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

1  線形代数
1.1  グラムシュミットの直交化
1.2  直交補空間
2  微積分
2.1  Taylor展開
2.2  積分の比較
2.2.1  積分の誤差
3  1-(2) 図形の面積
4  1-(2)改
5  1-(1) 放物線の接線の距離
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 線形代数\n", "\n", "## グラムシュミットの直交化\n" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from sympy import *\n", "\n", "init_printing()\n", "x1 = Matrix([1,1,1])\n", "x2 = Matrix([0,1,0])\n", "x3 = Matrix([-1,1,0])" ] }, { "cell_type": "code", "execution_count": 135, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOsAAABNCAMAAABwkWMUAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRO/NZt0ibIm72KKapwAAAAlwSFlzAAAOxAAADsQBlSsOGwAABa9JREFUeAHtXGuD\noyoM9VXvbp/X+v//64KamBMDrQ4Wd3b8UAmP4zkEgWaYFsXnr/K62zN3hN7Cuayr+5Z2b7R5BV32\nw1WNUGj5vPtY/saTZJUljig9TVo3QUeRiwkakdkq+7px12nkgpbPu/nSqhdU30kucUQr0roJOopM\nWhGZrbIvBQ20uOC2XqtEZZwxQVoHay10gOH0CAmNyN7CtmgxR2zG2eFEAGdsIAkVa6GjyOTX4TmI\nHNd6urQkBptRbuQeZXQ0rW1VV/0/otX5rPnRGhm5c1F8DF/miqnfVwGNL178fd3Jr4/63lcNq0VG\nnB1MxHoRoRE5i1alAhmpQsOMacXqiPyjlXonxRhur3e+rub2Anufnh2+s19fQiNyDr+O22v3OclB\nRmGNVMJaKWO+K2hEzqF1pjamkJEuXdoRraoyIufR2jV1xbuUtGtOVz9rhj6A1vpWFPWTfYCMODuY\niPm1exRFxYEARI75ta4u/bVyvPyFzca86GeEUd0B3lroCHJxd1rL3n0MFyLHtIIUbAZFthFj5Frs\n5Nf7uSgevfsYLiSdS2t7mZcfZDTRjNxe9GJx4l08ImfS2lznLeLq1+OV1jtjH0LrbmPYDYeu5jFx\nEK2PfopvwTTFLGOJuF9vs1Q1YnKM4eFdffTTBJ9Y69nN8WeaC3L6dVro/cLQbA/vxPxaXk+n05M2\nExm10kJfdk1X0RqY1q+XYUdMb0BGrbjQEyH1VnF2MBHzKzbKqRUWeqaFjDg7mPgrtHr280LPWr6t\n1nmhT6q1rbtbR9NvANn3KY4JaaWPhYuFPsCIs4MJyZAqXd1q3fr5AC4cMRGte8TC5ULPrJARZwcT\nhtbbECjt+Mvc1BaRI1pd/RTxJkkZFnouQEacHUwYWp/DH1QXcwEif1QrLvSsBRlxdjBhaO0Hred5\n3zk2RuSPasWFnrUgI84OJpZa237YBJ97/oaTXWuA/de1Pnq3C/aT7HCbH4PIH/XrTAJSyAiKTGPp\n18fo139C6/tj+L9fv0X/SUvMw///EnXeSUqcsb4KVDPIWuglcjHOTSc9NyGyt3BMSEtoXTvQFCoL\nMxJroSXDCe45RGBvHGiashHZW9hWWqm1YgycZCMjyg3fJUMSNewl6gPtJVQMnNQk0Frc/R5RxCZH\nbESO+TUSC2973YVEnO9G76sYOFUFRtuQi7ZzR7C+tPcnNu4OhNxZtjvFOUQdSBpafbmIgVN1hN6M\nTHDijsjeQlZocUNs5r6H6u8UXHNK2DjLcbboxo3I+vneRtKbtaodyvJJplaIgVMbZFRsQyYwuCPy\nVq2t2nnCIwbD1PrGGN6MvKSQyK8c3DWeMGYFtIoYODVVvU/ZwXsA2aivkLe+rwYyZi0Z6Rg41UdG\nlBu+L5FDdRHZW9gWLUbBZpwdThg4cwwcDm+vhTaQHQ2AnGgh8ke1UgxcHd5GRuHeoxJLq4LMr5XI\nuhVLnoFPoVVB/mj95Pv641e1Cs4dEkpZ7+vXx3D6WPjE/2jv6x6xcHLV0bQ6Xqm/q5PUI87D+2kV\nh7dTva8ScupUXM3ie4mdtOLh7SRaFeRxtPJQHhPY+6rQMO152KioejGLXxWt76P15eFt1fuqIwzz\nb/Krjokn8KuGnHoIkb/NGIYz0jQYjqAVY+LIiHiG79YYpqNT2AqRs/hVxcSREbK1LEurfXQKkWNa\nI7Fwi4LKsxhNVVRMHBkpGMO0kPGMNDVC5JhWajHcsRkU2YbFSNQUMfG10CHkxXEJNcPn0ipj4qm0\nLo9OIXImrRATR0bC94FkwK/G0SlEzqQVYuLIKCBQZNtaraNTiJxNq4iJIyMhKpA0tZpHpxA5h1Yd\nE0dGAYEi29JqH51C5FHrsMX63G9pzDFx/o0LIeVl0votjcXRKURmq/W/ldE00z/woOUfPPwQRfPy\nj1WK4xKHK1BMfCN0BJkfoUiThD9T9WHItlZrZwAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ \\left[\\begin{matrix}1\\\\1\\\\1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}- \\frac{1}{3}\\\\\\frac{2}{3}\\\\- \\frac{1}{3}\\end{matrix}\\right], \\quad \\left[\\begin{matrix}- \\frac{1}{2}\\\\0\\\\\\frac{1}{2}\\end{matrix}\\right]\\right ]$$" ], "text/plain": [ "⎡⎡1⎤, ⎡-1/3⎤, ⎡-1/2⎤⎤\n", "⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥\n", "⎢⎢1⎥ ⎢2/3 ⎥ ⎢ 0 ⎥⎥\n", "⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥\n", "⎣⎣1⎦ ⎣-1/3⎦ ⎣1/2 ⎦⎦" ] }, "execution_count": 135, "metadata": {}, "output_type": "execute_result" } ], "source": [ "GramSchmidt([x1,x2,x3])" ] }, { "cell_type": "code", "execution_count": 141, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAACsAAABYCAMAAABbNf6dAAAANlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABHL6OuAAAAEXRSTlMAMquZdlQQ\nQN0iRIm7Zs3vfEQ8efgAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAGhSURBVEgN7VfBsoMgDAyCVrTa\n8v8/+xLgRQwG2xlvlYNmYNmGuF0ATIitA9FSNz4BxhQDmGAdtl5Ap3nrWAjQ4SQTzNa7Rc8BZme7\ngXsWFds7sAuAfX6ARYzFLIgsN5V3QU5qn/Dmnx5e21oqXp/KkaviVpfo6Smx0xxo4QOvqJnDSmW1\nExFR84HrLnkBJiT28RvGXH3Ii6xzQKbVQucj54gv9+KPUfMi8ZSlYWY352k09wALa0i0kbt4HGFN\nUacCeshbjpex4GXJ7oI8QWBLmirWsHvlpmkKVii3jd0rt4mlwUI1Z9hSuSfYnXJPsF/lUCq3xSuV\n28KCUG4TK5TbxMZB1k+CtjRZel9EK3qgsZ/zvkd45AqiQzW8713tAbf3ceGUQOiMJbsL8lyBVRhT\nt4a9wvsGx1vNfw5i107dHvcGQ//wulXe14/aWeP2Pizf194nS67pgXCstTypgb19Emt0nxHx8Hn9\nGZFluAvONZkR20vT7xXed3AfOvQ+LQfMsvI+HfuL3he/fnmPld7H99iBLqnO8dVqUwBH8R7rHPwB\nl1MlDUDSQ34AAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left[\\begin{matrix}\\frac{\\sqrt{3}}{3}\\\\\\frac{\\sqrt{3}}{3}\\\\\\frac{\\sqrt{3}}{3}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡√3⎤\n", "⎢──⎥\n", "⎢3 ⎥\n", "⎢ ⎥\n", "⎢√3⎥\n", "⎢──⎥\n", "⎢3 ⎥\n", "⎢ ⎥\n", "⎢√3⎥\n", "⎢──⎥\n", "⎣3 ⎦" ] }, "execution_count": 141, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a1 = x1/x1.norm()\n", "a1" ] }, { "cell_type": "code", "execution_count": 149, "metadata": { "collapsed": true }, "outputs": [], "source": [ "y2 = x2-x2.dot(a1)*a1" ] }, { "cell_type": "code", "execution_count": 150, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAABYCAMAAAB8m391AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}- \\frac{\\sqrt{6}}{6}\\\\\\frac{\\sqrt{6}}{3}\\\\- \\frac{\\sqrt{6}}{6}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡-√6 ⎤\n", "⎢────⎥\n", "⎢ 6 ⎥\n", "⎢ ⎥\n", "⎢ √6 ⎥\n", "⎢ ── ⎥\n", "⎢ 3 ⎥\n", "⎢ ⎥\n", "⎢-√6 ⎥\n", "⎢────⎥\n", "⎣ 6 ⎦" ] }, "execution_count": 150, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a2 = y2/y2.norm()\n", "a2" ] }, { "cell_type": "code", "execution_count": 154, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC0AAABNCAMAAAAFMB3yAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left[\\begin{matrix}- \\frac{1}{2}\\\\0\\\\\\frac{1}{2}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡-1/2⎤\n", "⎢ ⎥\n", "⎢ 0 ⎥\n", "⎢ ⎥\n", "⎣1/2 ⎦" ] }, "execution_count": 154, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y3 = x3 - x3.dot(a1)*a1-x3.dot(a2)*a2\n", "y3" ] }, { "cell_type": "code", "execution_count": 155, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAABXCAMAAACNzc2gAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left[\\begin{matrix}- \\frac{\\sqrt{2}}{2}\\\\0\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$" ], "text/plain": [ "⎡-√2 ⎤\n", "⎢────⎥\n", "⎢ 2 ⎥\n", "⎢ ⎥\n", "⎢ 0 ⎥\n", "⎢ ⎥\n", "⎢ √2 ⎥\n", "⎢ ── ⎥\n", "⎣ 2 ⎦" ] }, "execution_count": 155, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a3 = y3/y3.norm()\n", "a3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 直交補空間" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAANUAAAAyBAMAAAA5L4OKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEImZRO/dMlQiu6vN\nZnZmcXX2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAADz0lEQVRYCbVYzUtUURT/vZnRmTfjzDwE3Rg6\nFS1C0KnoA0Wcv6APKKMIMoggEhsKWgUaEWSbBgoXUWibFm2aCAKhcmjRooUKUTtrTFpEkWNEWVLT\n/XxvBsf37n2DZ3HPvff8zvndL895I0CkOU3bDZQ7ozJ4aCAruxuk48ctEfmm17bMXQ/rXM34Kc5l\n/vHaTRsCP7ww7nZzldtjy+444DFwxAvjYe/nB5MseOAwAEzXeYgjRUaSzHtxzVp1c3WkGUmHJxeB\nnbeMb1P7u7xWVWFv/3L7Y9EeCxIVrtB3NONdcQiBRXJ5KmKk4sONBXT1WgytwdWYx2Z0W2PoQ48K\nExCwAsuRXHAuOKnLtQuw8Jp47cQHNS4DTYQlmIqXNLkS7Gb/MS/FfQHJIsVr7+sqzBzCv6ivcZS2\nKjKSpagW1kL5vsJpJHJXEiUsAF+LKjxAxFpCwkJihsOVuca3bH0aWomWzDniqHiGB0Yf4R6B657h\nbLn803ixsDhP1zgxSltPafvW3keXZq4wqPK+nMDktiZoBFVpyZp/GdYHF87ik6VKRHCNucQh31wX\nPk5pUCHc1ZljeJF0J1I63j6xyUnm+CHj01/HranE0GdYW5WWSfrTkeoScK6Wa4Q9kcBvahNpOVyg\nA7RwxfoKjSgBPTso1jhR02Oa3tu2DLXxtBzuHaQDXeElYDueuTjGDgObeD3iaRmRQQduvDpDZCiL\nZLla+NE7QFECzmKErl1IhQufuVZEK9s4GbK0XMklvVQ0LQGnMZFxwV5KAbuzHMDSsk8uXgJw01qf\nK0DOEEFWKkRa9sclSgCG16dCN9sSe/MiLQsur3fYUaiMKktAIE1na7/DEC+0SxmCEGlZcHn9fTWK\nT2Yam0QXJeANH57jqrqNlti4Mke5neGMdXFUBhiTnQodSoXYOdlTTQvhfXIgPnjFZyKbjRyUxrX6\nZPm7PZmye07n8vUbljMivYZyOSUn1tYUc89qUVrX6JcvcnLOqIU6WS5LO9dNnZ/tibVctqlWJ+9M\nxp2uSy9WYfPPVRHEpVsH1/P3e10C1zDF5hezclpzX/fdc5CM6uioZdo/ETW5gOigE0it90DCtLmC\nrNRJdxXdL5+uHlesoPurmZSz7oxYkR5XQwHBZZW92BhSoPstX1xB8rVXEJ5qKg0MSaTevsw03srT\nlxHc9S3Ej0mE4BKfiXJ2XX2rU5bwdSHVBnPnE3mEEEmXXMTGy0iKccS0X7KPpS1lmZP8/42PCMou\nhvw/1FJR2ccvsPWQ8Iyv2lfoN5aHX+SEzdCW98DWa747RyP8B1w1Dg0NeSxDAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left \\{ x_{1} : - \\frac{2 x_{3}}{5}, \\quad x_{2} : \\frac{x_{3}}{5}\\right \\}$$" ], "text/plain": [ "⎧ -2⋅x₃ x₃⎫\n", "⎨x₁: ──────, x₂: ──⎬\n", "⎩ 5 5 ⎭" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x1,x2,x3,a,b=symbols('x1,x2,x3,a,b')\n", "eq1=2*x1-x2+x3\n", "eq2=x1-3*x2+x3\n", "\n", "solve({eq1,eq2},{x1,x2,x3})" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": true }, "outputs": [], "source": [ "eq3 = expand(5/a*(-Rational(2,5)*a*x1+Rational(1,5)*a*x2+a*x3))" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE0AAAAyBAMAAADrU0uXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEIFwa9g3nQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ \\frac{a}{2} + \\frac{5 b}{2}\\right ]$$" ], "text/plain": [ "⎡a 5⋅b⎤\n", "⎢─ + ───⎥\n", "⎣2 2 ⎦" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(eq3.subs({x2:a,x3:b}),x1)" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAABMCAMAAADwSaEZAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}\\frac{a}{2} + \\frac{5 b}{2}\\\\a\\\\b\\end{matrix}\\right]$$" ], "text/plain": [ "⎡a 5⋅b⎤\n", "⎢─ + ───⎥\n", "⎢2 2 ⎥\n", "⎢ ⎥\n", "⎢ a ⎥\n", "⎢ ⎥\n", "⎣ b ⎦" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a*Matrix([Rational(1,2),1,0]) + b*Matrix([Rational(5,2),0,1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 微積分\n", "\n", "## Taylor展開" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "IPython console for SymPy 1.0 (Python 3.6.1-64-bit) (ground types: python)\n", "\n", "These commands were executed:\n", ">>> from __future__ import division\n", ">>> from sympy import *\n", ">>> x, y, z, t = symbols('x y z t')\n", ">>> k, m, n = symbols('k m n', integer=True)\n", ">>> f, g, h = symbols('f g h', cls=Function)\n", ">>> init_printing()\n", "\n", "Documentation can be found at http://docs.sympy.org/1.0/\n" ] } ], "source": [ "from sympy import *\n", "init_session()\n", "\n", "t = symbols('t')\n", "v = exp(-t)+1.0" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$2.0 - t + \\frac{t^{2}}{2} - \\frac{t^{3}}{6} + \\frac{t^{4}}{24} - \\frac{t^{5}}{120} + \\mathcal{O}\\left(t^{6}\\right)$$" ], "text/plain": [ " 2 3 4 5 \n", " t t t t ⎛ 6⎞\n", "2.0 - t + ── - ── + ── - ─── + O⎝t ⎠\n", " 2 6 24 120 " ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vs = v.series(t,0,6)\n", "vs" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "vsO = vs.removeO()\n", "p = plot(v, vsO, (t,0,2), ylim=[1,2], show=False)\n", "p[0].line_color = 'b'\n", "p[1].line_color = 'r'\n", "p.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 積分の比較\n" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " -2 \n", "- 1.0⋅ℯ + 3.0\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$$2.86466471676339$$" ], "text/plain": [ "2.86466471676339" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "i_v = integrate(v,(t,0,2))\n", "pprint(i_v)\n", "i_v.evalf()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2.84444444444444\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$$2.84444444444444$$" ], "text/plain": [ "2.84444444444444" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "i_vs = integrate(vsO,(t,0,2))\n", "pprint(i_vs)\n", "i_vs.evalf()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 積分の誤差\n", "\n", "誤差をわかりやすくするには,下のようにまとめれば良い.そうすると必要な次数は7(8)次であることがわかる." ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " 2 3 4 5 6 7 8 9 \n", " t t t t t t t t ⎛ 10⎞\n", "2.0 - t + ── - ── + ── - ─── + ─── - ──── + ───── - ────── + O⎝t ⎠\n", " 2 6 24 120 720 5040 40320 362880 \n", "2.86462081128748\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$$-4.39054759100443 \\cdot 10^{-5}$$" ], "text/plain": [ "-4.39054759100443e-5" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vs = v.series(t,0,10)\n", "pprint(vs)\n", "vsO = vs.removeO()\n", "i_vs = integrate(vsO,(t,0,2))\n", "pprint(i_vs)\n", "i_vs.evalf()-i_v.evalf()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 1-(2) 図形の面積\n", "\n", "2点Q($x_1,y_1$)とR($x_2,y_2$)を通る直線の方程式は\n", "$$\n", "y-y_1 = \\frac{y_1-y_2}{x_1-x_2}(x-x_1)\n", "$$\n", "で求められる." ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "image/png": 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yjDPfUQghpF2mhiK8PkGGY9dKoSiu4TWLVpV4cXUDztwow/ND7WFn\nbsx3HEIIeaznh9qDMWDDiSxec2hViW86nYPsslosHEujcEKIduvTS4yZAf0Rk5yP/Mp63nJoTYlX\nNzTjh3M3McXLFgP6mfIdhxBCnmjB6LsXX2w8yd9oXGtKfHvCLdQ0tmDR2IF8RyGEkE6xMzfGVF8p\ndiTmory2iZcMWlHijS1KfHcqGyNlfeFtL+E7DiGEdNqisc6ob1Ziy5kcXl5fK0r8l0v5KKlpxMIx\nNAonhAiLi7UZgj2sseVsDuqaWjT++ryXeGsrwze/ZcHTrjdGu/TjOw4hhDy1RWMHorKuGTsSc5+8\nsYrxXuJHMoqRVVqLhWMHas3UjoQQ8jT8HPsgYIAFNp7MQlOzUqOvzWuJM8aw4cQNOFgYI9TLhs8o\nhBDSLUvHyQAAB1I1u4QbryV+PqcCl25V4rXRzhDRAsiEEAEbPagfehmK8M2JLKh4dtgO8dqcG07c\ngEUvMV70o6XXCCHCxnEcXhvjjKtFNRpdNIK3Er9aWI1jV0swd4QTjMW09BohRPim+trByswQ0b9p\n7uYf3kp805kcjHC2wCvDaek18j9vv/023Nzc4OPjg+nTp6Oysv2pPp2cnODt7Q1fX1/I5XINpySk\nfYYifcwd6YSTijKkF1Rr5DV5KfGS6gbsvpgHF2szWvSBPCI4OBipqalISUnBoEGDsGrVqsduGx8f\nj+TkZFqth2iVWQGOMBHra+xWfF5KfOvZm2hpZXh1JC36QB41adIkiER311QNDAxEXl4ez4kIeToS\nEwO85O+AvZcLUFil/omxNF7idU0t+E/CTUzysIZTv16afnkiIN9//z2mTJnS7nMcxyEoKAh+fn6I\njo5+7D6io6Mhl8shl8tRWlqqrqiEPGLeyAFguDszq7ppfBn5XRfzUVnXjAWjabpZXRUUFISioqI2\n31+5ciWmTp364GuRSIRZs2a1u49Tp05BKpWipKQEwcHBcHNzw5gxY9psFxkZicjISACgc+dEYxws\nTBDqbYsfE25h6QQZehsZqO21NFrira0M35/KxmAHc8gd+2jypYkWiYuL6/D5zZs3Y//+/Th69Ohj\n7+KVSqUAACsrK0yfPh2JiYntljghfHlt9ADsu1yAnxJz8ZoaVyrT6OmUo1dLkF1WiwWjBtAt9qRd\nsbGxWLNmDfbu3QsTE5N2t6mtrUVNTc2Drw8fPgwvLy9NxiTkiXzszTHWpR8Ss2+jWdmqttfRaIl/\nezILUnNjTKFb7MljLF26FDU1NQgODoavry8WLVoEACgoKEBoaCgAoLi4GKNGjcLgwYMREBCAsLAw\nTJ48mc/YhLRr9nBHHMkowaHUtqcPVYVT8e2hj91ZSl4lwtedxt/D3Ol8OOGFXC6nyxGJRrW2MgR9\ndgJmxgb4ZcmIjs5AdPnUhMZG4htPZsPUUISX/OkWe0KIbtDT4/DqSCdczq3ExVsV6nkNtez1d/Ir\n63HgSiEi/B1gpsZPaQkhRNs8N9QevY1E+O5Utlr2r5ESv79s0dyRTpp4OUII0Rq9DEWYOaw/YlOL\nkFtep/L9q73EaxqasT3hFqZ42cC+T/tXGxBCSE82Z7gTOI5Tyzqcai/xXRfyILM2xWv0YSYhREfZ\nmRsj1NsWP53PxZ1G1a7DqdYSb21l2HwmBxyAwQ7m6nwpQgjRavNHDUBNYwt2Jql2HU61lvjx6yXI\nuV2HuTTRFSFEx/k6mMPPsQ82nc6BslV1l3artcQ3nc6BdW9DurmHEEJwdzR+q7wOcRnFKtun2ko8\ns6QGJxVlmB3oCANaP5MQQjDJwxpyxz6ISxdAiW8+kwOxSA8zA/qr6yUIIURQRPp6mORpjZ0X8lS2\n8o9aSryqrhm7LuRj6mA79DU1VMdLEEKIIM2QO8DIQE9llxuqpcT/m5SL+mYl5oxwUsfuCSFEsMxN\nxJg+xB6/JOejorap2/tTeYkrWxm2nM1BgJMFvKQSVe+eEEIEb+4IJzS2tGLH+e5fbqjyEo/LKEZe\nRT1epVvsCSGkXa42ZhgxsC+2nc1BSzfnGld5iW8+nQM7iRGCPaxVvWtCCOkx5o5wQkFVA45080oV\nlZZ4RmE1zmbdxuzhThDRZYWkC95//334+PjA19cXkyZNQkFBQbvbxcbGwtXVFTKZDKtXr9ZwSkK6\nb6K7Nez7GGNTNz/gVGnTbjmTAyMDPcwMoDnDSde8/fbbSElJQXJyMp555hksX768zTZKpRJRUVE4\ndOgQ0tPTsX37dqSnp/OQlpCu09fjMGe4ExKzy7u1H5WW+J5L+Zg+RApzE7Eqd0t0SO/evR98XVtb\n2+5KKImJiZDJZHB2doZYLEZERARiYmI0GZMQlZghd4CxgX639qHS1e7fDBqEIHcrVe6S6KD33nsP\nW7duhUQiQXx8fJvn8/Pz4eDwv3d79vb2SEhIaHdf0dHRiI6OBgDU19erJzAhXSQxMcDfQt26tQ+N\nrbFJyH1BQUEoKmq7cOzKlSsxderUB49XrVqFhoYGfPjhh49s9/PPPyM2NhYbN24EAGzbtg0JCQlY\nt26deoMToj5dXmNTpSNxQjojLi6uU9vNmjULoaGhbUpcKpUiN/d/19fm5eVBKpWqNCMhQkGXkBCt\nolAoHnwdExMDN7e2bzX9/f2hUCiQnZ2NpqYm7NixA+Hh4ZqMSYjWoJE40SrLli3DtWvXoKenB0dH\nR2zYsAEAUFBQgAULFuDgwYMQiURYt24dQkJCoFQqMW/ePHh6evKcnBB+0DlxQgjhX5fPidPpFEII\nETAqcUIIETAqcUIIETAqcUIIETBVX53S5ZPzhBBCnh6NxAkhRMCoxAkhRMCoxAkhRMCoxAkhRMCo\nxAkhRMCoxAkhRMCoxAkhRMCoxAkhRMCoxAkhRMCoxAkhRMD+H/NrEShUieKwAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import *\n", "init_printing()\n", "b, x, t = symbols('b,x,t')\n", "y_c = 1-x**2\n", "plot(y_c,(x,-2,2))" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAdt3NMolEEFTvq5lmIsfa\npuIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAB3SURBVCgVYxAyYSAEDqsxhBFSA5QPQ1bFMR2Ljs4C\nVFVVu55jqGJfvQ5NFQMjpioGBrlBrqpYCQSAgTuYXV9ngBb+6KHKvfLxGgYmdVRVu+ZZHUCNbbB8\nL6oqMA8lTYBFAohRxS5AjCoeLIqA7hJSwSaOIiakBgD0ZimSClhTrgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ -1, \\quad 1\\right ]$$" ], "text/plain": [ "[-1, 1]" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s_c = solve(y_c,x)\n", "s_c" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": true }, "outputs": [], "source": [ "x1=s_c[0]\n", "y1=0\n", "x2=s_c[1]-b\n", "y2=y_c.subs({x:x2})" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAELvv3c2rVESJdpkiZvtO\n0HUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABSSURBVAgdY2BUdmAAAtYEEMlXACL7BEDkFBDBELZq\nOwMD41MBuQMM7A8Z+CYwcAcwyE1gYN3A0FfAwHeBYZ8AA18Doy0DA9sFpg1Albv2MjAAAB2yD7/K\nbXugAAAAAElFTkSuQmCC\n", "text/latex": [ "$$b$$" ], "text/plain": [ "b" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = (y1-y2)/(x1-x2)\n", "simplify(m)" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAQBAMAAABAXPr7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAELvv3c2rVESJdpki\nZjI6QXVuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzElEQVQYGWNgVHZgwAKgwqwJWOQYGCDC/AUo\nkuwLIFyIcL8AVkmI8BQUOQaYTohw2KrtDIyrcrpmQRTBJEHCDIxfBeQPSDJcWeCKIgkWZmD/yMA/\nYSLDfIEaFEmwMAN3AIP8BAGGILAUZ2hoiGloaAIDRJiBdQNDP9AzvyD6GGAOggjzX2DYL8DA9w9N\nEiLM38Boy1DI9YFhIUQa6lqwMAPbBaYNHN+ZPvBcQJEECQO9smsvA2PWwtV3oeZCdYKEMQEsEDBl\ngCKMBxDCAKYKOkjYW34jAAAAAElFTkSuQmCC\n", "text/latex": [ "$$b x + b$$" ], "text/plain": [ "b⋅x + b" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y_m = expand(simplify(m)*(x+1))\n", "y_m" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(y_c,y_m.subs({b:0.7}),(x,-2,2))" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$- \\frac{b^{3}}{6} + b^{2} - 2 b + \\frac{4}{3}$$" ], "text/plain": [ " 3 \n", " b 2 4\n", "- ── + b - 2⋅b + ─\n", " 6 3" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s1=expand(integrate(y_c-y_m,(x,-1,1-b)))\n", "s1" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK4AAAAvBAMAAACIzuygAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n", "text/latex": [ "$$\\frac{2 b^{3}}{3} + 2 b^{2} - \\frac{5 b}{2} + \\frac{2}{3}$$" ], "text/plain": [ " 3 \n", "2⋅b 2 5⋅b 2\n", "──── + 2⋅b - ─── + ─\n", " 3 2 3" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s2=expand(integrate(y_m-y_c,(x,1-b,b)))\n", "s2" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAvBAMAAACWB9wTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\frac{b^{3}}{2} + 3 b^{2} - \\frac{9 b}{2} + 2$$" ], "text/plain": [ " 3 \n", "b 2 9⋅b \n", "── + 3⋅b - ─── + 2\n", "2 2 " ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "expand(s1+s2)" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(s1+s2,(b,0.5,1))" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\left [ -2 + \\sqrt{7}, \\quad - \\sqrt{7} - 2\\right ]$$" ], "text/plain": [ "[-2 + √7, -√7 - 2]" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(diff(s1+s2,b),b)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 1-(2)改\n", "\n", "放物線$C$の方程式を$y=1-0.5x^2$として問題を解く.\n", "放物線$C$上の2点Q($-\\sqrt{2},0$)とR($\\sqrt{2}-b,1-(\\sqrt{2}-b)^2$)と読み替える.また,$S_2$を求めるときの範囲は$\\sqrt{2}-b \\leq x \\leq b$と読み替える." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import *\n", "init_printing()\n", "b, x, t = symbols('b,x,t')\n", "# y_c = 1-0.5*x**2\n", "y_c = 1-Rational(1,2)*x**2\n", "plot(y_c,(x,-2,2))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHEAAAAmBAMAAAAB22msAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nlGFRenAfBAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ - \\sqrt{2}, \\quad \\sqrt{2}\\right ]$$" ], "text/plain": [ "[-√2, √2]" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s_c = solve(y_c,x)\n", "s_c" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "x1=s_c[0]\n", "y1=0\n", "x2=s_c[1]-b\n", "y2=y_c.subs({x:x2})" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\frac{\\left(x + \\sqrt{2}\\right) \\left(\\left(b - \\sqrt{2}\\right)^{2} - 2\\right)}{2 b - 4 \\sqrt{2}}$$" ], "text/plain": [ " ⎛ 2 ⎞\n", "(x + √2)⋅⎝(b - √2) - 2⎠\n", "────────────────────────\n", " 2⋅(b - 2⋅√2) " ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y_m = simplify((y1-y2)/(x1-x2)*(x-x1)+y1)\n", "y_m\n" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$- \\frac{1}{12 b - 24 \\sqrt{2}} \\left(b^{4} - 8 \\sqrt{2} b^{3} + 48 b^{2} - 64 \\sqrt{2} b + 64\\right)$$" ], "text/plain": [ " ⎛ 4 3 2 ⎞ \n", "-⎝b - 8⋅√2⋅b + 48⋅b - 64⋅√2⋅b + 64⎠ \n", "───────────────────────────────────────\n", " 12⋅b - 24⋅√2 " ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s1= expand(integrate(y_c-y_m,(x,x1,x2)))\n", "simplify(s1)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\frac{b^{3}}{3} + \\frac{2 \\sqrt{2} b^{3}}{2 b - 4 \\sqrt{2}} - \\frac{14 b^{2}}{2 b - 4 \\sqrt{2}} + \\frac{b}{2} + \\frac{14 \\sqrt{2} b}{2 b - 4 \\sqrt{2}} - \\frac{\\sqrt{2}}{3} - \\frac{8}{2 b - 4 \\sqrt{2}}$$" ], "text/plain": [ " 3 3 2 \n", "b 2⋅√2⋅b 14⋅b b 14⋅√2⋅b √2 8 \n", "── + ────────── - ────────── + ─ + ────────── - ── - ──────────\n", "3 2⋅b - 4⋅√2 2⋅b - 4⋅√2 2 2⋅b - 4⋅√2 3 2⋅b - 4⋅√2" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s2=expand(integrate(y_m-y_c,(x,x2,b)))\n", "s2" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "image/png": 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tPGlny0g8dvEO1ElrR7i3C0N8XYns40pgT+dmPRjU6/UEBASwZcsW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\nTEggCyGEmfh/y8ERgzagD2wAAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(s1+s2,(b,1/2,s_c[1]))" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAUBAMAAABCNWFYAAAAG1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAB4Gco9AAAACHRSTlMAdt3NMolEZgN4ymIAAAAJcEhZcwAADsQAAA7EAZUrDhsA\nAAAXSURBVAgdYxAyKVZjCGMAIpoQQipCagCy4Q1mVUJFwQAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left [ \\right ]$$" ], "text/plain": [ "[]" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(diff(s1+s2,b),b)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "ename": "IndexError", "evalue": "list index out of range", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mIndexError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdiff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms1\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0ms2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mevalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;31mIndexError\u001b[0m: list index out of range" ] } ], "source": [ "solve(diff(s1+s2,b),b)[1].evalf()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAARUAAAAmBAMAAAAYWRJmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nhVObe1f13axCAf4DPyME3OSNAhgAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left [ - \\sqrt{14} - 2 \\sqrt{2}, \\quad - 2 \\sqrt{2} + \\sqrt{14}\\right ]$$" ], "text/plain": [ "[-√14 - 2⋅√2, -2⋅√2 + √14]" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(simplify(diff(s1+s2,b)),b)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKkAAAAPBAMAAABtvvLvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nFzEBEB2JRHSpAAAAAElFTkSuQmCC\n", "text/latex": [ "$$0.913230262027751$$" ], "text/plain": [ "0.913230262027751" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(simplify(diff(s1+s2,b)),b)[1].evalf()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "与関数を\n", "$$y=1-\\frac{1}{2}x^2$$\n", "つまり\n", "``` python\n", "y=1-Rational(1,2)*x**2\n", "```\n", "\n", "とRationalを明示的に使えば.答えは,\n", "$$\n", "\\left [ - \\sqrt{14} - 2 \\sqrt{2}, \\quad - 2 \\sqrt{2} + \\sqrt{14}\\right ]\n", "$$\n", "と解析的に求められる.後ろ側が求めた数値解と一致する.\n", "その場合,\n", "``` python\n", "solve(simplify(diff(s1+s2,b)),b)\n", "```\n", "としないと求められない." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 1-(1) 放物線の接線の距離\n", "\n", "2015 年度大学入試センター試験 追試 数学 II・B 第 2 問 (1)の解答例を参考に示しておく(苦労して解いたんで)." ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "image/png": 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4aB6dRiFEUYZbmWDNZGfEpJci5nr3C9qQ+6jE+2D/5QIk5lbh3ZkjMNSCn3UD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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import *\n", "init_printing()\n", "a, x, t = symbols('a,x,t')\n", "y_c = 1-x**2\n", "plot(y_c,(x,-2,2))" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAACQAAAAOBAMAAAC1GaP7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZu6uJRO92\nVGZ6zyUAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjElEQVQYGWNggALWtHIHGBtKSzBw/kITamRg\n2I0mtJmBod8BVUxfAEMIqMBegPFM65o5SEq5fjJIMjw+YIQkxNzAMJFhvsBVBiFjEFABSqUxMAgw\nqCOpYeBJAPH+IAtdY2ANYGD5giTEksDAE3CR5wPDQbhg8MxpzVx/2T+wPoAL6f///5mx5+CxdwwA\ngUQiMYh5CboAAAAASUVORK5CYII=\n", "text/latex": [ "$$- 2 x$$" ], "text/plain": [ "-2⋅x" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = diff(y_c,x)\n", "m" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAAWBAMAAADultUCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABZklEQVQ4EWNgAANmLTUIg2TSkOEFyXogGjYy\nnBcgT+t2BvkC8nQyMNxPIFfnFnI1Miug6RTSWoMmgoM7CU2c7QGDPR6f84bB1LMbsB+AscE0XwID\nawCKCIjD6wARytP7BmEwMJy4eysBxgbTrBsY+P6giIA4MJ0MLHCd8f//oyrj+EZYJ6OyCtzRKLq5\nvzGA5dhcHpaYQGRQ7SwV4PiAogPGqTdgAMslM8x22IFFJ1skA9MCmGIUeh8DRM6MwT7hIhadTN8Y\nuB1gOjJ2g0APmMukwACRS2DoAgswd3R0R3R0PABxQCHEuoAh/wCQjQGcEHJfYJIo/sw3YLgPk0Cm\neRUYBKFyPPDoQdV5gWEFrwCyHgj7EAODYz5Y7hrLBwY3iCCKTg4Hzn3MmBoZNxvrbQDLsf/l+sA4\nAYtOtic2cu6YOjn+//+/ASzH9sTNaSZUAcxOpqc/XmLqwScC04lPDXY5NqQAAQBL2WAQ8WpbzwAA\nAABJRU5ErkJggg==\n", "text/latex": [ "$$a^{2} - 2 a x + 1$$" ], "text/plain": [ " 2 \n", "a - 2⋅a⋅x + 1" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y_l=collect(expand(m.subs({x: a})*(x-a)+y_c.subs({x: a})),x)\n", "y_l" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "image/png": 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Fu7+ksXxXBk+Pj2DmNT3lfowVWXQ6pU+fPsrVVfvrTAsLC/Hx8bF2Gb+hOLvD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\nCGHDJMSFEMKGSYgLIYQNkxAXQggb9v8BCxrINxu4JoAAAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(y_l.subs({a:1}),y_c,(x,-2,2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "点($x_0,y_0$)と直線($c_ax+c_by+c_c=0$)の距離($h$)の公式\n", "$$\n", "h = \\frac{|c_ax_0+c_by_0+c_c|}{\\sqrt{c_a^2+c_b^2}}\n", "$$" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": true }, "outputs": [], "source": [ "c_a=y_l.coeff(x)\n", "c_c=y_l.coeff(x,0)\n", "c_b=-1" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFQAAAA0BAMAAADmsQKoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3XKVpK4GSs/tH/mc0lUUqbG9AAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\frac{a^{2} + 1}{\\sqrt{4 a^{2} + 1}}$$" ], "text/plain": [ " 2 \n", " a + 1 \n", "─────────────\n", " __________\n", " ╱ 2 \n", "╲╱ 4⋅a + 1 " ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h = (c_a*0+c_b*0+c_c)/sqrt(c_a**2+c_b**2)\n", "h" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "collapsed": true }, "outputs": [], "source": [ "a_2 = solve(sqrt(4*a**2+1)**2-t**2,a**2)[0]" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAwBAMAAABDBS/wAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n", "text/latex": [ "$$\\frac{t^{2} + 3}{4 t}$$" ], "text/plain": [ " 2 \n", "t + 3\n", "──────\n", " 4⋅t " ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify((a_2+1)/t)" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "image/png": 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J25rCpbTV0NDA5s2bmTFjRudrXZ1PS/rf4wAKgYDv/hVc4PUudTtBtqZpXwAX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\nyfZr9qGQN8f5cUu8H4u35HD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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(h,(a,-1,1))" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\left [ 0, \\quad - \\frac{\\sqrt{2}}{2}, \\quad \\frac{\\sqrt{2}}{2}\\right ]$$" ], "text/plain": [ "⎡ -√2 √2⎤\n", "⎢0, ────, ──⎥\n", "⎣ 2 2 ⎦" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s1 = solve(diff(h,a),a)\n", "s1" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAvBAMAAAACzbekAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVQoFWNggAL5/yAA4zGYwFkgBmMACpet\nAIXbicJjSGdgXLUWroJjAoMYA/tvmBKuDQz6BQw/YVygVh8Hjq8wrjaIAVbM4gBkTQBxOxcACR2g\nA5gUgAypaQ1A5zR9YWDgBPIYGLiOAAmOvwwMu8BchjMNQDpfgGEGkBJkYLgPcod8AvcCIPW/AcJl\n/sHZAOSeZWCYLwCk+X6VAUmG3Qzsn0A0Q0wEiGRJy3UA0Qz1CWAKRrCCtNAcgOMCTlDdOsa0uw+Q\nDG1m4AKGJxxYMDCcgXMYGE4yMNg/QPDjG1C4QIn5DQhZYMD9QuYxcBqgcNNQeEwJKNwyBkak0OVL\nYGBC4kquWmaJpDr+/39IIgIA361IMdjdwjAAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\frac{\\sqrt{3}}{2}$$" ], "text/plain": [ "√3\n", "──\n", "2 " ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h.subs({a:s1[1]})" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": 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