer_latex": [
    "(\\textit{Exercises III.}):\n\\begin{itemize}\n\\item[(1)] (\\textit{a}) $\\dfrac{d^2 y}{dx^2} = \\dfrac{d^3 y}{dx^3} = 1 + x + \\frac{1}{2}x^2 + \\frac{1}{6} x^3 + \\ldots$. \\\\\n  (\\textit{b}) $2a$, $0$.\\hfil\n  (\\textit{c}) $2$, $0$.\\hfil\n  (\\textit{d}) $6x + 6a$, $6$.\n\n\\DPPageSep{268.png}{256}%\n\\item[(2)] $-b$, $0$.\\hfil (3) $2$, $0$.\\hfil\n\n\\item[(4)] $\\begin{gathered}[t]\n    56440x^3 - 196212x^2 - 4488x + 8192. \\\\\n    169320x^2 - 392424x - 4488.\n    \\end{gathered}$\n\n\\item[(5)] $2$, $0$. \\hfil (6) $371.80453x$, $371.80453$. \\hfil\n\n\\item[(7)] $\\dfrac{30}{(3x + 2)^3}$,\\quad $-\\dfrac{270}{(3x + 2)^4}$.\n\\end{itemize}\n\n\\paragraph{\\normalfont(\\textit{Examples}, \\Pageref{examples3}):}\n\\begin{itemize}\n\\item[(1)] $\\dfrac{6a}{b^2} x$,\\quad $\\dfrac{6a}{b^2}$.\\hfil\n(2) $\\dfrac{3a \\sqrt{b}} {2 \\sqrt{x}} - \\dfrac{6b \\sqrt[3]{a}}{x^3}$,\\quad\n$\\dfrac{18b \\sqrt[3]{a}}{x^4} - \\dfrac{3a \\sqrt{b}}{4 \\sqrt{x^3}}$\\DPtypo{}{.}\n\n\\Item{(3)}\n$\\dfrac{2}{\\sqrt[3]{\\theta^8}} - \\dfrac{1.056}{\\sqrt[5]{\\theta^{11}}}$,\\quad\n$\\dfrac{2.3232}{\\sqrt[5]{\\theta^{16}}} - \\dfrac{16}{3 \\sqrt[3]{\\theta^{11}}}$.\n\n\\Item{(4)} $\\begin{gathered}[t]\n  810t^4 - 648t^3 + 479.52t^2 - 139.968t + 26.64. \\\\\n  3240t^3 - 1944t^2 + 959.04t - 139.968.\n  \\end{gathered}$\n\n\\Item{(5)}  $12x + 2$, $12$.\\hfil\n(6) $6x^2 - 9x$,\\quad $12x - 9$.\\hfil\n\n\\Item{(7)}\n$\\begin{aligned}[t]\n&\\dfrac{3}{4} \\left(\\dfrac{1}{\\sqrt{\\theta}} + \\dfrac{1}{\\sqrt{\\theta^5}}\\right)\n+\\dfrac{1}{4} \\left(\\dfrac{15}{\\sqrt{\\theta^7}} - \\dfrac{1}{\\sqrt{\\theta^3}}\\right). \\\\\n&\\dfrac{3}{8} \\left(\\dfrac{1}{\\sqrt{\\theta^5}} - \\dfrac{1}{\\sqrt{\\theta^3}}\\right)\n-\\dfrac{15}{8}\\left(\\dfrac{7}{\\sqrt{\\theta^9}} + \\dfrac{1}{\\sqrt{\\theta^7}}\\right).\n\\end{aligned}$\n\\end{itemize}"
   ],
   "answer_markdown": [
    "(*Exercises III.*): itemize [(1)] (*a*) $\\dfrac{d^2 y}{dx^2} = \\dfrac{d^3 y}{dx^3} = 1 + x + \\frac{1}{2}x^2 + \\frac{1}{6} x^3 + \\ldots$. (*b*) $2a$, $0$. (*c*) $2$, $0$. (*d*) $6x + 6a$, $6$. 268.png256% [(2)] $-b$, $0$.(3) $2$, $0$. [(4)] $\\begin{gathered}[t] 56440x^3 - 196212x^2 - 4488x + 8192. \\\\ 169320x^2 - 392424x - 4488. \\end{gathered}$ [(5)] $2$, $0$. (6) $371.80453x$, $371.80453$. [(7)] $\\dfrac{30}{(3x + 2)^3}$, $-\\dfrac{270}{(3x + 2)^4}$. itemize (*Examples*, examples3): itemize [(1)] $\\dfrac{6a}{b^2} x$, $\\dfrac{6a}{b^2}$.(2) $\\dfrac{3a \\sqrt{b}} {2 \\sqrt{x}} - \\dfrac{6b \\sqrt[3]{a}}{x^3}$, $\\dfrac{18b \\sqrt[3]{a}}{x^4} - \\dfrac{3a \\sqrt{b}}{4 \\sqrt{x^3}}$ (3) $\\dfrac{2}{\\sqrt[3]{\\theta^8}} - \\dfrac{1.056}{\\sqrt[5]{\\theta^{11}}}$, $\\dfrac{2.3232}{\\sqrt[5]{\\theta^{16}}} - \\dfrac{16}{3 \\sqrt[3]{\\theta^{11}}}$. (4) $\\begin{gathered}[t] 810t^4 - 648t^3 + 479.52t^2 - 139.968t + 26.64. \\\\ 3240t^3 - 1944t^2 + 959.04t - 139.968. \\end{gathered}$ (5) $12x + 2$, $12$.(6) $6x^2 - 9x$, $12x - 9$. (7) $\\begin{aligned}[t] &\\dfrac{3}{4} \\left(\\dfrac{1}{\\sqrt{\\theta}} + \\dfrac{1}{\\sqrt{\\theta^5}}\\right) +\\dfrac{1}{4} \\left(\\dfrac{15}{\\sqrt{\\theta^7}} - \\dfrac{1}{\\sqrt{\\theta^3}}\\right). \\\\ &\\dfrac{3}{8} \\left(\\dfrac{1}{\\sqrt{\\theta^5}} - \\dfrac{1}{\\sqrt{\\theta^3}}\\right) -\\dfrac{15}{8}\\left(\\dfrac{7}{\\sqrt{\\theta^9}} + \\dfrac{1}{\\sqrt{\\theta^7}}\\right). \\end{aligned}$ itemize"
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-EXTRACTION",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-EXTRACTION"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/10",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 10,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "109",
   "location": "Exercise IX, problem 10",
   "problem_latex": "A spherical balloon is increasing in volume.\nIf, when its radius is $r$~feet, its volume is increasing\nat the rate of $4$~cubic feet per~second, at what rate is\nits surface then increasing?",
   "markdown": "A spherical balloon is increasing in volume. If, when its radius is $r$ feet, its volume is increasing at the rate of $4$ cubic feet per second, at what rate is its surface then increasing?",
   "answer_latex": [
    "At the rate of $\\dfrac{8}{r}$ square feet per second."
   ],
   "answer_markdown": [
    "At the rate of $\\dfrac{8}{r}$ square feet per second."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "4*pi*(3*(V0+4*t)/(4*pi))**Rational(2,3)",
     "answer_expr": "8/((3*(V0+4*t)/(4*pi))**Rational(1,3))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: 2**(2/3)*pi**(1/3)*(3*a + 12*x)**(2/3)"
   ],
   "shape": [
    "differentiate: pi**N*N**N*(N*a + N*x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst",
    "other:related rates (chain rule in t)"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/11",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 11,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "111",
   "location": "Exercise IX, problem 11",
   "problem_latex": "Inscribe in a given sphere a cone whose volume\nis a maximum.",
   "markdown": "Inscribe in a given sphere a cone whose volume is a maximum.",
   "answer_latex": [
    "$r = \\dfrac{R \\sqrt{8}}{3}$."
   ],
   "answer_markdown": [
    "$r = \\dfrac{R \\sqrt{8}}{3}$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "pi*r**2*(R+sqrt(R**2-r**2))/3",
     "answer_expr": "R*sqrt(8)/3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "extremum: pi*x**2*(a + sqrt(a**2 - x**2))/3"
   ],
   "shape": [
    "extremum: pi*N*x**N*(a + (a**N - x**N)**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.solve.nonpoly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/12",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 12,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "111",
   "location": "Exercise IX, problem 12",
   "problem_latex": "The current~$C$ given by a battery of $N$~similar\nvoltaic cells is $C=\\dfrac{n×E}{R+\\dfrac{rn^2}{N}}$, where $E$,~$R$,~$r$, are constants\nand $n$~is the number of cells coupled in series. Find\nthe proportion of $n$~to~$N$ for which the current is\ngreatest.",
   "markdown": "The current $C$ given by a battery of $N$ similar voltaic cells is $C=\\dfrac{n×E}{R+\\dfrac{rn^2}{N}}$, where $E$, $R$, $r$, are constants and $n$ is the number of cells coupled in series. Find the proportion of $n$ to $N$ for which the current is greatest.",
   "answer_latex": [
    "$n = \\sqrt{\\dfrac{NR}{r}}$."
   ],
   "answer_markdown": [
    "$n = \\sqrt{\\dfrac{NR}{r}}$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "n*E/(R+r*n**2/N)",
     "answer_expr": "sqrt(N*R/r)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: a*x/(c + d*x**2/b)"
   ],
   "shape": [
    "extremum: a*x/(c + d*x**N/b)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/1a",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 1,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "109",
   "location": "Exercise IX, problem 1a",
   "problem_latex": "What values of~$x$ will make $y$ a maximum\nand a minimum, if $y=\\dfrac{x^2}{x+1}$?",
   "markdown": "What values of $x$ will make $y$ a maximum and a minimum, if $y=\\dfrac{x^2}{x+1}$?",
   "answer_latex": [
    "Min.: $x = 0$, $y = 0$; max.: $x = -2$, $y = -4$."
   ],
   "answer_markdown": [
    "Min.: $x = 0$, $y = 0$; max.: $x = -2$, $y = -4$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**2/(x+1)",
     "answer_expr": "-2"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -4.0, printed -4 (half-unit 0.5; correctly rounded at the printed digits: -4.0)",
     "problem_expr": "x**2/(x+1)",
     "answer_expr": "-2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "extremum: x**2/(x + 1)",
    "evaluate: x**2/(x + 1) at x=-2"
   ],
   "shape": [
    "evaluate: x**N/(x + 1)",
    "extremum: x**N/(x + 1)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-ix/1b"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/1b",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 1,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "109",
   "location": "Exercise IX, problem 1b",
   "problem_latex": "What values of~$x$ will make $y$ a maximum\nand a minimum, if $y=\\dfrac{x^2}{x+1}$?",
   "markdown": "What values of $x$ will make $y$ a maximum and a minimum, if $y=\\dfrac{x^2}{x+1}$?",
   "answer_latex": [
    "Min.: $x = 0$, $y = 0$; max.: $x = -2$, $y = -4$."
   ],
   "answer_markdown": [
    "Min.: $x = 0$, $y = 0$; max.: $x = -2$, $y = -4$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**2/(x+1)",
     "answer_expr": "0"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.0, printed 0 (half-unit 0.5; correctly rounded at the printed digits: 0.0)",
     "problem_expr": "x**2/(x+1)",
     "answer_expr": "0"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "extremum: x**2/(x + 1)",
    "evaluate: x**2/(x + 1) at x=0"
   ],
   "shape": [
    "evaluate: x**N/(x + 1)",
    "extremum: x**N/(x + 1)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-ix/1a"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/2",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "109",
   "location": "Exercise IX, problem 2",
   "problem_latex": "What value of~$x$ will make $y$ a maximum in\nthe equation $y=\\dfrac{x}{a^2+x^2}$?\n\\DPPageSep{122.png}{110}%",
   "markdown": "What value of $x$ will make $y$ a maximum in the equation $y=\\dfrac{x}{a^2+x^2}$? 122.png110%",
   "answer_latex": [
    "$x = a$."
   ],
   "answer_markdown": [
    "$x = a$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x/(a**2+x**2)",
     "answer_expr": "a"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: x/(a**2 + x**2)"
   ],
   "shape": [
    "extremum: x/(a**N + x**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/3",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 3",
   "problem_latex": "A line of length~$p$ is to be cut up into $4$~parts\nand put together as a rectangle. Show that the area\nof the rectangle will be a maximum if each of its\nsides is equal to~$\\frac{1}{4}p$.",
   "markdown": "A line of length $p$ is to be cut up into $4$ parts and put together as a rectangle. Show that the area of the rectangle will be a maximum if each of its sides is equal to $\\frac{1}{4}p$.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-EXTRACTION",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-EXTRACTION"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/4",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": null,
   "location": "Exercise IX, problem 4",
   "problem_latex": "A piece of string $30$~inches long has its two\nends joined together and is stretched by $3$~pegs so\nas to form a triangle. What is the largest triangular\narea that can be enclosed by the string?",
   "markdown": "A piece of string $30$ inches long has its two ends joined together and is stretched by $3$ pegs so as to form a triangle. What is the largest triangular area that can be enclosed by the string?",
   "answer_latex": [
    "$25 \\sqrt{3}$ square inches.\n\n\\ResetCols{1}"
   ],
   "answer_markdown": [
    "$25 \\sqrt{3}$ square inches. % InMulticols% multicols% 11% InMulticolsfalse% % InMulticolstrue% multicols1[]% %"
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-EXTRACTION",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "25*sqrt(3)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-EXTRACTION"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith",
    "other:two-variable maximisation (Heron's formula)"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/5a",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 5,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 5a",
   "problem_latex": "Plot the curve corresponding to the equation\n\\[\ny = \\frac{10}{x} + \\frac{10}{8-x};\n\\]\nalso find~$\\dfrac{dy}{dx}$, and deduce the value of~$x$ that will\nmake $y$ a minimum; and find that minimum value\nof~$y$.",
   "markdown": "Plot the curve corresponding to the equation y = 10x + 108-x; also find $\\dfrac{dy}{dx}$, and deduce the value of $x$ that will make $y$ a minimum; and find that minimum value of $y$.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = - \\dfrac{10}{x^2} + \\dfrac{10}{(8 - x)^2}$; $x = 4$; $y = 5$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = - \\dfrac{10}{x^2} + \\dfrac{10}{(8 - x)^2}$; $x = 4$; $y = 5$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "10/x + 10/(8-x)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "core.graph",
    "core.table"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/5b",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 5,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 5b",
   "problem_latex": "Plot the curve corresponding to the equation\n\\[\ny = \\frac{10}{x} + \\frac{10}{8-x};\n\\]\nalso find~$\\dfrac{dy}{dx}$, and deduce the value of~$x$ that will\nmake $y$ a minimum; and find that minimum value\nof~$y$.",
   "markdown": "Plot the curve corresponding to the equation y = 10x + 108-x; also find $\\dfrac{dy}{dx}$, and deduce the value of $x$ that will make $y$ a minimum; and find that minimum value of $y$.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = - \\dfrac{10}{x^2} + \\dfrac{10}{(8 - x)^2}$; $x = 4$; $y = 5$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = - \\dfrac{10}{x^2} + \\dfrac{10}{(8 - x)^2}$; $x = 4$; $y = 5$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "10/x + 10/(8-x)",
     "answer_expr": "-10/x**2 + 10/(8-x)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 10/(-x + 8) + 10/x"
   ],
   "shape": [
    "differentiate: N/(N - x) + N/x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/5c",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 5,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 5c",
   "problem_latex": "Plot the curve corresponding to the equation\n\\[\ny = \\frac{10}{x} + \\frac{10}{8-x};\n\\]\nalso find~$\\dfrac{dy}{dx}$, and deduce the value of~$x$ that will\nmake $y$ a minimum; and find that minimum value\nof~$y$.",
   "markdown": "Plot the curve corresponding to the equation y = 10x + 108-x; also find $\\dfrac{dy}{dx}$, and deduce the value of $x$ that will make $y$ a minimum; and find that minimum value of $y$.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = - \\dfrac{10}{x^2} + \\dfrac{10}{(8 - x)^2}$; $x = 4$; $y = 5$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = - \\dfrac{10}{x^2} + \\dfrac{10}{(8 - x)^2}$; $x = 4$; $y = 5$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "10/x + 10/(8-x)",
     "answer_expr": "4"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 5.0, printed 5 (half-unit 0.5; correctly rounded at the printed digits: 5.0)",
     "problem_expr": "10/x + 10/(8-x)",
     "answer_expr": "4"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "extremum: 10/(-x + 8) + 10/x",
    "evaluate: 10/(-x + 8) + 10/x at x=4"
   ],
   "shape": [
    "evaluate: N/(N - x) + N/x",
    "extremum: N/(N - x) + N/x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/6a",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 6,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 6a",
   "problem_latex": "If $y = x^5-5x$, find what values of~$x$ will make\n$y$ a maximum or a minimum.",
   "markdown": "If $y = x^5-5x$, find what values of $x$ will make $y$ a maximum or a minimum.",
   "answer_latex": [
    "Max.\\ for $x = -1$; min.\\ for $x = 1$."
   ],
   "answer_markdown": [
    "Max. for $x = -1$; min. for $x = 1$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**5-5*x",
     "answer_expr": "-1"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: x**5 - 5*x"
   ],
   "shape": [
    "extremum: N*x + x**N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-ix/6b"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/6b",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 6,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 6b",
   "problem_latex": "If $y = x^5-5x$, find what values of~$x$ will make\n$y$ a maximum or a minimum.",
   "markdown": "If $y = x^5-5x$, find what values of $x$ will make $y$ a maximum or a minimum.",
   "answer_latex": [
    "Max.\\ for $x = -1$; min.\\ for $x = 1$."
   ],
   "answer_markdown": [
    "Max. for $x = -1$; min. for $x = 1$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**5-5*x",
     "answer_expr": "1"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: x**5 - 5*x"
   ],
   "shape": [
    "extremum: N*x + x**N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-ix/6a"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/7",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "109",
   "location": "Exercise IX, problem 7",
   "problem_latex": "What is the smallest square that can be inscribed\nin a given square?",
   "markdown": "What is the smallest square that can be inscribed in a given square?",
   "answer_latex": [
    "Join the middle points of the four sides."
   ],
   "answer_markdown": [
    "Join the middle points of the four sides."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "x**2+(s-x)**2",
     "answer_expr": "s/2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "extremum: x**2 + (a - x)**2"
   ],
   "shape": [
    "extremum: x**N + (a - x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.expand",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/8a",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 8,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 8a",
   "problem_latex": "Inscribe in a given cone, the height of which\nis equal to the radius of the base, a cylinder\n(\\textit{a})~whose volume is a maximum; (\\textit{b})~whose lateral\narea is a maximum; (\\textit{c})~whose total area is a\nmaximum.",
   "markdown": "Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder (*a*) whose volume is a maximum; (*b*) whose lateral area is a maximum; (*c*) whose total area is a maximum.",
   "answer_latex": [
    "$r = \\frac{2}{3} R$, $r = \\dfrac{R}{2}$, no max."
   ],
   "answer_markdown": [
    "$r = \\frac{2}{3} R$, $r = \\dfrac{R}{2}$, no max."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "pi*r**2*(R-r)",
     "answer_expr": "2*R/3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "extremum: pi*x**2*(a - x)"
   ],
   "shape": [
    "extremum: pi*x**N*(a - x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.expand",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/8b",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 8,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 8b",
   "problem_latex": "Inscribe in a given cone, the height of which\nis equal to the radius of the base, a cylinder\n(\\textit{a})~whose volume is a maximum; (\\textit{b})~whose lateral\narea is a maximum; (\\textit{c})~whose total area is a\nmaximum.",
   "markdown": "Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder (*a*) whose volume is a maximum; (*b*) whose lateral area is a maximum; (*c*) whose total area is a maximum.",
   "answer_latex": [
    "$r = \\frac{2}{3} R$, $r = \\dfrac{R}{2}$, no max."
   ],
   "answer_markdown": [
    "$r = \\frac{2}{3} R$, $r = \\dfrac{R}{2}$, no max."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "2*pi*r*(R-r)",
     "answer_expr": "R/2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "extremum: 2*pi*x*(a - x)"
   ],
   "shape": [
    "extremum: pi*N*x*(a - x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/8c",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 8,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 8c",
   "problem_latex": "Inscribe in a given cone, the height of which\nis equal to the radius of the base, a cylinder\n(\\textit{a})~whose volume is a maximum; (\\textit{b})~whose lateral\narea is a maximum; (\\textit{c})~whose total area is a\nmaximum.",
   "markdown": "Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder (*a*) whose volume is a maximum; (*b*) whose lateral area is a maximum; (*c*) whose total area is a maximum.",
   "answer_latex": [
    "$r = \\frac{2}{3} R$, $r = \\dfrac{R}{2}$, no max."
   ],
   "answer_markdown": [
    "$r = \\frac{2}{3} R$, $r = \\dfrac{R}{2}$, no max."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "2*pi*r*(R-r)+2*pi*r**2",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/9a",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 9,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 9a",
   "problem_latex": "Inscribe in a sphere, a cylinder (\\textit{a})~whose\nvolume is a maximum; (\\textit{b})~whose lateral area is a\nmaximum; (\\textit{c})~whose total area is a maximum.\n\\DPPageSep{123.png}{111}%",
   "markdown": "Inscribe in a sphere, a cylinder (*a*) whose volume is a maximum; (*b*) whose lateral area is a maximum; (*c*) whose total area is a maximum. 123.png111%",
   "answer_latex": [
    "$r = R \\sqrt{\\dfrac{2}{3}}$, $r = \\dfrac{R}{\\sqrt{2}}$, $r = 0.8506R$."
   ],
   "answer_markdown": [
    "$r = R \\sqrt{\\dfrac{2}{3}}$, $r = \\dfrac{R}{\\sqrt{2}}$, $r = 0.8506R$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "2*pi*r**2*sqrt(R**2-r**2)",
     "answer_expr": "R*sqrt(Rational(2,3))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "extremum: 2*pi*x**2*sqrt(a**2 - x**2)"
   ],
   "shape": [
    "extremum: pi*N*x**N*(a**N - x**N)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/9b",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 9,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 9b",
   "problem_latex": "Inscribe in a sphere, a cylinder (\\textit{a})~whose\nvolume is a maximum; (\\textit{b})~whose lateral area is a\nmaximum; (\\textit{c})~whose total area is a maximum.\n\\DPPageSep{123.png}{111}%",
   "markdown": "Inscribe in a sphere, a cylinder (*a*) whose volume is a maximum; (*b*) whose lateral area is a maximum; (*c*) whose total area is a maximum. 123.png111%",
   "answer_latex": [
    "$r = R \\sqrt{\\dfrac{2}{3}}$, $r = \\dfrac{R}{\\sqrt{2}}$, $r = 0.8506R$."
   ],
   "answer_markdown": [
    "$r = R \\sqrt{\\dfrac{2}{3}}$, $r = \\dfrac{R}{\\sqrt{2}}$, $r = 0.8506R$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "4*pi*r*sqrt(R**2-r**2)",
     "answer_expr": "R/sqrt(2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "extremum: 4*pi*x*sqrt(a**2 - x**2)"
   ],
   "shape": [
    "extremum: pi*N*x*(a**N - x**N)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-ix/9c",
   "set": "thompson-calculus-made-easy-1914/ex-ix",
   "number": 9,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "110",
   "location": "Exercise IX, problem 9c",
   "problem_latex": "Inscribe in a sphere, a cylinder (\\textit{a})~whose\nvolume is a maximum; (\\textit{b})~whose lateral area is a\nmaximum; (\\textit{c})~whose total area is a maximum.\n\\DPPageSep{123.png}{111}%",
   "markdown": "Inscribe in a sphere, a cylinder (*a*) whose volume is a maximum; (*b*) whose lateral area is a maximum; (*c*) whose total area is a maximum. 123.png111%",
   "answer_latex": [
    "$r = R \\sqrt{\\dfrac{2}{3}}$, $r = \\dfrac{R}{\\sqrt{2}}$, $r = 0.8506R$."
   ],
   "answer_markdown": [
    "$r = R \\sqrt{\\dfrac{2}{3}}$, $r = \\dfrac{R}{\\sqrt{2}}$, $r = 0.8506R$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "4*pi*r*sqrt(R**2-r**2)+2*pi*r**2",
     "answer_expr": "R*sqrt(Rational(1,2)+sqrt(5)/10)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS-LOOSE",
     "judge_why": "computed 0.850650808352, printed 0.8506 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.8507): one unit off in the last printed place",
     "problem_expr": "4*pi*r*sqrt(R**2-r**2)+2*pi*r**2",
     "answer_expr": "R*sqrt(Rational(1,2)+sqrt(5)/10)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED",
     "PASS-LOOSE"
    ]
   },
   "form": [
    "extremum: 2*pi*x**2 + 4*pi*x*sqrt(a**2 - x**2)",
    "evaluate: 2*pi*x**2 + 4*pi*x*sqrt(a**2 - x**2) at R=1"
   ],
   "shape": [
    "evaluate: pi*N*x*(a**N - x**N)**N + pi*N*x**N",
    "extremum: pi*N*x*(a**N - x**N)**N + pi*N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.solve.nonpoly",
    "core.arith",
    "core.solve.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/10a",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 10,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "66",
   "location": "Exercise V, problem 10a",
   "problem_latex": "A body moves in such a way that the spaces\ndescribed in the time~$t$ from starting is given by\n$s = t^n$, where $n$~is a constant. Find the value of~$n$\nwhen the velocity is doubled from the $5$th to the $10$th\nsecond; find it also when the velocity is numerically\nequal to the acceleration at the end of the $10$th~second.",
   "markdown": "A body moves in such a way that the spaces described in the time $t$ from starting is given by $s = t^n$, where $n$ is a constant. Find the value of $n$ when the velocity is doubled from the $5$th to the $10$th second; find it also when the velocity is numerically equal to the acceleration at the end of the $10$th second.",
   "answer_latex": [
    "$n = 2$, $n = 11$."
   ],
   "answer_markdown": [
    "$n = 2$, $n = 11$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "t**n",
     "answer_expr": "2"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.nonpoly",
    "core.eqn",
    "core.log",
    "core.solve.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/10b",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 10,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "66",
   "location": "Exercise V, problem 10b",
   "problem_latex": "A body moves in such a way that the spaces\ndescribed in the time~$t$ from starting is given by\n$s = t^n$, where $n$~is a constant. Find the value of~$n$\nwhen the velocity is doubled from the $5$th to the $10$th\nsecond; find it also when the velocity is numerically\nequal to the acceleration at the end of the $10$th~second.",
   "markdown": "A body moves in such a way that the spaces described in the time $t$ from starting is given by $s = t^n$, where $n$ is a constant. Find the value of $n$ when the velocity is doubled from the $5$th to the $10$th second; find it also when the velocity is numerically equal to the acceleration at the end of the $10$th second.",
   "answer_latex": [
    "$n = 2$, $n = 11$."
   ],
   "answer_markdown": [
    "$n = 2$, $n = 11$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "t**n",
     "answer_expr": "11"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.cancel",
    "cas.derive",
    "cas.solve.poly",
    "core.eqn"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/1a",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 1,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "64",
   "location": "Exercise V, problem 1a",
   "problem_latex": "If $y = a + bt^2 + ct^4$; find $\\dfrac{dy}{dt}$ and $\\dfrac{d^2y}{dt^2}$.\n\\[\n\\text{\\textit{Ans.} $\\dfrac{dy}{dt} = 2bt + 4ct^3$;\\quad $\\dfrac{d^2y}{dt^2} = 2b + 12ct^2$.}\n\\]",
   "markdown": "If $y = a + bt^2 + ct^4$; find $\\dfrac{dy}{dt}$ and $\\dfrac{d^2y}{dt^2}$. *Ans.* $\\dfrac{dy}{dt} = 2bt + 4ct^3$; $\\dfrac{d^2y}{dt^2} = 2b + 12ct^2$.",
   "answer_latex": [
    "\\text{\\textit{Ans.} $\\dfrac{dy}{dt} = 2bt + 4ct^3$;\\quad $\\dfrac{d^2y}{dt^2} = 2b + 12ct^2$.}"
   ],
   "answer_markdown": [
    "*Ans.* $\\dfrac{dy}{dt} = 2bt + 4ct^3$; $\\dfrac{d^2y}{dt^2} = 2b + 12ct^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "FLAG-EXTRACTION",
     "judge_why": null,
     "problem_expr": "a + b*t**2 + c*t**4",
     "answer_expr": "2*b*t + 4*c*t**3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-EXTRACTION"
    ]
   },
   "form": [
    "differentiate: a + b*x**2 + c*x**4"
   ],
   "shape": [
    "differentiate: a + b*x**N + c*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/1b",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 1,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "64",
   "location": "Exercise V, problem 1b",
   "problem_latex": "If $y = a + bt^2 + ct^4$; find $\\dfrac{dy}{dt}$ and $\\dfrac{d^2y}{dt^2}$.\n\\[\n\\text{\\textit{Ans.} $\\dfrac{dy}{dt} = 2bt + 4ct^3$;\\quad $\\dfrac{d^2y}{dt^2} = 2b + 12ct^2$.}\n\\]",
   "markdown": "If $y = a + bt^2 + ct^4$; find $\\dfrac{dy}{dt}$ and $\\dfrac{d^2y}{dt^2}$. *Ans.* $\\dfrac{dy}{dt} = 2bt + 4ct^3$; $\\dfrac{d^2y}{dt^2} = 2b + 12ct^2$.",
   "answer_latex": [
    "\\text{\\textit{Ans.} $\\dfrac{dy}{dt} = 2bt + 4ct^3$;\\quad $\\dfrac{d^2y}{dt^2} = 2b + 12ct^2$.}"
   ],
   "answer_markdown": [
    "*Ans.* $\\dfrac{dy}{dt} = 2bt + 4ct^3$; $\\dfrac{d^2y}{dt^2} = 2b + 12ct^2$."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "FLAG-EXTRACTION",
     "judge_why": null,
     "problem_expr": "a + b*t**2 + c*t**4",
     "answer_expr": "2*b + 12*c*t**2"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-EXTRACTION"
    ]
   },
   "form": [
    "differentiate2: a + b*x**2 + c*x**4"
   ],
   "shape": [
    "differentiate2: a + b*x**N + c*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/2",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "64",
   "location": "Exercise V, problem 2",
   "problem_latex": "A body falling freely in space describes in $t$~seconds\na space~$s$, in feet, expressed by the equation\n$s = 16t^2$. Draw a curve showing the relation between\n$s$~and~$t$. Also determine the velocity of the body at\nthe following times from its being let drop: $t = 2$\nseconds; $t = 4.6$ seconds; $t = 0.01$ second.",
   "markdown": "A body falling freely in space describes in $t$ seconds a space $s$, in feet, expressed by the equation $s = 16t^2$. Draw a curve showing the relation between $s$ and $t$. Also determine the velocity of the body at the following times from its being let drop: $t = 2$ seconds; $t = 4.6$ seconds; $t = 0.01$ second.",
   "answer_latex": [
    "64; 147.2; and 0.32 feet per second."
   ],
   "answer_markdown": [
    "64; 147.2; and 0.32 feet per second."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "16*t**2",
     "answer_expr": "32*t"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 64.0, printed 64 (half-unit 0.5; correctly rounded at the printed digits: 64.0)",
     "problem_expr": "16*t**2",
     "answer_expr": "32*t"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 147.2, printed 147.2 (half-unit 0.05; correctly rounded at the printed digits: 147.2)",
     "problem_expr": "16*t**2",
     "answer_expr": "32*t"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.32, printed 0.32 (half-unit 0.005; correctly rounded at the printed digits: 0.32)",
     "problem_expr": "16*t**2",
     "answer_expr": "32*t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: 16*x**2",
    "evaluate: 16*x**2 at t=2",
    "evaluate: 16*x**2 at t=4.6",
    "evaluate: 16*x**2 at t=0.01"
   ],
   "shape": [
    "differentiate: N*x**N",
    "evaluate: N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/3a",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 3,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "64",
   "location": "Exercise V, problem 3a",
   "problem_latex": "If $x = at - \\frac{1}{2}gt^2$; find $\\dot{x}$ and~$\\ddot{x}$.",
   "markdown": "If $x = at - \\frac{1}{2}gt^2$; find $\\dot{x}$ and $\\ddot{x}$.",
   "answer_latex": [
    "$x = a - gt$; $\\ddot{x} = -g$."
   ],
   "answer_markdown": [
    "$x = a - gt$; $\\ddot{x} = -g$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a*t - g*t**2/2",
     "answer_expr": "a - g*t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a*x - b*x**2/2"
   ],
   "shape": [
    "differentiate: N*b*x**N + a*x"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-iii/2"
   ],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/3b",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 3,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "64",
   "location": "Exercise V, problem 3b",
   "problem_latex": "If $x = at - \\frac{1}{2}gt^2$; find $\\dot{x}$ and~$\\ddot{x}$.",
   "markdown": "If $x = at - \\frac{1}{2}gt^2$; find $\\dot{x}$ and $\\ddot{x}$.",
   "answer_latex": [
    "$x = a - gt$; $\\ddot{x} = -g$."
   ],
   "answer_markdown": [
    "$x = a - gt$; $\\ddot{x} = -g$."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a*t - g*t**2/2",
     "answer_expr": "-g"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate2: a*x - b*x**2/2"
   ],
   "shape": [
    "differentiate2: N*b*x**N + a*x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/4",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "64",
   "location": "Exercise V, problem 4",
   "problem_latex": "If a body move according to the law\n\\[\ns = 12 - 4.5t + 6.2t^2,\n\\]\nfind its velocity when $t = 4$~seconds; $s$~being in feet.",
   "markdown": "If a body move according to the law s = 12 - 4.5t + 6.2t^2, find its velocity when $t = 4$ seconds; $s$ being in feet.",
   "answer_latex": [
    "$45.1$ feet per second."
   ],
   "answer_markdown": [
    "$45.1$ feet per second."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "12 - 4.5*t + 6.2*t**2",
     "answer_expr": "-4.5 + 12.4*t"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 45.1, printed 45.1 (half-unit 0.05; correctly rounded at the printed digits: 45.1)",
     "problem_expr": "12 - 4.5*t + 6.2*t**2",
     "answer_expr": "-4.5 + 12.4*t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: 31*x**2/5 - 9*x/2 + 12",
    "evaluate: 31*x**2/5 - 9*x/2 + 12 at t=4"
   ],
   "shape": [
    "differentiate: N*x + N*x**N + N",
    "evaluate: N*x + N*x**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/5",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "65",
   "location": "Exercise V, problem 5",
   "problem_latex": "Find the acceleration of the body mentioned in\nthe preceding example. Is the acceleration the same\nfor all values of~$t$?",
   "markdown": "Find the acceleration of the body mentioned in the preceding example. Is the acceleration the same for all values of $t$?",
   "answer_latex": [
    "$12.4$ feet per second per second.\\quad Yes."
   ],
   "answer_markdown": [
    "$12.4$ feet per second per second. Yes."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "12 - 4.5*t + 6.2*t**2",
     "answer_expr": "12.4"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 12.4, printed 12.4 (half-unit 0.05; correctly rounded at the printed digits: 12.4)",
     "problem_expr": "12 - 4.5*t + 6.2*t**2",
     "answer_expr": "12.4"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate2: 31*x**2/5 - 9*x/2 + 12",
    "evaluate: 31*x**2/5 - 9*x/2 + 12"
   ],
   "shape": [
    "differentiate2: N*x + N*x**N + N",
    "evaluate: N*x + N*x**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/6a",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 6,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "65",
   "location": "Exercise V, problem 6a",
   "problem_latex": "The angle~$\\theta$ (in radians) turned through by\na revolving wheel is connected with the time~$t$ (in\nseconds) that has elapsed since starting; by the law\n\\[\n\\theta = 2.1 - 3.2t + 4.8t^2.\n\\]\n\nFind the angular velocity (in radians per second) of\nthat wheel when $1\\frac{1}{2}$~seconds have elapsed. Find also\nits angular acceleration.",
   "markdown": "The angle $\\theta$ (in radians) turned through by a revolving wheel is connected with the time $t$ (in seconds) that has elapsed since starting; by the law = 2.1 - 3.2t + 4.8t^2. Find the angular velocity (in radians per second) of that wheel when $1\\frac{1}{2}$ seconds have elapsed. Find also its angular acceleration.",
   "answer_latex": [
    "Angular velocity ${} = 11.2$ radians per second; angular\n  acceleration ${}= 9.6$ radians per second per second."
   ],
   "answer_markdown": [
    "Angular velocity ${} = 11.2$ radians per second; angular acceleration ${}= 9.6$ radians per second per second."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "2.1 - 3.2*t + 4.8*t**2",
     "answer_expr": "-3.2 + 9.6*t"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 11.2, printed 11.2 (half-unit 0.05; correctly rounded at the printed digits: 11.2)",
     "problem_expr": "2.1 - 3.2*t + 4.8*t**2",
     "answer_expr": "-3.2 + 9.6*t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: 24*x**2/5 - 16*x/5 + 21/10",
    "evaluate: 24*x**2/5 - 16*x/5 + 21/10 at t=1.5"
   ],
   "shape": [
    "differentiate: N*x + N*x**N + N",
    "evaluate: N*x + N*x**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/6b",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 6,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "65",
   "location": "Exercise V, problem 6b",
   "problem_latex": "The angle~$\\theta$ (in radians) turned through by\na revolving wheel is connected with the time~$t$ (in\nseconds) that has elapsed since starting; by the law\n\\[\n\\theta = 2.1 - 3.2t + 4.8t^2.\n\\]\n\nFind the angular velocity (in radians per second) of\nthat wheel when $1\\frac{1}{2}$~seconds have elapsed. Find also\nits angular acceleration.",
   "markdown": "The angle $\\theta$ (in radians) turned through by a revolving wheel is connected with the time $t$ (in seconds) that has elapsed since starting; by the law = 2.1 - 3.2t + 4.8t^2. Find the angular velocity (in radians per second) of that wheel when $1\\frac{1}{2}$ seconds have elapsed. Find also its angular acceleration.",
   "answer_latex": [
    "Angular velocity ${} = 11.2$ radians per second; angular\n  acceleration ${}= 9.6$ radians per second per second."
   ],
   "answer_markdown": [
    "Angular velocity ${} = 11.2$ radians per second; angular acceleration ${}= 9.6$ radians per second per second."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "2.1 - 3.2*t + 4.8*t**2",
     "answer_expr": "9.6"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 9.6, printed 9.6 (half-unit 0.05; correctly rounded at the printed digits: 9.6)",
     "problem_expr": "2.1 - 3.2*t + 4.8*t**2",
     "answer_expr": "9.6"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate2: 24*x**2/5 - 16*x/5 + 21/10",
    "evaluate: 24*x**2/5 - 16*x/5 + 21/10"
   ],
   "shape": [
    "differentiate2: N*x + N*x**N + N",
    "evaluate: N*x + N*x**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/7a",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 7,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "65",
   "location": "Exercise V, problem 7a",
   "problem_latex": "A slider moves so that, during the first part of\nits motion, its distance~$s$ in inches from its starting\npoint is given by the expression\n\\[\ns = 6.8t^3 - 10.8t;\\quad\\text{$t$~being in seconds}.\n\\]\n\nFind the expression for the velocity and the acceleration\nat any time; and hence find the velocity and the\nacceleration after $3$~seconds.",
   "markdown": "A slider moves so that, during the first part of its motion, its distance $s$ in inches from its starting point is given by the expression s = 6.8t^3 - 10.8t; $t$ being in seconds. Find the expression for the velocity and the acceleration at any time; and hence find the velocity and the acceleration after $3$ seconds.",
   "answer_latex": [
    "$v = 20.4t^2 - 10.8$.\\quad $a = 40.8t$.\\quad $172.8$~in./sec., $122.4~\\text{in./sec}^2$."
   ],
   "answer_markdown": [
    "$v = 20.4t^2 - 10.8$. $a = 40.8t$. $172.8$ in./sec., $122.4~\\text{in./sec}^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "6.8*t**3 - 10.8*t",
     "answer_expr": "20.4*t**2 - 10.8"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 172.8, printed 172.8 (half-unit 0.05; correctly rounded at the printed digits: 172.8)",
     "problem_expr": "6.8*t**3 - 10.8*t",
     "answer_expr": "20.4*t**2 - 10.8"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: 34*x**3/5 - 54*x/5",
    "evaluate: 34*x**3/5 - 54*x/5 at t=3"
   ],
   "shape": [
    "differentiate: N*x + N*x**N",
    "evaluate: N*x + N*x**N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-v/7b"
   ],
   "needs": [
    "cas.derive",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/7b",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 7,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "65",
   "location": "Exercise V, problem 7b",
   "problem_latex": "A slider moves so that, during the first part of\nits motion, its distance~$s$ in inches from its starting\npoint is given by the expression\n\\[\ns = 6.8t^3 - 10.8t;\\quad\\text{$t$~being in seconds}.\n\\]\n\nFind the expression for the velocity and the acceleration\nat any time; and hence find the velocity and the\nacceleration after $3$~seconds.",
   "markdown": "A slider moves so that, during the first part of its motion, its distance $s$ in inches from its starting point is given by the expression s = 6.8t^3 - 10.8t; $t$ being in seconds. Find the expression for the velocity and the acceleration at any time; and hence find the velocity and the acceleration after $3$ seconds.",
   "answer_latex": [
    "$v = 20.4t^2 - 10.8$.\\quad $a = 40.8t$.\\quad $172.8$~in./sec., $122.4~\\text{in./sec}^2$."
   ],
   "answer_markdown": [
    "$v = 20.4t^2 - 10.8$. $a = 40.8t$. $172.8$ in./sec., $122.4~\\text{in./sec}^2$."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "6.8*t**3 - 10.8*t",
     "answer_expr": "40.8*t"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 122.4, printed 122.4 (half-unit 0.05; correctly rounded at the printed digits: 122.4)",
     "problem_expr": "6.8*t**3 - 10.8*t",
     "answer_expr": "40.8*t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate2: 34*x**3/5 - 54*x/5",
    "evaluate: 34*x**3/5 - 54*x/5 at t=3"
   ],
   "shape": [
    "differentiate2: N*x + N*x**N",
    "evaluate: N*x + N*x**N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-v/7a"
   ],
   "needs": [
    "cas.derive",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/8a",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 8,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "65",
   "location": "Exercise V, problem 8a",
   "problem_latex": "The motion of a rising balloon is such that its\nheight~$h$, in miles, is given at any instant by the\nexpression $h = 0.5 + \\frac{1}{10}\\sqrt[3]{t-125}$; $t$~being in seconds.\n\nFind an expression for the velocity and the acceleration\nat any time. Draw curves to show the variation\nof height, velocity and acceleration during the first\nten minutes of the ascent.",
   "markdown": "The motion of a rising balloon is such that its height $h$, in miles, is given at any instant by the expression $h = 0.5 + \\frac{1}{10}\\sqrt[3]{t-125}$; $t$ being in seconds. Find an expression for the velocity and the acceleration at any time. Draw curves to show the variation of height, velocity and acceleration during the first ten minutes of the ascent.",
   "answer_latex": [
    "$v = \\dfrac{1}{30 \\sqrt[3]{(t - 125)^2}}$,\\quad $a = - \\dfrac{1}{45 \\sqrt[3]{(t - 125)^5}}$."
   ],
   "answer_markdown": [
    "$v = \\dfrac{1}{30 \\sqrt[3]{(t - 125)^2}}$, $a = - \\dfrac{1}{45 \\sqrt[3]{(t - 125)^5}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "FLAG-DOMAIN",
     "judge_why": "not enough points inside the domain",
     "problem_expr": "Rational(1,2) + (t - 125)**Rational(1,3)/10",
     "answer_expr": "1/(30*((t - 125)**2)**Rational(1,3))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-DOMAIN"
    ]
   },
   "form": [
    "differentiate: (x - 125)**(1/3)/10 + 1/2"
   ],
   "shape": [
    "differentiate: N*(N + x)**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.graph",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/8b",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 8,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "65",
   "location": "Exercise V, problem 8b",
   "problem_latex": "The motion of a rising balloon is such that its\nheight~$h$, in miles, is given at any instant by the\nexpression $h = 0.5 + \\frac{1}{10}\\sqrt[3]{t-125}$; $t$~being in seconds.\n\nFind an expression for the velocity and the acceleration\nat any time. Draw curves to show the variation\nof height, velocity and acceleration during the first\nten minutes of the ascent.",
   "markdown": "The motion of a rising balloon is such that its height $h$, in miles, is given at any instant by the expression $h = 0.5 + \\frac{1}{10}\\sqrt[3]{t-125}$; $t$ being in seconds. Find an expression for the velocity and the acceleration at any time. Draw curves to show the variation of height, velocity and acceleration during the first ten minutes of the ascent.",
   "answer_latex": [
    "$v = \\dfrac{1}{30 \\sqrt[3]{(t - 125)^2}}$,\\quad $a = - \\dfrac{1}{45 \\sqrt[3]{(t - 125)^5}}$."
   ],
   "answer_markdown": [
    "$v = \\dfrac{1}{30 \\sqrt[3]{(t - 125)^2}}$, $a = - \\dfrac{1}{45 \\sqrt[3]{(t - 125)^5}}$."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "FLAG-DOMAIN",
     "judge_why": "not enough points inside the domain",
     "problem_expr": "Rational(1,2) + (t - 125)**Rational(1,3)/10",
     "answer_expr": "-1/(45*((t - 125)**5)**Rational(1,3))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-DOMAIN"
    ]
   },
   "form": [
    "differentiate2: (x - 125)**(1/3)/10 + 1/2"
   ],
   "shape": [
    "differentiate2: N*(N + x)**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.graph",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/9a",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 9,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "65",
   "location": "Exercise V, problem 9a",
   "problem_latex": "A stone is thrown downwards into water and\nits depth~$p$ in metres at any instant $t$~seconds after\nreaching the surface of the water is given by the\nexpression\n\\[\np = \\frac{4}{4+t^2} + 0.8t - 1.\n\\]\n\\DPPageSep{078.png}{66}%\n\nFind an expression for the velocity and the acceleration\nat any time. Find the velocity and acceleration\nafter $10$~seconds.",
   "markdown": "A stone is thrown downwards into water and its depth $p$ in metres at any instant $t$ seconds after reaching the surface of the water is given by the expression p = 44+t^2 + 0.8t - 1. 078.png66% Find an expression for the velocity and the acceleration at any time. Find the velocity and acceleration after $10$ seconds.",
   "answer_latex": [
    "$v = 0.8 - \\dfrac{8t}{(4 + t^2)^2}$,\\quad $a = \\dfrac{24t^2 - 32}{(4 + t^2)^3}$,\\quad $0.7926$ and $0.00211$."
   ],
   "answer_markdown": [
    "$v = 0.8 - \\dfrac{8t}{(4 + t^2)^2}$, $a = \\dfrac{24t^2 - 32}{(4 + t^2)^3}$, $0.7926$ and $0.00211$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "4/(4 + t**2) + 0.8*t - 1",
     "answer_expr": "0.8 - 8*t/(4 + t**2)**2"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.792603550296, printed 0.7926 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.7926)",
     "problem_expr": "4/(4 + t**2) + 0.8*t - 1",
     "answer_expr": "0.8 - 8*t/(4 + t**2)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: 4*x/5 - 1 + 4/(x**2 + 4)",
    "evaluate: 4*x/5 - 1 + 4/(x**2 + 4) at t=10"
   ],
   "shape": [
    "differentiate: N*x + N/(N + x**N) - 1",
    "evaluate: N*x + N/(N + x**N) - 1"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-v/9b"
   ],
   "needs": [
    "cas.derive",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-v/9b",
   "set": "thompson-calculus-made-easy-1914/ex-v",
   "number": 9,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "65",
   "location": "Exercise V, problem 9b",
   "problem_latex": "A stone is thrown downwards into water and\nits depth~$p$ in metres at any instant $t$~seconds after\nreaching the surface of the water is given by the\nexpression\n\\[\np = \\frac{4}{4+t^2} + 0.8t - 1.\n\\]\n\\DPPageSep{078.png}{66}%\n\nFind an expression for the velocity and the acceleration\nat any time. Find the velocity and acceleration\nafter $10$~seconds.",
   "markdown": "A stone is thrown downwards into water and its depth $p$ in metres at any instant $t$ seconds after reaching the surface of the water is given by the expression p = 44+t^2 + 0.8t - 1. 078.png66% Find an expression for the velocity and the acceleration at any time. Find the velocity and acceleration after $10$ seconds.",
   "answer_latex": [
    "$v = 0.8 - \\dfrac{8t}{(4 + t^2)^2}$,\\quad $a = \\dfrac{24t^2 - 32}{(4 + t^2)^3}$,\\quad $0.7926$ and $0.00211$."
   ],
   "answer_markdown": [
    "$v = 0.8 - \\dfrac{8t}{(4 + t^2)^2}$, $a = \\dfrac{24t^2 - 32}{(4 + t^2)^3}$, $0.7926$ and $0.00211$."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "4/(4 + t**2) + 0.8*t - 1",
     "answer_expr": "(24*t**2 - 32)/(4 + t**2)**3"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.00210514337733, printed 0.00211 (half-unit 5.0e-6; correctly rounded at the printed digits: 0.00211)",
     "problem_expr": "4/(4 + t**2) + 0.8*t - 1",
     "answer_expr": "(24*t**2 - 32)/(4 + t**2)**3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate2: 4*x/5 - 1 + 4/(x**2 + 4)",
    "evaluate: 4*x/5 - 1 + 4/(x**2 + 4) at t=10"
   ],
   "shape": [
    "differentiate2: N*x + N/(N + x**N) - 1",
    "evaluate: N*x + N/(N + x**N) - 1"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-v/9a"
   ],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi/1",
   "set": "thompson-calculus-made-easy-1914/ex-vi",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "73",
   "location": "Exercise VI, problem 1",
   "problem_latex": "$y = \\sqrt{x^2 + 1}$.",
   "markdown": "$y = \\sqrt{x^2 + 1}$.",
   "answer_latex": [
    "$\\dfrac{x}{\\sqrt{ x^2 + 1}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{x}{\\sqrt{ x^2 + 1}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sqrt(x**2 + 1)",
     "answer_expr": "x/sqrt(x**2 + 1)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(x**2 + 1)"
   ],
   "shape": [
    "differentiate: (x**N + 1)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi/2",
   "set": "thompson-calculus-made-easy-1914/ex-vi",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "73",
   "location": "Exercise VI, problem 2",
   "problem_latex": "$y = \\sqrt{x^2+a^2}$.",
   "markdown": "$y = \\sqrt{x^2+a^2}$.",
   "answer_latex": [
    "$\\dfrac{x}{\\sqrt{ x^2 + a^2}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{x}{\\sqrt{ x^2 + a^2}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sqrt(x**2 + a**2)",
     "answer_expr": "x/sqrt(x**2 + a**2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(a**2 + x**2)"
   ],
   "shape": [
    "differentiate: (a**N + x**N)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi/3",
   "set": "thompson-calculus-made-easy-1914/ex-vi",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "73",
   "location": "Exercise VI, problem 3",
   "problem_latex": "$y = \\dfrac{1}{\\sqrt{a+x}}$.",
   "markdown": "$y = \\dfrac{1}{\\sqrt{a+x}}$.",
   "answer_latex": [
    "$- \\dfrac{1}{2 \\sqrt{(a + x)^3}}$."
   ],
   "answer_markdown": [
    "$- \\dfrac{1}{2 \\sqrt{(a + x)^3}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/sqrt(a + x)",
     "answer_expr": "-1/(2*sqrt((a + x)**3))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 1/sqrt(a + x)"
   ],
   "shape": [
    "differentiate: (a + x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi/4",
   "set": "thompson-calculus-made-easy-1914/ex-vi",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "73",
   "location": "Exercise VI, problem 4",
   "problem_latex": "$y = \\dfrac{a}{\\sqrt{a-x^2}}$.",
   "markdown": "$y = \\dfrac{a}{\\sqrt{a-x^2}}$.",
   "answer_latex": [
    "$\\dfrac{ax}{\\sqrt{(a - x^2)^3}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{ax}{\\sqrt{(a - x^2)^3}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a/sqrt(a - x**2)",
     "answer_expr": "a*x/sqrt((a - x**2)**3)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a/sqrt(a - x**2)"
   ],
   "shape": [
    "differentiate: a*(a - x**N)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi/5",
   "set": "thompson-calculus-made-easy-1914/ex-vi",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "73",
   "location": "Exercise VI, problem 5",
   "problem_latex": "$y = \\dfrac{\\sqrt{x^2-a^2}}{x^2}$.",
   "markdown": "$y = \\dfrac{\\sqrt{x^2-a^2}}{x^2}$.",
   "answer_latex": [
    "$\\dfrac{2a^2 - x^2}{x^3 \\sqrt{ x^2 - a^2}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{2a^2 - x^2}{x^3 \\sqrt{ x^2 - a^2}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sqrt(x**2 - a**2)/x**2",
     "answer_expr": "(2*a**2 - x**2)/(x**3*sqrt(x**2 - a**2))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(-a**2 + x**2)/x**2"
   ],
   "shape": [
    "differentiate: x**N*(-a**N + x**N)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi/6",
   "set": "thompson-calculus-made-easy-1914/ex-vi",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "73",
   "location": "Exercise VI, problem 6",
   "problem_latex": "$y = \\dfrac{\\sqrt[3]{x^4+a}}{\\sqrt[2]{x^3+a}}$.",
   "markdown": "$y = \\dfrac{\\sqrt[3]{x^4+a}}{\\sqrt[2]{x^3+a}}$.",
   "answer_latex": [
    "$ \\dfrac{\\frac{3}{2} x^2 \\left[ \\frac{8}{9} x \\left( x^3 + a \\right) - \\left( x^4 + a \\right) \\right]}{(x^4 + a)^{\\efrac{2}{3}} (x^3 + a)^{\\efrac{3}{2}}}$"
   ],
   "answer_markdown": [
    "$ \\dfrac{\\frac{3}{2} x^2 \\left[ \\frac{8}{9} x \\left( x^3 + a \\right) - \\left( x^4 + a \\right) \\right]}{(x^4 + a)^{\\efrac{2}{3}} (x^3 + a)^{\\efrac{3}{2}}}$"
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x**4 + a)**Rational(1,3)/sqrt(x**3 + a)",
     "answer_expr": "Rational(3,2)*x**2*(Rational(8,9)*x*(x**3 + a) - (x**4 + a))/((x**4 + a)**Rational(2,3)*(x**3 + a)**Rational(3,2))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (a + x**4)**(1/3)/sqrt(a + x**3)"
   ],
   "shape": [
    "differentiate: (a + x**N)**(2*N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.factor",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi/7",
   "set": "thompson-calculus-made-easy-1914/ex-vi",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "73",
   "location": "Exercise VI, problem 7",
   "problem_latex": "$y = \\dfrac{a^2+x^2}{(a+x)^2}$.",
   "markdown": "$y = \\dfrac{a^2+x^2}{(a+x)^2}$.",
   "answer_latex": [
    "$\\dfrac{2a \\left(x - a \\right)}{(x + a)^3}$."
   ],
   "answer_markdown": [
    "$\\dfrac{2a \\left(x - a \\right)}{(x + a)^3}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(a**2 + x**2)/(a + x)**2",
     "answer_expr": "2*a*(x - a)/(x + a)**3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (a**2 + x**2)/(a + x)**2"
   ],
   "shape": [
    "differentiate: (a + x)**N*(a**N + x**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.cancel",
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi/8",
   "set": "thompson-calculus-made-easy-1914/ex-vi",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "74",
   "location": "Exercise VI, problem 8",
   "problem_latex": "Differentiate $y^5$ with respect to~$y^2$.",
   "markdown": "Differentiate $y^5$ with respect to $y^2$.",
   "answer_latex": [
    "$\\frac{5}{2} y^3$."
   ],
   "answer_markdown": [
    "$\\frac{5}{2} y^3$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "y**5",
     "answer_expr": "Rational(5,2)*y**3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.cancel",
    "cas.derive",
    "other:derivative with respect to a function of the variable (ratio dy5/dy2, chain rule)"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vi/9",
   "set": "thompson-calculus-made-easy-1914/ex-vi",
   "number": 9,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "74",
   "location": "Exercise VI, problem 9",
   "problem_latex": "Differentiate $y = \\dfrac{\\sqrt{1 - \\theta^2}}{1 - \\theta}$.",
   "markdown": "Differentiate $y = \\dfrac{\\sqrt{1 - \\theta^2}}{1 - \\theta}$.",
   "answer_latex": [
    "$\\dfrac{1}{(1 - \\theta) \\sqrt{1 - \\theta^2}}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{(1 - \\theta) \\sqrt{1 - \\theta^2}}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sqrt(1 - theta**2)/(1 - theta)",
     "answer_expr": "1/((1 - theta)*sqrt(1 - theta**2))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(-x**2 + 1)/(-x + 1)"
   ],
   "shape": [
    "differentiate: (-x**N + 1)**N/(-x + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vii/1",
   "set": "thompson-calculus-made-easy-1914/ex-vii",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "75",
   "location": "Exercise VII, problem 1",
   "problem_latex": "If $u = \\frac{1}{2}x^3$;\\quad $v = 3(u+u^2)$;\\quad and $w = \\dfrac{1}{v^2}$, find~$\\dfrac{dw}{dx}$.",
   "markdown": "If $u = \\frac{1}{2}x^3$; $v = 3(u+u^2)$; and $w = \\dfrac{1}{v^2}$, find $\\dfrac{dw}{dx}$.",
   "answer_latex": [
    "$\\dfrac{dw}{dx} = \\dfrac{3x^2 \\left( 3 + 3x^3 \\right)} {27 \\left(\\frac{1}{2} x^3 + \\frac{1}{4} x^6 \\right)^3}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dw}{dx} = \\dfrac{3x^2 \\left( 3 + 3x^3 \\right)} {27 \\left(\\frac{1}{2} x^3 + \\frac{1}{4} x^6 \\right)^3}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": [
      "-1023.9881515202734856",
      "1023.9881515202734856"
     ],
     "problem_expr": "1/(3*(x**3/2 + (x**3/2)**2))**2",
     "answer_expr": "3*x**2*(3 + 3*x**3)/(27*(Rational(1,2)*x**3 + Rational(1,4)*x**6)**3)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "differentiate: (3*x**6/4 + 3*x**3/2)**(-2)"
   ],
   "shape": [
    "differentiate: (2*N*x**N)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vii/2",
   "set": "thompson-calculus-made-easy-1914/ex-vii",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "75",
   "location": "Exercise VII, problem 2",
   "problem_latex": "If $y = 3x^2 + \\sqrt{2}$;\\quad $z = \\sqrt{1+y}$;\\quad and $v = \\dfrac{1}{\\sqrt{3}+4z}$,\nfind~$\\dfrac{dv}{dx}$.",
   "markdown": "If $y = 3x^2 + \\sqrt{2}$; $z = \\sqrt{1+y}$; and $v = \\dfrac{1}{\\sqrt{3}+4z}$, find $\\dfrac{dv}{dx}$.",
   "answer_latex": [
    "$\\dfrac{dv}{dx} = - \\dfrac{12x}{\\sqrt{1 + \\sqrt{2} + 3x^2} \\left(\\sqrt{3} + 4 \\sqrt{1 + \\sqrt{2} + 3x^2}\\right)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dv}{dx} = - \\dfrac{12x}{\\sqrt{1 + \\sqrt{2} + 3x^2} \\left(\\sqrt{3} + 4 \\sqrt{1 + \\sqrt{2} + 3x^2}\\right)^2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/(sqrt(3) + 4*sqrt(1 + 3*x**2 + sqrt(2)))",
     "answer_expr": "-12*x/(sqrt(1 + sqrt(2) + 3*x**2)*(sqrt(3) + 4*sqrt(1 + sqrt(2) + 3*x**2))**2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 1/(4*sqrt(3*x**2 + 1 + sqrt(2)) + sqrt(3))"
   ],
   "shape": [
    "differentiate: 1/(N*(N*x**N + N**N + 1)**N + N**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-vii/3",
   "set": "thompson-calculus-made-easy-1914/ex-vii",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "75",
   "location": "Exercise VII, problem 3",
   "problem_latex": "If $y = \\dfrac{x^3}{\\sqrt{3}}$;\\quad $z = (1+y)^2$;\\quad and $u = \\dfrac{1}{\\sqrt{1+z}}$, find~$\\dfrac{du}{dx}$.",
   "markdown": "If $y = \\dfrac{x^3}{\\sqrt{3}}$; $z = (1+y)^2$; and $u = \\dfrac{1}{\\sqrt{1+z}}$, find $\\dfrac{du}{dx}$.",
   "answer_latex": [
    "$\\dfrac{du}{dx} = - \\dfrac{x^2 \\left(\\sqrt{3} + x^3 \\right)} {\\sqrt{ \\left[ 1 + \\left( 1 + \\dfrac{x^3}{\\sqrt{3}} \\right) ^2 \\right]^3}} $"
   ],
   "answer_markdown": [
    "$\\dfrac{du}{dx} = - \\dfrac{x^2 \\left(\\sqrt{3} + x^3 \\right)} {\\sqrt{ \\left[ 1 + \\left( 1 + \\dfrac{x^3}{\\sqrt{3}} \\right) ^2 \\right]^3}} $"
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/sqrt(1 + (1 + x**3/sqrt(3))**2)",
     "answer_expr": "-x**2*(sqrt(3) + x**3)/sqrt((1 + (1 + x**3/sqrt(3))**2)**3)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 1/sqrt((sqrt(3)*x**3/3 + 1)**2 + 1)"
   ],
   "shape": [
    "differentiate: ((N*N**N*x**N + 1)**N + 1)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/1",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "91",
   "location": "Exercise VIII, problem 1",
   "problem_latex": "Plot the curve $y = \\tfrac{3}{4} x^2 - 5$, using a scale of\nmillimetres. Measure at points corresponding to\ndifferent values of~$x$, the angle of its slope.\n\nFind, by differentiating the equation, the expression\nfor slope; and see, from a Table of Natural Tangents,\nwhether this agrees with the measured angle.",
   "markdown": "Plot the curve $y = \\tfrac{3}{4} x^2 - 5$, using a scale of millimetres. Measure at points corresponding to different values of $x$, the angle of its slope. Find, by differentiating the equation, the expression for slope; and see, from a Table of Natural Tangents, whether this agrees with the measured angle.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.graph",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/10a",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 10,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "92",
   "location": "Exercise VIII, problem 10a",
   "problem_latex": "A straight line $y = 2x - b$ touches a curve\n$y = 3x^2 + 2$ at one point. What are the coordinates\nof the point of contact, and what is the value of~$b$?",
   "markdown": "A straight line $y = 2x - b$ touches a curve $y = 3x^2 + 2$ at one point. What are the coordinates of the point of contact, and what is the value of $b$?",
   "answer_latex": [
    "$x = \\frac{1}{3}$, $y = 2 \\frac{1}{3}$, $b = -\\frac{5}{3}$."
   ],
   "answer_markdown": [
    "$x = \\frac{1}{3}$, $y = 2 \\frac{1}{3}$, $b = -\\frac{5}{3}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "3*x**2 + 2",
     "answer_expr": "1/3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.eqn"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/10b",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 10,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "92",
   "location": "Exercise VIII, problem 10b",
   "problem_latex": "A straight line $y = 2x - b$ touches a curve\n$y = 3x^2 + 2$ at one point. What are the coordinates\nof the point of contact, and what is the value of~$b$?",
   "markdown": "A straight line $y = 2x - b$ touches a curve $y = 3x^2 + 2$ at one point. What are the coordinates of the point of contact, and what is the value of $b$?",
   "answer_latex": [
    "$x = \\frac{1}{3}$, $y = 2 \\frac{1}{3}$, $b = -\\frac{5}{3}$."
   ],
   "answer_markdown": [
    "$x = \\frac{1}{3}$, $y = 2 \\frac{1}{3}$, $b = -\\frac{5}{3}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "3*x**2 + 2",
     "answer_expr": "7/3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.subst",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/10c",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 10,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "92",
   "location": "Exercise VIII, problem 10c",
   "problem_latex": "A straight line $y = 2x - b$ touches a curve\n$y = 3x^2 + 2$ at one point. What are the coordinates\nof the point of contact, and what is the value of~$b$?",
   "markdown": "A straight line $y = 2x - b$ touches a curve $y = 3x^2 + 2$ at one point. What are the coordinates of the point of contact, and what is the value of $b$?",
   "answer_latex": [
    "$x = \\frac{1}{3}$, $y = 2 \\frac{1}{3}$, $b = -\\frac{5}{3}$."
   ],
   "answer_markdown": [
    "$x = \\frac{1}{3}$, $y = 2 \\frac{1}{3}$, $b = -\\frac{5}{3}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "3*x**2 + 2",
     "answer_expr": "-5/3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.subst",
    "core.eqn",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/2",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "91",
   "location": "Exercise VIII, problem 2",
   "problem_latex": "Find what will be the slope of the curve\n\\[\ny = 0.12x^3 - 2,\n\\]\nat the particular point that has as abscissa $x = 2$.",
   "markdown": "Find what will be the slope of the curve y = 0.12x^3 - 2, at the particular point that has as abscissa $x = 2$.",
   "answer_latex": [
    "$1.44$."
   ],
   "answer_markdown": [
    "$1.44$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "0.12*x**3 - 2",
     "answer_expr": "0.36*x**2"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 1.44, printed 1.44 (half-unit 0.005; correctly rounded at the printed digits: 1.44)",
     "problem_expr": "0.12*x**3 - 2",
     "answer_expr": "0.36*x**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: 3*x**3/25 - 2",
    "evaluate: 3*x**3/25 - 2 at x=2"
   ],
   "shape": [
    "differentiate: N*x**N + N",
    "evaluate: N*x**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/3",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "91",
   "location": "Exercise VIII, problem 3",
   "problem_latex": "If $y = (x - a)(x - b)$, show that at the particular\npoint of the curve where $\\dfrac{dy}{dx} = 0$, $x$ will have the value\n$\\tfrac{1}{2} (a + b)$.",
   "markdown": "If $y = (x - a)(x - b)$, show that at the particular point of the curve where $\\dfrac{dy}{dx} = 0$, $x$ will have the value $\\tfrac{1}{2} (a + b)$.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/4",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "91",
   "location": "Exercise VIII, problem 4",
   "problem_latex": "Find the $\\dfrac{dy}{dx}$ of the equation $y = x^3 + 3x$; and\ncalculate the numerical values of $\\dfrac{dy}{dx}$ for the points\ncorresponding to $x = 0$, $x = \\tfrac{1}{2}$, $x = 1$, $x = 2$.",
   "markdown": "Find the $\\dfrac{dy}{dx}$ of the equation $y = x^3 + 3x$; and calculate the numerical values of $\\dfrac{dy}{dx}$ for the points corresponding to $x = 0$, $x = \\tfrac{1}{2}$, $x = 1$, $x = 2$.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = 3x^2 + 3$; and the numerical values are:\n     $3$,~$3 \\frac{3}{4}$, $6$,~and~$15$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = 3x^2 + 3$; and the numerical values are: $3$, $3 \\frac{3}{4}$, $6$, and $15$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**3 + 3*x",
     "answer_expr": "3*x**2 + 3"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 3.0, printed 3 (half-unit 0.5; correctly rounded at the printed digits: 3.0)",
     "problem_expr": "x**3 + 3*x",
     "answer_expr": "3*x**2 + 3"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 3.75, printed 3 + 3/4 (half-unit 1.0e-40; correctly rounded at the printed digits: 3.75)",
     "problem_expr": "x**3 + 3*x",
     "answer_expr": "3*x**2 + 3"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 6.0, printed 6 (half-unit 0.5; correctly rounded at the printed digits: 6.0)",
     "problem_expr": "x**3 + 3*x",
     "answer_expr": "3*x**2 + 3"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 15.0, printed 15 (half-unit 0.5; correctly rounded at the printed digits: 15.0)",
     "problem_expr": "x**3 + 3*x",
     "answer_expr": "3*x**2 + 3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "PASS",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: x**3 + 3*x",
    "evaluate: x**3 + 3*x at x=0",
    "evaluate: x**3 + 3*x at x=1/2",
    "evaluate: x**3 + 3*x at x=1",
    "evaluate: x**3 + 3*x at x=2"
   ],
   "shape": [
    "differentiate: N*x + x**N",
    "evaluate: N*x + x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/5",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "91",
   "location": "Exercise VIII, problem 5",
   "problem_latex": "In the curve to which the equation is $x^2 + y^2 = 4$,\nfind the values of~$x$ at those points where the slope ${} = 1$.",
   "markdown": "In the curve to which the equation is $x^2 + y^2 = 4$, find the values of $x$ at those points where the slope ${} = 1$.",
   "answer_latex": [
    "$ ± \\sqrt{2}$."
   ],
   "answer_markdown": [
    "$ ± \\sqrt{2}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "sqrt(2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "other:implicit differentiation"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/6",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "91",
   "location": "Exercise VIII, problem 6",
   "problem_latex": "Find the slope, at any point, of the curve whose\nequation is $\\dfrac{x^2 }{3^2} + \\dfrac{y^2}{2^2} = 1$; and give the numerical value\nof the slope at the place where $x = 0$, and at that\nwhere $x = 1$.",
   "markdown": "Find the slope, at any point, of the curve whose equation is $\\dfrac{x^2 }{3^2} + \\dfrac{y^2}{2^2} = 1$; and give the numerical value of the slope at the place where $x = 0$, and at that where $x = 1$.",
   "answer_latex": [
    "$ \\dfrac{dy}{dx} = - \\dfrac{4}{9} \\dfrac{x}{y}$. Slope is zero where $x = 0$; and is $\\mp \\dfrac{1}{3 \\sqrt{2}}$ where $x = 1$."
   ],
   "answer_markdown": [
    "$ \\dfrac{dy}{dx} = - \\dfrac{4}{9} \\dfrac{x}{y}$. Slope is zero where $x = 0$; and is $\\mp \\dfrac{1}{3 \\sqrt{2}}$ where $x = 1$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "-4*x/(9*y)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst",
    "other:implicit differentiation"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/7a",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 7,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "91",
   "location": "Exercise VIII, problem 7a",
   "problem_latex": "The equation of a tangent to the curve\n$y = 5 - 2x + 0.5x^3$, being of the form $y = mx + n$, where\n$m$ and~$n$ are constants, find the value of $m$ and~$n$ if\n\\DPPageSep{104.png}{92}%\nthe point where the tangent touches the curve has\n$x=2$ for abscissa.",
   "markdown": "The equation of a tangent to the curve $y = 5 - 2x + 0.5x^3$, being of the form $y = mx + n$, where $m$ and $n$ are constants, find the value of $m$ and $n$ if 104.png92% the point where the tangent touches the curve has $x=2$ for abscissa.",
   "answer_latex": [
    "$m = 4$, $n = -3$."
   ],
   "answer_markdown": [
    "$m = 4$, $n = -3$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "5 - 2*x + 0.5*x**3",
     "answer_expr": "-2 + 1.5*x**2"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 4.0, printed 4 (half-unit 0.5; correctly rounded at the printed digits: 4.0)",
     "problem_expr": "5 - 2*x + 0.5*x**3",
     "answer_expr": "-2 + 1.5*x**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: x**3/2 - 2*x + 5",
    "evaluate: x**3/2 - 2*x + 5 at x=2"
   ],
   "shape": [
    "differentiate: N*x + N*x**N + N",
    "evaluate: N*x + N*x**N + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/7b",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 7,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "91",
   "location": "Exercise VIII, problem 7b",
   "problem_latex": "The equation of a tangent to the curve\n$y = 5 - 2x + 0.5x^3$, being of the form $y = mx + n$, where\n$m$ and~$n$ are constants, find the value of $m$ and~$n$ if\n\\DPPageSep{104.png}{92}%\nthe point where the tangent touches the curve has\n$x=2$ for abscissa.",
   "markdown": "The equation of a tangent to the curve $y = 5 - 2x + 0.5x^3$, being of the form $y = mx + n$, where $m$ and $n$ are constants, find the value of $m$ and $n$ if 104.png92% the point where the tangent touches the curve has $x=2$ for abscissa.",
   "answer_latex": [
    "$m = 4$, $n = -3$."
   ],
   "answer_markdown": [
    "$m = 4$, $n = -3$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "5 - 2*x + 0.5*x**3",
     "answer_expr": "-3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst",
    "core.eqn"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/8",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "92",
   "location": "Exercise VIII, problem 8",
   "problem_latex": "At what angle do the two curves\n\\[\ny = 3.5x^2 + 2 \\quad \\text{and} \\quad y = x^2 - 5x + 9.5\n\\]\ncut one another?",
   "markdown": "At what angle do the two curves y = 3.5x^2 + 2 and y = x^2 - 5x + 9.5 cut one another?",
   "answer_latex": [
    "Intersections at $x = 1$, $x = -3$. Angles $153°\\;26'$, $2°\\;28'$."
   ],
   "answer_markdown": [
    "Intersections at $x = 1$, $x = -3$. Angles $153°\\;26'$, $2°\\;28'$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/9a",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 9,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "91",
   "location": "Exercise VIII, problem 9a",
   "problem_latex": "Tangents to the curve $y = ± \\sqrt{25-x^2}$ are drawn\nat points for which $x = 3$ and $x = 4$. Find the coordinates\nof the point of intersection of the tangents\nand their mutual inclination.",
   "markdown": "Tangents to the curve $y = ± \\sqrt{25-x^2}$ are drawn at points for which $x = 3$ and $x = 4$. Find the coordinates of the point of intersection of the tangents and their mutual inclination.",
   "answer_latex": [
    "Intersection at $x = 3.57$, $y = 3.50$. Angle $16°\\;16'$."
   ],
   "answer_markdown": [
    "Intersection at $x = 3.57$, $y = 3.50$. Angle $16°\\;16'$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 3.57142857143, printed 25/7 (half-unit 1.0e-40; correctly rounded at the printed digits: 3.57142857143)",
     "problem_expr": "solve([3*x + 4*y - 25, 4*x + 3*y - 25], [x, y])[x]",
     "answer_expr": "25/7"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: 25/7"
   ],
   "shape": [
    "evaluate: N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst",
    "core.linsys",
    "other:implicit differentiation"
   ],
   "expectation": "X#3.5714285714285714285714285714285714,1E-30",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/9b",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 9,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "92",
   "location": "Exercise VIII, problem 9b",
   "problem_latex": "Tangents to the curve $y = ± \\sqrt{25-x^2}$ are drawn\nat points for which $x = 3$ and $x = 4$. Find the coordinates\nof the point of intersection of the tangents\nand their mutual inclination.",
   "markdown": "Tangents to the curve $y = ± \\sqrt{25-x^2}$ are drawn at points for which $x = 3$ and $x = 4$. Find the coordinates of the point of intersection of the tangents and their mutual inclination.",
   "answer_latex": [
    "Intersection at $x = 3.57$, $y = 3.50$. Angle $16°\\;16'$."
   ],
   "answer_markdown": [
    "Intersection at $x = 3.57$, $y = 3.50$. Angle $16°\\;16'$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "3.5"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.subst",
    "core.linsys",
    "core.trig",
    "other:implicit differentiation"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-viii/9c",
   "set": "thompson-calculus-made-easy-1914/ex-viii",
   "number": 9,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "92",
   "location": "Exercise VIII, problem 9c",
   "problem_latex": "Tangents to the curve $y = ± \\sqrt{25-x^2}$ are drawn\nat points for which $x = 3$ and $x = 4$. Find the coordinates\nof the point of intersection of the tangents\nand their mutual inclination.",
   "markdown": "Tangents to the curve $y = ± \\sqrt{25-x^2}$ are drawn at points for which $x = 3$ and $x = 4$. Find the coordinates of the point of intersection of the tangents and their mutual inclination.",
   "answer_latex": [
    "Intersection at $x = 3.57$, $y = 3.50$. Angle $16°\\;16'$."
   ],
   "answer_markdown": [
    "Intersection at $x = 3.57$, $y = 3.50$. Angle $16°\\;16'$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/101",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 10,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "120",
   "location": "Exercise X, problem 101",
   "problem_latex": "Suppose it to be known that consumption of\ncoal by a certain steamer may be represented by the\nformula $y = 0.3 + 0.001v^3$; where $y$~is the number of\ntons of coal burned per hour and $v$~is the speed\nexpressed in nautical miles per hour. The cost of\nwages, interest on capital, and depreciation of that\nship are together equal, per hour, to the cost of\n$1$~ton of coal. What speed will make the total cost\nof a voyage of $1000$ nautical miles a minimum?\nAnd, if coal costs $10$~shillings per ton, what will that\nminimum cost of the voyage amount to?",
   "markdown": "Suppose it to be known that consumption of coal by a certain steamer may be represented by the formula $y = 0.3 + 0.001v^3$; where $y$ is the number of tons of coal burned per hour and $v$ is the speed expressed in nautical miles per hour. The cost of wages, interest on capital, and depreciation of that ship are together equal, per hour, to the cost of $1$ ton of coal. What speed will make the total cost of a voyage of $1000$ nautical miles a minimum? And, if coal costs $10$ shillings per ton, what will that minimum cost of the voyage amount to?",
   "answer_latex": [
    "Speed $8.66$ nautical miles per hour. Time taken $115.47$~hours. \\\\\nMinimum cost £$112$.~$12$\\textit{s}."
   ],
   "answer_markdown": [
    "Speed $8.66$ nautical miles per hour. Time taken $115.47$ hours. Minimum cost £$112$. $12$*s*."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "1000*(0.3 + 0.001*v**3 + 1)/v",
     "answer_expr": "650**(Rational(1,3))"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 8.66239105341, printed 8.66 (half-unit 0.005; correctly rounded at the printed digits: 8.66)",
     "problem_expr": "1000*(0.3 + 0.001*v**3 + 1)/v",
     "answer_expr": "650**(Rational(1,3))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED",
     "PASS"
    ]
   },
   "form": [
    "extremum: (x**3 + 1300)/x",
    "evaluate: (x**3 + 1300)/x"
   ],
   "shape": [
    "evaluate: (N + x**N)/x",
    "extremum: (N + x**N)/x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.arith",
    "core.graph",
    "core.solve.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/102",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 10,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "118",
   "location": "Exercise X, problem 102",
   "problem_latex": "Suppose it to be known that consumption of\ncoal by a certain steamer may be represented by the\nformula $y = 0.3 + 0.001v^3$; where $y$~is the number of\ntons of coal burned per hour and $v$~is the speed\nexpressed in nautical miles per hour. The cost of\nwages, interest on capital, and depreciation of that\nship are together equal, per hour, to the cost of\n$1$~ton of coal. What speed will make the total cost\nof a voyage of $1000$ nautical miles a minimum?\nAnd, if coal costs $10$~shillings per ton, what will that\nminimum cost of the voyage amount to?",
   "markdown": "Suppose it to be known that consumption of coal by a certain steamer may be represented by the formula $y = 0.3 + 0.001v^3$; where $y$ is the number of tons of coal burned per hour and $v$ is the speed expressed in nautical miles per hour. The cost of wages, interest on capital, and depreciation of that ship are together equal, per hour, to the cost of $1$ ton of coal. What speed will make the total cost of a voyage of $1000$ nautical miles a minimum? And, if coal costs $10$ shillings per ton, what will that minimum cost of the voyage amount to?",
   "answer_latex": [
    "Speed $8.66$ nautical miles per hour. Time taken $115.47$~hours. \\\\\nMinimum cost £$112$.~$12$\\textit{s}."
   ],
   "answer_markdown": [
    "Speed $8.66$ nautical miles per hour. Time taken $115.47$ hours. Minimum cost £$112$. $12$*s*."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 8.66239105341, printed 8.66 (half-unit 0.005; correctly rounded at the printed digits: 8.66)",
     "problem_expr": "650**(Rational(1,3))",
     "answer_expr": "650**(Rational(1,3))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: 1000/x at v=650**(Rational(1,3))",
    "evaluate: 650**(1/3)"
   ],
   "shape": [
    "evaluate: N**N",
    "evaluate: N/x"
   ],
   "same_problem_in": [],
   "needs": [
    "calculus.optimize",
    "core.arith",
    "core.units"
   ],
   "expectation": "X#8.66,5E-3",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/103",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 10,
   "part": "3",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "120",
   "location": "Exercise X, problem 103",
   "problem_latex": "Suppose it to be known that consumption of\ncoal by a certain steamer may be represented by the\nformula $y = 0.3 + 0.001v^3$; where $y$~is the number of\ntons of coal burned per hour and $v$~is the speed\nexpressed in nautical miles per hour. The cost of\nwages, interest on capital, and depreciation of that\nship are together equal, per hour, to the cost of\n$1$~ton of coal. What speed will make the total cost\nof a voyage of $1000$ nautical miles a minimum?\nAnd, if coal costs $10$~shillings per ton, what will that\nminimum cost of the voyage amount to?",
   "markdown": "Suppose it to be known that consumption of coal by a certain steamer may be represented by the formula $y = 0.3 + 0.001v^3$; where $y$ is the number of tons of coal burned per hour and $v$ is the speed expressed in nautical miles per hour. The cost of wages, interest on capital, and depreciation of that ship are together equal, per hour, to the cost of $1$ ton of coal. What speed will make the total cost of a voyage of $1000$ nautical miles a minimum? And, if coal costs $10$ shillings per ton, what will that minimum cost of the voyage amount to?",
   "answer_latex": [
    "Speed $8.66$ nautical miles per hour. Time taken $115.47$~hours. \\\\\nMinimum cost £$112$.~$12$\\textit{s}."
   ],
   "answer_markdown": [
    "Speed $8.66$ nautical miles per hour. Time taken $115.47$ hours. Minimum cost £$112$. $12$*s*."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "10*1000*(0.3 + 0.001*v**3 + 1)/v",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [
    "evaluate: (10*x**3 + 13000)/x at v=650**(Rational(1,3))"
   ],
   "shape": [
    "evaluate: (N*x**N + N)/x"
   ],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/11",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 1,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "118",
   "location": "Exercise X, problem 11",
   "problem_latex": "Find the maxima and minima of\n\\[\ny = x^3 + x^2 - 10x + 8.\n\\]",
   "markdown": "Find the maxima and minima of y = x^3 + x^2 - 10x + 8.",
   "answer_latex": [
    "Max.: $x = -2.19$, $y = 24.19$; min.:, $x = 1.52$, $y = -1.38$."
   ],
   "answer_markdown": [
    "Max.: $x = -2.19$, $y = 24.19$; min.:, $x = 1.52$, $y = -1.38$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "f'(x0) is not 0: ('-2.1382117680737565169e-50', '0.0')",
     "problem_expr": "x**3 + x**2 - 10*x + 8",
     "answer_expr": "(-1 - sqrt(31))/3"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "the formula itself fails: f'(x0) is not 0: ('-2.1382117680737565169e-50', '0.0')",
     "problem_expr": "x**3 + x**2 - 10*x + 8",
     "answer_expr": "(-1 - sqrt(31))/3"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 24.1926440924, printed 24.19 (half-unit 0.005; correctly rounded at the printed digits: 24.19)",
     "problem_expr": "x**3 + x**2 - 10*x + 8",
     "answer_expr": "(-1 - sqrt(31))/3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH",
     "FLAG-MISMATCH",
     "PASS"
    ]
   },
   "form": [
    "extremum: x**3 + x**2 - 10*x + 8",
    "evaluate: x**3 + x**2 - 10*x + 8",
    "evaluate: x**3 + x**2 - 10*x + 8 at x=(-1 - sqrt(31))/3"
   ],
   "shape": [
    "evaluate: N*x + N + 2*x**N",
    "extremum: N*x + N + 2*x**N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-x/12"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph",
    "core.solve.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/111",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 11,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "120",
   "location": "Exercise X, problem 111",
   "problem_latex": "Find the maxima and minima of\\Pagelabel{Ex:X11}%\n\\[\ny = ±\\frac{x}{6}\\sqrt{x(10-x)}.\n\\]",
   "markdown": "Find the maxima and minima ofEx:X11% y = ±x6x(10-x).",
   "answer_latex": [
    "Max.\\ and min.\\ for $x = 7.5$, $y = ±5.414$. (See example\nno.~10, \\Pageref{Example10}.)"
   ],
   "answer_markdown": [
    "Max. and min. for $x = 7.5$, $y = ±5.414$. (See example no. 10, Example10.)"
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x/6*sqrt(x*(10 - x))",
     "answer_expr": "Rational(15,2)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 7.5, printed 7.5 (half-unit 0.05; correctly rounded at the printed digits: 7.5)",
     "problem_expr": "x/6*sqrt(x*(10 - x))",
     "answer_expr": "Rational(15,2)"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 5.41265877365, printed 5.414 (half-unit 0.0005; correctly rounded at the printed digits: 5.413)",
     "problem_expr": "x/6*sqrt(x*(10 - x))",
     "answer_expr": "Rational(15,2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: x*sqrt(x*(-x + 10))/6",
    "evaluate: x*sqrt(x*(-x + 10))/6",
    "evaluate: x*sqrt(x*(-x + 10))/6 at x=15/2"
   ],
   "shape": [
    "evaluate: N*x*(x*(N - x))**N",
    "extremum: N*x*(x*(N - x))**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/112",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 11,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "120",
   "location": "Exercise X, problem 112",
   "problem_latex": "Find the maxima and minima of\\Pagelabel{Ex:X11}%\n\\[\ny = ±\\frac{x}{6}\\sqrt{x(10-x)}.\n\\]",
   "markdown": "Find the maxima and minima ofEx:X11% y = ±x6x(10-x).",
   "answer_latex": [
    "Max.\\ and min.\\ for $x = 7.5$, $y = ±5.414$. (See example\nno.~10, \\Pageref{Example10}.)"
   ],
   "answer_markdown": [
    "Max. and min. for $x = 7.5$, $y = ±5.414$. (See example no. 10, Example10.)"
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "-x/6*sqrt(x*(10 - x))",
     "answer_expr": "Rational(15,2)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 7.5, printed 7.5 (half-unit 0.05; correctly rounded at the printed digits: 7.5)",
     "problem_expr": "-x/6*sqrt(x*(10 - x))",
     "answer_expr": "Rational(15,2)"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed -5.41265877365, printed -5.414 (half-unit 0.0005; correctly rounded at the printed digits: -5.413)",
     "problem_expr": "-x/6*sqrt(x*(10 - x))",
     "answer_expr": "Rational(15,2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: -x*sqrt(x*(-x + 10))/6",
    "evaluate: -x*sqrt(x*(-x + 10))/6",
    "evaluate: -x*sqrt(x*(-x + 10))/6 at x=15/2"
   ],
   "shape": [
    "evaluate: N*x*(x*(N - x))**N",
    "extremum: N*x*(x*(N - x))**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/12",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 1,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "118",
   "location": "Exercise X, problem 12",
   "problem_latex": "Find the maxima and minima of\n\\[\ny = x^3 + x^2 - 10x + 8.\n\\]",
   "markdown": "Find the maxima and minima of y = x^3 + x^2 - 10x + 8.",
   "answer_latex": [
    "Max.: $x = -2.19$, $y = 24.19$; min.:, $x = 1.52$, $y = -1.38$."
   ],
   "answer_markdown": [
    "Max.: $x = -2.19$, $y = 24.19$; min.:, $x = 1.52$, $y = -1.38$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**3 + x**2 - 10*x + 8",
     "answer_expr": "(-1 + sqrt(31))/3"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 1.52258812094, printed 1.52 (half-unit 0.005; correctly rounded at the printed digits: 1.52)",
     "problem_expr": "x**3 + x**2 - 10*x + 8",
     "answer_expr": "(-1 + sqrt(31))/3"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -1.37782927761, printed -1.38 (half-unit 0.005; correctly rounded at the printed digits: -1.38)",
     "problem_expr": "x**3 + x**2 - 10*x + 8",
     "answer_expr": "(-1 + sqrt(31))/3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "extremum: x**3 + x**2 - 10*x + 8",
    "evaluate: x**3 + x**2 - 10*x + 8",
    "evaluate: x**3 + x**2 - 10*x + 8 at x=(-1 + sqrt(31))/3"
   ],
   "shape": [
    "evaluate: N*x + N + 2*x**N",
    "extremum: N*x + N + 2*x**N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-x/11"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph",
    "core.solve.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/121",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 12,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "120",
   "location": "Exercise X, problem 121",
   "problem_latex": "Find the maxima and minima of\n\\[\ny= 4x^3 - x^2 - 2x + 1.\n\\]",
   "markdown": "Find the maxima and minima of y= 4x^3 - x^2 - 2x + 1.",
   "answer_latex": [
    "Min.: $x = \\frac{1}{2}$, $y= 0.25$; max.: $x = - \\frac{1}{3}$, $y= 1.408$."
   ],
   "answer_markdown": [
    "Min.: $x = \\frac{1}{2}$, $y= 0.25$; max.: $x = - \\frac{1}{3}$, $y= 1.408$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "4*x**3 - x**2 - 2*x + 1",
     "answer_expr": "Rational(1,2)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.5, printed 1/2 (half-unit 0.5; correctly rounded at the printed digits: 0.0)",
     "problem_expr": "4*x**3 - x**2 - 2*x + 1",
     "answer_expr": "Rational(1,2)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.25, printed 0.25 (half-unit 0.005; correctly rounded at the printed digits: 0.25)",
     "problem_expr": "4*x**3 - x**2 - 2*x + 1",
     "answer_expr": "Rational(1,2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "extremum: 4*x**3 - x**2 - 2*x + 1",
    "evaluate: 4*x**3 - x**2 - 2*x + 1",
    "evaluate: 4*x**3 - x**2 - 2*x + 1 at x=1/2"
   ],
   "shape": [
    "evaluate: N*x + N*x**N - x**N + 1",
    "extremum: N*x + N*x**N - x**N + 1"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-x/122"
   ],
   "needs": [
    "cas.derive",
    "cas.factor",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.frac",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/122",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 12,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "120",
   "location": "Exercise X, problem 122",
   "problem_latex": "Find the maxima and minima of\n\\[\ny= 4x^3 - x^2 - 2x + 1.\n\\]",
   "markdown": "Find the maxima and minima of y= 4x^3 - x^2 - 2x + 1.",
   "answer_latex": [
    "Min.: $x = \\frac{1}{2}$, $y= 0.25$; max.: $x = - \\frac{1}{3}$, $y= 1.408$."
   ],
   "answer_markdown": [
    "Min.: $x = \\frac{1}{2}$, $y= 0.25$; max.: $x = - \\frac{1}{3}$, $y= 1.408$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "4*x**3 - x**2 - 2*x + 1",
     "answer_expr": "-Rational(1,3)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -0.333333333333, printed -1/3 (half-unit 0.5; correctly rounded at the printed digits: 0.0)",
     "problem_expr": "4*x**3 - x**2 - 2*x + 1",
     "answer_expr": "-Rational(1,3)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS-LOOSE",
     "judge_why": "computed 1.40740740741, printed 1.408 (half-unit 0.0005; correctly rounded at the printed digits: 1.407): one unit off in the last printed place",
     "problem_expr": "4*x**3 - x**2 - 2*x + 1",
     "answer_expr": "-Rational(1,3)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "PASS-LOOSE"
    ]
   },
   "form": [
    "extremum: 4*x**3 - x**2 - 2*x + 1",
    "evaluate: 4*x**3 - x**2 - 2*x + 1",
    "evaluate: 4*x**3 - x**2 - 2*x + 1 at x=-1/3"
   ],
   "shape": [
    "evaluate: N*x + N*x**N - x**N + 1",
    "extremum: N*x + N*x**N - x**N + 1"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-x/121"
   ],
   "needs": [
    "cas.derive",
    "cas.factor",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.frac",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/21",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 2,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "118",
   "location": "Exercise X, problem 21",
   "problem_latex": "Given $y = \\dfrac{b}{a}x - cx^2$, find expressions for~$\\dfrac{dy}{dx}$, and\nfor~$\\dfrac{d^2y}{dx^2}$, also find the value of~$x$ which makes $y$ a\nmaximum or a minimum, and show whether it is\nmaximum or minimum.",
   "markdown": "Given $y = \\dfrac{b}{a}x - cx^2$, find expressions for $\\dfrac{dy}{dx}$, and for $\\dfrac{d^2y}{dx^2}$, also find the value of $x$ which makes $y$ a maximum or a minimum, and show whether it is maximum or minimum.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = \\dfrac{b}{a} - 2cx$; $\\dfrac{d^2 y}{dx^2} = -2c$; $x = \\dfrac{b}{2ac}$ (\\emph{a maximum})."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = \\dfrac{b}{a} - 2cx$; $\\dfrac{d^2 y}{dx^2} = -2c$; $x = \\dfrac{b}{2ac}$ (*a maximum*)."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "b/a*x - c*x**2",
     "answer_expr": "b/a - 2*c*x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: -c*x**2 + b*x/a"
   ],
   "shape": [
    "differentiate: -c*x**N + b*x/a"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/22",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 2,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "118",
   "location": "Exercise X, problem 22",
   "problem_latex": "Given $y = \\dfrac{b}{a}x - cx^2$, find expressions for~$\\dfrac{dy}{dx}$, and\nfor~$\\dfrac{d^2y}{dx^2}$, also find the value of~$x$ which makes $y$ a\nmaximum or a minimum, and show whether it is\nmaximum or minimum.",
   "markdown": "Given $y = \\dfrac{b}{a}x - cx^2$, find expressions for $\\dfrac{dy}{dx}$, and for $\\dfrac{d^2y}{dx^2}$, also find the value of $x$ which makes $y$ a maximum or a minimum, and show whether it is maximum or minimum.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = \\dfrac{b}{a} - 2cx$; $\\dfrac{d^2 y}{dx^2} = -2c$; $x = \\dfrac{b}{2ac}$ (\\emph{a maximum})."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = \\dfrac{b}{a} - 2cx$; $\\dfrac{d^2 y}{dx^2} = -2c$; $x = \\dfrac{b}{2ac}$ (*a maximum*)."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "b/a*x - c*x**2",
     "answer_expr": "-2*c"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate2: -c*x**2 + b*x/a"
   ],
   "shape": [
    "differentiate2: -c*x**N + b*x/a"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/23",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 2,
   "part": "3",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "118",
   "location": "Exercise X, problem 23",
   "problem_latex": "Given $y = \\dfrac{b}{a}x - cx^2$, find expressions for~$\\dfrac{dy}{dx}$, and\nfor~$\\dfrac{d^2y}{dx^2}$, also find the value of~$x$ which makes $y$ a\nmaximum or a minimum, and show whether it is\nmaximum or minimum.",
   "markdown": "Given $y = \\dfrac{b}{a}x - cx^2$, find expressions for $\\dfrac{dy}{dx}$, and for $\\dfrac{d^2y}{dx^2}$, also find the value of $x$ which makes $y$ a maximum or a minimum, and show whether it is maximum or minimum.",
   "answer_latex": [
    "$\\dfrac{dy}{dx} = \\dfrac{b}{a} - 2cx$; $\\dfrac{d^2 y}{dx^2} = -2c$; $x = \\dfrac{b}{2ac}$ (\\emph{a maximum})."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dx} = \\dfrac{b}{a} - 2cx$; $\\dfrac{d^2 y}{dx^2} = -2c$; $x = \\dfrac{b}{2ac}$ (*a maximum*)."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "b/a*x - c*x**2",
     "answer_expr": "b/(2*a*c)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: -c*x**2 + b*x/a"
   ],
   "shape": [
    "extremum: -c*x**N + b*x/a"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/3a",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 3,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "118",
   "location": "Exercise X, problem 3a",
   "problem_latex": "Find how many maxima and how many minima\nthere are in the curve, the equation to which is\n\\[\ny = 1 - \\frac{x^2}{2} + \\frac{x^4}{24};\n\\]\nand how many in that of which the equation is\n\\[\ny = 1 - \\frac{x^2}{2} + \\frac{x^4}{24} - \\frac{x^6}{720}.\n\\]",
   "markdown": "Find how many maxima and how many minima there are in the curve, the equation to which is y = 1 - x^22 + x^424; and how many in that of which the equation is y = 1 - x^22 + x^424 - x^6720.",
   "answer_latex": [
    "(\\textit{a}) One maximum and two minima.  \\\\\n(\\textit{b}) One maximum. ($x = 0$; other points unreal.)"
   ],
   "answer_markdown": [
    "(*a*) One maximum and two minima. (*b*) One maximum. ($x = 0$; other points unreal.)"
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "1 - x**2/2 + x**4/24",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/3b",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 3,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "118",
   "location": "Exercise X, problem 3b",
   "problem_latex": "Find how many maxima and how many minima\nthere are in the curve, the equation to which is\n\\[\ny = 1 - \\frac{x^2}{2} + \\frac{x^4}{24};\n\\]\nand how many in that of which the equation is\n\\[\ny = 1 - \\frac{x^2}{2} + \\frac{x^4}{24} - \\frac{x^6}{720}.\n\\]",
   "markdown": "Find how many maxima and how many minima there are in the curve, the equation to which is y = 1 - x^22 + x^424; and how many in that of which the equation is y = 1 - x^22 + x^424 - x^6720.",
   "answer_latex": [
    "(\\textit{a}) One maximum and two minima.  \\\\\n(\\textit{b}) One maximum. ($x = 0$; other points unreal.)"
   ],
   "answer_markdown": [
    "(*a*) One maximum and two minima. (*b*) One maximum. ($x = 0$; other points unreal.)"
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "1 - x**2/2 + x**4/24 - x**6/720",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/4",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "119",
   "location": "Exercise X, problem 4",
   "problem_latex": "Find the maxima and minima of\n\\[\ny=2x+1+\\frac{5}{x^2}.\n\\]",
   "markdown": "Find the maxima and minima of y=2x+1+5x^2.",
   "answer_latex": [
    "Min.: $x = 1.71$, $y = 6.14$."
   ],
   "answer_markdown": [
    "Min.: $x = 1.71$, $y = 6.14$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "2*x + 1 + 5/x**2",
     "answer_expr": "5**(Rational(1,3))"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 1.70997594668, printed 1.71 (half-unit 0.005; correctly rounded at the printed digits: 1.71)",
     "problem_expr": "2*x + 1 + 5/x**2",
     "answer_expr": "5**(Rational(1,3))"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 6.12992784003, printed 6.14 (half-unit 0.005; correctly rounded at the printed digits: 6.13)",
     "problem_expr": "2*x + 1 + 5/x**2",
     "answer_expr": "5**(Rational(1,3))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: 2*x + 1 + 5/x**2",
    "evaluate: 2*x + 1 + 5/x**2",
    "evaluate: 2*x + 1 + 5/x**2 at x=5**(Rational(1,3))"
   ],
   "shape": [
    "evaluate: N*x + N*x**N + 1",
    "extremum: N*x + N*x**N + 1"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph",
    "core.solve.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/5",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "119",
   "location": "Exercise X, problem 5",
   "problem_latex": "Find the maxima and minima of\n\\[\ny=\\frac{3}{x^2+x+1}.\n\\]",
   "markdown": "Find the maxima and minima of y=3x^2+x+1.",
   "answer_latex": [
    "Max: $x = -.5$, $y = 4$."
   ],
   "answer_markdown": [
    "Max: $x = -.5$, $y = 4$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "3/(x**2 + x + 1)",
     "answer_expr": "-Rational(1,2)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -0.5, printed -.5 (half-unit 0.05; correctly rounded at the printed digits: -0.5)",
     "problem_expr": "3/(x**2 + x + 1)",
     "answer_expr": "-Rational(1,2)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 4.0, printed 4 (half-unit 0.5; correctly rounded at the printed digits: 4.0)",
     "problem_expr": "3/(x**2 + x + 1)",
     "answer_expr": "-Rational(1,2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "extremum: 3/(x**2 + x + 1)",
    "evaluate: 3/(x**2 + x + 1)",
    "evaluate: 3/(x**2 + x + 1) at x=-Rational(1,2)"
   ],
   "shape": [
    "evaluate: N/(x + x**N + 1)",
    "extremum: N/(x + x**N + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/61",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 6,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "119",
   "location": "Exercise X, problem 61",
   "problem_latex": "Find the maxima and minima of\n\\[\ny=\\frac{5x}{2+x^2}.\n\\]",
   "markdown": "Find the maxima and minima of y=5x2+x^2.",
   "answer_latex": [
    "Max.: $x = 1.414$, $y = 1.7675$.  \\\\\nMin.: $x = -1.414$, $y = 1.7675$."
   ],
   "answer_markdown": [
    "Max.: $x = 1.414$, $y = 1.7675$. Min.: $x = -1.414$, $y = 1.7675$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "5*x/(2 + x**2)",
     "answer_expr": "sqrt(2)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 1.41421356237, printed 1.414 (half-unit 0.0005; correctly rounded at the printed digits: 1.414)",
     "problem_expr": "5*x/(2 + x**2)",
     "answer_expr": "sqrt(2)"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 1.76776695297, printed 1.7675 (half-unit 5.0e-5; correctly rounded at the printed digits: 1.7678)",
     "problem_expr": "5*x/(2 + x**2)",
     "answer_expr": "sqrt(2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: 5*x/(x**2 + 2)",
    "evaluate: 5*x/(x**2 + 2)",
    "evaluate: 5*x/(x**2 + 2) at x=sqrt(2)"
   ],
   "shape": [
    "evaluate: N*x/(N + x**N)",
    "extremum: N*x/(N + x**N)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-x/62"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/62",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 6,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "119",
   "location": "Exercise X, problem 62",
   "problem_latex": "Find the maxima and minima of\n\\[\ny=\\frac{5x}{2+x^2}.\n\\]",
   "markdown": "Find the maxima and minima of y=5x2+x^2.",
   "answer_latex": [
    "Max.: $x = 1.414$, $y = 1.7675$.  \\\\\nMin.: $x = -1.414$, $y = 1.7675$."
   ],
   "answer_markdown": [
    "Max.: $x = 1.414$, $y = 1.7675$. Min.: $x = -1.414$, $y = 1.7675$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "5*x/(2 + x**2)",
     "answer_expr": "-sqrt(2)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -1.41421356237, printed -1.414 (half-unit 0.0005; correctly rounded at the printed digits: -1.414)",
     "problem_expr": "5*x/(2 + x**2)",
     "answer_expr": "-sqrt(2)"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed -1.76776695297, printed 1.7675 (half-unit 5.0e-5; correctly rounded at the printed digits: -1.7678)",
     "problem_expr": "5*x/(2 + x**2)",
     "answer_expr": "-sqrt(2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: 5*x/(x**2 + 2)",
    "evaluate: 5*x/(x**2 + 2)",
    "evaluate: 5*x/(x**2 + 2) at x=-sqrt(2)"
   ],
   "shape": [
    "evaluate: N*x/(N + x**N)",
    "extremum: N*x/(N + x**N)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-x/61"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/71",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 7,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "119",
   "location": "Exercise X, problem 71",
   "problem_latex": "Find the maxima and minima of\n\\[\ny=\\frac{3x}{x^2-3} + \\frac{x}{2} + 5.\n\\]",
   "markdown": "Find the maxima and minima of y=3xx^2-3 + x2 + 5.",
   "answer_latex": [
    "Max.: $x = -3.565$, $y = 2.12$.  \\\\\nMin.: $x = +3.565$, $y = 7.88$."
   ],
   "answer_markdown": [
    "Max.: $x = -3.565$, $y = 2.12$. Min.: $x = +3.565$, $y = 7.88$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "3*x/(x**2 - 3) + x/2 + 5",
     "answer_expr": "-sqrt(6 + 3*sqrt(5))"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -3.5648567899, printed -3.565 (half-unit 0.0005; correctly rounded at the printed digits: -3.565)",
     "problem_expr": "3*x/(x**2 - 3) + x/2 + 5",
     "answer_expr": "-sqrt(6 + 3*sqrt(5))"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 2.11597027446, printed 2.12 (half-unit 0.005; correctly rounded at the printed digits: 2.12)",
     "problem_expr": "3*x/(x**2 - 3) + x/2 + 5",
     "answer_expr": "-sqrt(6 + 3*sqrt(5))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "extremum: x/2 + 3*x/(x**2 - 3) + 5",
    "evaluate: x/2 + 3*x/(x**2 - 3) + 5",
    "evaluate: x/2 + 3*x/(x**2 - 3) + 5 at x=-sqrt(6 + 3*sqrt(5))"
   ],
   "shape": [
    "evaluate: N*x + N*x/(N + x**N) + N",
    "extremum: N*x + N*x/(N + x**N) + N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-x/72"
   ],
   "needs": [
    "cas.cancel",
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph",
    "core.solve.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/72",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 7,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "119",
   "location": "Exercise X, problem 72",
   "problem_latex": "Find the maxima and minima of\n\\[\ny=\\frac{3x}{x^2-3} + \\frac{x}{2} + 5.\n\\]",
   "markdown": "Find the maxima and minima of y=3xx^2-3 + x2 + 5.",
   "answer_latex": [
    "Max.: $x = -3.565$, $y = 2.12$.  \\\\\nMin.: $x = +3.565$, $y = 7.88$."
   ],
   "answer_markdown": [
    "Max.: $x = -3.565$, $y = 2.12$. Min.: $x = +3.565$, $y = 7.88$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "3*x/(x**2 - 3) + x/2 + 5",
     "answer_expr": "sqrt(6 + 3*sqrt(5))"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 3.5648567899, printed +3.565 (half-unit 0.0005; correctly rounded at the printed digits: 3.565)",
     "problem_expr": "3*x/(x**2 - 3) + x/2 + 5",
     "answer_expr": "sqrt(6 + 3*sqrt(5))"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 7.88402972554, printed 7.88 (half-unit 0.005; correctly rounded at the printed digits: 7.88)",
     "problem_expr": "3*x/(x**2 - 3) + x/2 + 5",
     "answer_expr": "sqrt(6 + 3*sqrt(5))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "extremum: x/2 + 3*x/(x**2 - 3) + 5",
    "evaluate: x/2 + 3*x/(x**2 - 3) + 5",
    "evaluate: x/2 + 3*x/(x**2 - 3) + 5 at x=sqrt(6 + 3*sqrt(5))"
   ],
   "shape": [
    "evaluate: N*x + N*x/(N + x**N) + N",
    "extremum: N*x + N*x/(N + x**N) + N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-x/71"
   ],
   "needs": [
    "cas.cancel",
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph",
    "core.solve.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/8",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "119",
   "location": "Exercise X, problem 8",
   "problem_latex": "Divide a number~$N$ into two parts in such a\nway that three times the square of one part plus\ntwice the square of the other part shall be a\nminimum.",
   "markdown": "Divide a number $N$ into two parts in such a way that three times the square of one part plus twice the square of the other part shall be a minimum.",
   "answer_latex": [
    "$0.4N$, $0.6N$."
   ],
   "answer_markdown": [
    "$0.4N$, $0.6N$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "3*x**2 + 2*(N - x)**2",
     "answer_expr": "2*N/5"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "extremum: 3*x**2 + 2*(a - x)**2"
   ],
   "shape": [
    "extremum: N*x**N + N*(a - x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.solve.poly",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-x/9",
   "set": "thompson-calculus-made-easy-1914/ex-x",
   "number": 9,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "119",
   "location": "Exercise X, problem 9",
   "problem_latex": "The efficiency~$u$ of an electric generator at\ndifferent values of output~$x$ is expressed by the\ngeneral equation:\n\\[\nu=\\frac{x}{a+bx+cx^2};\n\\]\nwhere $a$ is a constant depending chiefly on the energy\nlosses in the iron and $c$~a constant depending chiefly\non the resistance of the copper parts. Find an expression\nfor that value of the output at which the\nefficiency will be a maximum.",
   "markdown": "The efficiency $u$ of an electric generator at different values of output $x$ is expressed by the general equation: u=xa+bx+cx^2; where $a$ is a constant depending chiefly on the energy losses in the iron and $c$ a constant depending chiefly on the resistance of the copper parts. Find an expression for that value of the output at which the efficiency will be a maximum.",
   "answer_latex": [
    "$x = \\sqrt{\\dfrac{a}{c}}$."
   ],
   "answer_markdown": [
    "$x = \\sqrt{\\dfrac{a}{c}}$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x/(a + b*x + c*x**2)",
     "answer_expr": "sqrt(a/c)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: x/(a + b*x + c*x**2)"
   ],
   "shape": [
    "extremum: x/(a + b*x + c*x**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.cancel",
    "cas.derive",
    "cas.solve.poly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/1",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "130",
   "location": "Exercise XI, problem 1",
   "problem_latex": "$\\dfrac{3x + 5}{(x - 3)(x + 4)}$.",
   "markdown": "$\\dfrac{3x + 5}{(x - 3)(x + 4)}$.",
   "answer_latex": [
    "$\\dfrac{2}{ x - 3} + \\dfrac{1}{ x + 4}$."
   ],
   "answer_markdown": [
    "$\\dfrac{2}{ x - 3} + \\dfrac{1}{ x + 4}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(3*x+5)/((x-3)*(x+4))",
     "answer_expr": "2/(x-3)+1/(x+4)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.partfrac",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/10",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 10,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 10",
   "problem_latex": "$\\dfrac{x^4 + 1}{x^3 + 1}$.",
   "markdown": "$\\dfrac{x^4 + 1}{x^3 + 1}$.",
   "answer_latex": [
    "$x + \\dfrac{2}{3(x + 1)} + \\dfrac{1 - 2x}{3(x^2 - x + 1)}$."
   ],
   "answer_markdown": [
    "$x + \\dfrac{2}{3(x + 1)} + \\dfrac{1 - 2x}{3(x^2 - x + 1)}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(x**4+1)/(x**3+1)",
     "answer_expr": "x+2/(3*(x+1))+(1-2*x)/(3*(x**2-x+1))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.factor",
    "cas.partfrac",
    "cas.pdiv",
    "core.linsys"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/11",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 11,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 11",
   "problem_latex": "$\\dfrac{5x^2 + 6x + 4}{(x +1)(x^2 + x + 1)}$.",
   "markdown": "$\\dfrac{5x^2 + 6x + 4}{(x +1)(x^2 + x + 1)}$.",
   "answer_latex": [
    "$\\dfrac{3}{(x + 1)} + \\dfrac{2x + 1}{x^2 + x + 1}$."
   ],
   "answer_markdown": [
    "$\\dfrac{3}{(x + 1)} + \\dfrac{2x + 1}{x^2 + x + 1}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(5*x**2+6*x+4)/((x+1)*(x**2+x+1))",
     "answer_expr": "3/(x+1)+(2*x+1)/(x**2+x+1)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.partfrac",
    "core.linsys"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/12",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 12,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 12",
   "problem_latex": "$\\dfrac{x}{(x - 1)(x - 2)^2}$.",
   "markdown": "$\\dfrac{x}{(x - 1)(x - 2)^2}$.",
   "answer_latex": [
    "$\\dfrac{1}{ x - 1} - \\dfrac{1}{ x - 2} + \\dfrac{2}{(x - 2)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{ x - 1} - \\dfrac{1}{ x - 2} + \\dfrac{2}{(x - 2)^2}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "x/((x-1)*(x-2)**2)",
     "answer_expr": "1/(x-1)-1/(x-2)+2/(x-2)**2"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.partfrac",
    "core.linsys"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/13",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 13,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 13",
   "problem_latex": "$\\dfrac{x}{(x^2 - 1)(x + 1)}$.",
   "markdown": "$\\dfrac{x}{(x^2 - 1)(x + 1)}$.",
   "answer_latex": [
    "$\\dfrac{1}{4(x - 1)} - \\dfrac{1}{4(x + 1)} + \\dfrac{1}{2(x + 1)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{4(x - 1)} - \\dfrac{1}{4(x + 1)} + \\dfrac{1}{2(x + 1)^2}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "x/((x**2-1)*(x+1))",
     "answer_expr": "1/(4*(x-1))-1/(4*(x+1))+1/(2*(x+1)**2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.factor",
    "cas.partfrac",
    "core.linsys"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/14",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 14,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 14",
   "problem_latex": "$\\dfrac{x + 3}{ (x +2)^2(x - 1)}$.",
   "markdown": "$\\dfrac{x + 3}{ (x +2)^2(x - 1)}$.",
   "answer_latex": [
    "$\\dfrac{4}{9(x - 1)} - \\dfrac{4}{9(x + 2)} - \\dfrac{1}{3(x + 2)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{4}{9(x - 1)} - \\dfrac{4}{9(x + 2)} - \\dfrac{1}{3(x + 2)^2}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(x+3)/((x+2)**2*(x-1))",
     "answer_expr": "4/(9*(x-1))-4/(9*(x+2))-1/(3*(x+2)**2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.partfrac",
    "core.linsys"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/15",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 15,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 15",
   "problem_latex": "$\\dfrac{3x^2 + 2x + 1}{(x + 2)(x^2 + x + 1)^2}$.",
   "markdown": "$\\dfrac{3x^2 + 2x + 1}{(x + 2)(x^2 + x + 1)^2}$.",
   "answer_latex": [
    "$\\dfrac{1}{ x + 2} - \\dfrac{x - 1}{ x^2 + x + 1} - \\dfrac{1}{(x^2 + x + 1)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{ x + 2} - \\dfrac{x - 1}{ x^2 + x + 1} - \\dfrac{1}{(x^2 + x + 1)^2}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(3*x**2+2*x+1)/((x+2)*(x**2+x+1)**2)",
     "answer_expr": "1/(x+2)-(x-1)/(x**2+x+1)-1/(x**2+x+1)**2"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.expand",
    "cas.partfrac",
    "core.linsys"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/16",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 16,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 16",
   "problem_latex": "$\\dfrac{5x^2 + 8x - 12}{(x + 4)^3}$.",
   "markdown": "$\\dfrac{5x^2 + 8x - 12}{(x + 4)^3}$.",
   "answer_latex": [
    "$\\dfrac{5}{ x + 4} -\\dfrac{32}{(x + 4)^2} + \\dfrac{36}{(x + 4)^3}$."
   ],
   "answer_markdown": [
    "$\\dfrac{5}{ x + 4} -\\dfrac{32}{(x + 4)^2} + \\dfrac{36}{(x + 4)^3}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(5*x**2+8*x-12)/(x+4)**3",
     "answer_expr": "5/(x+4)-32/(x+4)**2+36/(x+4)**3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.expand",
    "cas.partfrac",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/17",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 17,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 17",
   "problem_latex": "$\\dfrac{7x^2 + 9x - 1}{(3x - 2)^4}$.",
   "markdown": "$\\dfrac{7x^2 + 9x - 1}{(3x - 2)^4}$.",
   "answer_latex": [
    "$\\dfrac{7}{9(3x - 2)^2} + \\dfrac{55}{9(3x - 2)^3} + \\dfrac{73}{9(3x - 2)^4}$."
   ],
   "answer_markdown": [
    "$\\dfrac{7}{9(3x - 2)^2} + \\dfrac{55}{9(3x - 2)^3} + \\dfrac{73}{9(3x - 2)^4}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(7*x**2+9*x-1)/(3*x-2)**4",
     "answer_expr": "7/(9*(3*x-2)**2)+55/(9*(3*x-2)**3)+73/(9*(3*x-2)**4)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.expand",
    "cas.partfrac",
    "cas.subst",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/18",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 18,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 18",
   "problem_latex": "$\\dfrac{x^2}{(x^3 - 8)(x - 2)}$.",
   "markdown": "$\\dfrac{x^2}{(x^3 - 8)(x - 2)}$.",
   "answer_latex": [
    "$\\dfrac{1}{6(x - 2)} + \\dfrac{1}{3(x - 2)^2} - \\dfrac{x}{6(x^2 + 2x + 4)}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{6(x - 2)} + \\dfrac{1}{3(x - 2)^2} - \\dfrac{x}{6(x^2 + 2x + 4)}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "x**2/((x**3-8)*(x-2))",
     "answer_expr": "1/(6*(x-2))+1/(3*(x-2)**2)-x/(6*(x**2+2*x+4))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.factor",
    "cas.partfrac",
    "core.linsys"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/2",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "130",
   "location": "Exercise XI, problem 2",
   "problem_latex": "$\\dfrac{3x - 4}{(x - 1)(x - 2)}$.",
   "markdown": "$\\dfrac{3x - 4}{(x - 1)(x - 2)}$.",
   "answer_latex": [
    "$\\dfrac{1}{ x - 1} + \\dfrac{2}{ x - 2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{ x - 1} + \\dfrac{2}{ x - 2}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(3*x-4)/((x-1)*(x-2))",
     "answer_expr": "1/(x-1)+2/(x-2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.partfrac",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/3",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "130",
   "location": "Exercise XI, problem 3",
   "problem_latex": "$\\dfrac{3x + 5}{x^2 + x - 12}$.",
   "markdown": "$\\dfrac{3x + 5}{x^2 + x - 12}$.",
   "answer_latex": [
    "$\\dfrac{2}{ x - 3} + \\dfrac{1}{ x + 4}$."
   ],
   "answer_markdown": [
    "$\\dfrac{2}{ x - 3} + \\dfrac{1}{ x + 4}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(3*x+5)/(x**2+x-12)",
     "answer_expr": "2/(x-3)+1/(x+4)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.factor",
    "cas.partfrac",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/4",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "130",
   "location": "Exercise XI, problem 4",
   "problem_latex": "$\\dfrac{x + 1}{x^2 - 7x + 12}$.",
   "markdown": "$\\dfrac{x + 1}{x^2 - 7x + 12}$.",
   "answer_latex": [
    "$\\dfrac{5}{ x - 4} - \\dfrac{4}{ x - 3}$."
   ],
   "answer_markdown": [
    "$\\dfrac{5}{ x - 4} - \\dfrac{4}{ x - 3}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(x+1)/(x**2-7*x+12)",
     "answer_expr": "5/(x-4)-4/(x-3)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.factor",
    "cas.partfrac",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/5",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "130",
   "location": "Exercise XI, problem 5",
   "problem_latex": "$\\dfrac{x - 8}{(2x + 3)(3x - 2)}$.",
   "markdown": "$\\dfrac{x - 8}{(2x + 3)(3x - 2)}$.",
   "answer_latex": [
    "$\\dfrac{19}{13(2x + 3)} - \\dfrac{22}{13(3x - 2)}$."
   ],
   "answer_markdown": [
    "$\\dfrac{19}{13(2x + 3)} - \\dfrac{22}{13(3x - 2)}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(x-8)/((2*x+3)*(3*x-2))",
     "answer_expr": "19/(13*(2*x+3))-22/(13*(3*x-2))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.partfrac",
    "cas.subst",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/6",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "130",
   "location": "Exercise XI, problem 6",
   "problem_latex": "$\\dfrac{x^2 - 13x + 26}{(x - 2)(x - 3)(x - 4)}$.",
   "markdown": "$\\dfrac{x^2 - 13x + 26}{(x - 2)(x - 3)(x - 4)}$.",
   "answer_latex": [
    "$\\dfrac{2}{ x - 2} + \\dfrac{4}{ x - 3} - \\dfrac{5}{ x - 4}$."
   ],
   "answer_markdown": [
    "$\\dfrac{2}{ x - 2} + \\dfrac{4}{ x - 3} - \\dfrac{5}{ x - 4}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(x**2-13*x+26)/((x-2)*(x-3)*(x-4))",
     "answer_expr": "2/(x-2)+4/(x-3)-5/(x-4)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.partfrac",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/7",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 7",
   "problem_latex": "$\\dfrac{x^2 - 3x + 1}{(x - 1)(x + 2)(x - 3)}$.",
   "markdown": "$\\dfrac{x^2 - 3x + 1}{(x - 1)(x + 2)(x - 3)}$.",
   "answer_latex": [
    "$\\dfrac{1}{6(x - 1)} + \\dfrac{11}{15(x + 2)} + \\dfrac{1}{10(x - 3)}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{6(x - 1)} + \\dfrac{11}{15(x + 2)} + \\dfrac{1}{10(x - 3)}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(x**2-3*x+1)/((x-1)*(x+2)*(x-3))",
     "answer_expr": "1/(6*(x-1))+11/(15*(x+2))+1/(10*(x-3))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.partfrac",
    "cas.subst",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/8",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 8",
   "problem_latex": "$\\dfrac{5x^2 + 7x + 1}{(2x + 1)(3x - 2)(3x + 1)}$.",
   "markdown": "$\\dfrac{5x^2 + 7x + 1}{(2x + 1)(3x - 2)(3x + 1)}$.",
   "answer_latex": [
    "$\\dfrac{7}{9(3x + 1)} + \\dfrac{71}{63(3x - 2)} - \\dfrac{5}{7(2x + 1)}$."
   ],
   "answer_markdown": [
    "$\\dfrac{7}{9(3x + 1)} + \\dfrac{71}{63(3x - 2)} - \\dfrac{5}{7(2x + 1)}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "(5*x**2+7*x+1)/((2*x+1)*(3*x-2)*(3*x+1))",
     "answer_expr": "7/(9*(3*x+1))+71/(63*(3*x-2))-5/(7*(2*x+1))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.partfrac",
    "cas.subst",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xi/9",
   "set": "thompson-calculus-made-easy-1914/ex-xi",
   "number": 9,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "131",
   "location": "Exercise XI, problem 9",
   "problem_latex": "$\\dfrac{x^2}{x^3 - 1}$.",
   "markdown": "$\\dfrac{x^2}{x^3 - 1}$.",
   "answer_latex": [
    "$\\dfrac{1}{3(x - 1)} + \\dfrac{2x + 1}{3(x^2 + x + 1)}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{3(x - 1)} + \\dfrac{2x + 1}{3(x^2 + x + 1)}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "x**2/(x**3-1)",
     "answer_expr": "1/(3*(x-1))+(2*x+1)/(3*(x**2+x+1))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.factor",
    "cas.partfrac",
    "core.linsys"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/1",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "153",
   "location": "Exercise XII, problem 1",
   "problem_latex": "Differentiate $y=b(\\epsilon^{ax} -\\epsilon^{-ax})$.",
   "markdown": "Differentiate $y=b(\\epsilon^{ax} -\\epsilon^{-ax})$.",
   "answer_latex": [
    "$ab(\\epsilon^{ax} + \\epsilon^{-ax})$."
   ],
   "answer_markdown": [
    "$ab(\\epsilon^{ax} + \\epsilon^{-ax})$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "b*(exp(a*x) - exp(-a*x))",
     "answer_expr": "a*b*(exp(a*x) + exp(-a*x))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: b*(exp(a*x) - exp(-a*x))"
   ],
   "shape": [
    "differentiate: b*(exp(a*x) - exp(-a*x))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/10",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 10,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 10",
   "problem_latex": "$y=(3x^2-1)(\\sqrt{x}+1)$.",
   "markdown": "$y=(3x^2-1)(\\sqrt{x}+1)$.",
   "answer_latex": [
    "$\\left(\\dfrac{6x}{3x^2-1} + \\dfrac{1}{2\\left(\\sqrt x + x\\right)}\\right) \\left(3x^2-1\\right)\\left(\\sqrt x + 1\\right)$."
   ],
   "answer_markdown": [
    "$\\left(\\dfrac{6x}{3x^2-1} + \\dfrac{1}{2\\left(\\sqrt x + x\\right)}\\right) \\left(3x^2-1\\right)\\left(\\sqrt x + 1\\right)$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(3*x**2-1)*(sqrt(x)+1)",
     "answer_expr": "(6*x/(3*x**2-1) + 1/(2*(sqrt(x)+x)))*(3*x**2-1)*(sqrt(x)+1)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (sqrt(x) + 1)*(3*x**2 - 1)"
   ],
   "shape": [
    "differentiate: (x**N + 1)*(N*x**N - 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.factor",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/11",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 11,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 11",
   "problem_latex": "$y=\\dfrac{\\log_\\epsilon(x+3)}{x+3}$.",
   "markdown": "$y=\\dfrac{\\log_\\epsilon(x+3)}{x+3}$.",
   "answer_latex": [
    "$\\dfrac{1 - \\log_\\epsilon \\left(x + 3\\right)}{\\left(x + 3\\right)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1 - \\log_\\epsilon \\left(x + 3\\right)}{\\left(x + 3\\right)^2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "log(x+3)/(x+3)",
     "answer_expr": "(1 - log(x+3))/(x+3)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: log(x + 3)/(x + 3)"
   ],
   "shape": [
    "differentiate: log(N + x)/(N + x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/12",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 12,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 12",
   "problem_latex": "$y=a^x × x^a$.",
   "markdown": "$y=a^x × x^a$.",
   "answer_latex": [
    "$a^x\\left(ax^{a-1} + x^a \\log_\\epsilon a\\right)$."
   ],
   "answer_markdown": [
    "$a^x\\left(ax^{a-1} + x^a \\log_\\epsilon a\\right)$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a**x*x**a",
     "answer_expr": "a**x*(a*x**(a-1) + x**a*log(a))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a**x*x**a"
   ],
   "shape": [
    "differentiate: a**x*x**a"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/13",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 13,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 13",
   "problem_latex": "It was shown by Lord Kelvin that the speed of\nsignalling through a submarine cable depends on the\nvalue of the ratio of the external diameter of the core\nto the diameter of the enclosed copper wire. If this\nratio is called~$y$, then the number of signals~$s$ that can\nbe sent per minute can be expressed by the formula\n\\[\ns=ay^2 \\log_\\epsilon \\frac{1}{y};\n\\]\nwhere $a$ is a constant depending on the length and\nthe quality of the materials. Show that if these are\ngiven, $s$~will be a maximum if $y=1 ÷ \\sqrt{\\epsilon}$.",
   "markdown": "It was shown by Lord Kelvin that the speed of signalling through a submarine cable depends on the value of the ratio of the external diameter of the core to the diameter of the enclosed copper wire. If this ratio is called $y$, then the number of signals $s$ that can be sent per minute can be expressed by the formula s=ay^2 _1y; where $a$ is a constant depending on the length and the quality of the materials. Show that if these are given, $s$ will be a maximum if $y=1 ÷ \\sqrt{\\epsilon}$.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "a*y**2*log(1/y)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.nonpoly",
    "core.graph",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/14",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 14,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 14",
   "problem_latex": "Find the maximum or minimum of\n\\[\ny=x^3-\\log_\\epsilon x.\n\\]",
   "markdown": "Find the maximum or minimum of y=x^3-_x.",
   "answer_latex": [
    "Min.: $y = 0.7$ for $x = 0.694$."
   ],
   "answer_markdown": [
    "Min.: $y = 0.7$ for $x = 0.694$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "x**3 - log(x)",
     "answer_expr": null
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.699538702475, printed 0.7 (half-unit 0.05; correctly rounded at the printed digits: 0.7)",
     "problem_expr": "x**3 - log(x)",
     "answer_expr": "0.7"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING",
     "PASS"
    ]
   },
   "form": [
    "evaluate: x**3 - log(x) at x=0.694"
   ],
   "shape": [
    "evaluate: x**N - log(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst",
    "core.arith",
    "core.graph",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/15",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 15,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 15",
   "problem_latex": "Differentiate $y=\\log_\\epsilon(ax\\epsilon^x)$.",
   "markdown": "Differentiate $y=\\log_\\epsilon(ax\\epsilon^x)$.",
   "answer_latex": [
    "$\\dfrac{1 + x}{x}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1 + x}{x}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "log(a*x*exp(x))",
     "answer_expr": "(1 + x)/x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: log(a*x*exp(x))"
   ],
   "shape": [
    "differentiate: log(a*x*exp(x))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/16",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 16,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 16",
   "problem_latex": "Differentiate $y=(\\log_\\epsilon ax)^3$.",
   "markdown": "Differentiate $y=(\\log_\\epsilon ax)^3$.",
   "answer_latex": [
    "$\\dfrac{3}{x} (\\log_\\epsilon ax)^2$."
   ],
   "answer_markdown": [
    "$\\dfrac{3}{x} (\\log_\\epsilon ax)^2$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "log(a*x)**3",
     "answer_expr": "3/x*log(a*x)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: log(a*x)**3"
   ],
   "shape": [
    "differentiate: log(a*x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/2",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "153",
   "location": "Exercise XII, problem 2",
   "problem_latex": "Find the differential coefficient with respect to~$t$\nof the expression $u=at^2+2\\log_\\epsilon t$.",
   "markdown": "Find the differential coefficient with respect to $t$ of the expression $u=at^2+2\\log_\\epsilon t$.",
   "answer_latex": [
    "$2at + \\dfrac{2}{t}$."
   ],
   "answer_markdown": [
    "$2at + \\dfrac{2}{t}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a*t**2 + 2*log(t)",
     "answer_expr": "2*a*t + 2/t"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a*x**2 + 2*log(x)"
   ],
   "shape": [
    "differentiate: N*log(x) + a*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/3",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "153",
   "location": "Exercise XII, problem 3",
   "problem_latex": "If $y=n^t$, find $\\dfrac{d(\\log_\\epsilon y)}{dt}$.",
   "markdown": "If $y=n^t$, find $\\dfrac{d(\\log_\\epsilon y)}{dt}$.",
   "answer_latex": [
    "$\\log_\\epsilon n$."
   ],
   "answer_markdown": [
    "$\\log_\\epsilon n$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "log(n**t)",
     "answer_expr": "log(n)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: log(a**x)"
   ],
   "shape": [
    "differentiate: log(a**x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/4",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "153",
   "location": "Exercise XII, problem 4",
   "problem_latex": "Show that if $y=\\dfrac{1}{b}·\\dfrac{a^{bx}}{\\log_\\epsilon a}$,\\quad $\\dfrac{dy}{dx}=a^{bx}$.",
   "markdown": "Show that if $y=\\dfrac{1}{b}·\\dfrac{a^{bx}}{\\log_\\epsilon a}$, $\\dfrac{dy}{dx}=a^{bx}$.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "(1/b)*a**(b*x)/log(a)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [
    "differentiate: a**(b*x)/(b*log(a))"
   ],
   "shape": [
    "differentiate: a**(b*x)/(b*log(a))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/5",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "153",
   "location": "Exercise XII, problem 5",
   "problem_latex": "If $w=pn^v$, find $\\dfrac{dw}{dv}$.",
   "markdown": "If $w=pn^v$, find $\\dfrac{dw}{dv}$.",
   "answer_latex": [
    "$npv^{n-1}$."
   ],
   "answer_markdown": [
    "$npv^{n-1}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": [
      "3.8659504910179441807",
      "6.9563727941624194329"
     ],
     "problem_expr": "p*n**v",
     "answer_expr": "n*p*v**(n-1)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "differentiate: a**x*b"
   ],
   "shape": [
    "differentiate: a**x*b"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/6",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 6",
   "problem_latex": "$y=\\log_\\epsilon x^n$.",
   "markdown": "$y=\\log_\\epsilon x^n$.",
   "answer_latex": [
    "$\\dfrac{n}{x}$."
   ],
   "answer_markdown": [
    "$\\dfrac{n}{x}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "log(x**n)",
     "answer_expr": "n/x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: log(x**a)"
   ],
   "shape": [
    "differentiate: log(x**a)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/7",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 7",
   "problem_latex": "$y=3\\epsilon^{-\\efrac{x}{x-1}}$.",
   "markdown": "$y=3\\epsilon^{-\\efrac{x}{x-1}}$.",
   "answer_latex": [
    "$\\dfrac{3\\epsilon^{- \\frac{x}{x-1}}}{(x - 1)^2}$."
   ],
   "answer_markdown": [
    "$\\dfrac{3\\epsilon^{- \\frac{x}{x-1}}}{(x - 1)^2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "3*exp(-x/(x-1))",
     "answer_expr": "3*exp(-x/(x-1))/(x-1)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 3*exp(-x/(x - 1))"
   ],
   "shape": [
    "differentiate: N*exp(-x/(x - 1))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/8",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 8",
   "problem_latex": "$y=(3x^2+1)\\epsilon^{-5x}$.",
   "markdown": "$y=(3x^2+1)\\epsilon^{-5x}$.",
   "answer_latex": [
    "$6x \\epsilon^{-5x} - 5(3x^2 + 1)\\epsilon^{-5x}$."
   ],
   "answer_markdown": [
    "$6x \\epsilon^{-5x} - 5(3x^2 + 1)\\epsilon^{-5x}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(3*x**2+1)*exp(-5*x)",
     "answer_expr": "6*x*exp(-5*x) - 5*(3*x**2+1)*exp(-5*x)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: (3*x**2 + 1)*exp(-5*x)"
   ],
   "shape": [
    "differentiate: (N*x**N + 1)*exp(N*x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xii/9",
   "set": "thompson-calculus-made-easy-1914/ex-xii",
   "number": 9,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "154",
   "location": "Exercise XII, problem 9",
   "problem_latex": "$y=\\log_\\epsilon(x^a+a)$.",
   "markdown": "$y=\\log_\\epsilon(x^a+a)$.",
   "answer_latex": [
    "$\\dfrac{ax^{a-1}}{x^a + a}$."
   ],
   "answer_markdown": [
    "$\\dfrac{ax^{a-1}}{x^a + a}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "log(x**a + a)",
     "answer_expr": "a*x**(a-1)/(x**a + a)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: log(a + x**a)"
   ],
   "shape": [
    "differentiate: log(a + x**a)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/1",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "162",
   "location": "Exercise XIII, problem 1",
   "problem_latex": "Draw the curve $y = b \\epsilon^{-\\efrac{t}{T}}$; where $b = 12$, $T = 8$,\nand $t$ is given various values from $0$~to~$20$.",
   "markdown": "Draw the curve $y = b \\epsilon^{-\\efrac{t}{T}}$; where $b = 12$, $T = 8$, and $t$ is given various values from $0$ to $20$.",
   "answer_latex": [
    "Let $\\dfrac{t}{T} = x$ ($\\therefore t = 8x$), and use the Table on \\Pageref[page]{littletable}."
   ],
   "answer_markdown": [
    "Let $\\dfrac{t}{T} = x$ ($\\therefore t = 8x$), and use the Table on [page]littletable."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "12*exp(-t/8)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.graph",
    "core.table"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/101",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 10,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "162",
   "location": "Exercise XIII, problem 101",
   "problem_latex": "The pressure~$p$ of the atmosphere at an altitude\n$h$~kilometres is given by $p=p_0 \\epsilon^{-kh}$; $p_0$~being the\npressure at sea-level ($760$~millimetres).\n\nThe pressures at $10$,~$20$ and~$50$ kilometres being\n$199.2$, $42.2$, $0.32$ respectively, find~$k$ in each case.\nUsing the mean value of~$k$, find the percentage error\nin each case.",
   "markdown": "The pressure $p$ of the atmosphere at an altitude $h$ kilometres is given by $p=p_0 \\epsilon^{-kh}$; $p_0$ being the pressure at sea-level ($760$ millimetres). The pressures at $10$, $20$ and $50$ kilometres being $199.2$, $42.2$, $0.32$ respectively, find $k$ in each case. Using the mean value of $k$, find the percentage error in each case.",
   "answer_latex": [
    "$0.133$, $0.145$, $0.155$, mean $0.144$; $-10.2$\\%, $-0.9$\\%, $+77.2$\\%."
   ],
   "answer_markdown": [
    "$0.133$, $0.145$, $0.155$, mean $0.144$; $-10.2$%, $-0.9$%, $+77.2$%."
   ],
   "checks": [
    {
     "task": "solve",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "Eq(Rational(1992,10), 760*exp(-10*k))",
     "answer_expr": "log(760/Rational(1992,10))/10"
    },
    {
     "task": "evaluate",
     "verdict": "PASS-LOOSE",
     "judge_why": "computed 0.133900908813, printed 0.133 (half-unit 0.0005; correctly rounded at the printed digits: 0.134): one unit off in the last printed place",
     "problem_expr": "log(760/199.2)/10",
     "answer_expr": "log(760/199.2)/10"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed -9.6058752016, printed -10.2 (half-unit 0.05; correctly rounded at the printed digits: -9.6)",
     "problem_expr": "100*(760*exp(-0.144*10) - 199.2)/199.2",
     "answer_expr": "100*(760*exp(-0.144*10) - 199.2)/199.2"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS-LOOSE",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "solve: Eq(996/5, 760*exp(-10*x))",
    "evaluate: log(950/249)/10",
    "evaluate: -100 + 95000*exp(-36/25)/249"
   ],
   "shape": [
    "evaluate: N*exp(N) + N",
    "evaluate: N*log(N)",
    "solve: Eq(N, N*exp(N*x))"
   ],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.eqn",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/102",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 10,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "164",
   "location": "Exercise XIII, problem 102",
   "problem_latex": "The pressure~$p$ of the atmosphere at an altitude\n$h$~kilometres is given by $p=p_0 \\epsilon^{-kh}$; $p_0$~being the\npressure at sea-level ($760$~millimetres).\n\nThe pressures at $10$,~$20$ and~$50$ kilometres being\n$199.2$, $42.2$, $0.32$ respectively, find~$k$ in each case.\nUsing the mean value of~$k$, find the percentage error\nin each case.",
   "markdown": "The pressure $p$ of the atmosphere at an altitude $h$ kilometres is given by $p=p_0 \\epsilon^{-kh}$; $p_0$ being the pressure at sea-level ($760$ millimetres). The pressures at $10$, $20$ and $50$ kilometres being $199.2$, $42.2$, $0.32$ respectively, find $k$ in each case. Using the mean value of $k$, find the percentage error in each case.",
   "answer_latex": [
    "$0.133$, $0.145$, $0.155$, mean $0.144$; $-10.2$\\%, $-0.9$\\%, $+77.2$\\%."
   ],
   "answer_markdown": [
    "$0.133$, $0.145$, $0.155$, mean $0.144$; $-10.2$%, $-0.9$%, $+77.2$%."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "log(760/42.2)/20"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.144544910612, printed 0.145 (half-unit 0.0005; correctly rounded at the printed digits: 0.145)",
     "problem_expr": "log(760/42.2)/20",
     "answer_expr": "log(760/42.2)/20"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 1.09578140744, printed -0.9 (half-unit 0.05; correctly rounded at the printed digits: 1.1)",
     "problem_expr": "100*(760*exp(-0.144*20) - 42.2)/42.2",
     "answer_expr": "100*(760*exp(-0.144*20) - 42.2)/42.2"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING",
     "PASS",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: log(3800/211)/20",
    "evaluate: -100 + 380000*exp(-72/25)/211"
   ],
   "shape": [
    "evaluate: N*exp(N) + N",
    "evaluate: N*log(N)"
   ],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.eqn",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/103",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 10,
   "part": "3",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "162",
   "location": "Exercise XIII, problem 103",
   "problem_latex": "The pressure~$p$ of the atmosphere at an altitude\n$h$~kilometres is given by $p=p_0 \\epsilon^{-kh}$; $p_0$~being the\npressure at sea-level ($760$~millimetres).\n\nThe pressures at $10$,~$20$ and~$50$ kilometres being\n$199.2$, $42.2$, $0.32$ respectively, find~$k$ in each case.\nUsing the mean value of~$k$, find the percentage error\nin each case.",
   "markdown": "The pressure $p$ of the atmosphere at an altitude $h$ kilometres is given by $p=p_0 \\epsilon^{-kh}$; $p_0$ being the pressure at sea-level ($760$ millimetres). The pressures at $10$, $20$ and $50$ kilometres being $199.2$, $42.2$, $0.32$ respectively, find $k$ in each case. Using the mean value of $k$, find the percentage error in each case.",
   "answer_latex": [
    "$0.133$, $0.145$, $0.155$, mean $0.144$; $-10.2$\\%, $-0.9$\\%, $+77.2$\\%."
   ],
   "answer_markdown": [
    "$0.133$, $0.145$, $0.155$, mean $0.144$; $-10.2$%, $-0.9$%, $+77.2$%."
   ],
   "checks": [
    {
     "task": "solve",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "Eq(Rational(32,100), 760*exp(-50*k))",
     "answer_expr": "log(760/Rational(32,100))/50"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.155455054329, printed 0.155 (half-unit 0.0005; correctly rounded at the printed digits: 0.155)",
     "problem_expr": "log(760/0.32)/50",
     "answer_expr": "log(760/0.32)/50"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 77.3141294895, printed +77.2 (half-unit 0.05; correctly rounded at the printed digits: 77.3)",
     "problem_expr": "100*(760*exp(-0.144*50) - 0.32)/0.32",
     "answer_expr": "100*(760*exp(-0.144*50) - 0.32)/0.32"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "solve: Eq(8/25, 760*exp(-50*x))",
    "evaluate: log(2375)/50",
    "evaluate: -100 + 237500*exp(-36/5)"
   ],
   "shape": [
    "evaluate: N*exp(N) + N",
    "evaluate: N*log(N)",
    "solve: Eq(N, N*exp(N*x))"
   ],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.eqn",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/104",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 10,
   "part": "4",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "164",
   "location": "Exercise XIII, problem 104",
   "problem_latex": "The pressure~$p$ of the atmosphere at an altitude\n$h$~kilometres is given by $p=p_0 \\epsilon^{-kh}$; $p_0$~being the\npressure at sea-level ($760$~millimetres).\n\nThe pressures at $10$,~$20$ and~$50$ kilometres being\n$199.2$, $42.2$, $0.32$ respectively, find~$k$ in each case.\nUsing the mean value of~$k$, find the percentage error\nin each case.",
   "markdown": "The pressure $p$ of the atmosphere at an altitude $h$ kilometres is given by $p=p_0 \\epsilon^{-kh}$; $p_0$ being the pressure at sea-level ($760$ millimetres). The pressures at $10$, $20$ and $50$ kilometres being $199.2$, $42.2$, $0.32$ respectively, find $k$ in each case. Using the mean value of $k$, find the percentage error in each case.",
   "answer_latex": [
    "$0.133$, $0.145$, $0.155$, mean $0.144$; $-10.2$\\%, $-0.9$\\%, $+77.2$\\%."
   ],
   "answer_markdown": [
    "$0.133$, $0.145$, $0.155$, mean $0.144$; $-10.2$%, $-0.9$%, $+77.2$%."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "(log(760/199.2)/10 + log(760/42.2)/20 + log(760/0.32)/50)/3"
    },
    {
     "task": "evaluate",
     "verdict": "PASS-LOOSE",
     "judge_why": "computed 0.144633624585, printed 0.144 (half-unit 0.0005; correctly rounded at the printed digits: 0.145): one unit off in the last printed place",
     "problem_expr": "(log(760/199.2)/10 + log(760/42.2)/20 + log(760/0.32)/50)/3",
     "answer_expr": "(log(760/199.2)/10 + log(760/42.2)/20 + log(760/0.32)/50)/3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING",
     "PASS-LOOSE"
    ]
   },
   "form": [
    "evaluate: log(950/249)/30 + log(3800/211)/60 + log(2375)/150"
   ],
   "shape": [
    "evaluate: 3*N*log(N)"
   ],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.log",
    "core.stat"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/11",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 11,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "164",
   "location": "Exercise XIII, problem 11",
   "problem_latex": "Find the minimum or maximum of $y = x^x$.",
   "markdown": "Find the minimum or maximum of $y = x^x$.",
   "answer_latex": [
    "Min.\\ for $x = \\dfrac{1}{\\epsilon}$."
   ],
   "answer_markdown": [
    "Min. for $x = \\dfrac{1}{\\epsilon}$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**x",
     "answer_expr": "exp(-1)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: x**x"
   ],
   "shape": [
    "extremum: x**x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.nonpoly",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/12",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 12,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "164",
   "location": "Exercise XIII, problem 12",
   "problem_latex": "Find the minimum or maximum of $y = x^{\\efrac{1}{x}}$.",
   "markdown": "Find the minimum or maximum of $y = x^{\\efrac{1}{x}}$.",
   "answer_latex": [
    "Max.\\ for $x = \\epsilon$."
   ],
   "answer_markdown": [
    "Max. for $x = \\epsilon$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**(1/x)",
     "answer_expr": "E"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: x**(1/x)"
   ],
   "shape": [
    "extremum: x**(1/x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.nonpoly",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/13",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 13,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "164",
   "location": "Exercise XIII, problem 13",
   "problem_latex": "Find the minimum or maximum of $y = xa^{\\efrac{1}{x}}$.",
   "markdown": "Find the minimum or maximum of $y = xa^{\\efrac{1}{x}}$.",
   "answer_latex": [
    "Min.\\ for $x = \\log_\\epsilon a$."
   ],
   "answer_markdown": [
    "Min. for $x = \\log_\\epsilon a$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "f''(x0) = -6.0547511 contradicts 'minimum'",
     "problem_expr": "x*a**(1/x)",
     "answer_expr": "log(a)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: a**(1/x)*x"
   ],
   "shape": [
    "extremum: a**(1/x)*x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.nonpoly",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/2",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "162",
   "location": "Exercise XIII, problem 2",
   "problem_latex": "If a hot body cools so that in $24$~minutes its\nexcess of temperature has fallen to half the initial\namount, deduce the time-constant, and find how long\nit will be in cooling down to $1$~per~cent.\\ of the original\nexcess.",
   "markdown": "If a hot body cools so that in $24$ minutes its excess of temperature has fallen to half the initial amount, deduce the time-constant, and find how long it will be in cooling down to $1$ per cent. of the original excess.",
   "answer_latex": [
    "$T = 34.627$; $159.46$ minutes."
   ],
   "answer_markdown": [
    "$T = 34.627$; $159.46$ minutes."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": null
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 34.6246809813, printed 34.627 (half-unit 0.0005; correctly rounded at the printed digits: 34.625)",
     "problem_expr": "24/log(2)",
     "answer_expr": "24/log(2)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS-LOOSE",
     "judge_why": "computed 159.452548555, printed 159.46 (half-unit 0.005; correctly rounded at the printed digits: 159.45): one unit off in the last printed place",
     "problem_expr": "24*log(100)/log(2)",
     "answer_expr": "24*log(100)/log(2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING",
     "FLAG-MISMATCH",
     "PASS-LOOSE"
    ]
   },
   "form": [
    "evaluate: 24/log(2)",
    "evaluate: 24*log(100)/log(2)"
   ],
   "shape": [
    "evaluate: N",
    "evaluate: N/log(N)"
   ],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.eqn",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/3",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "163",
   "location": "Exercise XIII, problem 3",
   "problem_latex": "Plot the curve $y = 100(1-\\epsilon^{-2t})$.",
   "markdown": "Plot the curve $y = 100(1-\\epsilon^{-2t})$.",
   "answer_latex": [
    "Take $2t = x$; and use the Table on \\Pageref[page]{littletable}."
   ],
   "answer_markdown": [
    "Take $2t = x$; and use the Table on [page]littletable."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-EXTRACTION",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "100*(1 - exp(-2*t))",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-EXTRACTION"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.graph",
    "core.table"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/41",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 4,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": null,
   "location": "Exercise XIII, problem 41",
   "problem_latex": "The following equations give very similar curves:\n\\begin{align*}\n\\text{(i)}\\   y &= \\frac{ax}{x + b}; \\\\\n\\text{(ii)}\\  y &= a(1 - \\epsilon^{-\\efrac{x}{b}}); \\\\\n\\text{(iii)}\\ y &= \\frac{a}{90°} \\arctan \\left(\\frac{x}{b}\\right).\n\\end{align*}\n\nDraw all three curves, taking $a= 100$ millimetres;\n$b = 30$ millimetres.",
   "markdown": "The following equations give very similar curves: align* (i) y &= axx + b; (ii) y &= a(1 - ^-xb); (iii) y &= a90° (xb). align* Draw all three curves, taking $a= 100$ millimetres; $b = 30$ millimetres.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-EXTRACTION",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "a*x/(x + b)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-EXTRACTION"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "core.graph",
    "core.table"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/42",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 4,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": null,
   "location": "Exercise XIII, problem 42",
   "problem_latex": "The following equations give very similar curves:\n\\begin{align*}\n\\text{(i)}\\   y &= \\frac{ax}{x + b}; \\\\\n\\text{(ii)}\\  y &= a(1 - \\epsilon^{-\\efrac{x}{b}}); \\\\\n\\text{(iii)}\\ y &= \\frac{a}{90°} \\arctan \\left(\\frac{x}{b}\\right).\n\\end{align*}\n\nDraw all three curves, taking $a= 100$ millimetres;\n$b = 30$ millimetres.",
   "markdown": "The following equations give very similar curves: align* (i) y &= axx + b; (ii) y &= a(1 - ^-xb); (iii) y &= a90° (xb). align* Draw all three curves, taking $a= 100$ millimetres; $b = 30$ millimetres.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-EXTRACTION",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "a*(1 - exp(-x/b))",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-EXTRACTION"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "core.graph",
    "core.table"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/43",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 4,
   "part": "3",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": null,
   "location": "Exercise XIII, problem 43",
   "problem_latex": "The following equations give very similar curves:\n\\begin{align*}\n\\text{(i)}\\   y &= \\frac{ax}{x + b}; \\\\\n\\text{(ii)}\\  y &= a(1 - \\epsilon^{-\\efrac{x}{b}}); \\\\\n\\text{(iii)}\\ y &= \\frac{a}{90°} \\arctan \\left(\\frac{x}{b}\\right).\n\\end{align*}\n\nDraw all three curves, taking $a= 100$ millimetres;\n$b = 30$ millimetres.",
   "markdown": "The following equations give very similar curves: align* (i) y &= axx + b; (ii) y &= a(1 - ^-xb); (iii) y &= a90° (xb). align* Draw all three curves, taking $a= 100$ millimetres; $b = 30$ millimetres.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-EXTRACTION",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "a*atan(x/b)/(pi/2)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-EXTRACTION"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "core.graph",
    "core.table",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/5a",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 5,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "163",
   "location": "Exercise XIII, problem 5a",
   "problem_latex": "Find the differential coefficient of~$y$ with respect\nto~$x$, if\n\\[\n(\\textit{a})~y = x^x;\\quad\n(\\textit{b})~y = (\\epsilon^x)^x;\\quad\n(\\textit{c})~y = \\epsilon^{x^x}.\n\\]",
   "markdown": "Find the differential coefficient of $y$ with respect to $x$, if (*a*) y = x^x; (*b*) y = (^x)^x; (*c*) y = ^x^x.",
   "answer_latex": [
    "(\\textit{a}) $x^x \\left(1 + \\log_\\epsilon x\\right)$;\\quad\n(\\textit{b}) $2x(\\epsilon^x)^x$;\\quad\n(\\textit{c}) $\\epsilon^{x^x} × x^x \\left(1 + \\log_\\epsilon x\\right)$."
   ],
   "answer_markdown": [
    "(*a*) $x^x \\left(1 + \\log_\\epsilon x\\right)$; (*b*) $2x(\\epsilon^x)^x$; (*c*) $\\epsilon^{x^x} × x^x \\left(1 + \\log_\\epsilon x\\right)$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**x",
     "answer_expr": "x**x*(1 + log(x))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: x**x"
   ],
   "shape": [
    "differentiate: x**x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/5b",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 5,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "163",
   "location": "Exercise XIII, problem 5b",
   "problem_latex": "Find the differential coefficient of~$y$ with respect\nto~$x$, if\n\\[\n(\\textit{a})~y = x^x;\\quad\n(\\textit{b})~y = (\\epsilon^x)^x;\\quad\n(\\textit{c})~y = \\epsilon^{x^x}.\n\\]",
   "markdown": "Find the differential coefficient of $y$ with respect to $x$, if (*a*) y = x^x; (*b*) y = (^x)^x; (*c*) y = ^x^x.",
   "answer_latex": [
    "(\\textit{a}) $x^x \\left(1 + \\log_\\epsilon x\\right)$;\\quad\n(\\textit{b}) $2x(\\epsilon^x)^x$;\\quad\n(\\textit{c}) $\\epsilon^{x^x} × x^x \\left(1 + \\log_\\epsilon x\\right)$."
   ],
   "answer_markdown": [
    "(*a*) $x^x \\left(1 + \\log_\\epsilon x\\right)$; (*b*) $2x(\\epsilon^x)^x$; (*c*) $\\epsilon^{x^x} × x^x \\left(1 + \\log_\\epsilon x\\right)$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "exp(x)**x",
     "answer_expr": "2*x*exp(x)**x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: exp(x**2)"
   ],
   "shape": [
    "differentiate: exp(x**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/5c",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 5,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "163",
   "location": "Exercise XIII, problem 5c",
   "problem_latex": "Find the differential coefficient of~$y$ with respect\nto~$x$, if\n\\[\n(\\textit{a})~y = x^x;\\quad\n(\\textit{b})~y = (\\epsilon^x)^x;\\quad\n(\\textit{c})~y = \\epsilon^{x^x}.\n\\]",
   "markdown": "Find the differential coefficient of $y$ with respect to $x$, if (*a*) y = x^x; (*b*) y = (^x)^x; (*c*) y = ^x^x.",
   "answer_latex": [
    "(\\textit{a}) $x^x \\left(1 + \\log_\\epsilon x\\right)$;\\quad\n(\\textit{b}) $2x(\\epsilon^x)^x$;\\quad\n(\\textit{c}) $\\epsilon^{x^x} × x^x \\left(1 + \\log_\\epsilon x\\right)$."
   ],
   "answer_markdown": [
    "(*a*) $x^x \\left(1 + \\log_\\epsilon x\\right)$; (*b*) $2x(\\epsilon^x)^x$; (*c*) $\\epsilon^{x^x} × x^x \\left(1 + \\log_\\epsilon x\\right)$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "exp(x**x)",
     "answer_expr": "exp(x**x)*x**x*(1 + log(x))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: exp(x**x)"
   ],
   "shape": [
    "differentiate: exp(x**x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/6",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "162",
   "location": "Exercise XIII, problem 6",
   "problem_latex": "For ``Thorium~$A$,'' the value of~$\\lambda$ is~$5$; find the\n``mean life,'' that is, the time taken by the transformation\nof a quantity~$Q$ of ``Thorium~$A$'' equal to\nhalf the initial quantity~$Q_0$ in the expression\n\\[\nQ = Q_0 \\epsilon^{-\\lambda t};\n\\]\n$t$~being in seconds.",
   "markdown": "For “Thorium $A$,” the value of $\\lambda$ is $5$; find the “mean life,” that is, the time taken by the transformation of a quantity $Q$ of “Thorium $A$” equal to half the initial quantity $Q_0$ in the expression Q = Q_0 ^-t; $t$ being in seconds.",
   "answer_latex": [
    "$0.14$ second."
   ],
   "answer_markdown": [
    "$0.14$ second."
   ],
   "checks": [
    {
     "task": "solve",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "Eq(exp(-5*t), Rational(1, 2))",
     "answer_expr": "log(2)/5"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.138629436112, printed 0.14 (half-unit 0.005; correctly rounded at the printed digits: 0.14)",
     "problem_expr": "log(2)/5",
     "answer_expr": "log(2)/5"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "evaluate: log(2)/5",
    "solve: Eq(exp(-5*x), 1/2)"
   ],
   "shape": [
    "evaluate: N*log(N)",
    "solve: Eq(exp(N*x), N)"
   ],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.eqn",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/7a",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 7,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "163",
   "location": "Exercise XIII, problem 7a",
   "problem_latex": "A condenser of capacity $K = 4 × 10^{-6}$, charged\nto a potential $V_0 = 20$, is discharging through a resistance\nof $10,000$~ohms. Find the potential~$V$ after (\\textit{a})~$0.1$\nsecond; (\\textit{b})~$0.01$ second; assuming that the fall of\npotential follows the rule $V = V_0 \\epsilon^{-\\efrac{t}{KR}}$.",
   "markdown": "A condenser of capacity $K = 4 × 10^{-6}$, charged to a potential $V_0 = 20$, is discharging through a resistance of $10,000$ ohms. Find the potential $V$ after (*a*) $0.1$ second; (*b*) $0.01$ second; assuming that the fall of potential follows the rule $V = V_0 \\epsilon^{-\\efrac{t}{KR}}$.",
   "answer_latex": [
    "(\\textit{a}) $1.642$;\\quad (\\textit{b}) $15.58$."
   ],
   "answer_markdown": [
    "(*a*) $1.642$; (*b*) $15.58$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "V0*exp(-t/(K*R))",
     "answer_expr": null
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 1.64169997248, printed 1.642 (half-unit 0.0005; correctly rounded at the printed digits: 1.642)",
     "problem_expr": "V0*exp(-t/(K*R))",
     "answer_expr": "V0*exp(-t/(K*R))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING",
     "PASS"
    ]
   },
   "form": [
    "evaluate: c*exp(-x/(a*b)) at K=4e-6, R=10000, V0=20, t=0.1"
   ],
   "shape": [
    "evaluate: c*exp(-x/(a*b))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.subst",
    "core.arith",
    "core.const",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/7b",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 7,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "163",
   "location": "Exercise XIII, problem 7b",
   "problem_latex": "A condenser of capacity $K = 4 × 10^{-6}$, charged\nto a potential $V_0 = 20$, is discharging through a resistance\nof $10,000$~ohms. Find the potential~$V$ after (\\textit{a})~$0.1$\nsecond; (\\textit{b})~$0.01$ second; assuming that the fall of\npotential follows the rule $V = V_0 \\epsilon^{-\\efrac{t}{KR}}$.",
   "markdown": "A condenser of capacity $K = 4 × 10^{-6}$, charged to a potential $V_0 = 20$, is discharging through a resistance of $10,000$ ohms. Find the potential $V$ after (*a*) $0.1$ second; (*b*) $0.01$ second; assuming that the fall of potential follows the rule $V = V_0 \\epsilon^{-\\efrac{t}{KR}}$.",
   "answer_latex": [
    "(\\textit{a}) $1.642$;\\quad (\\textit{b}) $15.58$."
   ],
   "answer_markdown": [
    "(*a*) $1.642$; (*b*) $15.58$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "V0*exp(-t/(K*R))",
     "answer_expr": null
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 15.5760156614, printed 15.58 (half-unit 0.005; correctly rounded at the printed digits: 15.58)",
     "problem_expr": "V0*exp(-t/(K*R))",
     "answer_expr": "V0*exp(-t/(K*R))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING",
     "PASS"
    ]
   },
   "form": [
    "evaluate: c*exp(-x/(a*b)) at K=4e-6, R=10000, V0=20, t=0.01"
   ],
   "shape": [
    "evaluate: c*exp(-x/(a*b))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.subst",
    "core.arith",
    "core.const",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/81",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 8,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "162",
   "location": "Exercise XIII, problem 81",
   "problem_latex": "The charge~$Q$ of an electrified insulated metal\nsphere is reduced from $20$ to $16$~units in $10$~minutes.\nFind the coefficient~$\\mu$ of leakage, if $Q = Q_0 × \\epsilon^{-\\mu t}$; $Q_0$\nbeing the initial charge and $t$~being in seconds. Hence\nfind the time taken by half the charge to leak away.",
   "markdown": "The charge $Q$ of an electrified insulated metal sphere is reduced from $20$ to $16$ units in $10$ minutes. Find the coefficient $\\mu$ of leakage, if $Q = Q_0 × \\epsilon^{-\\mu t}$; $Q_0$ being the initial charge and $t$ being in seconds. Hence find the time taken by half the charge to leak away.",
   "answer_latex": [
    "$\\mu = 0.00037$",
    "$\\mu = 0.00037$, $31^m \\frac{1}{4}$. %[** Time units]"
   ],
   "answer_markdown": [
    "$\\mu = 0.00037$",
    "$\\mu = 0.00037$, $31^m \\frac{1}{4}$. %[** Time units]"
   ],
   "checks": [
    {
     "task": "solve",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "Eq(16, 20*exp(-mu*600))",
     "answer_expr": "log(Rational(5,4))/600"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.000371905918857, printed 0.00037 (half-unit 5.0e-6; correctly rounded at the printed digits: 0.00037)",
     "problem_expr": "log(Rational(20,16))/600",
     "answer_expr": "log(Rational(20,16))/600"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "solve: Eq(16, 20*exp(-600*x))",
    "evaluate: log(5/4)/600"
   ],
   "shape": [
    "evaluate: N*log(N)",
    "solve: Eq(N, N*exp(N*x))"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xiii/82"
   ],
   "needs": [
    "core.arith",
    "core.eqn",
    "core.log",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/82",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 8,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "162",
   "location": "Exercise XIII, problem 82",
   "problem_latex": "The charge~$Q$ of an electrified insulated metal\nsphere is reduced from $20$ to $16$~units in $10$~minutes.\nFind the coefficient~$\\mu$ of leakage, if $Q = Q_0 × \\epsilon^{-\\mu t}$; $Q_0$\nbeing the initial charge and $t$~being in seconds. Hence\nfind the time taken by half the charge to leak away.",
   "markdown": "The charge $Q$ of an electrified insulated metal sphere is reduced from $20$ to $16$ units in $10$ minutes. Find the coefficient $\\mu$ of leakage, if $Q = Q_0 × \\epsilon^{-\\mu t}$; $Q_0$ being the initial charge and $t$ being in seconds. Hence find the time taken by half the charge to leak away.",
   "answer_latex": [
    "$\\mu = 0.00037$, $31^m \\frac{1}{4}$. %[** Time units]"
   ],
   "answer_markdown": [
    "$\\mu = 0.00037$, $31^m \\frac{1}{4}$. %[** Time units]"
   ],
   "checks": [
    {
     "task": "solve",
     "verdict": "PASS-ERRATUM",
     "judge_why": "the corrected answer holds; the printed one is the book's misprint",
     "problem_expr": "Eq(16, 20*exp(-mu*600))",
     "answer_expr": "log(Rational(5,4))/600"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 31.0628371951, printed 31.25 (half-unit 0.005; correctly rounded at the printed digits: 31.06)",
     "problem_expr": "10*log(2)/log(Rational(5,4))",
     "answer_expr": "10*log(2)/log(Rational(5,4))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ERRATUM",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "solve: Eq(16, 20*exp(-600*x))",
    "evaluate: 10*log(2)/log(5/4)"
   ],
   "shape": [
    "evaluate: N",
    "solve: Eq(N, N*exp(N*x))"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xiii/81"
   ],
   "needs": [
    "core.arith",
    "core.eqn",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/91",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 9,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "164",
   "location": "Exercise XIII, problem 91",
   "problem_latex": "The damping on a telephone line can be ascertained\nfrom the relation $i = i_0 \\epsilon^{-\\beta l}$, where $i$~is the\nstrength, after $t$~seconds, of a telephonic current of\ninitial strength~$i_0$; $l$~is the length of the line in kilometres,\nand $\\beta$~is a constant. For the Franco-English\nsubmarine cable laid in 1910, $\\beta = 0.0114$. Find the\ndamping at the end of the cable ($40$~kilometres), and\nthe length along which $i$~is still $8$\\%~of the original\ncurrent (limiting value of very good audition).",
   "markdown": "The damping on a telephone line can be ascertained from the relation $i = i_0 \\epsilon^{-\\beta l}$, where $i$ is the strength, after $t$ seconds, of a telephonic current of initial strength $i_0$; $l$ is the length of the line in kilometres, and $\\beta$ is a constant. For the Franco-English submarine cable laid in 1910, $\\beta = 0.0114$. Find the damping at the end of the cable ($40$ kilometres), and the length along which $i$ is still $8$% of the original current (limiting value of very good audition).",
   "answer_latex": [
    "$i$ is $63.4$\\% of~$i_0$, $220$~kilometres."
   ],
   "answer_markdown": [
    "$i$ is $63.4$% of $i_0$, $220$ kilometres."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "100*exp(-Rational(114,10000)*40)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 63.3813837099, printed 63.4 (half-unit 0.05; correctly rounded at the printed digits: 63.4)",
     "problem_expr": "100*exp(-Rational(114,10000)*40)",
     "answer_expr": "100*exp(-Rational(114,10000)*40)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING",
     "PASS"
    ]
   },
   "form": [
    "evaluate: 100*exp(-57/125)"
   ],
   "shape": [
    "evaluate: N*exp(N)"
   ],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.const",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiii/92",
   "set": "thompson-calculus-made-easy-1914/ex-xiii",
   "number": 9,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "164",
   "location": "Exercise XIII, problem 92",
   "problem_latex": "The damping on a telephone line can be ascertained\nfrom the relation $i = i_0 \\epsilon^{-\\beta l}$, where $i$~is the\nstrength, after $t$~seconds, of a telephonic current of\ninitial strength~$i_0$; $l$~is the length of the line in kilometres,\nand $\\beta$~is a constant. For the Franco-English\nsubmarine cable laid in 1910, $\\beta = 0.0114$. Find the\ndamping at the end of the cable ($40$~kilometres), and\nthe length along which $i$~is still $8$\\%~of the original\ncurrent (limiting value of very good audition).",
   "markdown": "The damping on a telephone line can be ascertained from the relation $i = i_0 \\epsilon^{-\\beta l}$, where $i$ is the strength, after $t$ seconds, of a telephonic current of initial strength $i_0$; $l$ is the length of the line in kilometres, and $\\beta$ is a constant. For the Franco-English submarine cable laid in 1910, $\\beta = 0.0114$. Find the damping at the end of the cable ($40$ kilometres), and the length along which $i$ is still $8$% of the original current (limiting value of very good audition).",
   "answer_latex": [
    "$i$ is $63.4$\\% of~$i_0$, $220$~kilometres."
   ],
   "answer_markdown": [
    "$i$ is $63.4$% of $i_0$, $220$ kilometres."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "log(Rational(100,8))/Rational(114,10000)"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 221.555144238, printed 220 (half-unit 0.5; correctly rounded at the printed digits: 222.0)",
     "problem_expr": "log(Rational(100,8))/Rational(114,10000)",
     "answer_expr": "log(Rational(100,8))/Rational(114,10000)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: 5000*log(25/2)/57"
   ],
   "shape": [
    "evaluate: N*log(N)"
   ],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.eqn",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/101",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 10,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 101",
   "problem_latex": "Differentiate $y=\\epsilon^x \\sin^2 x$.",
   "markdown": "Differentiate $y=\\epsilon^x \\sin^2 x$.",
   "answer_latex": [
    "$\\epsilon^x \\left(\\sin^2 x + \\sin2x\\right)$;\\quad $\\epsilon^x \\left(\\sin^2 x + 2\\sin2x + 2\\cos2x\\right)$."
   ],
   "answer_markdown": [
    "$\\epsilon^x \\left(\\sin^2 x + \\sin2x\\right)$; $\\epsilon^x \\left(\\sin^2 x + 2\\sin2x + 2\\cos2x\\right)$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "exp(x)*sin(x)**2",
     "answer_expr": "exp(x)*(sin(x)**2 + sin(2*x))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: exp(x)*sin(x)**2"
   ],
   "shape": [
    "differentiate: exp(x)*sin(x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.trig",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/102",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 10,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 102",
   "problem_latex": "Differentiate $y=\\epsilon^x \\sin^2 x$.",
   "markdown": "Differentiate $y=\\epsilon^x \\sin^2 x$.",
   "answer_latex": [
    "$\\epsilon^x \\left(\\sin^2 x + \\sin2x\\right)$;\\quad $\\epsilon^x \\left(\\sin^2 x + 2\\sin2x + 2\\cos2x\\right)$."
   ],
   "answer_markdown": [
    "$\\epsilon^x \\left(\\sin^2 x + \\sin2x\\right)$; $\\epsilon^x \\left(\\sin^2 x + 2\\sin2x + 2\\cos2x\\right)$."
   ],
   "checks": [
    {
     "task": "differentiate2",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "exp(x)*sin(x)**2",
     "answer_expr": "exp(x)*(sin(x)**2 + 2*sin(2*x) + 2*cos(2*x))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate2: exp(x)*sin(x)**2"
   ],
   "shape": [
    "differentiate2: exp(x)*sin(x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.trig",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/111",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 11,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 111",
   "problem_latex": "Differentiate the three equations of Exercises~XIII.\n(\\Pageref{XIII:4}), No.~4, and compare their differential\ncoefficients, as to whether they are equal, or nearly\nequal, for very small values of~$x$, or for very large\nvalues of~$x$, or for values of~$x$ in the neighbourhood\nof $x=30$.",
   "markdown": "Differentiate the three equations of Exercises XIII. (XIII:4), No. 4, and compare their differential coefficients, as to whether they are equal, or nearly equal, for very small values of $x$, or for very large values of $x$, or for values of $x$ in the neighbourhood of $x=30$.",
   "answer_latex": [
    "$\\left(i\\right) \\dfrac{dy}{dx} = \\dfrac{ab}{\\left(x + b\\right)^2}$;\\quad\n(ii)~$\\dfrac{a}{b} \\epsilon^{-\\efrac{x}{b}}$;\\quad\n(iii)~$\\dfrac{1}{90}° × \\dfrac{ab}{\\left(b^2 + x^2\\right)}$."
   ],
   "answer_markdown": [
    "$\\left(i\\right) \\dfrac{dy}{dx} = \\dfrac{ab}{\\left(x + b\\right)^2}$; (ii) $\\dfrac{a}{b} \\epsilon^{-\\efrac{x}{b}}$; (iii) $\\dfrac{1}{90}° × \\dfrac{ab}{\\left(b^2 + x^2\\right)}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "a*b/(x + b)**2"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith",
    "core.table"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/112",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 11,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 112",
   "problem_latex": "Differentiate the three equations of Exercises~XIII.\n(\\Pageref{XIII:4}), No.~4, and compare their differential\ncoefficients, as to whether they are equal, or nearly\nequal, for very small values of~$x$, or for very large\nvalues of~$x$, or for values of~$x$ in the neighbourhood\nof $x=30$.",
   "markdown": "Differentiate the three equations of Exercises XIII. (XIII:4), No. 4, and compare their differential coefficients, as to whether they are equal, or nearly equal, for very small values of $x$, or for very large values of $x$, or for values of $x$ in the neighbourhood of $x=30$.",
   "answer_latex": [
    "$\\left(i\\right) \\dfrac{dy}{dx} = \\dfrac{ab}{\\left(x + b\\right)^2}$;\\quad\n(ii)~$\\dfrac{a}{b} \\epsilon^{-\\efrac{x}{b}}$;\\quad\n(iii)~$\\dfrac{1}{90}° × \\dfrac{ab}{\\left(b^2 + x^2\\right)}$."
   ],
   "answer_markdown": [
    "$\\left(i\\right) \\dfrac{dy}{dx} = \\dfrac{ab}{\\left(x + b\\right)^2}$; (ii) $\\dfrac{a}{b} \\epsilon^{-\\efrac{x}{b}}$; (iii) $\\dfrac{1}{90}° × \\dfrac{ab}{\\left(b^2 + x^2\\right)}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "a/b*exp(-x/b)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith",
    "core.table"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/113",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 11,
   "part": "3",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 113",
   "problem_latex": "Differentiate the three equations of Exercises~XIII.\n(\\Pageref{XIII:4}), No.~4, and compare their differential\ncoefficients, as to whether they are equal, or nearly\nequal, for very small values of~$x$, or for very large\nvalues of~$x$, or for values of~$x$ in the neighbourhood\nof $x=30$.",
   "markdown": "Differentiate the three equations of Exercises XIII. (XIII:4), No. 4, and compare their differential coefficients, as to whether they are equal, or nearly equal, for very small values of $x$, or for very large values of $x$, or for values of $x$ in the neighbourhood of $x=30$.",
   "answer_latex": [
    "$\\left(i\\right) \\dfrac{dy}{dx} = \\dfrac{ab}{\\left(x + b\\right)^2}$;\\quad\n(ii)~$\\dfrac{a}{b} \\epsilon^{-\\efrac{x}{b}}$;\\quad\n(iii)~$\\dfrac{1}{90}° × \\dfrac{ab}{\\left(b^2 + x^2\\right)}$."
   ],
   "answer_markdown": [
    "$\\left(i\\right) \\dfrac{dy}{dx} = \\dfrac{ab}{\\left(x + b\\right)^2}$; (ii) $\\dfrac{a}{b} \\epsilon^{-\\efrac{x}{b}}$; (iii) $\\dfrac{1}{90}° × \\dfrac{ab}{\\left(b^2 + x^2\\right)}$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": "a*b/(90*(b**2 + x**2))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith",
    "core.table",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/12i",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 12,
   "part": "i",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 12i",
   "problem_latex": "Differentiate the following:\n\\begin{align*}%[** TN: Reformatted in two columns]\n\\text{(i)}\\quad   y &= \\sec x.    &\n\\text{(ii)}\\quad  y &= \\arccos x. \\\\\n\\text{(iii)}\\quad y &= \\arctan x. &\n\\text{(iv)}\\quad  y &= \\arcsec x. \\\\\n\\text{(v)}\\quad   y &= \\tan x × \\sqrt{3 \\sec x}. &&\n\\end{align*}",
   "markdown": "Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$;\n\n(ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$;\n\n(iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$;\n\n(iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$;\n\n(v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$; (ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$; (iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$; (iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$; (v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/cos(x)",
     "answer_expr": "tan(x)/cos(x)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: 1/cos(x)"
   ],
   "shape": [
    "differentiate: 1/cos(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/12ii",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 12,
   "part": "ii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 12ii",
   "problem_latex": "Differentiate the following:\n\\begin{align*}%[** TN: Reformatted in two columns]\n\\text{(i)}\\quad   y &= \\sec x.    &\n\\text{(ii)}\\quad  y &= \\arccos x. \\\\\n\\text{(iii)}\\quad y &= \\arctan x. &\n\\text{(iv)}\\quad  y &= \\arcsec x. \\\\\n\\text{(v)}\\quad   y &= \\tan x × \\sqrt{3 \\sec x}. &&\n\\end{align*}",
   "markdown": "Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$;\n\n(ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$;\n\n(iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$;\n\n(iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$;\n\n(v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$; (ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$; (iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$; (iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$; (v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "acos(x)",
     "answer_expr": "-1/sqrt(1 - x**2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: acos(x)"
   ],
   "shape": [
    "differentiate: acos(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/12iii",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 12,
   "part": "iii",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 12iii",
   "problem_latex": "Differentiate the following:\n\\begin{align*}%[** TN: Reformatted in two columns]\n\\text{(i)}\\quad   y &= \\sec x.    &\n\\text{(ii)}\\quad  y &= \\arccos x. \\\\\n\\text{(iii)}\\quad y &= \\arctan x. &\n\\text{(iv)}\\quad  y &= \\arcsec x. \\\\\n\\text{(v)}\\quad   y &= \\tan x × \\sqrt{3 \\sec x}. &&\n\\end{align*}",
   "markdown": "Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$;\n\n(ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$;\n\n(iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$;\n\n(iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$;\n\n(v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$; (ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$; (iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$; (iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$; (v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "atan(x)",
     "answer_expr": "1/(1 + x**2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: atan(x)"
   ],
   "shape": [
    "differentiate: atan(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/12iv",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 12,
   "part": "iv",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 12iv",
   "problem_latex": "Differentiate the following:\n\\begin{align*}%[** TN: Reformatted in two columns]\n\\text{(i)}\\quad   y &= \\sec x.    &\n\\text{(ii)}\\quad  y &= \\arccos x. \\\\\n\\text{(iii)}\\quad y &= \\arctan x. &\n\\text{(iv)}\\quad  y &= \\arcsec x. \\\\\n\\text{(v)}\\quad   y &= \\tan x × \\sqrt{3 \\sec x}. &&\n\\end{align*}",
   "markdown": "Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$;\n\n(ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$;\n\n(iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$;\n\n(iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$;\n\n(v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$; (ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$; (iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$; (iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$; (v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "asec(x)",
     "answer_expr": "1/(x*sqrt(x**2 - 1))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: asec(x)"
   ],
   "shape": [
    "differentiate: asec(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/12v",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 12,
   "part": "v",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 12v",
   "problem_latex": "Differentiate the following:\n\\begin{align*}%[** TN: Reformatted in two columns]\n\\text{(i)}\\quad   y &= \\sec x.    &\n\\text{(ii)}\\quad  y &= \\arccos x. \\\\\n\\text{(iii)}\\quad y &= \\arctan x. &\n\\text{(iv)}\\quad  y &= \\arcsec x. \\\\\n\\text{(v)}\\quad   y &= \\tan x × \\sqrt{3 \\sec x}. &&\n\\end{align*}",
   "markdown": "Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$;\n\n(ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$;\n\n(iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$;\n\n(iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$;\n\n(v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{dx} = \\sec x \\tan x$; (ii) $\\dfrac{dy}{dx} = - \\dfrac{1}{\\sqrt{ 1 - x^2}}$; (iii) $\\dfrac{dy}{dx} = \\dfrac{1}{ 1 + x^2}$; (iv) $\\dfrac{dy}{dx} = \\dfrac{1}{x \\sqrt{ x^2 - 1}}$; (v) $\\dfrac{dy}{dx} = \\dfrac{\\sqrt{ 3\\sec x} \\left(3\\sec^2 x - 1\\right)}{2}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "tan(x)*sqrt(3/cos(x))",
     "answer_expr": "sqrt(3/cos(x))*(3/cos(x)**2 - 1)/2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sqrt(3)*sqrt(1/cos(x))*tan(x)"
   ],
   "shape": [
    "differentiate: N**N*(1/cos(x))**N*tan(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.trig",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/13",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 13,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 13",
   "problem_latex": "Differentiate $y=\\sin(2\\theta +3)^{2.3}$.",
   "markdown": "Differentiate $y=\\sin(2\\theta +3)^{2.3}$.",
   "answer_latex": [
    "$\\dfrac{dy}{d\\theta} = 4.6\\left(2\\theta + 3\\right)^{1.3} \\cos\\left(2\\theta + 3\\right)^{2.3}$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{d\\theta} = 4.6\\left(2\\theta + 3\\right)^{1.3} \\cos\\left(2\\theta + 3\\right)^{2.3}$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin((2*theta + 3)**Rational(23,10))",
     "answer_expr": "Rational(23,5)*(2*theta + 3)**Rational(13,10)*cos((2*theta + 3)**Rational(23,10))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sin((2*x + 3)**(23/10))"
   ],
   "shape": [
    "differentiate: sin((N*x + N)**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/14",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 14,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 14",
   "problem_latex": "Differentiate $y=\\theta^3+3 \\sin(\\theta+3)-3^{\\sin \\theta} - 3^\\theta$.",
   "markdown": "Differentiate $y=\\theta^3+3 \\sin(\\theta+3)-3^{\\sin \\theta} - 3^\\theta$.",
   "answer_latex": [
    "$\\dfrac{dy}{d\\theta} = 3\\theta^2 + 3\\cos \\left( \\theta + 3 \\right) - \\log_\\epsilon 3 \\left( \\cos\\theta × 3^{\\sin\\theta} + 3\\theta \\right)$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{d\\theta} = 3\\theta^2 + 3\\cos \\left( \\theta + 3 \\right) - \\log_\\epsilon 3 \\left( \\cos\\theta × 3^{\\sin\\theta} + 3\\theta \\right)$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": [
      "-3.0359957020041965318",
      "-2.9758952354769154793"
     ],
     "problem_expr": "theta**3 + 3*sin(theta + 3) - 3**sin(theta) - 3**theta",
     "answer_expr": "3*theta**2 + 3*cos(theta + 3) - log(3)*(cos(theta)*3**sin(theta) + 3*theta)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "differentiate: -3**x - 3**sin(x) + x**3 + 3*sin(x + 3)"
   ],
   "shape": [
    "differentiate: N*sin(N + x) - N**x - N**sin(x) + x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/15a",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 15,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 15a",
   "problem_latex": "Find the maximum or minimum of $y=\\theta \\cos \\theta$.",
   "markdown": "Find the maximum or minimum of $y=\\theta \\cos \\theta$.",
   "answer_latex": [
    "$\\theta = \\cot\\theta; \\theta = ±0.86$; is max.~for $+\\theta$, min.~for $-\\theta$."
   ],
   "answer_markdown": [
    "$\\theta = \\cot\\theta; \\theta = ±0.86$; is max. for $+\\theta$, min. for $-\\theta$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "f'(x0) is not 0: ('-7.7929082032528131995e-17', '0.0')",
     "problem_expr": "theta*cos(theta)",
     "answer_expr": "0.8603335890193798"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "the formula itself fails: f'(x0) is not 0: ('-7.7929082032528131995e-17', '0.0')",
     "problem_expr": "theta*cos(theta)",
     "answer_expr": "0.8603335890193798"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: x*cos(x)",
    "evaluate: x*cos(x)"
   ],
   "shape": [
    "evaluate: x*cos(x)",
    "extremum: x*cos(x)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xiv/15b"
   ],
   "needs": [
    "cas.derive",
    "core.graph",
    "core.solve.num",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/15b",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 15,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 15b",
   "problem_latex": "Find the maximum or minimum of $y=\\theta \\cos \\theta$.",
   "markdown": "Find the maximum or minimum of $y=\\theta \\cos \\theta$.",
   "answer_latex": [
    "$\\theta = \\cot\\theta; \\theta = ±0.86$; is max.~for $+\\theta$, min.~for $-\\theta$."
   ],
   "answer_markdown": [
    "$\\theta = \\cot\\theta; \\theta = ±0.86$; is max. for $+\\theta$, min. for $-\\theta$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "f'(x0) is not 0: ('-7.7929082032528131995e-17', '0.0')",
     "problem_expr": "theta*cos(theta)",
     "answer_expr": "-0.8603335890193798"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "the formula itself fails: f'(x0) is not 0: ('-7.7929082032528131995e-17', '0.0')",
     "problem_expr": "theta*cos(theta)",
     "answer_expr": "-0.8603335890193798"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH",
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: x*cos(x)",
    "evaluate: x*cos(x)"
   ],
   "shape": [
    "evaluate: x*cos(x)",
    "extremum: x*cos(x)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xiv/15a"
   ],
   "needs": [
    "cas.derive",
    "core.graph",
    "core.solve.num",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/1a",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 1,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "173",
   "location": "Exercise XIV, problem 1a",
   "problem_latex": "Differentiate the following:\n\\begin{align*}\n\\text{(i)}\\quad   y &= A \\sin\\left(\\theta - \\frac{\\pi}{2}\\right).\\\\\n\\text{(ii)}\\quad  y &= \\sin^2 \\theta;\\quad \\text{and } y = \\sin 2\\theta.\\\\\n\\text{(iii)}\\quad y &= \\sin^3 \\theta;\\quad \\text{and } y = \\sin 3\\theta.\n\\end{align*}",
   "markdown": "Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$;\n\n(ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$;\n\n(iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$; (ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$; (iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "A*sin(theta - pi/2)",
     "answer_expr": "A*cos(theta - pi/2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: -a*cos(x)"
   ],
   "shape": [
    "differentiate: -a*cos(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/1b",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 1,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "173",
   "location": "Exercise XIV, problem 1b",
   "problem_latex": "Differentiate the following:\n\\begin{align*}\n\\text{(i)}\\quad   y &= A \\sin\\left(\\theta - \\frac{\\pi}{2}\\right).\\\\\n\\text{(ii)}\\quad  y &= \\sin^2 \\theta;\\quad \\text{and } y = \\sin 2\\theta.\\\\\n\\text{(iii)}\\quad y &= \\sin^3 \\theta;\\quad \\text{and } y = \\sin 3\\theta.\n\\end{align*}",
   "markdown": "Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$;\n\n(ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$;\n\n(iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$; (ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$; (iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin(theta)**2",
     "answer_expr": "2*sin(theta)*cos(theta)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sin(x)**2"
   ],
   "shape": [
    "differentiate: sin(x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.trig",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/1c",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 1,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "173",
   "location": "Exercise XIV, problem 1c",
   "problem_latex": "Differentiate the following:\n\\begin{align*}\n\\text{(i)}\\quad   y &= A \\sin\\left(\\theta - \\frac{\\pi}{2}\\right).\\\\\n\\text{(ii)}\\quad  y &= \\sin^2 \\theta;\\quad \\text{and } y = \\sin 2\\theta.\\\\\n\\text{(iii)}\\quad y &= \\sin^3 \\theta;\\quad \\text{and } y = \\sin 3\\theta.\n\\end{align*}",
   "markdown": "Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$;\n\n(ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$;\n\n(iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$; (ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$; (iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin(2*theta)",
     "answer_expr": "2*cos(2*theta)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sin(2*x)"
   ],
   "shape": [
    "differentiate: sin(N*x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/1d",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 1,
   "part": "d",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "173",
   "location": "Exercise XIV, problem 1d",
   "problem_latex": "Differentiate the following:\n\\begin{align*}\n\\text{(i)}\\quad   y &= A \\sin\\left(\\theta - \\frac{\\pi}{2}\\right).\\\\\n\\text{(ii)}\\quad  y &= \\sin^2 \\theta;\\quad \\text{and } y = \\sin 2\\theta.\\\\\n\\text{(iii)}\\quad y &= \\sin^3 \\theta;\\quad \\text{and } y = \\sin 3\\theta.\n\\end{align*}",
   "markdown": "Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$;\n\n(ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$;\n\n(iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$; (ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$; (iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin(theta)**3",
     "answer_expr": "3*sin(theta)**2*cos(theta)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sin(x)**3"
   ],
   "shape": [
    "differentiate: sin(x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/1e",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 1,
   "part": "e",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "173",
   "location": "Exercise XIV, problem 1e",
   "problem_latex": "Differentiate the following:\n\\begin{align*}\n\\text{(i)}\\quad   y &= A \\sin\\left(\\theta - \\frac{\\pi}{2}\\right).\\\\\n\\text{(ii)}\\quad  y &= \\sin^2 \\theta;\\quad \\text{and } y = \\sin 2\\theta.\\\\\n\\text{(iii)}\\quad y &= \\sin^3 \\theta;\\quad \\text{and } y = \\sin 3\\theta.\n\\end{align*}",
   "markdown": "Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*",
   "answer_latex": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$;\n\n(ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$;\n\n(iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "answer_markdown": [
    "(i) $\\dfrac{dy}{d\\theta} = A \\cos \\left( \\theta - \\dfrac{\\pi}{2} \\right)$; (ii) $\\dfrac{dy}{d\\theta} = 2\\sin\\theta \\cos\\theta = \\sin2\\theta$ and $\\dfrac{dy}{d\\theta} = 2\\cos2\\theta$; (iii) $\\dfrac{dy}{d\\theta} = 3\\sin^2 \\theta \\cos\\theta$ and $\\dfrac{dy}{d\\theta} = 3\\cos3\\theta$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin(3*theta)",
     "answer_expr": "3*cos(3*theta)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sin(3*x)"
   ],
   "shape": [
    "differentiate: sin(N*x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/2",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "173",
   "location": "Exercise XIV, problem 2",
   "problem_latex": "Find the value of~$\\theta$ for which $\\sin\\theta × \\cos\\theta$ is a\nmaximum.",
   "markdown": "Find the value of $\\theta$ for which $\\sin\\theta × \\cos\\theta$ is a maximum.",
   "answer_latex": [
    "$\\theta = 45°$ or $\\dfrac{\\pi}{4}$ radians."
   ],
   "answer_markdown": [
    "$\\theta = 45°$ or $\\dfrac{\\pi}{4}$ radians."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin(theta)*cos(theta)",
     "answer_expr": "pi/4"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: sin(x)*cos(x)"
   ],
   "shape": [
    "extremum: sin(x)*cos(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.trig",
    "core.graph",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/3",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "173",
   "location": "Exercise XIV, problem 3",
   "problem_latex": "Differentiate $y=\\dfrac{1}{2\\pi} \\cos 2\\pi nt$.",
   "markdown": "Differentiate $y=\\dfrac{1}{2\\pi} \\cos 2\\pi nt$.",
   "answer_latex": [
    "$\\dfrac{dy}{dt} = -n \\sin 2\\pi nt$."
   ],
   "answer_markdown": [
    "$\\dfrac{dy}{dt} = -n \\sin 2\\pi nt$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "cos(2*pi*n*t)/(2*pi)",
     "answer_expr": "-n*sin(2*pi*n*t)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: cos(2*pi*a*x)/(2*pi)"
   ],
   "shape": [
    "differentiate: N*cos(pi*N*a*x)/pi"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/4",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 4",
   "problem_latex": "If $y = \\sin a^x$, find~$\\dfrac{dy}{dx}$.",
   "markdown": "If $y = \\sin a^x$, find $\\dfrac{dy}{dx}$.",
   "answer_latex": [
    "$a^x \\log_\\epsilon a \\cos a^x$."
   ],
   "answer_markdown": [
    "$a^x \\log_\\epsilon a \\cos a^x$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin(a**x)",
     "answer_expr": "a**x*log(a)*cos(a**x)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sin(a**x)"
   ],
   "shape": [
    "differentiate: sin(a**x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/5",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 5",
   "problem_latex": "Differentiate $y=\\log_\\epsilon \\cos x$.",
   "markdown": "Differentiate $y=\\log_\\epsilon \\cos x$.",
   "answer_latex": [
    "$\\dfrac{\\cos x}{\\sin x} = \\cotan x$"
   ],
   "answer_markdown": [
    "$\\dfrac{\\cos x}{\\sin x} = \\cotan x$"
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": [
      "0.20314335661707171961",
      "-4.9226320597085290345"
     ],
     "problem_expr": "log(cos(x))",
     "answer_expr": "cos(x)/sin(x)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "differentiate: log(cos(x))"
   ],
   "shape": [
    "differentiate: log(cos(x))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.log",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/6",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 6",
   "problem_latex": "Differentiate $y=18.2 \\sin(x+26°)$.",
   "markdown": "Differentiate $y=18.2 \\sin(x+26°)$.",
   "answer_latex": [
    "$18.2 \\cos \\left(x + 26° \\right)$."
   ],
   "answer_markdown": [
    "$18.2 \\cos \\left(x + 26° \\right)$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "18.2*sin(x + 26*pi/180)",
     "answer_expr": "18.2*cos(x + 26*pi/180)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "differentiate: 91*sin(x + 13*pi/90)/5"
   ],
   "shape": [
    "differentiate: N*sin(pi*N + x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/7a",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 7,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 7a",
   "problem_latex": "Plot the curve $y=100 \\sin(\\theta-15°)$; and show\nthat the slope of the curve at $\\theta = 75°$ is half the\nmaximum slope.",
   "markdown": "Plot the curve $y=100 \\sin(\\theta-15°)$; and show that the slope of the curve at $\\theta = 75°$ is half the maximum slope.",
   "answer_latex": [
    "{\\loosen The slope is $\\dfrac{dy}{d\\theta} = 100\\cos\\left(\\theta - 15° \\right)$, which is a maximum\n when $(\\theta -15°) = 0$, or $\\theta = 15°$; the value of the slope\n being then ${}= 100$. When $\\theta = 75°$ the slope is\n $100\\cos(75°  - 15°) = 100\\cos 60° = 100 × \\frac{1}{2} = 50$.}"
   ],
   "answer_markdown": [
    "0.5em plus 0.5em minus 0.25emThe slope is $\\dfrac{dy}{d\\theta} = 100\\cos\\left(\\theta - 15° \\right)$, which is a maximum when $(\\theta -15°) = 0$, or $\\theta = 15°$; the value of the slope being then ${}= 100$. When $\\theta = 75°$ the slope is $100\\cos(75° - 15°) = 100\\cos 60° = 100 × \\frac{1}{2} = 50$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "100*sin(theta - 15*pi/180)",
     "answer_expr": "100*cos(theta - 15*pi/180)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 100.0, printed 100 (half-unit 0.5; correctly rounded at the printed digits: 100.0)",
     "problem_expr": "100*sin(theta - 15*pi/180)",
     "answer_expr": "100*cos(theta - 15*pi/180)"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 50.0, printed 50 (half-unit 0.5; correctly rounded at the printed digits: 50.0)",
     "problem_expr": "100*sin(theta - 15*pi/180)",
     "answer_expr": "100*cos(theta - 15*pi/180)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "differentiate: -100*cos(x + 5*pi/12)",
    "evaluate: -100*cos(x + 5*pi/12) at theta=15*pi/180",
    "evaluate: -100*cos(x + 5*pi/12) at theta=75*pi/180"
   ],
   "shape": [
    "differentiate: N*cos(pi*N + x)",
    "evaluate: N*cos(pi*N + x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.arith",
    "core.trig",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/7b",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 7,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 7b",
   "problem_latex": "Plot the curve $y=100 \\sin(\\theta-15°)$; and show\nthat the slope of the curve at $\\theta = 75°$ is half the\nmaximum slope.",
   "markdown": "Plot the curve $y=100 \\sin(\\theta-15°)$; and show that the slope of the curve at $\\theta = 75°$ is half the maximum slope.",
   "answer_latex": [
    "{\\loosen The slope is $\\dfrac{dy}{d\\theta} = 100\\cos\\left(\\theta - 15° \\right)$, which is a maximum\n when $(\\theta -15°) = 0$, or $\\theta = 15°$; the value of the slope\n being then ${}= 100$. When $\\theta = 75°$ the slope is\n $100\\cos(75°  - 15°) = 100\\cos 60° = 100 × \\frac{1}{2} = 50$.}"
   ],
   "answer_markdown": [
    "0.5em plus 0.5em minus 0.25emThe slope is $\\dfrac{dy}{d\\theta} = 100\\cos\\left(\\theta - 15° \\right)$, which is a maximum when $(\\theta -15°) = 0$, or $\\theta = 15°$; the value of the slope being then ${}= 100$. When $\\theta = 75°$ the slope is $100\\cos(75° - 15°) = 100\\cos 60° = 100 × \\frac{1}{2} = 50$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "100*sin(theta - 15*pi/180)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "core.arith",
    "core.graph",
    "core.trig",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/8",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 8",
   "problem_latex": "If $y=\\sin \\theta·\\sin 2\\theta$, find~$\\dfrac{dy}{d\\theta}$.",
   "markdown": "If $y=\\sin \\theta·\\sin 2\\theta$, find $\\dfrac{dy}{d\\theta}$.",
   "answer_latex": [
    "$\\begin{aligned}[t]\n  \\cos\\theta \\sin2\\theta + 2\\cos2\\theta \\sin\\theta\n  &= 2\\sin\\theta\\left(\\cos^2 \\theta + \\cos2\\theta\\right) \\\\\n  &= 2\\sin\\theta\\left(3\\cos^2 \\theta - 1\\right).\n  \\end{aligned}$"
   ],
   "answer_markdown": [
    "$\\begin{aligned}[t] \\cos\\theta \\sin2\\theta + 2\\cos2\\theta \\sin\\theta &= 2\\sin\\theta\\left(\\cos^2 \\theta + \\cos2\\theta\\right) \\\\ &= 2\\sin\\theta\\left(3\\cos^2 \\theta - 1\\right). \\end{aligned}$"
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin(theta)*sin(2*theta)",
     "answer_expr": "2*sin(theta)*(3*cos(theta)**2 - 1)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: sin(x)*sin(2*x)"
   ],
   "shape": [
    "differentiate: sin(x)*sin(N*x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify",
    "cas.trig",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xiv/9",
   "set": "thompson-calculus-made-easy-1914/ex-xiv",
   "number": 9,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "174",
   "location": "Exercise XIV, problem 9",
   "problem_latex": "If $y=a·\\tan^m(\\theta^n)$, find the differential coefficient\nof~$y$ with respect to~$\\theta$.",
   "markdown": "If $y=a·\\tan^m(\\theta^n)$, find the differential coefficient of $y$ with respect to $\\theta$.",
   "answer_latex": [
    "$amn\\theta^{n-1} \\tan^{m-1}\\left(\\theta^n\\right)\\sec^2 \\theta^n$."
   ],
   "answer_markdown": [
    "$amn\\theta^{n-1} \\tan^{m-1}\\left(\\theta^n\\right)\\sec^2 \\theta^n$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a*tan(theta**n)**m",
     "answer_expr": "a*m*n*theta**(n-1)*tan(theta**n)**(m-1)/cos(theta**n)**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a*tan(x**c)**b"
   ],
   "shape": [
    "differentiate: a*tan(x**c)**b"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/1",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 1",
   "problem_latex": "Find $\\ds\\int \\sqrt {a^2 - x^2}\\, dx$.",
   "markdown": "Find $\\ds\\int \\sqrt {a^2 - x^2}\\, dx$.",
   "answer_latex": [
    "$\\dfrac{x\\sqrt{a^2 - x^2}}{2} + \\dfrac{a^2}{2} \\sin^{-1} \\dfrac{x}{a} + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{x\\sqrt{a^2 - x^2}}{2} + \\dfrac{a^2}{2} \\sin^{-1} \\dfrac{x}{a} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sqrt(a**2 - x**2)",
     "answer_expr": "x*sqrt(a**2 - x**2)/2 + a**2/2*asin(x/a)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: sqrt(a**2 - x**2)"
   ],
   "shape": [
    "integrate: (a**N - x**N)**N"
   ],
   "same_problem_in": [
    "hardy-course-of-pure-mathematics-1921/ex-lxiii/6b",
    "hardy-course-of-pure-mathematics-1921/ex-xlix/2b"
   ],
   "needs": [
    "cas.integrate.subst",
    "cas.trig",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/10",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 10,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 10",
   "problem_latex": "Find $\\ds\\int \\dfrac{(x^2 -3)\\, dx}{x^3 - 7x+6}$.",
   "markdown": "Find $\\ds\\int \\dfrac{(x^2 -3)\\, dx}{x^3 - 7x+6}$.",
   "answer_latex": [
    "$\\frac{1}{2} \\log_\\epsilon(x - 1) + \\frac{1}{5} \\log_\\epsilon(x - 2) + \\frac{3}{10} \\log_\\epsilon(x + 3) + C$."
   ],
   "answer_markdown": [
    "$\\frac{1}{2} \\log_\\epsilon(x - 1) + \\frac{1}{5} \\log_\\epsilon(x - 2) + \\frac{3}{10} \\log_\\epsilon(x + 3) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS-ABS-CONVENTION",
     "judge_why": "answer not real on [-41/14, -26/15] where the integrand is; with log|u| for log(u) it holds on ['[212/97, 165/58]', '[-41/14, -26/15]']",
     "problem_expr": "(x**2 - 3)/(x**3 - 7*x + 6)",
     "answer_expr": "Rational(1, 2)*log(x - 1) + Rational(1, 5)*log(x - 2) + Rational(3, 10)*log(x + 3)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ABS-CONVENTION"
    ]
   },
   "form": [
    "integrate: (x**2 - 3)/(x**3 - 7*x + 6)"
   ],
   "shape": [
    "integrate: (N + x**N)/(N*x + N + x**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.factor",
    "cas.integrate",
    "cas.partfrac",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/11",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 11,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 11",
   "problem_latex": "Find $\\ds\\int \\dfrac{b\\, dx}{x^2 -a^2}$.",
   "markdown": "Find $\\ds\\int \\dfrac{b\\, dx}{x^2 -a^2}$.",
   "answer_latex": [
    "$\\dfrac{b}{2a} \\log_\\epsilon \\dfrac{x - a}{x + a} + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{b}{2a} \\log_\\epsilon \\dfrac{x - a}{x + a} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS-ABS-CONVENTION",
     "judge_why": "answer not real on [4/7, 103/94] where the integrand is; with log|u| for log(u) it holds on ['[4/7, 103/94]', '[-361/98, -43/17]']",
     "problem_expr": "b/(x**2 - a**2)",
     "answer_expr": "b/(2*a)*log((x - a)/(x + a))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ABS-CONVENTION"
    ]
   },
   "form": [
    "integrate: b/(-a**2 + x**2)"
   ],
   "shape": [
    "integrate: b/(-a**N + x**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.partfrac",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/12",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 12,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 12",
   "problem_latex": "Find $\\ds\\int \\dfrac{4x\\, dx}{x^4 -1}$.",
   "markdown": "Find $\\ds\\int \\dfrac{4x\\, dx}{x^4 -1}$.",
   "answer_latex": [
    "$\\log_\\epsilon \\dfrac{x^2 - 1}{x^2 + 1} + C$."
   ],
   "answer_markdown": [
    "$\\log_\\epsilon \\dfrac{x^2 - 1}{x^2 + 1} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "4*x/(x**4 - 1)",
     "answer_expr": "log((x**2 - 1)/(x**2 + 1))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: 4*x/(x**4 - 1)"
   ],
   "shape": [
    "integrate: N*x/(x**N - 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.integrate.subst",
    "cas.partfrac",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/13",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 13,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 13",
   "problem_latex": "Find $\\ds\\int \\dfrac{dx}{1-x^4}$.",
   "markdown": "Find $\\ds\\int \\dfrac{dx}{1-x^4}$.",
   "answer_latex": [
    "$\\frac{1}{4} \\log_\\epsilon \\dfrac{1 + x}{1 - x} + \\frac{1}{2} \\arctan x + C$."
   ],
   "answer_markdown": [
    "$\\frac{1}{4} \\log_\\epsilon \\dfrac{1 + x}{1 - x} + \\frac{1}{2} \\arctan x + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS-ABS-CONVENTION",
     "judge_why": "answer not real on [83/56, 293/106] where the integrand is; with log|u| for log(u) it holds on ['[83/56, 293/106]', '[-23/6, -145/54]']",
     "problem_expr": "1/(1 - x**4)",
     "answer_expr": "Rational(1, 4)*log((1 + x)/(1 - x)) + Rational(1, 2)*atan(x)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ABS-CONVENTION"
    ]
   },
   "form": [
    "integrate: 1/(-x**4 + 1)"
   ],
   "shape": [
    "integrate: 1/(-x**N + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.factor",
    "cas.integrate",
    "cas.partfrac",
    "core.log",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/14",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 14,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 14",
   "problem_latex": "Find $\\ds\\int \\dfrac{dx}{x \\sqrt {a-bx^2}}$.",
   "markdown": "Find $\\ds\\int \\dfrac{dx}{x \\sqrt {a-bx^2}}$.",
   "answer_latex": [
    "$\\dfrac{1}{\\sqrt{a}} \\log_\\epsilon \\dfrac{\\sqrt{a} - \\sqrt{a - bx^2}}{x\\sqrt{a}}$"
   ],
   "answer_markdown": [
    "$\\dfrac{1}{\\sqrt{a}} \\log_\\epsilon \\dfrac{\\sqrt{a} - \\sqrt{a - bx^2}}{x\\sqrt{a}}$"
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/(x*sqrt(a - b*x**2))",
     "answer_expr": "1/sqrt(a)*log((sqrt(a) - sqrt(a - b*x**2))/(x*sqrt(a)))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: 1/(x*sqrt(a - b*x**2))"
   ],
   "shape": [
    "integrate: (a - b*x**N)**N/x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate.subst",
    "cas.simplify",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/2",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 2",
   "problem_latex": "Find $\\ds\\int x \\log_\\epsilon x\\, dx$.",
   "markdown": "Find $\\ds\\int x \\log_\\epsilon x\\, dx$.",
   "answer_latex": [
    "$\\dfrac{x^2}{2}(\\log_\\epsilon x - \\tfrac{1}{2}) + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{x^2}{2}(\\log_\\epsilon x - \\tfrac{1}{2}) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x*log(x)",
     "answer_expr": "x**2/2*(log(x) - Rational(1, 2))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: x*log(x)"
   ],
   "shape": [
    "integrate: x*log(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate.parts",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/3",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 3",
   "problem_latex": "Find $\\ds\\int x^a \\log_\\epsilon x\\, dx$.",
   "markdown": "Find $\\ds\\int x^a \\log_\\epsilon x\\, dx$.",
   "answer_latex": [
    "$\\dfrac{x^{a+1}}{a + 1} \\left(\\log_\\epsilon x - \\dfrac{1}{a + 1}\\right) + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{x^{a+1}}{a + 1} \\left(\\log_\\epsilon x - \\dfrac{1}{a + 1}\\right) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**a*log(x)",
     "answer_expr": "x**(a + 1)/(a + 1)*(log(x) - 1/(a + 1))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: x**a*log(x)"
   ],
   "shape": [
    "integrate: x**a*log(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate.parts",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/4",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 4",
   "problem_latex": "Find $\\ds\\int \\epsilon^x \\cos \\epsilon^x\\, dx$.",
   "markdown": "Find $\\ds\\int \\epsilon^x \\cos \\epsilon^x\\, dx$.",
   "answer_latex": [
    "$\\sin \\DPtypo{e}{\\epsilon}^x + C$."
   ],
   "answer_markdown": [
    "$\\sin \\DPtypo{e}{\\epsilon}^x + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "exp(x)*cos(exp(x))",
     "answer_expr": "sin(exp(x))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: exp(x)*cos(exp(x))"
   ],
   "shape": [
    "integrate: exp(x)*cos(exp(x))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate.subst",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/5",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 5",
   "problem_latex": "Find $\\ds\\int \\dfrac{1}{x} \\cos (\\log_\\epsilon x)\\, dx$.",
   "markdown": "Find $\\ds\\int \\dfrac{1}{x} \\cos (\\log_\\epsilon x)\\, dx$.",
   "answer_latex": [
    "$\\sin(\\log_\\epsilon x) + C$."
   ],
   "answer_markdown": [
    "$\\sin(\\log_\\epsilon x) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "cos(log(x))/x",
     "answer_expr": "sin(log(x))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: cos(log(x))/x"
   ],
   "shape": [
    "integrate: cos(log(x))/x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate.subst",
    "core.log",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/6",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 6",
   "problem_latex": "Find $\\ds\\int x^2 \\epsilon^x\\, dx$.",
   "markdown": "Find $\\ds\\int x^2 \\epsilon^x\\, dx$.",
   "answer_latex": [
    "$\\DPtypo{e}{\\epsilon}^x (x^2 - 2x + 2) + C$."
   ],
   "answer_markdown": [
    "$\\DPtypo{e}{\\epsilon}^x (x^2 - 2x + 2) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**2*exp(x)",
     "answer_expr": "exp(x)*(x**2 - 2*x + 2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: x**2*exp(x)"
   ],
   "shape": [
    "integrate: x**N*exp(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate.parts"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/7",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 7",
   "problem_latex": "Find $\\ds\\int \\dfrac{(\\log_\\epsilon x)^a}{x}\\, dx$.",
   "markdown": "Find $\\ds\\int \\dfrac{(\\log_\\epsilon x)^a}{x}\\, dx$.",
   "answer_latex": [
    "$\\dfrac{1}{a + 1} (\\log_\\epsilon x)^{a+1} + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{a + 1} (\\log_\\epsilon x)^{a+1} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "log(x)**a/x",
     "answer_expr": "log(x)**(a + 1)/(a + 1)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: log(x)**a/x"
   ],
   "shape": [
    "integrate: log(x)**a/x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate.subst",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/8",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 8",
   "problem_latex": "Find $\\ds\\int \\dfrac{dx}{x \\log_\\epsilon x}$.",
   "markdown": "Find $\\ds\\int \\dfrac{dx}{x \\log_\\epsilon x}$.",
   "answer_latex": [
    "$\\log_\\epsilon(\\log_\\epsilon x) + C$."
   ],
   "answer_markdown": [
    "$\\log_\\epsilon(\\log_\\epsilon x) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "1/(x*log(x))",
     "answer_expr": "log(log(x))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: 1/(x*log(x))"
   ],
   "shape": [
    "integrate: 1/(x*log(x))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate.subst",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xix/9",
   "set": "thompson-calculus-made-easy-1914/ex-xix",
   "number": 9,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "233",
   "location": "Exercise XIX, problem 9",
   "problem_latex": "Find $\\ds\\int \\dfrac{5x+1}{x^2 +x-2}\\, dx$.",
   "markdown": "Find $\\ds\\int \\dfrac{5x+1}{x^2 +x-2}\\, dx$.",
   "answer_latex": [
    "$2\\log_\\epsilon(x - 1) + 3\\log_\\epsilon(x + 2) + C$."
   ],
   "answer_markdown": [
    "$2\\log_\\epsilon(x - 1) + 3\\log_\\epsilon(x + 2) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS-ABS-CONVENTION",
     "judge_why": "answer not real on [-133/37, -40/13] where the integrand is; with log|u| for log(u) it holds on ['[11/9, 47/12]', '[-133/37, -40/13]']",
     "problem_expr": "(5*x + 1)/(x**2 + x - 2)",
     "answer_expr": "2*log(x - 1) + 3*log(x + 2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ABS-CONVENTION"
    ]
   },
   "form": [
    "integrate: (5*x + 1)/(x**2 + x - 2)"
   ],
   "shape": [
    "integrate: (N*x + 1)/(N + x + x**N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.factor",
    "cas.integrate",
    "cas.partfrac",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/10x",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 10,
   "part": "x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 10x",
   "problem_latex": "Find the maximum or minimum of $u = \\dfrac{\\epsilon^{x+y}}{xy}$.",
   "markdown": "Find the maximum or minimum of $u = \\dfrac{\\epsilon^{x+y}}{xy}$.",
   "answer_latex": [
    "Minimum for $x = y = 1$."
   ],
   "answer_markdown": [
    "Minimum for $x = y = 1$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "exp(x + y)/(x*y)",
     "answer_expr": "1"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: exp(a + x)/(a*x)"
   ],
   "shape": [
    "extremum: exp(a + x)/(a*x)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xv/10y"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/10y",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 10,
   "part": "y",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 10y",
   "problem_latex": "Find the maximum or minimum of $u = \\dfrac{\\epsilon^{x+y}}{xy}$.",
   "markdown": "Find the maximum or minimum of $u = \\dfrac{\\epsilon^{x+y}}{xy}$.",
   "answer_latex": [
    "Minimum for $x = y = 1$."
   ],
   "answer_markdown": [
    "Minimum for $x = y = 1$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "exp(x + y)/(x*y)",
     "answer_expr": "1"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: exp(a + x)/(a*x)"
   ],
   "shape": [
    "extremum: exp(a + x)/(a*x)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xv/10x"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/11x",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 11,
   "part": "x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 11x",
   "problem_latex": "Find maximum and minimum of\n\\[\nu = y + 2x - 2 \\log_\\epsilon y - \\log_\\epsilon x.\n\\]",
   "markdown": "Find maximum and minimum of u = y + 2x - 2 _y - _x.",
   "answer_latex": [
    "Min.: $x = \\frac{1}{2}$ and $y = 2$."
   ],
   "answer_markdown": [
    "Min.: $x = \\frac{1}{2}$ and $y = 2$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "y + 2*x - 2*log(y) - log(x)",
     "answer_expr": "Rational(1,2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: a + 2*x - 2*log(a) - log(x)"
   ],
   "shape": [
    "extremum: N*x + N*log(a) + a - log(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/11y",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 11,
   "part": "y",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 11y",
   "problem_latex": "Find maximum and minimum of\n\\[\nu = y + 2x - 2 \\log_\\epsilon y - \\log_\\epsilon x.\n\\]",
   "markdown": "Find maximum and minimum of u = y + 2x - 2 _y - _x.",
   "answer_latex": [
    "Min.: $x = \\frac{1}{2}$ and $y = 2$."
   ],
   "answer_markdown": [
    "Min.: $x = \\frac{1}{2}$ and $y = 2$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "y + 2*x - 2*log(y) - log(x)",
     "answer_expr": "2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "extremum: 2*a + x - log(a) - 2*log(x)"
   ],
   "shape": [
    "extremum: N*a + N*log(x) + x - log(a)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/121",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 12,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 121",
   "problem_latex": "A telpherage bucket of given capacity has\nthe shape of a horizontal isosceles triangular prism\nwith the apex underneath, and the opposite face open.\nFind its dimensions in order that the least amount\nof iron sheet may be used in its construction.",
   "markdown": "A telpherage bucket of given capacity has the shape of a horizontal isosceles triangular prism with the apex underneath, and the opposite face open. Find its dimensions in order that the least amount of iron sheet may be used in its construction.",
   "answer_latex": [
    "Angle at apex $= 90°$; equal sides = length = $\\sqrt[3]{2V}$."
   ],
   "answer_markdown": [
    "Angle at apex $= 90°$; equal sides = length = $\\sqrt[3]{2V}$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "3*(2*V)**Rational(2,3)*sin(alpha)**Rational(-1,3)",
     "answer_expr": "pi/2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "extremum: 3*2**(2/3)*a**(2/3)/sin(x)**(1/3)"
   ],
   "shape": [
    "extremum: N*N**N*a**N*sin(x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.nonpoly",
    "cas.subst",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/122",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 12,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 122",
   "problem_latex": "A telpherage bucket of given capacity has\nthe shape of a horizontal isosceles triangular prism\nwith the apex underneath, and the opposite face open.\nFind its dimensions in order that the least amount\nof iron sheet may be used in its construction.",
   "markdown": "A telpherage bucket of given capacity has the shape of a horizontal isosceles triangular prism with the apex underneath, and the opposite face open. Find its dimensions in order that the least amount of iron sheet may be used in its construction.",
   "answer_latex": [
    "Angle at apex $= 90°$; equal sides = length = $\\sqrt[3]{2V}$."
   ],
   "answer_markdown": [
    "Angle at apex $= 90°$; equal sides = length = $\\sqrt[3]{2V}$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "a**2 + 4*V/a",
     "answer_expr": "(2*V)**Rational(1,3)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "extremum: 4*a/x + x**2"
   ],
   "shape": [
    "extremum: N*a/x + x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.solve.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/1a",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 1,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Exercise XV, problem 1a",
   "problem_latex": "Differentiate the expression $\\dfrac{x^3}{3} - 2x^3y - 2y^2x + \\dfrac{y}{3}$\nwith respect to $x$~alone, and with respect to $y$~alone.",
   "markdown": "Differentiate the expression $\\dfrac{x^3}{3} - 2x^3y - 2y^2x + \\dfrac{y}{3}$ with respect to $x$ alone, and with respect to $y$ alone.",
   "answer_latex": [
    "$x^3 - 6x^2 y - 2y^2;\\quad \\frac{1}{3} - 2x^3 - 4xy$."
   ],
   "answer_markdown": [
    "$x^3 - 6x^2 y - 2y^2;\\quad \\frac{1}{3} - 2x^3 - 4xy$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": [
      "-90.670617283950617284",
      "-82.606617283950617284"
     ],
     "problem_expr": "x**3/3 - 2*x**3*y - 2*y**2*x + y/3",
     "answer_expr": "x**3 - 6*x**2*y - 2*y**2"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "differentiate: -2*a**2*x - 2*a*x**3 + a/3 + x**3/3"
   ],
   "shape": [
    "differentiate: N*a*x**N + N*a + N*a**N*x + N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/1b",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 1,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Exercise XV, problem 1b",
   "problem_latex": "Differentiate the expression $\\dfrac{x^3}{3} - 2x^3y - 2y^2x + \\dfrac{y}{3}$\nwith respect to $x$~alone, and with respect to $y$~alone.",
   "markdown": "Differentiate the expression $\\dfrac{x^3}{3} - 2x^3y - 2y^2x + \\dfrac{y}{3}$ with respect to $x$ alone, and with respect to $y$ alone.",
   "answer_latex": [
    "$x^3 - 6x^2 y - 2y^2;\\quad \\frac{1}{3} - 2x^3 - 4xy$."
   ],
   "answer_markdown": [
    "$x^3 - 6x^2 y - 2y^2;\\quad \\frac{1}{3} - 2x^3 - 4xy$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**3/3 - 2*x**3*y - 2*y**2*x + y/3",
     "answer_expr": "Rational(1,3) - 2*x**3 - 4*x*y"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: -2*a**3*x + a**3/3 - 2*a*x**2 + x/3"
   ],
   "shape": [
    "differentiate: N*a*x**N + N*a**N*x + N*a**N + N*x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/2x",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 2,
   "part": "x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Exercise XV, problem 2x",
   "problem_latex": "Find the partial differential coefficients with\nrespect to $x$,~$y$ and~$z$, of the expression\n\\[\nx^2yz + xy^2z + xyz^2 + x^2y^2z^2.\n\\]",
   "markdown": "Find the partial differential coefficients with respect to $x$, $y$ and $z$, of the expression x^2yz + xy^2z + xyz^2 + x^2y^2z^2.",
   "answer_latex": [
    "$2xyz + y^2 z + z^2 y + 2xy^2 z^2$;\\\\\n     $2xyz + x^2 z + xz^2 + 2x^2 yz^2$;\\\\\n     $2xyz + x^2 y + xy^2 + 2x^2 y^2 z$."
   ],
   "answer_markdown": [
    "$2xyz + y^2 z + z^2 y + 2xy^2 z^2$; $2xyz + x^2 z + xz^2 + 2x^2 yz^2$; $2xyz + x^2 y + xy^2 + 2x^2 y^2 z$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**2*y*z + x*y**2*z + x*y*z**2 + x**2*y**2*z**2",
     "answer_expr": "2*x*y*z + y**2*z + z**2*y + 2*x*y**2*z**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a**2*b**2*x**2 + a**2*b*x + a*b**2*x + a*b*x**2"
   ],
   "shape": [
    "differentiate: a*b*x**N + a*b**N*x + a**N*b*x + a**N*b**N*x**N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xv/2y",
    "thompson-calculus-made-easy-1914/ex-xv/2z"
   ],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/2y",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 2,
   "part": "y",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Exercise XV, problem 2y",
   "problem_latex": "Find the partial differential coefficients with\nrespect to $x$,~$y$ and~$z$, of the expression\n\\[\nx^2yz + xy^2z + xyz^2 + x^2y^2z^2.\n\\]",
   "markdown": "Find the partial differential coefficients with respect to $x$, $y$ and $z$, of the expression x^2yz + xy^2z + xyz^2 + x^2y^2z^2.",
   "answer_latex": [
    "$2xyz + y^2 z + z^2 y + 2xy^2 z^2$;\\\\\n     $2xyz + x^2 z + xz^2 + 2x^2 yz^2$;\\\\\n     $2xyz + x^2 y + xy^2 + 2x^2 y^2 z$."
   ],
   "answer_markdown": [
    "$2xyz + y^2 z + z^2 y + 2xy^2 z^2$; $2xyz + x^2 z + xz^2 + 2x^2 yz^2$; $2xyz + x^2 y + xy^2 + 2x^2 y^2 z$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**2*y*z + x*y**2*z + x*y*z**2 + x**2*y**2*z**2",
     "answer_expr": "2*x*y*z + x**2*z + x*z**2 + 2*x**2*y*z**2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a**2*b**2*x**2 + a**2*b*x + a*b**2*x + a*b*x**2"
   ],
   "shape": [
    "differentiate: a*b*x**N + a*b**N*x + a**N*b*x + a**N*b**N*x**N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xv/2x",
    "thompson-calculus-made-easy-1914/ex-xv/2z"
   ],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/2z",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 2,
   "part": "z",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Exercise XV, problem 2z",
   "problem_latex": "Find the partial differential coefficients with\nrespect to $x$,~$y$ and~$z$, of the expression\n\\[\nx^2yz + xy^2z + xyz^2 + x^2y^2z^2.\n\\]",
   "markdown": "Find the partial differential coefficients with respect to $x$, $y$ and $z$, of the expression x^2yz + xy^2z + xyz^2 + x^2y^2z^2.",
   "answer_latex": [
    "$2xyz + y^2 z + z^2 y + 2xy^2 z^2$;\\\\\n     $2xyz + x^2 z + xz^2 + 2x^2 yz^2$;\\\\\n     $2xyz + x^2 y + xy^2 + 2x^2 y^2 z$."
   ],
   "answer_markdown": [
    "$2xyz + y^2 z + z^2 y + 2xy^2 z^2$; $2xyz + x^2 z + xz^2 + 2x^2 yz^2$; $2xyz + x^2 y + xy^2 + 2x^2 y^2 z$."
   ],
   "checks": [
    {
     "task": "differentiate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**2*y*z + x*y**2*z + x*y*z**2 + x**2*y**2*z**2",
     "answer_expr": "2*x*y*z + x**2*y + x*y**2 + 2*x**2*y**2*z"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "differentiate: a**2*b**2*x**2 + a**2*b*x + a*b**2*x + a*b*x**2"
   ],
   "shape": [
    "differentiate: a*b*x**N + a*b**N*x + a**N*b*x + a**N*b**N*x**N"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xv/2x",
    "thompson-calculus-made-easy-1914/ex-xv/2y"
   ],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/3a",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 3,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Exercise XV, problem 3a",
   "problem_latex": "Let $r^2 = (x-a)^2 + (y-b)^2 + (z-c)^2$.\n\nFind the value of $\\dfrac{\\partial r}{\\partial x} +\n                   \\dfrac{\\partial r}{\\partial y} +\n                   \\dfrac{\\partial r}{\\partial z}$. Also find the value\nof $\\dfrac{\\partial^2r}{\\partial x^2} +\n    \\dfrac{\\partial^2r}{\\partial y^2} +\n    \\dfrac{\\partial^2r}{\\partial z^2}$.",
   "markdown": "Let $r^2 = (x-a)^2 + (y-b)^2 + (z-c)^2$. Find the value of $\\dfrac{\\partial r}{\\partial x} + \\dfrac{\\partial r}{\\partial y} + \\dfrac{\\partial r}{\\partial z}$. Also find the value of $\\dfrac{\\partial^2r}{\\partial x^2} + \\dfrac{\\partial^2r}{\\partial y^2} + \\dfrac{\\partial^2r}{\\partial z^2}$.",
   "answer_latex": [
    "$\\dfrac{1}{r} \\{ \\left(x - a\\right) + \\left( y - b \\right) + \\left( z - c \\right) \\} = \\dfrac{ \\left( x + y + z \\right) - \\left( a + b + c \\right) }{r}$; $\\dfrac{3}{r}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{r} \\{ \\left(x - a\\right) + \\left( y - b \\right) + \\left( z - c \\right) \\} = \\dfrac{ \\left( x + y + z \\right) - \\left( a + b + c \\right) }{r}$; $\\dfrac{3}{r}$."
   ],
   "checks": [
    {
     "task": "identity",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "diff(sqrt((x-a)**2 + (y-b)**2 + (z-c)**2), x) + diff(sqrt((x-a)**2 + (y-b)**2 + (z-c)**2), y) + diff(sqrt((x-a)**2 + (y-b)**2 + (z-c)**2), z)",
     "answer_expr": "((x-a) + (y-b) + (z-c))/sqrt((x-a)**2 + (y-b)**2 + (z-c)**2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "identity: (-a + d)/sqrt((-a + d)**2 + (-b + e)**2 + (-c + f)**2) + (-b + e)/sqrt((-a + d)**2 + (-b + e)**2 + (-c + f)**2) + (-c + f)/sqrt((-a + d)**2 + (-b + e)**2 + (-c + f)**2)"
   ],
   "shape": [
    "identity: (-a + d)*((-a + d)**N + (-b + e)**N + (-c + f)**N)**N + (-b + e)*((-a + d)**N + (-b + e)**N + (-c + f)**N)**N + (-c + f)*((-a + d)**N + (-b + e)**N + (-c + f)**N)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/3b",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 3,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Exercise XV, problem 3b",
   "problem_latex": "Let $r^2 = (x-a)^2 + (y-b)^2 + (z-c)^2$.\n\nFind the value of $\\dfrac{\\partial r}{\\partial x} +\n                   \\dfrac{\\partial r}{\\partial y} +\n                   \\dfrac{\\partial r}{\\partial z}$. Also find the value\nof $\\dfrac{\\partial^2r}{\\partial x^2} +\n    \\dfrac{\\partial^2r}{\\partial y^2} +\n    \\dfrac{\\partial^2r}{\\partial z^2}$.",
   "markdown": "Let $r^2 = (x-a)^2 + (y-b)^2 + (z-c)^2$. Find the value of $\\dfrac{\\partial r}{\\partial x} + \\dfrac{\\partial r}{\\partial y} + \\dfrac{\\partial r}{\\partial z}$. Also find the value of $\\dfrac{\\partial^2r}{\\partial x^2} + \\dfrac{\\partial^2r}{\\partial y^2} + \\dfrac{\\partial^2r}{\\partial z^2}$.",
   "answer_latex": [
    "$\\dfrac{1}{r} \\{ \\left(x - a\\right) + \\left( y - b \\right) + \\left( z - c \\right) \\} = \\dfrac{ \\left( x + y + z \\right) - \\left( a + b + c \\right) }{r}$; $\\dfrac{3}{r}$."
   ],
   "answer_markdown": [
    "$\\dfrac{1}{r} \\{ \\left(x - a\\right) + \\left( y - b \\right) + \\left( z - c \\right) \\} = \\dfrac{ \\left( x + y + z \\right) - \\left( a + b + c \\right) }{r}$; $\\dfrac{3}{r}$."
   ],
   "checks": [
    {
     "task": "identity",
     "verdict": "FLAG-MISMATCH",
     "judge_why": [
      "1.3744307877906782497",
      "2.0616461816860173745"
     ],
     "problem_expr": "diff(sqrt((x-a)**2 + (y-b)**2 + (z-c)**2), x, 2) + diff(sqrt((x-a)**2 + (y-b)**2 + (z-c)**2), y, 2) + diff(sqrt((x-a)**2 + (y-b)**2 + (z-c)**2), z, 2)",
     "answer_expr": "3/sqrt((x-a)**2 + (y-b)**2 + (z-c)**2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "identity: (-(a - d)**2/((a - d)**2 + (b - e)**2 + (c - f)**2) + 1)/sqrt((a - d)**2 + (b - e)**2 + (c - f)**2) + (-(b - e)**2/((a - d)**2 + (b - e)**2 + (c - f)**2) + 1)/sqrt((a - d)**2 + (b - e)**2 + (c - f)**2) + (-(c - f)**2/((a - d)**2 + (b - e)**2 + (c - f)**2) + 1)/sqrt((a - d)**2 + (b - e)**2 + (c - f)**2)"
   ],
   "shape": [
    "identity: (-(a - d)**N/((a - d)**N + (b - e)**N + (c - f)**N) + 1)*((a - d)**N + (b - e)**N + (c - f)**N)**N + (-(b - e)**N/((a - d)**N + (b - e)**N + (c - f)**N) + 1)*((a - d)**N + (b - e)**N + (c - f)**N)**N + (-(c - f)**N/((a - d)**N + (b - e)**N + (c - f)**N) + 1)*((a - d)**N + (b - e)**N + (c - f)**N)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/4",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "180",
   "location": "Exercise XV, problem 4",
   "problem_latex": "Find the total differential of~$y=u^v$.",
   "markdown": "Find the total differential of $y=u^v$.",
   "answer_latex": [
    "$dy = vu^{v-1}\\, du + u^v \\log_\\epsilon u\\, dv$."
   ],
   "answer_markdown": [
    "$dy = vu^{v-1}\\, du + u^v \\log_\\epsilon u\\, dv$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "u**v",
     "answer_expr": "v*u**(v-1)*du + u**v*log(u)*dv"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/51",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 5,
   "part": "1",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 51",
   "problem_latex": "Find the total differential of $y=u^3 \\sin v$; of\n$y = (\\sin x)^u$; and of $y = \\dfrac{\\log_\\epsilon u}{v}$.",
   "markdown": "Find the total differential of $y=u^3 \\sin v$; of $y = (\\sin x)^u$; and of $y = \\dfrac{\\log_\\epsilon u}{v}$.",
   "answer_latex": [
    "$dy = 3\\sin v u^2\\, du + u^3 \\cos v\\, dv$,\\\\\n     $dy = u \\sin x^{u-1} \\cos x\\, dx + (\\sin x)^u \\log_\\epsilon \\sin x du$,\\\\\n     $dy = \\dfrac{1}{v}\\, \\dfrac{1}{u}\\, du - \\log_\\epsilon u \\dfrac{1}{v^2}\\, dv$."
   ],
   "answer_markdown": [
    "$dy = 3\\sin v u^2\\, du + u^3 \\cos v\\, dv$, $dy = u \\sin x^{u-1} \\cos x\\, dx + (\\sin x)^u \\log_\\epsilon \\sin x du$, $dy = \\dfrac{1}{v}\\, \\dfrac{1}{u}\\, du - \\log_\\epsilon u \\dfrac{1}{v^2}\\, dv$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "u**3*sin(v)",
     "answer_expr": "3*sin(v)*u**2*du + u**3*cos(v)*dv"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/52",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 5,
   "part": "2",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 52",
   "problem_latex": "Find the total differential of $y=u^3 \\sin v$; of\n$y = (\\sin x)^u$; and of $y = \\dfrac{\\log_\\epsilon u}{v}$.",
   "markdown": "Find the total differential of $y=u^3 \\sin v$; of $y = (\\sin x)^u$; and of $y = \\dfrac{\\log_\\epsilon u}{v}$.",
   "answer_latex": [
    "$dy = 3\\sin v u^2\\, du + u^3 \\cos v\\, dv$,\\\\\n     $dy = u \\sin x^{u-1} \\cos x\\, dx + (\\sin x)^u \\log_\\epsilon \\sin x du$,\\\\\n     $dy = \\dfrac{1}{v}\\, \\dfrac{1}{u}\\, du - \\log_\\epsilon u \\dfrac{1}{v^2}\\, dv$."
   ],
   "answer_markdown": [
    "$dy = 3\\sin v u^2\\, du + u^3 \\cos v\\, dv$, $dy = u \\sin x^{u-1} \\cos x\\, dx + (\\sin x)^u \\log_\\epsilon \\sin x du$, $dy = \\dfrac{1}{v}\\, \\dfrac{1}{u}\\, du - \\log_\\epsilon u \\dfrac{1}{v^2}\\, dv$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "sin(x)**u",
     "answer_expr": "u*sin(x)**(u-1)*cos(x)*dx + sin(x)**u*log(sin(x))*du"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/53",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 5,
   "part": "3",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 53",
   "problem_latex": "Find the total differential of $y=u^3 \\sin v$; of\n$y = (\\sin x)^u$; and of $y = \\dfrac{\\log_\\epsilon u}{v}$.",
   "markdown": "Find the total differential of $y=u^3 \\sin v$; of $y = (\\sin x)^u$; and of $y = \\dfrac{\\log_\\epsilon u}{v}$.",
   "answer_latex": [
    "$dy = 3\\sin v u^2\\, du + u^3 \\cos v\\, dv$,\\\\\n     $dy = u \\sin x^{u-1} \\cos x\\, dx + (\\sin x)^u \\log_\\epsilon \\sin x du$,\\\\\n     $dy = \\dfrac{1}{v}\\, \\dfrac{1}{u}\\, du - \\log_\\epsilon u \\dfrac{1}{v^2}\\, dv$."
   ],
   "answer_markdown": [
    "$dy = 3\\sin v u^2\\, du + u^3 \\cos v\\, dv$, $dy = u \\sin x^{u-1} \\cos x\\, dx + (\\sin x)^u \\log_\\epsilon \\sin x du$, $dy = \\dfrac{1}{v}\\, \\dfrac{1}{u}\\, du - \\log_\\epsilon u \\dfrac{1}{v^2}\\, dv$."
   ],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-TASK",
     "judge_why": "task 'other' is not judged in the pilot",
     "problem_expr": "log(u)/v",
     "answer_expr": "(1/v)*(1/u)*du - log(u)*(1/v**2)*dv"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-TASK"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/6",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 6",
   "problem_latex": "Verify that the sum of three quantities $x$,~$y$,~$z$,\nwhose product is a constant~$k$, is maximum when\nthese three quantities are equal.",
   "markdown": "Verify that the sum of three quantities $x$, $y$, $z$, whose product is a constant $k$, is maximum when these three quantities are equal.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "x + y + k/(x*y)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "cas.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/7x",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 7,
   "part": "x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 7x",
   "problem_latex": "Find the maximum or minimum of the function\n\\[\nu = x + 2xy + y.\n\\]",
   "markdown": "Find the maximum or minimum of the function u = x + 2xy + y.",
   "answer_latex": [
    "Minimum for $x = y = -\\frac{1}{2}$."
   ],
   "answer_markdown": [
    "Minimum for $x = y = -\\frac{1}{2}$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "f'(x0) is not 0: ('8.0', '0.0')",
     "problem_expr": "x + 2*x*y + y",
     "answer_expr": "-Rational(1,2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: 2*a*x + a + x"
   ],
   "shape": [
    "extremum: N*a*x + a + x"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xv/7y"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/7y",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 7,
   "part": "y",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 7y",
   "problem_latex": "Find the maximum or minimum of the function\n\\[\nu = x + 2xy + y.\n\\]",
   "markdown": "Find the maximum or minimum of the function u = x + 2xy + y.",
   "answer_latex": [
    "Minimum for $x = y = -\\frac{1}{2}$."
   ],
   "answer_markdown": [
    "Minimum for $x = y = -\\frac{1}{2}$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "f'(x0) is not 0: ('6.047619047619047619', '0.0')",
     "problem_expr": "x + 2*x*y + y",
     "answer_expr": "-Rational(1,2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: 2*a*x + a + x"
   ],
   "shape": [
    "extremum: N*a*x + a + x"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xv/7x"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.graph"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/8a",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 8,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 8a",
   "problem_latex": "The post-office regulations state that no parcel\nis to be of such a size that its length plus its girth\nexceeds $6$~feet. What is the greatest volume that\ncan be sent by post (\\textit{a})~in the case of a package of\nrectangular cross section; (\\textit{b})~in the case of a package\nof circular cross section.",
   "markdown": "The post-office regulations state that no parcel is to be of such a size that its length plus its girth exceeds $6$ feet. What is the greatest volume that can be sent by post (*a*) in the case of a package of rectangular cross section; (*b*) in the case of a package of circular cross section.",
   "answer_latex": [
    "(\\textit{a}) Length $2$~feet, width = depth = $1$~foot, vol.\\ = $2$~cubic\n feet.\n\n (\\textit{b}) Radius = $\\dfrac{2}{\\pi}$ feet = $7.46$~in., length = $2$~feet, vol.\\ = $2.54$."
   ],
   "answer_markdown": [
    "(*a*) Length $2$ feet, width = depth = $1$ foot, vol. = $2$ cubic feet. (*b*) Radius = $\\dfrac{2}{\\pi}$ feet = $7.46$ in., length = $2$ feet, vol. = $2.54$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "s**2*(6 - 4*s)",
     "answer_expr": "1"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 2.0, printed 2 (half-unit 0.5; correctly rounded at the printed digits: 2.0)",
     "problem_expr": "6 - 4*s",
     "answer_expr": "1"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 2.0, printed 2 (half-unit 0.5; correctly rounded at the printed digits: 2.0)",
     "problem_expr": "s**2*(6 - 4*s)",
     "answer_expr": "1"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED",
     "PASS",
     "PASS"
    ]
   },
   "form": [
    "extremum: x**2*(-4*x + 6)",
    "evaluate: -4*x + 6 at s=1",
    "evaluate: x**2*(-4*x + 6) at s=1"
   ],
   "shape": [
    "evaluate: N*x + N",
    "evaluate: x**N*(N*x + N)",
    "extremum: x**N*(N*x + N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.arith"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/8b",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 8,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 8b",
   "problem_latex": "The post-office regulations state that no parcel\nis to be of such a size that its length plus its girth\nexceeds $6$~feet. What is the greatest volume that\ncan be sent by post (\\textit{a})~in the case of a package of\nrectangular cross section; (\\textit{b})~in the case of a package\nof circular cross section.",
   "markdown": "The post-office regulations state that no parcel is to be of such a size that its length plus its girth exceeds $6$ feet. What is the greatest volume that can be sent by post (*a*) in the case of a package of rectangular cross section; (*b*) in the case of a package of circular cross section.",
   "answer_latex": [
    "(\\textit{a}) Length $2$~feet, width = depth = $1$~foot, vol.\\ = $2$~cubic\n feet.\n\n (\\textit{b}) Radius = $\\dfrac{2}{\\pi}$ feet = $7.46$~in., length = $2$~feet, vol.\\ = $2.54$."
   ],
   "answer_markdown": [
    "(*a*) Length $2$ feet, width = depth = $1$ foot, vol. = $2$ cubic feet. (*b*) Radius = $\\dfrac{2}{\\pi}$ feet = $7.46$ in., length = $2$ feet, vol. = $2.54$."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "pi*r**2*(6 - 2*pi*r)",
     "answer_expr": "2/pi"
    },
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 7.63943726841, printed 7.46 (half-unit 0.005; correctly rounded at the printed digits: 7.64)",
     "problem_expr": "12*r",
     "answer_expr": "2/pi"
    },
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 2.0, printed 2 (half-unit 0.5; correctly rounded at the printed digits: 2.0)",
     "problem_expr": "6 - 2*pi*r",
     "answer_expr": "2/pi"
    },
    {
     "task": "evaluate",
     "verdict": "PASS-LOOSE",
     "judge_why": "computed 2.54647908947, printed 2.54 (half-unit 0.005; correctly rounded at the printed digits: 2.55): one unit off in the last printed place",
     "problem_expr": "pi*r**2*(6 - 2*pi*r)",
     "answer_expr": "2/pi"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED",
     "FLAG-MISMATCH",
     "PASS",
     "PASS-LOOSE"
    ]
   },
   "form": [
    "extremum: pi*x**2*(-2*pi*x + 6)",
    "evaluate: 12*x at r=2/pi",
    "evaluate: -2*pi*x + 6 at r=2/pi",
    "evaluate: pi*x**2*(-2*pi*x + 6) at r=2/pi"
   ],
   "shape": [
    "evaluate: N*x",
    "evaluate: pi*N*x + N",
    "evaluate: pi*x**N*(pi*N*x + N)",
    "extremum: pi*x**N*(pi*N*x + N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.derive",
    "cas.solve.poly",
    "core.arith",
    "core.const",
    "core.units"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/9x",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 9,
   "part": "x",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 9x",
   "problem_latex": "Divide $\\pi$ into $3$~parts such that the continued\nproduct of their sines may be a maximum or minimum.",
   "markdown": "Divide $\\pi$ into $3$ parts such that the continued product of their sines may be a maximum or minimum.",
   "answer_latex": [
    "All three parts equal; the product is maximum."
   ],
   "answer_markdown": [
    "All three parts equal; the product is maximum."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "f'(x0) is not 0: ('-0.55447389546524276799', '0.0')",
     "problem_expr": "sin(x)*sin(y)*sin(x + y)",
     "answer_expr": "pi/3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: sin(a)*sin(x)*sin(a + x)"
   ],
   "shape": [
    "extremum: sin(a)*sin(x)*sin(a + x)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xv/9y"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.nonpoly",
    "cas.trig",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xv/9y",
   "set": "thompson-calculus-made-easy-1914/ex-xv",
   "number": 9,
   "part": "y",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "181",
   "location": "Exercise XV, problem 9y",
   "problem_latex": "Divide $\\pi$ into $3$~parts such that the continued\nproduct of their sines may be a maximum or minimum.",
   "markdown": "Divide $\\pi$ into $3$ parts such that the continued product of their sines may be a maximum or minimum.",
   "answer_latex": [
    "All three parts equal; the product is maximum."
   ],
   "answer_markdown": [
    "All three parts equal; the product is maximum."
   ],
   "checks": [
    {
     "task": "extremum",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "f'(x0) is not 0: ('-0.089716881482863410823', '0.0')",
     "problem_expr": "sin(x)*sin(y)*sin(x + y)",
     "answer_expr": "pi/3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "extremum: sin(a)*sin(x)*sin(a + x)"
   ],
   "shape": [
    "extremum: sin(a)*sin(x)*sin(a + x)"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xv/9x"
   ],
   "needs": [
    "cas.derive",
    "cas.solve.nonpoly",
    "cas.trig",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvi/1",
   "set": "thompson-calculus-made-easy-1914/ex-xvi",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "190",
   "location": "Exercise XVI, problem 1",
   "problem_latex": "Find the ultimate sum of $\\frac{2}{3} + \\frac{1}{3} + \\frac{1}{6} + \\frac{1}{12} + \\frac{1}{24} + \\text{etc}$.",
   "markdown": "Find the ultimate sum of $\\frac{2}{3} + \\frac{1}{3} + \\frac{1}{6} + \\frac{1}{12} + \\frac{1}{24} + \\text{etc}$.",
   "answer_latex": [
    "$1\\frac{1}{3}$."
   ],
   "answer_markdown": [
    "$1\\frac{1}{3}$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 1.33333333333, printed 4/3 (half-unit 1.0e-40; correctly rounded at the printed digits: 1.33333333333)",
     "problem_expr": "Rational(2,3) + Rational(1,3)/(1 - Rational(1,2))",
     "answer_expr": "Rational(4,3)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: 4/3"
   ],
   "shape": [
    "evaluate: N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.sum",
    "core.arith",
    "core.frac"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvi/2",
   "set": "thompson-calculus-made-easy-1914/ex-xvi",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "190",
   "location": "Exercise XVI, problem 2",
   "problem_latex": "Show that the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\frac{1}{5} - \\frac{1}{6} + \\frac{1}{7}$\\DPnote{[** TN: [sic], no +]}~etc.,\nis convergent, and find its sum to $8$~terms.",
   "markdown": "Show that the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\frac{1}{5} - \\frac{1}{6} + \\frac{1}{7}$ etc., is convergent, and find its sum to $8$ terms.",
   "answer_latex": [
    "$0.6344$."
   ],
   "answer_markdown": [
    "$0.6344$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 0.634523809524, printed 0.6344 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.6345)",
     "problem_expr": "1 - Rational(1,2) + Rational(1,3) - Rational(1,4) + Rational(1,5) - Rational(1,6) + Rational(1,7) - Rational(1,8)",
     "answer_expr": "0.6344"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: 533/840"
   ],
   "shape": [
    "evaluate: N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.sum",
    "core.arith",
    "core.frac",
    "other:convergence test (alternating series)"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvi/3",
   "set": "thompson-calculus-made-easy-1914/ex-xvi",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "190",
   "location": "Exercise XVI, problem 3",
   "problem_latex": "If $\\log_\\epsilon(1+x) = x - \\dfrac{x^2}{2} + \\dfrac{x^3}{3} - \\dfrac{x^4}{4} + \\text{etc}$., find $\\log_\\epsilon 1.3$.",
   "markdown": "If $\\log_\\epsilon(1+x) = x - \\dfrac{x^2}{2} + \\dfrac{x^3}{3} - \\dfrac{x^4}{4} + \\text{etc}$., find $\\log_\\epsilon 1.3$.",
   "answer_latex": [
    "$0.2624$."
   ],
   "answer_markdown": [
    "$0.2624$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.262364264467, printed 0.2624 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.2624)",
     "problem_expr": "log(Rational(13,10))",
     "answer_expr": "0.2624"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: log(13/10)"
   ],
   "shape": [
    "evaluate: log(N)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.sum",
    "core.arith",
    "core.frac",
    "core.log"
   ],
   "expectation": "X=0.2624",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvi/4a",
   "set": "thompson-calculus-made-easy-1914/ex-xvi",
   "number": 4,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "190",
   "location": "Exercise XVI, problem 4a",
   "problem_latex": "Following a reasoning similar to that explained\nin this chapter, find~$y$,\n\\[\n\\text{(\\textit{a}) if $\\frac{dy}{dx} = \\tfrac{1}{4} x$;\\quad\n(\\textit{b}) if $\\frac{dy}{dx} = \\cos x$.}\n\\]",
   "markdown": "Following a reasoning similar to that explained in this chapter, find $y$, (*a*) if $\\frac{dy}{dx} = \\tfrac{1}{4} x$; (*b*) if $\\frac{dy}{dx} = \\cos x$.",
   "answer_latex": [
    "(\\textit{a}) $y = \\frac{1}{8} x^2 + C$;\\quad\n     (\\textit{b}) $y = \\sin x + C$."
   ],
   "answer_markdown": [
    "(*a*) $y = \\frac{1}{8} x^2 + C$; (*b*) $y = \\sin x + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x/4",
     "answer_expr": "x**2/8"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: x/4"
   ],
   "shape": [
    "integrate: N*x"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvi/4b",
   "set": "thompson-calculus-made-easy-1914/ex-xvi",
   "number": 4,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "190",
   "location": "Exercise XVI, problem 4b",
   "problem_latex": "Following a reasoning similar to that explained\nin this chapter, find~$y$,\n\\[\n\\text{(\\textit{a}) if $\\frac{dy}{dx} = \\tfrac{1}{4} x$;\\quad\n(\\textit{b}) if $\\frac{dy}{dx} = \\cos x$.}\n\\]",
   "markdown": "Following a reasoning similar to that explained in this chapter, find $y$, (*a*) if $\\frac{dy}{dx} = \\tfrac{1}{4} x$; (*b*) if $\\frac{dy}{dx} = \\cos x$.",
   "answer_latex": [
    "(\\textit{a}) $y = \\frac{1}{8} x^2 + C$;\\quad\n     (\\textit{b}) $y = \\sin x + C$."
   ],
   "answer_markdown": [
    "(*a*) $y = \\frac{1}{8} x^2 + C$; (*b*) $y = \\sin x + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "cos(x)",
     "answer_expr": "sin(x)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: cos(x)"
   ],
   "shape": [
    "integrate: cos(x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvi/5",
   "set": "thompson-calculus-made-easy-1914/ex-xvi",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "190",
   "location": "Exercise XVI, problem 5",
   "problem_latex": "If $\\dfrac{dy}{dx} = 2x + 3$, find~$y$.",
   "markdown": "If $\\dfrac{dy}{dx} = 2x + 3$, find $y$.",
   "answer_latex": [
    "$y = x^2 + 3x + C$."
   ],
   "answer_markdown": [
    "$y = x^2 + 3x + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "2*x + 3",
     "answer_expr": "x**2 + 3*x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: 2*x + 3"
   ],
   "shape": [
    "integrate: N*x + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/1",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 1,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 1",
   "problem_latex": "Find  $\\ds\\int y\\, dx$ when  $y^2 = 4 ax$.",
   "markdown": "Find $\\ds\\int y\\, dx$ when $y^2 = 4 ax$.",
   "answer_latex": [
    "$\\dfrac{4\\sqrt{a} x^{\\efrac{3}{2}}}{3} + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{4\\sqrt{a} x^{\\efrac{3}{2}}}{3} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "2*sqrt(a)*sqrt(x)",
     "answer_expr": "4*sqrt(a)*x**Rational(3,2)/3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "integrate: 2*sqrt(a)*sqrt(x)"
   ],
   "shape": [
    "integrate: N*a**N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.solve.nonpoly"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/10",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 10,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 10",
   "problem_latex": "Find $\\ds\\int (x + 2)(x - a)\\, dx$.",
   "markdown": "Find $\\ds\\int (x + 2)(x - a)\\, dx$.",
   "answer_latex": [
    "$\\dfrac{x^3}{3} + \\dfrac{2 - a}{2} x^2 - 2ax + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{x^3}{3} + \\dfrac{2 - a}{2} x^2 - 2ax + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x + 2)*(x - a)",
     "answer_expr": "x**3/3 + (2 - a)/2*x**2 - 2*a*x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: (-a + x)*(x + 2)"
   ],
   "shape": [
    "integrate: (N + x)*(-a + x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.expand",
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/11",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 11,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 11",
   "problem_latex": "Find $\\ds\\int (\\sqrt x + \\sqrt[3] x) 3a^2\\, dx$.",
   "markdown": "Find $\\ds\\int (\\sqrt x + \\sqrt[3] x) 3a^2\\, dx$.",
   "answer_latex": [
    "$a^2(2x^{\\efrac{3}{2}} + \\tfrac{9}{4} x^{\\efrac{4}{3}}) + C$."
   ],
   "answer_markdown": [
    "$a^2(2x^{\\efrac{3}{2}} + \\tfrac{9}{4} x^{\\efrac{4}{3}}) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(sqrt(x) + x**Rational(1,3))*3*a**2",
     "answer_expr": "a**2*(2*x**Rational(3,2) + Rational(9,4)*x**Rational(4,3))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: a**2*(3*x**(1/3) + 3*sqrt(x))"
   ],
   "shape": [
    "integrate: 2*N*a**N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/12",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 12,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 12",
   "problem_latex": "Find $\\ds\\int (\\sin \\theta - \\tfrac{1}{2})\\, \\frac{d\\theta}{3}$.",
   "markdown": "Find $\\ds\\int (\\sin \\theta - \\tfrac{1}{2})\\, \\frac{d\\theta}{3}$.",
   "answer_latex": [
    "$-\\tfrac{1}{3} \\cos\\theta - \\tfrac{1}{6} \\theta + C$."
   ],
   "answer_markdown": [
    "$-\\tfrac{1}{3} \\cos\\theta - \\tfrac{1}{6} \\theta + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS-INTERPRETED",
     "judge_why": null,
     "problem_expr": "(sin(theta) - Rational(1,2))/3",
     "answer_expr": "-cos(theta)/3 - theta/6"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-INTERPRETED"
    ]
   },
   "form": [
    "integrate: sin(x)/3 - 1/6"
   ],
   "shape": [
    "integrate: N*sin(x) + N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/13",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 13,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 13",
   "problem_latex": "Find  $\\ds\\int \\cos^2 a \\theta\\, d\\theta$.",
   "markdown": "Find $\\ds\\int \\cos^2 a \\theta\\, d\\theta$.",
   "answer_latex": [
    "$\\dfrac{\\theta}{2} + \\dfrac{\\sin 2a\\theta}{4a} + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{\\theta}{2} + \\dfrac{\\sin 2a\\theta}{4a} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "cos(a*theta)**2",
     "answer_expr": "theta/2 + sin(2*a*theta)/(4*a)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: cos(a*x)**2"
   ],
   "shape": [
    "integrate: cos(a*x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.integrate.subst",
    "cas.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/14",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 14,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 14",
   "problem_latex": "Find  $\\ds\\int \\sin^2 \\theta\\, d\\theta$.",
   "markdown": "Find $\\ds\\int \\sin^2 \\theta\\, d\\theta$.",
   "answer_latex": [
    "$\\dfrac{\\theta}{2} - \\dfrac{\\sin 2\\theta}{4} + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{\\theta}{2} - \\dfrac{\\sin 2\\theta}{4} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin(theta)**2",
     "answer_expr": "theta/2 - sin(2*theta)/4"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: sin(x)**2"
   ],
   "shape": [
    "integrate: sin(x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/15",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 15,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 15",
   "problem_latex": "Find  $\\ds\\int \\sin^2 a \\theta\\, d\\theta$.",
   "markdown": "Find $\\ds\\int \\sin^2 a \\theta\\, d\\theta$.",
   "answer_latex": [
    "$\\dfrac{\\theta}{2} - \\dfrac{\\sin 2a\\theta}{4a} + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{\\theta}{2} - \\dfrac{\\sin 2a\\theta}{4a} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "sin(a*theta)**2",
     "answer_expr": "theta/2 - sin(2*a*theta)/(4*a)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: sin(a*x)**2"
   ],
   "shape": [
    "integrate: sin(a*x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.integrate.subst",
    "cas.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/16",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 16,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 16",
   "problem_latex": "Find  $\\ds\\int \\epsilon^{3x}\\, dx$.",
   "markdown": "Find $\\ds\\int \\epsilon^{3x}\\, dx$.",
   "answer_latex": [
    "$\\tfrac{1}{3} \\epsilon^{3x}$. % [F1: +C?]"
   ],
   "answer_markdown": [
    "$\\tfrac{1}{3} \\epsilon^{3x}$. % [F1: +C?]"
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "exp(3*x)",
     "answer_expr": "exp(3*x)/3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: exp(3*x)"
   ],
   "shape": [
    "integrate: exp(N*x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.integrate.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/17",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 17,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 17",
   "problem_latex": "Find  $\\ds\\int \\dfrac{dx}{1 + x}$.",
   "markdown": "Find $\\ds\\int \\dfrac{dx}{1 + x}$.",
   "answer_latex": [
    "$\\log(1 + x) + C$."
   ],
   "answer_markdown": [
    "$\\log(1 + x) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS-ABS-CONVENTION",
     "judge_why": "answer not real on [-55/14, -124/59] where the integrand is; with log|u| for log(u) it holds on ['[29/19, 57/22]', '[-55/14, -124/59]']",
     "problem_expr": "1/(1 + x)",
     "answer_expr": "log(1 + x)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ABS-CONVENTION"
    ]
   },
   "form": [
    "integrate: 1/(x + 1)"
   ],
   "shape": [
    "integrate: 1/(x + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.integrate.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/18",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 18,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 18",
   "problem_latex": "Find  $\\ds\\int \\dfrac{dx}{1 - x}$.",
   "markdown": "Find $\\ds\\int \\dfrac{dx}{1 - x}$.",
   "answer_latex": [
    "$-\\log_\\epsilon (1 - x) + C$."
   ],
   "answer_markdown": [
    "$-\\log_\\epsilon (1 - x) + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS-ABS-CONVENTION",
     "judge_why": "answer not real on [9/4, 271/80] where the integrand is; with log|u| for log(u) it holds on ['[9/4, 271/80]', '[-39/11, -35/44]']",
     "problem_expr": "1/(1 - x)",
     "answer_expr": "-log(1 - x)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-ABS-CONVENTION"
    ]
   },
   "form": [
    "integrate: 1/(-x + 1)"
   ],
   "shape": [
    "integrate: 1/(-x + 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.integrate.subst"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/2",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 2",
   "problem_latex": "Find  $\\ds\\int \\frac{3}{x^4}\\, dx$.",
   "markdown": "Find $\\ds\\int \\frac{3}{x^4}\\, dx$.",
   "answer_latex": [
    "$-\\dfrac{1}{x^3} + C$."
   ],
   "answer_markdown": [
    "$-\\dfrac{1}{x^3} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "3/x**4",
     "answer_expr": "-1/x**3"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: 3/x**4"
   ],
   "shape": [
    "integrate: N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/3",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 3,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 3",
   "problem_latex": "Find  $\\ds\\int \\frac{1}{a} x^3\\, dx$.",
   "markdown": "Find $\\ds\\int \\frac{1}{a} x^3\\, dx$.",
   "answer_latex": [
    "$\\dfrac{x^4}{4a} + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{x^4}{4a} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**3/a",
     "answer_expr": "x**4/(4*a)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: x**3/a"
   ],
   "shape": [
    "integrate: x**N/a"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/4",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 4,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 4",
   "problem_latex": "Find  $\\ds\\int (x^2 + a)\\, dx$.",
   "markdown": "Find $\\ds\\int (x^2 + a)\\, dx$.",
   "answer_latex": [
    "$\\tfrac{1}{3} x^3 + ax + C$."
   ],
   "answer_markdown": [
    "$\\tfrac{1}{3} x^3 + ax + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "x**2 + a",
     "answer_expr": "x**3/3 + a*x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: a + x**2"
   ],
   "shape": [
    "integrate: a + x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/5",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 5,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 5",
   "problem_latex": "Integrate $5x^{-\\efrac{7}{2}}$.",
   "markdown": "Integrate $5x^{-\\efrac{7}{2}}$.",
   "answer_latex": [
    "$-2x^{-\\efrac{5}{2}} + C$."
   ],
   "answer_markdown": [
    "$-2x^{-\\efrac{5}{2}} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "5*x**(-Rational(7,2))",
     "answer_expr": "-2*x**(-Rational(5,2))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: 5/x**(7/2)"
   ],
   "shape": [
    "integrate: N*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/6",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 6",
   "problem_latex": "Find  $\\ds\\int (4x^3 + 3x^2 + 2x + 1)\\, dx$.",
   "markdown": "Find $\\ds\\int (4x^3 + 3x^2 + 2x + 1)\\, dx$.",
   "answer_latex": [
    "$x^4 + x^3 + x^2 + x + C$."
   ],
   "answer_markdown": [
    "$x^4 + x^3 + x^2 + x + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "4*x**3 + 3*x**2 + 2*x + 1",
     "answer_expr": "x**4 + x**3 + x**2 + x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: 4*x**3 + 3*x**2 + 2*x + 1"
   ],
   "shape": [
    "integrate: N*x + 2*N*x**N + 1"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/7",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 7",
   "problem_latex": "If $\\dfrac{dy}{dx} = \\dfrac{ax}{2} + \\dfrac{bx^2}{3} + \\dfrac{cx^3}{4}$; find~$y$.",
   "markdown": "If $\\dfrac{dy}{dx} = \\dfrac{ax}{2} + \\dfrac{bx^2}{3} + \\dfrac{cx^3}{4}$; find $y$.",
   "answer_latex": [
    "$\\dfrac{ax^2}{4} + \\dfrac{bx^3}{9} + \\dfrac{cx^4}{16} + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{ax^2}{4} + \\dfrac{bx^3}{9} + \\dfrac{cx^4}{16} + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "a*x/2 + b*x**2/3 + c*x**3/4",
     "answer_expr": "a*x**2/4 + b*x**3/9 + c*x**4/16"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: a*x/2 + b*x**2/3 + c*x**3/4"
   ],
   "shape": [
    "integrate: N*a*x + N*b*x**N + N*c*x**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/8",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 8",
   "problem_latex": "Find $\\ds\\int \\left(\\frac{x^2 + a}{x + a}\\right) dx$.",
   "markdown": "Find $\\ds\\int \\left(\\frac{x^2 + a}{x + a}\\right) dx$.",
   "answer_latex": [
    "{\\loosen $\\dfrac{x^2 + a}{x + a} = x - a + \\dfrac{a^2 + a}{x + a}$ by division. Therefore the answer\nis $\\dfrac{x^2}{2} - ax + (a^2 + a)\\log_\\epsilon (x + a) + C$.}\n  (See pages \\pageref{intex1} and~\\pageref{intex2}.)"
   ],
   "answer_markdown": [
    "0.5em plus 0.5em minus 0.25em$\\dfrac{x^2 + a}{x + a} = x - a + \\dfrac{a^2 + a}{x + a}$ by division. Therefore the answer is $\\dfrac{x^2}{2} - ax + (a^2 + a)\\log_\\epsilon (x + a) + C$. (See pages and .)"
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x**2 + a)/(x + a)",
     "answer_expr": "x**2/2 - a*x + (a**2 + a)*log(x + a)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: (a + x**2)/(a + x)"
   ],
   "shape": [
    "integrate: (a + x**N)/(a + x)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.integrate",
    "cas.pdiv"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xvii/9",
   "set": "thompson-calculus-made-easy-1914/ex-xvii",
   "number": 9,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "205",
   "location": "Exercise XVII, problem 9",
   "problem_latex": "Find $\\ds\\int (x + 3)^3\\, dx$.",
   "markdown": "Find $\\ds\\int (x + 3)^3\\, dx$.",
   "answer_latex": [
    "$\\dfrac{x^4}{4} + 3x^3 + \\dfrac{27}{2} x^2 + 27x + C$."
   ],
   "answer_markdown": [
    "$\\dfrac{x^4}{4} + 3x^3 + \\dfrac{27}{2} x^2 + 27x + C$."
   ],
   "checks": [
    {
     "task": "integrate",
     "verdict": "PASS",
     "judge_why": null,
     "problem_expr": "(x + 3)**3",
     "answer_expr": "x**4/4 + 3*x**3 + Rational(27,2)*x**2 + 27*x"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "integrate: (x + 3)**3"
   ],
   "shape": [
    "integrate: (N + x)**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.expand",
    "cas.integrate"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/10a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 10,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 10a",
   "problem_latex": "Find the area of the portion of the curve\n$xy=a$ included between $x=1$ and $x = a$. Find the\nmean ordinate between these limits.",
   "markdown": "Find the area of the portion of the curve $xy=a$ included between $x=1$ and $x = a$. Find the mean ordinate between these limits.",
   "answer_latex": [
    "$a\\log_\\epsilon a$,\\quad $\\dfrac{a}{a - 1} \\log_\\epsilon a$."
   ],
   "answer_markdown": [
    "$a\\log_\\epsilon a$, $\\dfrac{a}{a - 1} \\log_\\epsilon a$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "values do not cover ['a']",
     "problem_expr": "Integral(a/x, (x, 1, a))",
     "answer_expr": "a*log(a)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: Integral(a/x, (x, 1, a))"
   ],
   "shape": [
    "evaluate: Integral(a/x, (x, 1, a))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/10b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 10,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 10b",
   "problem_latex": "Find the area of the portion of the curve\n$xy=a$ included between $x=1$ and $x = a$. Find the\nmean ordinate between these limits.",
   "markdown": "Find the area of the portion of the curve $xy=a$ included between $x=1$ and $x = a$. Find the mean ordinate between these limits.",
   "answer_latex": [
    "$a\\log_\\epsilon a$,\\quad $\\dfrac{a}{a - 1} \\log_\\epsilon a$."
   ],
   "answer_markdown": [
    "$a\\log_\\epsilon a$, $\\dfrac{a}{a - 1} \\log_\\epsilon a$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "values do not cover ['a']",
     "problem_expr": "Integral(a/x, (x, 1, a))/(a - 1)",
     "answer_expr": "a*log(a)/(a - 1)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: Integral(a/x, (x, 1, a))/(a - 1)"
   ],
   "shape": [
    "evaluate: Integral(a/x, (x, 1, a))/(a - 1)"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate",
    "core.log"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/11a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 11,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 11a",
   "problem_latex": "Show that the quadratic mean of the function\n$y=\\sin x$, between the limits of $0$~and~$\\pi$ radians, is~$\\dfrac{\\sqrt2}{2}$.\nFind also the arithmetical mean of the same\nfunction between the same limits; and show that the\nform-factor is~$=1.11$.",
   "markdown": "Show that the quadratic mean of the function $y=\\sin x$, between the limits of $0$ and $\\pi$ radians, is $\\dfrac{\\sqrt2}{2}$. Find also the arithmetical mean of the same function between the same limits; and show that the form-factor is $=1.11$.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "sqrt(Integral(sin(x)**2, (x, 0, pi))/pi)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [
    "evaluate: sqrt(Integral(sin(x)**2, (x, 0, pi)))/sqrt(pi)"
   ],
   "shape": [
    "evaluate: pi**N*Integral(sin(x)**N, (x, 0, pi))**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.trig",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/11b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 11,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 11b",
   "problem_latex": "Show that the quadratic mean of the function\n$y=\\sin x$, between the limits of $0$~and~$\\pi$ radians, is~$\\dfrac{\\sqrt2}{2}$.\nFind also the arithmetical mean of the same\nfunction between the same limits; and show that the\nform-factor is~$=1.11$.",
   "markdown": "Show that the quadratic mean of the function $y=\\sin x$, between the limits of $0$ and $\\pi$ radians, is $\\dfrac{\\sqrt2}{2}$. Find also the arithmetical mean of the same function between the same limits; and show that the form-factor is $=1.11$.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "Integral(sin(x), (x, 0, pi))/pi",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [
    "evaluate: Integral(sin(x), (x, 0, pi))/pi"
   ],
   "shape": [
    "evaluate: Integral(sin(x), (x, 0, pi))/pi"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xviii/3b"
   ],
   "needs": [
    "cas.defint",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/11c",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 11,
   "part": "c",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 11c",
   "problem_latex": "Show that the quadratic mean of the function\n$y=\\sin x$, between the limits of $0$~and~$\\pi$ radians, is~$\\dfrac{\\sqrt2}{2}$.\nFind also the arithmetical mean of the same\nfunction between the same limits; and show that the\nform-factor is~$=1.11$.",
   "markdown": "Show that the quadratic mean of the function $y=\\sin x$, between the limits of $0$ and $\\pi$ radians, is $\\dfrac{\\sqrt2}{2}$. Find also the arithmetical mean of the same function between the same limits; and show that the form-factor is $=1.11$.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": "sqrt(Integral(sin(x)**2, (x, 0, pi))/pi)/(Integral(sin(x), (x, 0, pi))/pi)",
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [
    "evaluate: sqrt(pi)*sqrt(Integral(sin(x)**2, (x, 0, pi)))/Integral(sin(x), (x, 0, pi))"
   ],
   "shape": [
    "evaluate: pi**N*Integral(sin(x)**N, (x, 0, pi))**N/Integral(sin(x), (x, 0, pi))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.trig",
    "core.frac",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/12a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 12,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 12a",
   "problem_latex": "Find the arithmetical and quadratic means of\nthe function $x^2+3x+2$, from $x=0$ to $x=3$.",
   "markdown": "Find the arithmetical and quadratic means of the function $x^2+3x+2$, from $x=0$ to $x=3$.",
   "answer_latex": [
    "$\\text{Arithmetical mean} = 9.5$; $\\text{quadratic mean} = 10.85$."
   ],
   "answer_markdown": [
    "$\\text{Arithmetical mean} = 9.5$; $\\text{quadratic mean} = 10.85$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 9.5, printed 9.5 (half-unit 0.05; correctly rounded at the printed digits: 9.5)",
     "problem_expr": "Integral(x**2 + 3*x + 2, (x, 0, 3))/3",
     "answer_expr": "Rational(19,2)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(x**2 + 3*x + 2, (x, 0, 3))/3"
   ],
   "shape": [
    "evaluate: N*Integral(N*x + N + x**N, (x, 0, N))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "core.arith",
    "core.integ.num"
   ],
   "expectation": "X#9.5,5E-2",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/12b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 12,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 12b",
   "problem_latex": "Find the arithmetical and quadratic means of\nthe function $x^2+3x+2$, from $x=0$ to $x=3$.",
   "markdown": "Find the arithmetical and quadratic means of the function $x^2+3x+2$, from $x=0$ to $x=3$.",
   "answer_latex": [
    "$\\text{Arithmetical mean} = 9.5$; $\\text{quadratic mean} = 10.85$."
   ],
   "answer_markdown": [
    "$\\text{Arithmetical mean} = 9.5$; $\\text{quadratic mean} = 10.85$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 10.8489630841, printed 10.85 (half-unit 0.005; correctly rounded at the printed digits: 10.85)",
     "problem_expr": "sqrt(Integral((x**2 + 3*x + 2)**2, (x, 0, 3))/3)",
     "answer_expr": "sqrt(Rational(1177,10))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: sqrt(3)*sqrt(Integral((x**2 + 3*x + 2)**2, (x, 0, 3)))/3"
   ],
   "shape": [
    "evaluate: N*N**N*Integral((N*x + N + x**N)**N, (x, 0, N))**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.expand",
    "core.arith",
    "core.integ.num"
   ],
   "expectation": "X#10.85,5E-3",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/13a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 13,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 13a",
   "problem_latex": "Find the quadratic mean and the arithmetical\nmean of the function $y=A_1 \\sin x + A_1 \\sin 3x$.",
   "markdown": "Find the quadratic mean and the arithmetical mean of the function $y=A_1 \\sin x + A_1 \\sin 3x$.",
   "answer_latex": [
    "$\\text{Quadratic mean} = \\dfrac{1}{\\sqrt{2}} \\sqrt{A_1^2 + A_3^2}$; $\\text{arithmetical mean} = 0$.\n\nThe first involves a somewhat difficult integral, and may be\nstated thus: By definition the quadratic mean will be\n\\[\n\\sqrt{\\dfrac{1}{2\\pi} \\int_0^{2\\pi} (A_1 \\sin x + A_3 \\sin 3x)^2\\, dx}. %[** TN: Moved period out of radicand]\n\\]\nNow the integration indicated by\n\\[\n\\int (A_1^2 \\sin^2 x + 2A_1 A_3 \\sin x \\sin 3x + A_3^2 \\sin^2 3x)\\, dx\n\\]\nis more readily obtained if for $\\sin^2 x$ we write\n\\[\n\\dfrac{1 - \\cos 2x}{2}.\n\\]\nFor $2\\sin x \\sin 3x$ we write $\\cos 2x - \\cos 4x$; and, for $\\sin^2 3x$,\n\\[\n\\dfrac{1 - \\cos 6x}{2}.\n\\]\n\nMaking these substitutions, and integrating, we get (see\n\\Pageref{cosax})\n\\[\n\\dfrac{A_1^2}{2} \\left( x - \\dfrac{\\sin 2x}{2} \\right)\n + A_1 A_3 \\left( \\dfrac{\\sin 2x}{2} - \\dfrac{\\sin 4x}{4} \\right)\n + \\dfrac{A_3^2}{2} \\left( x - \\dfrac{\\sin 6x}{6} \\right).\n\\]\n\nAt the lower limit the substitution of $0$ for~$x$ causes all\nthis to vanish, whilst at the upper limit the substitution\nof $2\\pi$ for~$x$ gives $A_1^2 \\pi + A_3^2 \\pi$. And hence the\nanswer follows."
   ],
   "answer_markdown": [
    "$\\text{Quadratic mean} = \\dfrac{1}{\\sqrt{2}} \\sqrt{A_1^2 + A_3^2}$; $\\text{arithmetical mean} = 0$. The first involves a somewhat difficult integral, and may be stated thus: By definition the quadratic mean will be 12 _0^2 (A_1 x + A_3 3x)^2  dx. %[** TN: Moved period out of radicand] Now the integration indicated by (A_1^2 ^2 x + 2A_1 A_3 x 3x + A_3^2 ^2 3x)  dx is more readily obtained if for $\\sin^2 x$ we write 1 - 2x2. For $2\\sin x \\sin 3x$ we write $\\cos 2x - \\cos 4x$; and, for $\\sin^2 3x$, 1 - 6x2. Making these substitutions, and integrating, we get (see cosax) A_1^22 ( x - 2x2 ) + A_1 A_3 ( 2x2 - 4x4 ) + A_3^22 ( x - 6x6 ). At the lower limit the substitution of $0$ for $x$ causes all this to vanish, whilst at the upper limit the substitution of $2\\pi$ for $x$ gives $A_1^2 \\pi + A_3^2 \\pi$. And hence the answer follows."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "values do not cover ['A_1', 'A_3']",
     "problem_expr": "sqrt(Integral((A_1*sin(x) + A_1*sin(3*x))**2, (x, 0, 2*pi))/(2*pi))",
     "answer_expr": "sqrt(A_1**2 + A_3**2)/sqrt(2)"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: sqrt(2)*sqrt(Integral((a*sin(x) + a*sin(3*x))**2, (x, 0, 2*pi)))/(2*sqrt(pi))"
   ],
   "shape": [
    "evaluate: pi**N*N*N**N*Integral((a*sin(x) + a*sin(N*x))**N, (x, 0, pi*N))**N"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.expand",
    "cas.trig",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/13b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 13,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 13b",
   "problem_latex": "Find the quadratic mean and the arithmetical\nmean of the function $y=A_1 \\sin x + A_1 \\sin 3x$.",
   "markdown": "Find the quadratic mean and the arithmetical mean of the function $y=A_1 \\sin x + A_1 \\sin 3x$.",
   "answer_latex": [
    "$\\text{arithmetical mean} = 0$."
   ],
   "answer_markdown": [
    "$\\text{arithmetical mean} = 0$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed -4.36015087617e-106, printed 0 (half-unit 0.5; correctly rounded at the printed digits: 0.0)",
     "problem_expr": "Integral(A_1*sin(x) + A_1*sin(3*x), (x, 0, 2*pi))/(2*pi)",
     "answer_expr": "0"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(a*sin(x) + a*sin(3*x), (x, 0, 2*pi))/(2*pi)",
    "evaluate: Integral(a*sin(x) + a*sin(3*x), (x, 0, 2*pi))/(2*pi) at A_1=2"
   ],
   "shape": [
    "evaluate: N*Integral(a*sin(x) + a*sin(N*x), (x, 0, pi*N))/pi"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": "X#0,5E-1",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/14a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 14,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 14a",
   "problem_latex": "A certain curve has the equation $y=3.42\\epsilon^{0.21x}$.\nFind the area included between the curve and the\naxis of~$x$, from the ordinate at $x=2$ to the ordinate\nat $x = 8$. Find also the height of the mean ordinate\nof the curve between these points.",
   "markdown": "A certain curve has the equation $y=3.42\\epsilon^{0.21x}$. Find the area included between the curve and the axis of $x$, from the ordinate at $x=2$ to the ordinate at $x = 8$. Find also the height of the mean ordinate of the curve between these points.",
   "answer_latex": [
    "Area is $62.6$~square units. Mean ordinate is $10.42$."
   ],
   "answer_markdown": [
    "Area is $62.6$ square units. Mean ordinate is $10.42$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 62.5956804811, printed 62.6 (half-unit 0.05; correctly rounded at the printed digits: 62.6)",
     "problem_expr": "Integral(3.42*exp(0.21*x), (x, 2, 8))",
     "answer_expr": "3.42/0.21*(exp(1.68) - exp(0.42))"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(171*exp(21*x/100)/50, (x, 2, 8))"
   ],
   "shape": [
    "evaluate: Integral(N*exp(N*x), (x, N, N))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate",
    "core.arith",
    "core.const",
    "core.integ.num"
   ],
   "expectation": "X#62.6,5E-2",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/14b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 14,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 14b",
   "problem_latex": "A certain curve has the equation $y=3.42\\epsilon^{0.21x}$.\nFind the area included between the curve and the\naxis of~$x$, from the ordinate at $x=2$ to the ordinate\nat $x = 8$. Find also the height of the mean ordinate\nof the curve between these points.",
   "markdown": "A certain curve has the equation $y=3.42\\epsilon^{0.21x}$. Find the area included between the curve and the axis of $x$, from the ordinate at $x=2$ to the ordinate at $x = 8$. Find also the height of the mean ordinate of the curve between these points.",
   "answer_latex": [
    "Area is $62.6$~square units. Mean ordinate is $10.42$."
   ],
   "answer_markdown": [
    "Area is $62.6$ square units. Mean ordinate is $10.42$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 10.4326134135, printed 10.42 (half-unit 0.005; correctly rounded at the printed digits: 10.43)",
     "problem_expr": "Integral(3.42*exp(0.21*x), (x, 2, 8))/6",
     "answer_expr": "3.42/0.21*(exp(1.68) - exp(0.42))/6"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: Integral(171*exp(21*x/100)/50, (x, 2, 8))/6"
   ],
   "shape": [
    "evaluate: N*Integral(N*exp(N*x), (x, N, N))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "core.arith",
    "core.const",
    "core.integ.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/15",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 15,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 15",
   "problem_latex": "Show that the radius of a circle, the area of\nwhich is twice the area of a polar diagram, is equal\nto the quadratic mean of all the values of~$r$ for that\npolar diagram.",
   "markdown": "Show that the radius of a circle, the area of which is twice the area of a polar diagram, is equal to the quadratic mean of all the values of $r$ for that polar diagram.",
   "answer_latex": [],
   "answer_markdown": [],
   "checks": [
    {
     "task": "other",
     "verdict": "FLAG-MISSING",
     "judge_why": "problem or answer expression is null",
     "problem_expr": null,
     "answer_expr": null
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISSING"
    ]
   },
   "form": [],
   "shape": [],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "core.integ.num",
    "other:polar-coordinate area"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/16",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 16,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 16",
   "problem_latex": "Find the volume generated by the curve\n$y=±\\dfrac{x}{6}\\sqrt{x(10-x)}$ rotating about the axis of~$x$.",
   "markdown": "Find the volume generated by the curve $y=±\\dfrac{x}{6}\\sqrt{x(10-x)}$ rotating about the axis of $x$.",
   "answer_latex": [
    "$436.3$. (This solid is pear shaped.)"
   ],
   "answer_markdown": [
    "$436.3$. (This solid is pear shaped.)"
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 436.332312999, printed 436.3 (half-unit 0.05; correctly rounded at the printed digits: 436.3)",
     "problem_expr": "Integral(pi*(x/6)**2*x*(10 - x), (x, 0, 10))",
     "answer_expr": "1250*pi/9"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(pi*x**3*(-x + 10)/36, (x, 0, 10))"
   ],
   "shape": [
    "evaluate: Integral(pi*N*x**N*(N - x), (x, 0, N))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.expand",
    "cas.integrate",
    "core.const",
    "core.integ.num"
   ],
   "expectation": "X#436.3,5E-2",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/1a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 1,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 1a",
   "problem_latex": "Find the area of the curve $y=x^2+x-5$ between\n$x=0$ and $x=6$, and the mean ordinates between\nthese limits.",
   "markdown": "Find the area of the curve $y=x^2+x-5$ between $x=0$ and $x=6$, and the mean ordinates between these limits.",
   "answer_latex": [
    "$\\text{Area} = 60$; $\\text{mean ordinate} = 10$."
   ],
   "answer_markdown": [
    "$\\text{Area} = 60$; $\\text{mean ordinate} = 10$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 60.0, printed 60 (half-unit 0.5; correctly rounded at the printed digits: 60.0)",
     "problem_expr": "Integral(x**2 + x - 5, (x, 0, 6))",
     "answer_expr": "60"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(x**2 + x - 5, (x, 0, 6))"
   ],
   "shape": [
    "evaluate: Integral(N + x + x**N, (x, 0, N))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "core.arith",
    "core.integ.num"
   ],
   "expectation": "X#60,5E-1",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/1b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 1,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 1b",
   "problem_latex": "Find the area of the curve $y=x^2+x-5$ between\n$x=0$ and $x=6$, and the mean ordinates between\nthese limits.",
   "markdown": "Find the area of the curve $y=x^2+x-5$ between $x=0$ and $x=6$, and the mean ordinates between these limits.",
   "answer_latex": [
    "$\\text{Area} = 60$; $\\text{mean ordinate} = 10$."
   ],
   "answer_markdown": [
    "$\\text{Area} = 60$; $\\text{mean ordinate} = 10$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 10.0, printed 10 (half-unit 0.5; correctly rounded at the printed digits: 10.0)",
     "problem_expr": "Integral(x**2 + x - 5, (x, 0, 6))/6",
     "answer_expr": "10"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(x**2 + x - 5, (x, 0, 6))/6"
   ],
   "shape": [
    "evaluate: N*Integral(N + x + x**N, (x, 0, N))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "core.arith",
    "core.integ.num"
   ],
   "expectation": "X#10,5E-1",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/2",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 2,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 2",
   "problem_latex": "Find the area of the parabola $y=2a\\sqrt x$ between\n$x=0$ and $x=a$. Show that it is two-thirds of the\nrectangle of the limiting ordinate and of its abscissa.",
   "markdown": "Find the area of the parabola $y=2a\\sqrt x$ between $x=0$ and $x=a$. Show that it is two-thirds of the rectangle of the limiting ordinate and of its abscissa.",
   "answer_latex": [
    "$\\text{Area} = \\frac{2}{3}$ of $a × 2a \\sqrt{a}$."
   ],
   "answer_markdown": [
    "$\\text{Area} = \\frac{2}{3}$ of $a × 2a \\sqrt{a}$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "values do not cover ['a']",
     "problem_expr": "Integral(2*a*sqrt(x), (x, 0, a))",
     "answer_expr": "Rational(2,3)*(a*(2*a*sqrt(a)))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: Integral(2*a*sqrt(x), (x, 0, a))"
   ],
   "shape": [
    "evaluate: Integral(N*a*x**N, (x, 0, a))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/3a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 3,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 3a",
   "problem_latex": "Find the area of the positive portion of a sine\ncurve and the mean ordinate.",
   "markdown": "Find the area of the positive portion of a sine curve and the mean ordinate.",
   "answer_latex": [
    "$\\text{Area} = 2$; $\\text{mean ordinate} = \\dfrac{2}{\\pi} = 0.637$."
   ],
   "answer_markdown": [
    "$\\text{Area} = 2$; $\\text{mean ordinate} = \\dfrac{2}{\\pi} = 0.637$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 2.0, printed 2 (half-unit 0.5; correctly rounded at the printed digits: 2.0)",
     "problem_expr": "Integral(sin(x), (x, 0, pi))",
     "answer_expr": "2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(sin(x), (x, 0, pi))"
   ],
   "shape": [
    "evaluate: Integral(sin(x), (x, 0, pi))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": "X#2,5E-1",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/3b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 3,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 3b",
   "problem_latex": "Find the area of the positive portion of a sine\ncurve and the mean ordinate.",
   "markdown": "Find the area of the positive portion of a sine curve and the mean ordinate.",
   "answer_latex": [
    "$\\text{Area} = 2$; $\\text{mean ordinate} = \\dfrac{2}{\\pi} = 0.637$."
   ],
   "answer_markdown": [
    "$\\text{Area} = 2$; $\\text{mean ordinate} = \\dfrac{2}{\\pi} = 0.637$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.636619772368, printed 0.637 (half-unit 0.0005; correctly rounded at the printed digits: 0.637)",
     "problem_expr": "Integral(sin(x), (x, 0, pi))/pi",
     "answer_expr": "2/pi"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(sin(x), (x, 0, pi))/pi"
   ],
   "shape": [
    "evaluate: Integral(sin(x), (x, 0, pi))/pi"
   ],
   "same_problem_in": [
    "thompson-calculus-made-easy-1914/ex-xviii/11b"
   ],
   "needs": [
    "cas.defint",
    "core.frac",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": "X#0.637,5E-4",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/4a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 4,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 4a",
   "problem_latex": "Find the area of the positive portion of the\ncurve $y=\\sin^2 x$, and find the mean ordinate.",
   "markdown": "Find the area of the positive portion of the curve $y=\\sin^2 x$, and find the mean ordinate.",
   "answer_latex": [
    "$\\text{Area} = 1.57$; $\\text{mean ordinate} = 0.5$."
   ],
   "answer_markdown": [
    "$\\text{Area} = 1.57$; $\\text{mean ordinate} = 0.5$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 1.57079632679, printed 1.57 (half-unit 0.005; correctly rounded at the printed digits: 1.57)",
     "problem_expr": "Integral(sin(x)**2, (x, 0, pi))",
     "answer_expr": "pi/2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(sin(x)**2, (x, 0, pi))"
   ],
   "shape": [
    "evaluate: Integral(sin(x)**N, (x, 0, pi))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.trig",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": "X#1.57,5E-3",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/4b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 4,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 4b",
   "problem_latex": "Find the area of the positive portion of the\ncurve $y=\\sin^2 x$, and find the mean ordinate.",
   "markdown": "Find the area of the positive portion of the curve $y=\\sin^2 x$, and find the mean ordinate.",
   "answer_latex": [
    "$\\text{Area} = 1.57$; $\\text{mean ordinate} = 0.5$."
   ],
   "answer_markdown": [
    "$\\text{Area} = 1.57$; $\\text{mean ordinate} = 0.5$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.5, printed 0.5 (half-unit 0.05; correctly rounded at the printed digits: 0.5)",
     "problem_expr": "Integral(sin(x)**2, (x, 0, pi))/pi",
     "answer_expr": "1/2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(sin(x)**2, (x, 0, pi))/pi"
   ],
   "shape": [
    "evaluate: Integral(sin(x)**N, (x, 0, pi))/pi"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.trig",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": "X#0.5,5E-2",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/5a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 5,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 5a",
   "problem_latex": "Find the area included between the two branches\nof the curve $y=x^2 ± x^{\\efrac{5}{2}}$ from $x=0$ to $x=1$, also the\narea of the positive portion of the lower branch of\nthe curve (see \\Fig{30}, \\Pageref{fig:30}). %[ ** Page, xref]",
   "markdown": "Find the area included between the two branches of the curve $y=x^2 ± x^{\\efrac{5}{2}}$ from $x=0$ to $x=1$, also the area of the positive portion of the lower branch of the curve (see [fig:30]Fig. 30, fig:30). %[ ** Page, xref]",
   "answer_latex": [
    "$0.572$, $0.0476$."
   ],
   "answer_markdown": [
    "$0.572$, $0.0476$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS-LOOSE",
     "judge_why": "computed 0.571428571429, printed 0.572 (half-unit 0.0005; correctly rounded at the printed digits: 0.571): one unit off in the last printed place",
     "problem_expr": "Integral((x**2 + x**Rational(5,2)) - (x**2 - x**Rational(5,2)), (x, 0, 1))",
     "answer_expr": "Rational(4,7)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS-LOOSE"
    ]
   },
   "form": [
    "evaluate: Integral(2*x**(5/2), (x, 0, 1))"
   ],
   "shape": [
    "evaluate: Integral(N*x**N, (x, 0, 1))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate",
    "core.graph",
    "core.integ.num"
   ],
   "expectation": "X#0.572,5E-4",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/5b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 5,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 5b",
   "problem_latex": "Find the area included between the two branches\nof the curve $y=x^2 ± x^{\\efrac{5}{2}}$ from $x=0$ to $x=1$, also the\narea of the positive portion of the lower branch of\nthe curve (see \\Fig{30}, \\Pageref{fig:30}). %[ ** Page, xref]",
   "markdown": "Find the area included between the two branches of the curve $y=x^2 ± x^{\\efrac{5}{2}}$ from $x=0$ to $x=1$, also the area of the positive portion of the lower branch of the curve (see [fig:30]Fig. 30, fig:30). %[ ** Page, xref]",
   "answer_latex": [
    "$0.572$, $0.0476$."
   ],
   "answer_markdown": [
    "$0.572$, $0.0476$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 0.047619047619, printed 0.0476 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.0476)",
     "problem_expr": "Integral(x**2 - x**Rational(5,2), (x, 0, 1))",
     "answer_expr": "Rational(1,21)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(-x**(5/2) + x**2, (x, 0, 1))"
   ],
   "shape": [
    "evaluate: Integral(0, (x, 0, 1))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate",
    "core.graph",
    "core.integ.num"
   ],
   "expectation": "X#0.0476,5E-5",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/6",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 6,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 6",
   "problem_latex": "Find the volume of a cone of radius of base~$r$,\nand of height~$h$.",
   "markdown": "Find the volume of a cone of radius of base $r$, and of height $h$.",
   "answer_latex": [
    "$\\text{Volume} = \\pi r^2 \\dfrac{h}{3}$."
   ],
   "answer_markdown": [
    "$\\text{Volume} = \\pi r^2 \\dfrac{h}{3}$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "values do not cover ['h', 'r']",
     "problem_expr": "Integral(pi*(r*x/h)**2, (x, 0, h))",
     "answer_expr": "pi*r**2*h/3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: Integral(pi*b**2*x**2/a**2, (x, 0, a))"
   ],
   "shape": [
    "evaluate: Integral(pi*a**N*b**N*x**N, (x, 0, a))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate",
    "cas.simplify"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/7",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 7,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 7",
   "problem_latex": "Find the area of the curve $y=x^3-\\log_\\epsilon x$ between\n$x=0$ and $x=1$.",
   "markdown": "Find the area of the curve $y=x^3-\\log_\\epsilon x$ between $x=0$ and $x=1$.",
   "answer_latex": [
    "$1.25$."
   ],
   "answer_markdown": [
    "$1.25$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 1.25, printed 1.25 (half-unit 0.005; correctly rounded at the printed digits: 1.25)",
     "problem_expr": "Integral(x**3 - log(x), (x, 0, 1))",
     "answer_expr": "Rational(5,4)"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(x**3 - log(x), (x, 0, 1))"
   ],
   "shape": [
    "evaluate: Integral(x**N - log(x), (x, 0, 1))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate.parts",
    "core.log"
   ],
   "expectation": "X#1.25,5E-3",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/8",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 8,
   "part": null,
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "224",
   "location": "Exercise XVIII, problem 8",
   "problem_latex": "Find the volume generated by the curve\n$y=\\sqrt{1+x^2}$, as it revolves about the axis of~$x$, between\n$x=0$ and $x=4$.",
   "markdown": "Find the volume generated by the curve $y=\\sqrt{1+x^2}$, as it revolves about the axis of $x$, between $x=0$ and $x=4$.",
   "answer_latex": [
    "$79.4$."
   ],
   "answer_markdown": [
    "$79.4$."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 79.5870138909, printed 79.4 (half-unit 0.05; correctly rounded at the printed digits: 79.6)",
     "problem_expr": "Integral(pi*(1 + x**2), (x, 0, 4))",
     "answer_expr": "76*pi/3"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: Integral(pi*(x**2 + 1), (x, 0, 4))"
   ],
   "shape": [
    "evaluate: Integral(pi*(x**N + 1), (x, 0, N))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate",
    "core.const",
    "core.integ.num"
   ],
   "expectation": null,
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/9a",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 9,
   "part": "a",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 9a",
   "problem_latex": "Find the volume generated by a sine curve\nrevolving about the axis of~$x$. Find also the area of\nits surface.",
   "markdown": "Find the volume generated by a sine curve revolving about the axis of $x$. Find also the area of its surface.",
   "answer_latex": [
    "$\\text{Volume} = 4.9348$; $\\text{area of surface} = 12.57$ (from $0$ to~$\\pi$)."
   ],
   "answer_markdown": [
    "$\\text{Volume} = 4.9348$; $\\text{area of surface} = 12.57$ (from $0$ to $\\pi$)."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "PASS",
     "judge_why": "computed 4.93480220054, printed 4.9348 (half-unit 5.0e-5; correctly rounded at the printed digits: 4.9348)",
     "problem_expr": "Integral(pi*sin(x)**2, (x, 0, pi))",
     "answer_expr": "pi**2/2"
    }
   ],
   "verdict": "verified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "PASS"
    ]
   },
   "form": [
    "evaluate: Integral(pi*sin(x)**2, (x, 0, pi))"
   ],
   "shape": [
    "evaluate: Integral(pi*sin(x)**N, (x, 0, pi))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.trig",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": "X#4.9348,5E-5",
   "keys": []
  },
  {
   "id": "thompson-calculus-made-easy-1914/ex-xviii/9b",
   "set": "thompson-calculus-made-easy-1914/ex-xviii",
   "number": 9,
   "part": "b",
   "book": "thompson-calculus-made-easy-1914",
   "edition": "Macmillan, 2nd ed., 1914",
   "page": "225",
   "location": "Exercise XVIII, problem 9b",
   "problem_latex": "Find the volume generated by a sine curve\nrevolving about the axis of~$x$. Find also the area of\nits surface.",
   "markdown": "Find the volume generated by a sine curve revolving about the axis of $x$. Find also the area of its surface.",
   "answer_latex": [
    "$\\text{Volume} = 4.9348$; $\\text{area of surface} = 12.57$ (from $0$ to~$\\pi$)."
   ],
   "answer_markdown": [
    "$\\text{Volume} = 4.9348$; $\\text{area of surface} = 12.57$ (from $0$ to $\\pi$)."
   ],
   "checks": [
    {
     "task": "evaluate",
     "verdict": "FLAG-MISMATCH",
     "judge_why": "computed 14.4235994484, printed 12.57 (half-unit 0.005; correctly rounded at the printed digits: 14.42)",
     "problem_expr": "Integral(2*pi*sin(x)*sqrt(1 + cos(x)**2), (x, 0, pi))",
     "answer_expr": "2*pi*(sqrt(2) + asinh(1))"
    }
   ],
   "verdict": "unverified",
   "judge": {
    "commit": "4a88328",
    "verdicts": [
     "FLAG-MISMATCH"
    ]
   },
   "form": [
    "evaluate: Integral(2*pi*sqrt(cos(x)**2 + 1)*sin(x), (x, 0, pi))"
   ],
   "shape": [
    "evaluate: Integral(pi*N*(cos(x)**N + 1)**N*sin(x), (x, 0, pi))"
   ],
   "same_problem_in": [],
   "needs": [
    "cas.defint",
    "cas.integrate.subst",
    "core.hyp",
    "core.integ.num",
    "core.trig"
   ],
   "expectation": null,
   "keys": []
  }
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   "id": "thompson-calculus-made-easy-1914/T012",
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   "class": "content",
   "context": "\\sqrt[2]{-(.5)^5}$, and if $x = +0.5$, $y = 0.25 ± \\sqrt[2]{",
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   "context": "s~$+$; hence it is a minimum.} \\\\ % \\text{(\\textit{b})}\\quad",
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   "id": "thompson-calculus-made-easy-1914/T014",
   "kind": "transcriber-marked",
   "page": "136",
   "printed": "2.59375",
   "corrected": "2.59374",
   "class": "content",
   "context": "ctor ten times over) will multiply the original capital by~$",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
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  {
   "id": "thompson-calculus-made-easy-1914/T015",
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   "page": "136",
   "printed": "2.654",
   "corrected": "2.653",
   "class": "content",
   "context": "extit{s}.~$7$\\textit{d}.}; for \\[ (1 + \\tfrac{1}{20})^{20} =",
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   "page": "139",
   "printed": "2.593",
   "corrected": "2.594",
   "class": "content",
   "context": "be \\DPPageSep{151.png}{139}% $(1 + \\tfrac{1}{10})^{10}$ or $",
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  {
   "id": "thompson-calculus-made-easy-1914/T017",
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   "printed": "2.489",
   "corrected": "2.488",
   "class": "content",
   "context": ")^2 &&= 2.25. \\\\ &(1 + \\tfrac{1}{5})^5 &&=",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
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   "class": "content",
   "context": "})^{20} &&= 2.653. \\\\ &(1 + \\tfrac{1}{100})^{100} &&=",
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  {
   "id": "thompson-calculus-made-easy-1914/T019",
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   "class": "content",
   "context": " \\DPtypo{2.704}{2.705}. \\\\ &(1 + \\tfrac{1}{1000})^{1000} &&=",
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  {
   "id": "thompson-calculus-made-easy-1914/T020",
   "kind": "transcriber-marked",
   "page": "140",
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   "class": "content",
   "context": "po{2.7171}{2.7169}. \\\\ &(1 + \\tfrac{1}{10,000})^{10,000} &&=",
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   "status": "transcriber_marked"
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  {
   "id": "thompson-calculus-made-easy-1914/T021",
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   "page": "146",
   "printed": "7.69",
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  {
   "id": "thompson-calculus-made-easy-1914/T022",
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   "page": "147",
   "printed": "7.6010",
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   "context": "\\\\ 3.0 & 1.0986 && 1,000 & 6.9078 \\\\ 3.5 & 1.2528 && 2,000 &",
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   "id": "thompson-calculus-made-easy-1914/T023",
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   "page": "147",
   "printed": "9.2104",
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   "class": "content",
   "context": "\\ 4.0 & 1.3863 && 5,000 & 8.5172 \\\\ 4.5 & 1.5041 && 10,000 &",
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  {
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   "class": "content",
   "context": "}}};\\quad \\frac{dv}{dx} = 2x;\\quad \\frac{du}{dx} = \\frac{x}{",
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  {
   "id": "thompson-calculus-made-easy-1914/T025",
   "kind": "transcriber-marked",
   "page": "159",
   "printed": "5.754",
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   "class": "content",
   "context": "0.7135 \\\\ 1.50 & 4.4817 & 0.2231 & 0.7769 \\\\ 1.75 &",
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  {
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   "page": "159",
   "printed": "12.183",
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   "class": "content",
   "context": "0.8262 \\\\ 2.00 & 7.389 & 0.1353 & 0.8647 \\\\ 2.50 &",
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   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T027",
   "kind": "transcriber-marked",
   "page": "159",
   "printed": "20.085",
   "corrected": "20.086",
   "class": "content",
   "context": "& \\DPtypo{12.183}{12.182} & 0.0821 & 0.9179 \\\\ 3.00 &",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T028",
   "kind": "transcriber-marked",
   "page": "159",
   "printed": "0.00053",
   "corrected": "0.00055",
   "class": "content",
   "context": "\\\\ 6.00 & 403.43 & 0.00248 & 0.99752 \\\\ 7.50 & 1808.04 &",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T029",
   "kind": "transcriber-marked",
   "page": "162",
   "printed": "34.5",
   "corrected": "34.7",
   "class": "content",
   "context": "frac{\\DPchg{\\overset{-}{1}.6990}{-0.3010}}{-0.02 × 0.4343} =",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T030",
   "kind": "transcriber-marked",
   "page": "170",
   "printed": "\\cos \\theta",
   "corrected": "\\sin \\theta",
   "class": "content",
   "context": "ta$ it becomes $-\\sin \\theta$; or, in symbols, \\[ \\frac{d^2(",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T031",
   "kind": "transcriber-marked",
   "page": "171",
   "printed": "",
   "corrected": ".",
   "class": "spacing-or-punctuation",
   "context": "{1}{\\sqrt{1-x^2}}, \\end{DPalign*} a rather unexpected result",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T032",
   "kind": "transcriber-marked",
   "page": "173",
   "printed": "",
   "corrected": "(",
   "class": "content",
   "context": "ec^2 \\theta; \\\\ % \\lintertext{hence} \\frac{dv}{d\\theta} &= 6",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T033",
   "kind": "transcriber-marked",
   "page": "202",
   "printed": "2\\cos 2\\theta - 1",
   "corrected": "2\\cos^2 \\theta - 1",
   "class": "content",
   "context": "{DPgather*} \\cos 2\\theta = \\cos^2\\theta - \\sin^2\\theta =",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T034",
   "kind": "transcriber-marked",
   "page": "202",
   "printed": "\\cos^2 \\theta + 1",
   "corrected": "\\cos 2\\theta + 1",
   "class": "content",
   "context": "eta - 1}; \\\\ \\lintertext{hence} \\cos^2\\theta = \\tfrac{1}{2}(",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T035",
   "kind": "transcriber-marked",
   "page": "254",
   "printed": "-\\efrac{8}{5}",
   "corrected": "-\\efrac{8}{3}",
   "class": "content",
   "context": "efrac{2}{3}}$. \\Item{(6)} $\\dfrac{dy}{dx} = -\\dfrac{5}{3}x^{",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T036",
   "kind": "transcriber-marked",
   "page": "255",
   "printed": "0.000828",
   "corrected": "0.001024",
   "class": "content",
   "context": "\\dfrac{R^2 (a + 2bt)}{R_0}$. \\Item{(13)} $1.4340(0.000014t -",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T037",
   "kind": "transcriber-marked",
   "page": "256",
   "printed": "",
   "corrected": ".",
   "class": "spacing-or-punctuation",
   "context": "c{18b \\sqrt[3]{a}}{x^4} - \\dfrac{3a \\sqrt{b}}{4 \\sqrt{x^3}}$",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T038",
   "kind": "transcriber-marked",
   "page": "264",
   "printed": "e",
   "corrected": "\\epsilon",
   "class": "content",
   "context": "_\\epsilon x - \\dfrac{1}{a + 1}\\right) + C$. \\Item{(4)} $\\sin",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/T039",
   "kind": "transcriber-marked",
   "page": "264",
   "printed": "e",
   "corrected": "\\epsilon",
   "class": "content",
   "context": "Cols{2} \\Item{(5)} $\\sin(\\log_\\epsilon x) + C$. \\Item{(6)} $",
   "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)",
   "status": "transcriber_marked"
  },
  {
   "id": "thompson-calculus-made-easy-1914/F001",
   "kind": "found-by-us",
   "status": "confirmed",
   "page": "255",
   "location": "Answers to Exercises III, item (12), third formula: the alternative form",
   "printed": "\\dfrac{R^2 (a + 2bt)}{R_0}",
   "corrected": "-\\dfrac{R^2 (a + 2bt)}{R_0}",
   "class": "content (sign)",
   "evidence": "Under the book's own relation R = R_0/(1 + at + bt^2), the printed alternative equals +R_0(a + 2bt)/(1 + at + bt^2)^2, the main answer with its sign lost. Numeric check at seeded rational points, 50 digits (claudiverse judge.py, verdict PASS-ALT-ERRATUM on record III.12 part 3); confirmed by hand algebra (abacus).",
   "found_by": "claudiverse pilot (qwenniverse-fermion); confirmed by abacus",
   "date": "2026-10-09"
  },
  {
   "id": "thompson-calculus-made-easy-1914/C001",
   "kind": "candidate",
   "status": "candidate",
   "location": "Answers to Exercises XII, item (14): the minimum",
   "printed": "x = 0.694",
   "corrected": "x = 0.693",
   "class": "content (last digit)",
   "evidence": "Status: to be checked by the judge at set XII. 3^(-1/3) = 0.693361..., 0.693 at the printed digits; the STU-32 core and mpmath agree (tutor's acceptance run). y = 0.7 is right.",
   "found_by": "tutor (stu32-calc acceptance)",
   "date": "2026-10-09"
  },
  {
   "id": "thompson-calculus-made-easy-1914/C002",
   "kind": "candidate",
   "status": "candidate",
   "location": "Answers to Exercises VII, item (1)",
   "printed": "\\dfrac{dw}{dx} = \\dfrac{3x^2 (3 + 3x^3)}{27 (\\frac{1}{2} x^3 + \\frac{1}{4} x^6)^3}",
   "corrected": "\\dfrac{dw}{dx} = -\\dfrac{3x^2 (3 + 3x^3)}{27 (\\frac{1}{2} x^3 + \\frac{1}{4} x^6)^3}",
   "class": "content (sign)",
   "evidence": "Status: found by the pilot's judge; awaiting abacus. w = 1/v^2 with v = 3(u + u^2), u = x^3/2, so dw/dx = -2 v^-3 dv/dx: negative where the printed answer is positive. The judge's values at a seeded point are exactly opposite (-1023.988... against +1023.988...). The extractor flagged it in its notes before the judge ran.",
   "found_by": "claudiverse pilot (Sonnet 5.5 extractor, judge.py)",
   "date": "2026-10-09"
  },
  {
   "id": "thompson-calculus-made-easy-1914/C003",
   "kind": "candidate",
   "status": "candidate",
   "location": "Exercises XII, item (5), and its answer",
   "printed": "problem: w = p n^v, find dw/dv; answer: n p v^{n-1}",
   "corrected": "either the answer p n^v log_e n, or the problem w = p v^n",
   "class": "content (problem and answer do not match)",
   "evidence": "Status: found by the pilot's judge; awaiting abacus. n p v^(n-1) is the derivative of p v^n, not of p n^v (whose derivative is p n^v ln n). The judge's values disagree at seeded points (3.8659... against 6.9563...). Which of the two lines is misprinted cannot be told from the text alone.",
   "found_by": "claudiverse pilot (Sonnet 5.5 extractor, judge.py)",
   "date": "2026-10-09"
  }
 ]
}