{ "cells": [ { "cell_type": "markdown", "metadata": { "toc": "true" }, "source": [ "# Table of Contents\n", "

1  微積分
1.1  微分
1.2  2重積分
2  線形代数
2.1  写像のIm, Ker
2.2  行列の対角化
3  数式変形
3.1  (1-a) センター試験オリジナル
3.2  (1-b)
4  3-(a) 以降のpythonによる導出
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 微積分\n", "## 微分" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from sympy import *\n", "init_printing()" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\operatorname{asin}{\\left (\\frac{x^{2} - 1}{x^{2} + 1} \\right )}$$" ], "text/plain": [ " ⎛ 2 ⎞\n", " ⎜x - 1⎟\n", "asin⎜──────⎟\n", " ⎜ 2 ⎟\n", " ⎝x + 1⎠" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = symbols('x',positive = True)\n", "# x = symbols('x')\n", "\n", "y = asin((x**2-1)/(x**2+1))\n", "y" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\frac{- \\frac{2 x \\left(x^{2} - 1\\right)}{\\left(x^{2} + 1\\right)^{2}} + \\frac{2 x}{x^{2} + 1}}{\\sqrt{- \\frac{\\left(x^{2} - 1\\right)^{2}}{\\left(x^{2} + 1\\right)^{2}} + 1}}$$" ], "text/plain": [ " ⎛ 2 ⎞ \n", " 2⋅x⋅⎝x - 1⎠ 2⋅x \n", " - ──────────── + ────── \n", " 2 2 \n", " ⎛ 2 ⎞ x + 1 \n", " ⎝x + 1⎠ \n", "─────────────────────────\n", " _________________\n", " ╱ 2 \n", " ╱ ⎛ 2 ⎞ \n", " ╱ ⎝x - 1⎠ \n", " ╱ - ───────── + 1 \n", " ╱ 2 \n", " ╱ ⎛ 2 ⎞ \n", "╲╱ ⎝x + 1⎠ " ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dy = diff(y,x)\n", "dy" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAsBAMAAAAtLQ2eAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLElEQVQ4EWNgQAZCyibIXFQ2awJDfQOqEBKP\nLYCBcwISH5XJuYCB7QuqEBKP+ys+WaBCrq9IqjGY/QUYQkgCy5DYGEwmBQwhJIFDSGwMJq8CgyCG\nIFzAiYHhIJyDzmBcXq62AF0Qzuf+//8/blm4smHMAAYATvCBGv4uMgrAbQy7ANsG3LJcAiwf8cg2\nMP7ALcvAwIuekHmnIinnOgDh8ELpMDVk5UVQlTBZBh4kWT4F7LIxxySOH2BoZ2iBSKPqZS3gW8O1\ngWXvXVNssuwB7F95BTj//4cGOZpeBm4DqJUMDOy7d++avXt3AkgA6ir+AyAODKDqZWDwd4DJgGhU\nWd6A8ww8SPGDKtt/wZ5BDkkzqmzYmRi9B1hkmTJ/5yEJQ5lwvZhSQBFWASRhADekY+qdGUXdAAAA\nAElFTkSuQmCC\n", "text/latex": [ "$$\\frac{2}{x^{2} + 1}$$" ], "text/plain": [ " 2 \n", "──────\n", " 2 \n", "x + 1" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify(dy)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2重積分" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAaBAMAAABVzS+SAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nGYznev3BRh71LxAtjfz6s/xJAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\sqrt{x^{2} + 4 y^{2}}$$" ], "text/plain": [ " ___________\n", " ╱ 2 2 \n", "╲╱ x + 4⋅y " ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x,y = symbols('x,y')\n", "\n", "f = sqrt(x**2+4*y**2)\n", "f" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$2 y^{2} \\operatorname{asinh}{\\left (\\frac{1}{2} \\right )} + \\frac{\\sqrt{5} y^{2}}{2}$$" ], "text/plain": [ " 2\n", " 2 √5⋅y \n", "2⋅y ⋅asinh(1/2) + ─────\n", " 2 " ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dy = integrate(f,(x,0,y))\n", "dy" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAA0CAMAAAAtxFODAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n4/FH9lzIT9BNjcFD8fpa1z+fdyVBumg5zQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\frac{2}{3} \\operatorname{asinh}{\\left (\\frac{1}{2} \\right )} + \\frac{\\sqrt{5}}{6}$$" ], "text/plain": [ "2⋅asinh(1/2) √5\n", "──────────── + ──\n", " 3 6 " ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "integrate(dy,(y,0,1))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "# 線形代数\n", "## 写像のIm, Ker\n", "\n", "これ以降init_printing()していますが，Kernel->restartも忘れずに実行してください．" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHgAAABLCAMAAACMVLPjAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nBsNRugFfQFFK/AOhVSxGwtHvlAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left[\\begin{matrix}1 & 1 & 3 & 3\\\\0 & 1 & 1 & 2\\\\1 & 0 & 2 & 1\\end{matrix}\\right]$$" ], "text/plain": [ "⎡1 1 3 3⎤\n", "⎢ ⎥\n", "⎢0 1 1 2⎥\n", "⎢ ⎥\n", "⎣1 0 2 1⎦" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import *\n", "init_printing()\n", "A = Matrix([[1,1,3,3],[0,1,1,2],[1,0,2,1]])\n", "A" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAABkCAMAAABAUnk2AAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nLnGVa8dtWPvr5hnX9fIXxBtfHmatYckAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left [ \\left[\\begin{matrix}-2\\\\-1\\\\1\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-1\\\\-2\\\\0\\\\1\\end{matrix}\\right]\\right ]$$" ], "text/plain": [ "⎡⎡-2⎤, ⎡-1⎤⎤\n", "⎢⎢ ⎥ ⎢ ⎥⎥\n", "⎢⎢-1⎥ ⎢-2⎥⎥\n", "⎢⎢ ⎥ ⎢ ⎥⎥\n", "⎢⎢1 ⎥ ⎢0 ⎥⎥\n", "⎢⎢ ⎥ ⎢ ⎥⎥\n", "⎣⎣0 ⎦ ⎣1 ⎦⎦" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.nullspace()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAABLCAMAAABqfXinAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRO/NZondIrtsRPdGkQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAndJREFUaAXtWtt2\ngyAQ9ErTGC8p//+vZUFxB8USTLAP+JC4XGZkXMDxWJRSH1WhD4yoaDT1pjr0dwuDOBa0lHWjjtYA\nY0RlPdVWMpTXB4M4FrSUJUPGyFb0L7NzUAtTAA4FyIeR7Qa9bKn/xAPzCnvbiQX/DDuDCWYXVV3J\n8+wIE8yuRt28gR1hMjtPJkxPHmXltU5nZhyTMGddzjq1ex7sMnXVyUfVG5nisw5gXsm65f6o/3h2\nBuLiHI0dumV2kOMw4As2bwgaZuX5A6BHMlCMS+k598Dg3MnKZ+VN/qzpIuqhHxZZkmfdQzk7MU7m\nos6wx7iJviPe4XGSPdJNPLWlbmdDcWbsMc91UrPfpXHWidmFrEnzu2y09InZJzkQbWn+cIXU13P8\ns04dsIOIQ0PiDddoMmO/iD2V8l+3b6aijUzWtXPW/dxYm4BTC6Pa8pwHHAr2lS+eTyLpL5pxZrWp\nz642zthh7vizTr2no5W2mxd66CXkfE2kzv7BBeXKA84RuxjUa7r9XWYc7QudfXJ2Oz/gJtp58/Fw\nu8m0Ngse+9pFnUEvtflA5U7AlefVgHOkPO/ksAuz/EILDD7JPltLJITok+xAtBtk9q0s78i6Lapb\nkpV3FXHWjcj5vkV1S/6L8mBKcNU6HjuzAZCr7kB34nXsYEqC2dEGRLOjKQlmVwNiG3M0O5qS1Oxo\nShKzO4/GidkdU5KcXRuyxZQkZr9W+QJNSeKxO6YkNTuaktTsaErC2cEGRK91BZiScHbYO+LZASaz\nL3KQoOtWTKUYLe1QMVvqP/HAIE5mv/JrH0Ff8zTN3dxEjKhMf5jT/OlaMQe2MIizgP4CFOJAxZbP\nHKEAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left [ \\left[\\begin{matrix}1\\\\0\\\\1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\1\\\\0\\end{matrix}\\right]\\right ]$$" ], "text/plain": [ "⎡⎡1⎤, ⎡1⎤⎤\n", "⎢⎢ ⎥ ⎢ ⎥⎥\n", "⎢⎢0⎥ ⎢1⎥⎥\n", "⎢⎢ ⎥ ⎢ ⎥⎥\n", "⎣⎣1⎦ ⎣0⎦⎦" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.columnspace()" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPwAAABLCAMAAABjuhqYAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nlPmRoCYZl72uyQ76t67yL6uGTkHnAXHgAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left ( \\left[\\begin{matrix}1 & 0 & 2 & 1\\\\0 & 1 & 1 & 2\\\\0 & 0 & 0 & 0\\end{matrix}\\right], \\quad \\left [ 0, \\quad 1\\right ]\\right )$$" ], "text/plain": [ "⎛⎡1 0 2 1⎤, [0, 1]⎞\n", "⎜⎢ ⎥ ⎟\n", "⎜⎢0 1 1 2⎥ ⎟\n", "⎜⎢ ⎥ ⎟\n", "⎝⎣0 0 0 0⎦ ⎠" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A.rref()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 行列の対角化" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIkAAABLCAMAAABz2lREAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nX9YNg/sLzdA6vNqHUjUAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left[\\begin{matrix}0.5 & 0 & 0.5\\\\0 & 0.5 & 0.5\\\\0.5 & 0.5 & 0\\end{matrix}\\right]$$" ], "text/plain": [ "⎡0.5 0 0.5⎤\n", "⎢ ⎥\n", "⎢ 0 0.5 0.5⎥\n", "⎢ ⎥\n", "⎣0.5 0.5 0 ⎦" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import *\n", "init_printing()\n", "A = Matrix([[1/2,0,1/2],[0,1/2,1/2],[1/2,1/2,0]])\n", "A" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": true }, "outputs": [], "source": [ "P,D = A.diagonalize()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAABMCAMAAABH/cxSAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRM3diWYi77tskp5FzwAAAAlwSFlzAAAOxAAADsQBlSsOGwAAB2ZJREFUeAHtnel2\ntDYMhs3adtjL/d9rvUs2WGB7GL7TY35kvOiVpWcYQ4iSMFaOpwhUg/A8dT0/8BrVLo8aj5X2FwhU\nTT0KN7XkuzE2KtKMVXvT8qP7wiLFhUugU8g//Nj4aT4LzvUukFeuYel9i4BCLk7m7mN8zhTyvpnm\nCb0dQ9uzqrn+NPg6s9jFq+8e+tCKdCHNl6bZNpnGMk3T5scP08yuU411w22nibG+bafG8gqsH8pY\nIeeinnvSB4l84NH142Js2Sq2ItDacb/h6/z5QN93D31o8dNltRciaJoWNjSWvbhITTuntjS81e2z\nWl9rYJpBgrPadHcuFRpW+++TcmGWYKGMLfLaBs0o5PMqHE/yoiuXGKa6hTdAr3p8OeiOJqcjvnvo\n21ZfN/WuoocmtJg1ZDCosl15Mu0uyO1ifz2bZiBvVJqceCffoAUgQOjgI5ixQV7xa6c5KOSbvIfp\ndIpcIftGGX496MKmzozvHvrQEtzsCQNN08KGxnLVdw0V+6z8VO93jU9rYBolqD7KYlOdJKxe+nCi\nlR3tAzLuNnPIKA3ySX5WlJ5CLj5WjH3kqSGtnYSU/uzrQXdmdDLmu4c+tBKQD/I9qs09gt1YNC48\njdfhmbc8yM8ubjUmvRn5UWsfwYwN8gFtxwTyfpdvzWcXK8ujnue2QZdTM+6+HnXufLDnu4c+tK6Q\n4wg1D7XeaPaj0WR/Mo3XYUxtBvymeg4R1x+kcMad3Jv5RQK9ZQTyRV0pK7hgivd7Edch8jjqSHOY\n9N1DH1oXyLGh2VjkAh+VxTKNjdmXMHI97cgN52nfB6OBYFVL+QhlvDTjXssTdkTQSOTyLEfI5TJb\nYFuz0Szq0+Hr7Dzd8N1DX7cQKWhCi3s3Ejw42j1jEhu6OM6nrbznt8/i6OrPuBsN7zv0lY+4jDHy\nfhjtMVT8OuNvLDKEZr+4aQnopBh/8ZbTU7576OsWIgVNaHE3RoIGG3T5WvVdZmDayFt1lZ3Fe9Xu\n9s7UXn1lvMrH3YxVihi5GoGv6qLQ2ctnLYOQd7hgdNLydScmZ0O+e+hDi+sQKWjqlmOILFtFfJEn\ny6YTQmr9hjjyUW3lq9RUFgKDzwsEE5UxhXyTi872pmyVyBvbP+MmxnxdyM4b991DH1pcAqRQUw86\nhjDdCaSc9y5DPyA30wzLe4XRbC81uv6huPW6URlTyNUNfqNvY/X3YWy0fbS00/R1zmS4o041cA99\naHE1hdwxtJaVHObfxK3yHBr1zmgc2Wn1jaZOkD/rk6Gqs5xt57up9hGVMYWcjeIb/pXfFaoNTH6/\nPOsYwuj4M0qjo4wOc+DeXw5muMiQcpp60DE0lsvIn5XIx6izOFU7c9NrNHZaPRDQCXYa+Ufercz2\nRtkN2wQTkzGJvJ/4o0Z5H642sKWZanuP5a7t9EDnDF91wL2/HMw09boP6lMOTWiBIbODg3peIm60\nPg1/wqVuWM6mOXOb4LLqxyoVf8AVeKxlffCnVobUVZL8CS71JPFaXiziCRTk8cwyFQV5JsB4eUEe\nzyxTUZBnAoyXF+TxzDIVBXkmwHh5QR7PLFNRkGcCjJcX5PHMMhUFeSbAeDmNPFQRE1jHN7eVOAH7\nN4b9GH8fA408VBETiNM3dwp5AppfD/sx/nr9i8dawYqY8zgP5lDIcy54YfQQ4+9jIM9yqIi5FdjB\n3P6I95b8J0aHGH+yqrMIiVz9LArVDjnSQ+dg/gciP8R4yOLxAQp53A+uTyoC3Eqcx3O5sUBkSjc8\nxptQyEMVMYFVjuZOJU5A9dvhY4y/XV+sRiOXP3G9WwMUKKAxhTy/z+1kxUCMJ5bPDVHIIz+FAXNT\nyPNcDhGeAzFGeMg3Fcj/+vufc0fqWgO1Q+dWdtQ3dypxrNW7DT/GF6L592/id4WiKmKOJUO4EueF\n1E6XjEzp1EfmILWxsKiKGH5dkJW98aVGmSlEyf0Yo8TfMSaRx9YA2QIav/bnO7F+xYuN8SveUpzQ\nyGMqYvjqYO7X/qSE9owGYnzG/7VXGvm1vlhEEyjIo5HlCgryXILR+oI8GlmuoCDPJRitL8ijkeUK\nCvJcgtH6gjwaWa6gIM8lGK0vyKOR5QoK8lyC0XoaeWKdTaIsOvgcwXsx0sgT62wSZTkEb2jNXyZS\npu/FSCJPrLNJlN2glm4CfyFI+XgxRhJ5Yp1Noiyd5z2l+bVYaf1ijCTyxDqbRNk9cOlWDvIXY6SQ\nJ/44PFGWzvKmEiN/M0YKeWKdTaLsJrh0M4z8zRhp5FGlQwbGn1CeY2LBry7ypNSwu+Q2hTzx05co\nS07hrhAjfzNGCjlLrLNJlN1Fl2qHkaemlro21pHIE+tsEmU4rCfaDvIXYySRJ9bZJMqewIx9Oshf\njJFEHls6ZBJ8vzzHRIJfNXL9J+Dei5FGnlhnkyjDfL7ehr8QpMqa3ouRRv71xItDuqS/8HmEQDnL\nH8FKOS3IKTqPzBXkj2ClnBbkFJ1H5hRy+ccb/8BfjH0k5Red2n/lxP+NizjQXzV/Mar/9dLyXzm1\nLfsP8tZagpD3324AAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left[\\begin{matrix}-0.5 & 0 & -5.55111512312578 \\cdot 10^{-17}\\\\0 & 0.5 & 0\\\\0 & 0 & 1.0\\end{matrix}\\right]$$" ], "text/plain": [ "⎡-0.5 0 -5.55111512312578e-17⎤\n", "⎢ ⎥\n", "⎢ 0 0.5 0 ⎥\n", "⎢ ⎥\n", "⎣ 0 0 1.0 ⎦" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P.inv()*A*P" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 数式変形" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## (1-a) センター試験オリジナル" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "from sympy import *\n", "init_printing()\n", "\n", "a,x,t = symbols('a,x,t')" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAWBAMAAABnDkJrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyElEQVQoFWNggIDcvQJQFibF5sA8AVMUKsLh\nwPMFt2QAy1eckgwMXB/xSHIo4JHMxSPHfgFdkushXCSMIQbC5lKA0MKX4Y7gmbVqI6okAxNcku//\n/w/YJIWUXdWgRoGkYcaCdTImsDdxIAUdiiSbANtHLgeQHghAkWRk4NwAk2BgmzlzxsuZMw1AAhAH\n8SvAJYEMFJ0MDOcP4JTkEtBnYBJASKPojF+wn8EdIYdqrLCS0PUCLJK8dj/NkYQhTLixGDJAAUYH\nhCgAIVMq0b091h4AAAAASUVORK5CYII=\n", "text/latex": [ "$$x^{2} + 1$$" ], "text/plain": [ " 2 \n", "x + 1" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y_1 = x**2+1\n", "y_1 #放物線Cの関数" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAf0lEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChz\nFgNDvgOEvT8AzgQKrA9gPZPYUwNkcXxnCGd4dWA1kMmVwFDKUB9wEchUYmAIYNgMZDDwKIDIPyDi\nEgOjAAPLFyCLRYGBR+AqzweGowxS5UVZHH+ZPzA+YNj///8v1pyjh94xAABbhSE1xl33oAAAAABJ\nRU5ErkJggg==\n", "text/latex": [ "$$2 x$$" ], "text/plain": [ "2⋅x" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "l_a = 2*x\n", "l_a #Pの軌跡" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAf0lEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChz\nFgNDvgOEvT8AzgQKrA9gPZPYUwNkcXxnCGd4dWA1kMmVwFDKUB9wEchUYmAIYNgMZDDwKIDIPyDi\nEgOjAAPLFyCLRYGBR+AqzweGowxS5UVZHH+ZPzA+YNj///8v1pyjh94xAABbhSE1xl33oAAAAABJ\nRU5ErkJggg==\n", "text/latex": [ "$$2 x$$" ], "text/plain": [ "2⋅x" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = diff(y_1,x)\n", "m" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAAAWBAMAAAAWdi71AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXxoyr3hxAAAAAElFTkSuQmCC\n", "text/latex": [ "$$- t^{2} + 2 t x + 1$$" ], "text/plain": [ " 2 \n", "- t + 2⋅t⋅x + 1" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x0 = t\n", "y0 = y_1.subs({x:x0})\n", "l0 = m.subs({x:t})*(x-x0)+y0\n", "expand(l0) #アイ, 接線の方程式" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKcAAAAWBAMAAABXrpEPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n", "text/latex": [ "$$- 2 a t + 2 a + t^{2} - 1$$" ], "text/plain": [ " 2 \n", "-2⋅a⋅t + 2⋅a + t - 1" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eq1 = -(l0.subs({x:a})-l_a.subs({x:a}))\n", "expand(eq1) #ウエオ，tの方程式" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAUBAMAAAB8EA4WAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEVO8Qq5lm\nIrurE6D6AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHklEQVQ4EWMQMmEgEfAYJTCEYddze98T7BJA\nUVaIrooJ6Co4FzDEJ6ALgvjsnVBdnKvXY+jimsDA0YBF16xdP+F2yWPo4lBg4PqCRRcDIz5dzD/J\n0QW0huknA+/WjUBvoAC8dgFV5gcwZFxg/oCihwG/C4FqtRg4exi4H5Cmi3sDA/dPBiYBoC5OIyUg\nUHYAGUDAhdIMDBwPGOaDlYKUQwF+XewbGO7OD2Coh6mG0fh1eTMwyM4vYHjMfgGmHkKj6uL8jCLL\nqxq6S4FZgEeLFUUY2V+7+q0OMCihpA/m////K3AuDLklhqqLe8XnNbDUC5JBdwqqahQeNM2DxNCd\ngqIOlYOkayKqDD4ekq4D+NShygF1CamgChHmMSolAAD7Wk8hgSaNhwAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ 1, \\quad 2 a - 1\\right ]$$" ], "text/plain": [ "[1, 2⋅a - 1]" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s1 = solve(eq1,t)\n", "s1 #カキク, tの値" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMwAAAAZBAMAAACP/9xdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$- 4 a^{2} + 4 a + x \\left(4 a - 2\\right)$$" ], "text/plain": [ " 2 \n", "- 4⋅a + 4⋅a + x⋅(4⋅a - 2)" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "l_1 = collect(expand(l0.subs({t:s1[1]})),x)\n", "l_1" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAf0lEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChz\nFgNDvgOEvT8AzgQKrA9gPZPYUwNkcXxnCGd4dWA1kMmVwFDKUB9wEchUYmAIYNgMZDDwKIDIPyDi\nEgOjAAPLFyCLRYGBR+AqzweGowxS5UVZHH+ZPzA+YNj///8v1pyjh94xAABbhSE1xl33oAAAAABJ\nRU5ErkJggg==\n", "text/latex": [ "$$2 x$$" ], "text/plain": [ "2⋅x" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "l_0 = l0.subs({t:s1[0]})\n", "l_0 #" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## (1-b)" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "from sympy import *\n", "init_printing()" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": true }, "outputs": [], "source": [ "a,x,t = symbols('a,x,t')" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAWBAMAAACIzClVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA90lEQVQoFWNggIDcvQJQFhaKzYF5AhZhqBCH\nA88XPLIBLF9xyzIwcH3EJ8uhgE82F02S5e7uA3Ah9gtwJoQhzMCGcEkYQwxElEsBQhsyMLyDsBgY\neGat2ogq+4KBwf4AVJrv//8PqLLzBcCyQsquagpQRUAKZjKQ2S/AwJjA3sSBFIoIWdZvDAxsAmwf\nuRyw6eUwYGBgZODcAJdjmzlzxsuZM4HCQHAXTPIrgCkoATeZ6QJY5PwBrLJrGVgcGLgE9BmYBBDy\nML08FxiYHBjiF+xncEdIwn3kkZZrwcAgrCR0vQCL7Pz//38gCUOZMJMxZUAijA5I4gD8Ii+sTC4R\n3wAAAABJRU5ErkJggg==\n", "text/latex": [ "$$x^{2} + 2$$" ], "text/plain": [ " 2 \n", "x + 2" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y_1 = x**2+2\n", "y_1" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAAVBAMAAADyTXAxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLklEQVQoFXWRPUvDUBSGn9SmtmkqKrjoEiq4\nCYL+gOIf8E4KLtbBRQSLIF3UZnQQdNUuWdTBQcQfUC10K/gB7t0EB6nFoohSz01vMSA9kHPf530C\nuRCIjNXR04w0Jib8/13YTPTpme4j7KCPiN1HhZWdzRlehtHJuZ5cZPDT5D3sPCXf0Bocd6MdkFCk\njow4gc1cmN0GKXHvRlSUFvtCDqTbf0KaS+WW5aiHrzpt7PrqwY5A8oOxOx+CUBwWWOLl5kLAycPK\nDEOFUJzBNiX1IJCVJ/aGo7SIeaC41tGViNXiWQPVcH/r/Yg1LFdWpxoyHiMQ11eOe7giFnY9Lebh\nlie3SY3x4ta6NOmy/oR1XpwKkj8DTatBpdP5kirTkkVa/mxgb9Sqr5q6c9UL5vwFdhBJsUJoGO8A\nAAAASUVORK5CYII=\n", "text/latex": [ "$$2 \\sqrt{2} x$$" ], "text/plain": [ "2⋅√2⋅x" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "l_a = sqrt(8)*x\n", "l_a" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAf0lEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChz\nFgNDvgOEvT8AzgQKrA9gPZPYUwNkcXxnCGd4dWA1kMmVwFDKUB9wEchUYmAIYNgMZDDwKIDIPyDi\nEgOjAAPLFyCLRYGBR+AqzweGowxS5UVZHH+ZPzA+YNj///8v1pyjh94xAABbhSE1xl33oAAAAABJ\nRU5ErkJggg==\n", "text/latex": [ "$$2 x$$" ], "text/plain": [ "2⋅x" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = diff(y_1,x)\n", "m" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHoAAAAWBAMAAAD9QZX2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n", "text/latex": [ "$$- t^{2} + 2 t x + 2$$" ], "text/plain": [ " 2 \n", "- t + 2⋅t⋅x + 2" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x0 = t\n", "y0 = y_1.subs({x:x0})\n", "l0 = m.subs({x:t})*(x-x0)+y0\n", "expand(l0) #アイ, 接線の方程式" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAAAWBAMAAABtfi58AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\npLbSB9TCmnsgK+svdvdHgL7QXyLqrxbuTli3AAAAAElFTkSuQmCC\n", "text/latex": [ "$$- 2 a t + 2 \\sqrt{2} a + t^{2} - 2$$" ], "text/plain": [ " 2 \n", "-2⋅a⋅t + 2⋅√2⋅a + t - 2" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eq1 = -(l0.subs({x:a})-l_a.subs({x:a}))\n", "expand(eq1) #ウエオ，tの方程式" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaEAAAAyBAMAAAAHA3moAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n97j/AAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left [ a - \\sqrt{a^{2} - 2 \\sqrt{2} a + 2}, \\quad a + \\sqrt{a^{2} - 2 \\sqrt{2} a + 2}\\right ]$$" ], "text/plain": [ "⎡ _________________ _________________⎤\n", "⎢ ╱ 2 ╱ 2 ⎥\n", "⎣a - ╲╱ a - 2⋅√2⋅a + 2 , a + ╲╱ a - 2⋅√2⋅a + 2 ⎦" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s1 = solve(eq1,t)\n", "s1 #カキク, tの値" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 3-(a) 以降のpythonによる導出\n", "\n", "順番が前後しているので，入力の順序を間違えないように．\n", "ここは3-(a)からの続きになる．" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFwAAAAWBAMAAAC/AuZhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMpm7du8iZolU\nq0RaI+fpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHklEQVQ4EWNgwARbUwUwBXGKcE/gVcApiSnB\nPIHlJ6YoThHmB5z/cEpik+D5iE0UpxhzAU4phARLAIy9FcaA0wg5uBBzApTJBdeHKQcXYaiAKX/N\n8AQiygN3FFwOrpxzEVQ5i86ZRDTlcDm4agZuJqhyjv//P6Aph8shlK8EKZ+6tUIAIQR3DKYcYwFQ\nOV8ngz5CNQNMORY5bgagcokFDGFYlCPJCRmDgArDEZDyeAEGc5hybiUl5SYlpQtAPoYcA+MGkPI2\nBsYvMOVAGuoYLHI8d+/e7y38zcADDRSwJqhyrHIMHAmMfxjYFE4ijId5lQGLHAN/AoM6w+ILG7Ap\nx5Tjzv+7QLba8/oDLMqxyCFUwVkIx8CF8DEYJ6DLAgCNmFD8cjmZAgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$- 4 a^{2} + 4 a$$" ], "text/plain": [ " 2 \n", "- 4⋅a + 4⋅a" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# ll = -4*a**2+4*a+x*(4*a−2)\n", " \n", "rr = l_1.subs({x:0})\n", "rr #シス" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "image/png": 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BrW6o0BLffqoAhro6mCehaYV8WrNmDV577TWkpaXhzTffxJo1a/iO\nRAj5F7GOCItHuOFMXu2gnkehJR6fdg2zI5xgbqSryKcl/cRxHBobGwEADQ0NcHR05DkRIaQn9w1z\ngYHu4GqYY0xx9/DHHsllMQG2NCuFZ1lZWZgyZQoYY5DL5Th16hTc3Hpeyz0uLg5xcXEAgLa2NmRm\nZqoyKiFa79vTBVg0wn3A488KLXEAytvSmfxDTEwMystv3nx1/fr1OHToEMaOHYu5c+fi559/Rlxc\nHA4ePMhDSkJIH1GJk/8xNzdHfX09OI4DYwzm5uY3hlcIIWppwCVOK1NpIEdHRxw9ehQAcPjwYfj4\n+PCciBCiLLRrsQbavHkznnzySXR1dcHAwODGmDchRPPQcAohhPCPhlMIIUQbUYkTQoiAUYkTQoiA\nUYkTQoiAKXp2Cq16RQghKkRX4oQQImBU4oQQImBU4oQQImBU4oQQImBU4oQQImBU4oQQImBU4oQQ\nImBU4oQQImBU4oQQImBU4oQQImD/D1WHJAB10IPGAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%matplotlib inline\n", "\n", "plot(rr,(a,-1,2))" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAAZBAMAAAAiUgaxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABqUlEQVQ4EY2UvUvDUBTFT2hj2tiWIHRxkFJB\nR79w17/AuBTEpQouIlS6OLh0EAQnxUkX66gIdihFqNJu4uTHpigUR7soKBVB4n3P95qXttrcIe/c\nk/N7TS4vBdzKjttu418ZVk/Bf9pNmlbwze38KzOnffpPq8noh9r512aVZwe6ETEL0JITUzKX5UIb\nlf2f6wKQgiHfKpbgwV7a7Z+6t4F5YAnYE6lNbDD13BF6/HX1xQxxRhX7QGaKe8GLp0kmDnnXehEc\ncE2cvo2KLbmw47xSWsu3Irz3cDhj3qmNvuwVbcMrkhCCLa7v5dgEQw2EdlCR6UBNKrrl+l6uThkz\njbkaBmXavJEKqu/lHiiTBEZsnMh0OEdzPmZVbPrr5fJRuVziETYXzADsfXahv3OTLoyTpfje3yPu\nDlq8gSgbJS/lOXXF93JbCCYQiX8jULgVnDIXXfG9XB39a9llnOMlvSo4IyEELYrv5YZRcZwvpC6n\nV3IiHsoLQYviS27ooFQDim6mqbqcM5breKY6n+vZ5r4k6Fy3V8Bq91oc+o7ay8d3O9ZOkePnf+IH\nQ5Zz19819xsAAAAASUVORK5CYII=\n", "text/latex": [ "$$2 a^{2} \\left(- a + 1\\right)$$" ], "text/plain": [ " 2 \n", "2⋅a ⋅(-a + 1)" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S = simplify(a*rr/2)\n", "S #チツテ" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "image/png": 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eZBaUk110wuoogJ0LRESmi0imiGSJyF2neHywiGwQkRoR+UNr5lWO9WF6Hou/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\nznMTrU8AAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%matplotlib inline\n", "\n", "plot(S,(a,0,1))" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAyBAMAAAAXRmBdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAwONW9fOoCiHAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left [ 0, \\quad \\frac{2}{3}\\right ]$$" ], "text/plain": [ "[0, 2/3]" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s2 = solve(diff(S,a),a)\n", "s2 " ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqElEQVQYGZ2PPQ4BURSFvyEm3vgZNNoXQq1Q\naDRWYAcsQTehMbEBGolyKtNaAjYyswQSU06e+94KxC3Od3PuOcWF3nAK1NbsY/CXqBOoBP8DjcJB\nzkEhAseNw9VpRTs8nbY0XVkW8AAvjcaJ1I0xgn9H2sa8fmr3o7nkPE2WQ7MkTKB+YXXDjjXlw9Sq\ntxtZwDZ2UGfxOlRLCAsHNSF4Qzsnu0vkMJjBFyxOKQHdWH1vAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\frac{2}{3}$$" ], "text/plain": [ "2/3" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s2[1] # ト，ナ" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAqBAMAAACuFQ3dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaJmUSrVGYizbvd\n7zLJaKAlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7ElEQVQYGWNgAAIhZVcQBQYODGwFUCZ7AwPn\nAiibeQMD4wEom/W3AHMDlM2Q/8MCxmRg/r8Qzi6L+RsA5XAkMOhfhLJ5JzDwfYey5YF0BkKcYSeU\nzbqQgUMBymZ45oxwG0yMFvR/OPhAmfFTvPcx8B1zNjZmYGBMYLAP4AAa/I2BgU2AgfcCywQGhgQG\nBt4DDGz/mBkYmB8wMHB9BLKBNgZBbOX5CKQ3QNjxBkAlDRD2MSAFCi8gYHEAElogFgODCog4DSIY\n2B0YZjIwAm0CgicMDJoMfF9ATM7Dxm4HGNjBbC6gUw4wcF4HCgMA6vo8VDetzDwAAAAASUVORK5C\nYII=\n", "text/latex": [ "$$\\frac{8}{27}$$" ], "text/plain": [ "8/27" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S.subs({a:s2[1]}) # ニヌネ" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAvBAMAAADX32PpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n82e98gAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\frac{7 a^{3}}{3} - 3 a^{2} + a$$" ], "text/plain": [ " 3 \n", "7⋅a 2 \n", "──── - 3⋅a + a\n", " 3 " ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T = expand(integrate(y_1-l_1,(x,0,a)))\n", "T # ノハヒフ" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "image/png": 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