{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "# Implicit Differentiation Assignment" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Caution: Make sure you run the parts of each question *in order*!" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Note\n", "\n", "You may use $ -5 < x < 5 $ and $ -5 < y < 5 $ for each implicit_plot" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "## Question 1\n", "\n", "[2 points] Consider the curve defined by $ y^4-4y^2-x^4+9x^2=0$\n", "\n", "### Part a\n", "\n", "Calculate the derivative $\\frac{dy}{dx}$.\n", "\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "4*y(x)^3*diff(y(x), x) - 8*y(x)*diff(y(x), x) - 2*x == 0" ] }, "execution_count": 1, "metadata": { }, "output_type": "execute_result" } ], "source": [ "y=function ('y')(x)\n", "derivative (y^4-4*y^2-x^2==0, x)\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 4 \\, y\\left(x\\right)^{3} \\frac{\\partial}{\\partial x}y\\left(x\\right) - 8 \\, y\\left(x\\right) \\frac{\\partial}{\\partial x}y\\left(x\\right) - 2 \\, x = 0\$" ], "text/latex": [ "$\\displaystyle 4 \\, y\\left(x\\right)^{3} \\frac{\\partial}{\\partial x}y\\left(x\\right) - 8 \\, y\\left(x\\right) \\frac{\\partial}{\\partial x}y\\left(x\\right) - 2 \\, x = 0$" ], "text/plain": [ "4*y(x)^3*diff(y(x), x) - 8*y(x)*diff(y(x), x) - 2*x == 0" ] }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" } ], "source": [ "show(_)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 4 \\, y\\left(x\\right)^{3} \\frac{\\partial}{\\partial x}y\\left(x\\right) - 8 \\, y\\left(x\\right) \\frac{\\partial}{\\partial x}y\\left(x\\right) - 2 \\, x = 0\$" ], "text/latex": [ "$\\displaystyle 4 \\, y\\left(x\\right)^{3} \\frac{\\partial}{\\partial x}y\\left(x\\right) - 8 \\, y\\left(x\\right) \\frac{\\partial}{\\partial x}y\\left(x\\right) - 2 \\, x = 0$" ], "text/plain": [ "4*y(x)^3*diff(y(x), x) - 8*y(x)*diff(y(x), x) - 2*x == 0" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "solve(derivative(y^4-4*y^2-x^4+9*x^2--0,x),derivative(y,x))\n", "show(_)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part b\n", "\n", "Calculate the slope $m$ at the point $(0.5888,1)$.\n", "\n", "[Check: The slope should be approximately 2.445]" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "2.44547161292800" ] }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "( (2*.5888^3) - (9*.5888))/ (2* (1^3-(2*1)))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part c\n", "\n", "Calculate the tangent line at the given point $(x_0,y_0):\\quad TL(x)=y_0+m\\cdot(x-x_0)$." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "2.44500000000000*x - 0.439616000000000" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('x')\n", "TL (x)=1+2.445*(x-.5888)\n", "TL(x)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part d\n", "\n", "Graph the original equation and the tangent line on the same window.\n" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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ejNKlS+Ps2bMoXry47NA802+/AUePAl9+KTsSj8CJx6Q+++wzBAYGYuHChenvq1ixoryAPJ3VSrfPX3xRdiQegS+1TGr9+vVo0KABunXrhtKlS6N+/fqYN29etp+fkJCAuLi4TG8sjdJBYsgQ7iDhIP4pmdS5c+dgtVpRtWpVbNmyBUOGDMHw4cOxePHiLD8/NDQUfn5+6W+BgYEaR6xj3EHCabxJ1KQKFCiABg0aYO/evenvGz58OCIjI7Fv375HPj8hIQEJCQnp/46Li0NgYCBvEk1NpSZ9TZvSplDmEJ7jMamyZcuiRo0amd731FNP4Ycffsjy8318fODj46NFaJ5l2zbuIJEHfKllUk2bNsXJkyczve/UqVOoUKGCpIg8FHeQyBNOPCb17rvvYv/+/ZgyZQrOnDmD5cuXY+7cuQgJCZEdmueIjgY2bKBJZe4g4RSe4zGxjRs3Yty4cTh9+jSCgoIwatQoBAcHO/S1XAgMwIQJwPTpQEwMF3N3EiceliemTzxJSUCFCkDnzsA338iOxuPwpRZjeaF0kOCaynnCiYexvLBagSZNgDp1ZEfikfh2OmPOUjpILFkiOxKPxSMexpw1Zw7g7w907So7Eo/FiYcxZ9y/T1sk+vXjDhIu4MTDmDO+/546SAweLDsSj8aJhznF9J1EuYOEW/A6HpYnplzHc/gwUL8+sGYN8NprsqPxaDziYcxRViuVNe3YUXYkHo8TD2OO4A4SbsWJhzFHcAcJt+LEw1hulA4Sr77KHSTchBMPY7nZu5c6SAwdKjsSw+DEw1hurFagcmXuIOFGnHgYy8n167RokDtIuBX/JBnLSXg4VRfs1092JIbCiYex7KSm0obQN96gTaHMbXhBAmPZ2bYNOHuW29aogEc8zCmm2qtltVKhr+eekx2J4fBeLZYnht+rdekS1VSePZvLm6qARzyMZWXePKBwYaBnT9mRGBInHsYelpREieett7htjUo48TD2sA0bqIMEr1RWDScexh7GHSRUx7fTGcvo9Gng55/5FrrKXBrxhIcDf//tpkgY04OwMFos2K2b7Eh068ED4OOP6TGv8px4VqygVeStWwPnzuU9AMZ04/59+mvKHSSylZwMdO8OTJoE9OiR9+PkOfG0awfUrQtcu0bJ58qVvAfBmC58/z1w8yYwaJDsSHRJCFrStG4d4OMDDB+e92PlOfEULw5s3gwEBdGq8nbtAJst74EwJp3VCrz8MlC1quxIdGncOODbb2mT/ooVQIsWeT+WS3M8ZcvSPNzjjwN//UUF2u7fd+WIjEly+DCwfz/fQs/GF18An31Gz+fPd73Jhsu30ytVopFPsWLA7t103Zec7OpRGdNYWBgQEMAdJLKwZAnw/vv0PDTUPRVC3LKOp359WnPl4wOsXUtNFnkHmDEZcpNoXBwVc+cOEo/YuNGeaEaNAsaOdc9x3bpJNCKC+tinpgIffghMnuyuIzO9MdQmUasVeOcd4MIFLuaewb59VO31/n3aPbJokfuKMLp15fJrr9GIFQCmTAFmznTn0RlTAXeQyNLx40D79pR02rUDFixwb+VXt2+ZCA4GPv2Uno8cCaxa5e4zMOZGe/cCR45w6YsMLl0C2rQBbt0CGjemVQb587v3HKrs1frwQyAkhP6Y9OoF/PKLGmdhzA2UDhIvvSQ7El24eZOSzqVLQPXqNMdTpIj7z6NK4rFYgK++olK1SUl0CXbokBpnYswFN25wB4kM7t+nK87jx+kG308/qVdqWrWftrc37bNr2RKIj6frRN5awXRl4UL6K9m3r+xIpEtOpqUwv/0G+PlR0qlQQb3zqZrmfXzoTledOrS1om1b+iPDmHRKB4lu3YBSpWRHI5UQtP1h7Vr6nV2/HqhdW91zqj6+9POjBYYVKlDFgfbtgbt31T4rY7n4+Wfa68MrlTF5Mk11WSzAsmXACy+of05NLmwDAij5lCwJ/PEH8OabvLpZT0JDQ2GxWDBy5EjZoWjHaqU/6ybvILFoEfDRR/R85kzg9de1Oa9mM2pPPWVf3bxhA/D227y6WQ8iIyMxd+5c1DFTtb1Ll+h6YuhQ+jNvUj/9BAwYQM/HjAGGDdPu3JpO5TdpQrtavbyolnZoqJZnZw+7c+cOevbsiXnz5qFEiRKyw9HO/PnUQeKtt2RHIs3hwzS9lZJCP4YpU7Q9v+b3EF97jW61A8D48TTUY3KEhISgffv2eMmBNSwJCQmIi4vL9OaRuIMEzp+nGz137gCtWtlLXWhJyuKFYcNoaAfQSmdeYKi9lStX4uDBgwh1cNgZGhoKPz+/9LfAwECVI1TJhg1ATIxpVyrfukU3eK5do0J+a9YABQpoH4e0TqKpqbRuYNUq+sPz22/q38JjJDo6Gg0aNMDWrVtRt25dAECLFi1Qr149zJgxI8uvSUhIQEJCQvq/4+LiEBgY6HmbRF9+mW6r7t0rOxLNJSRQtdDdu2lb2v79QLlykoIREj14IETz5kIAQgQGCnHpksxozCMiIkIAEN7e3ulvAITFYhHe3t4iOTk512PYbDYBQNhsNg0idpNTp+jFtnix7Eg0l5IiRM+e9O0XKybEkSNy45FafMTHh4Z6TZoAJ09SDabdu4GiRWVGZXwvvvgijhw5kul9/fr1Q/Xq1TFmzBh4e3tLikxlc+bQmg4TdpCYMIHW6OTLR7tEatWSG4/0qkclS9JtvUaNaD9X9+60gpLrManH19cXtR565RUpUgT+/v6PvN8w7t+nLRIm7CCxcKG9NtbcuXS5JZsudsZVrEjLKgoVAjZtonIavMaHudXq1bT1evBg2ZFo6pdf7E0zxo93T9lSd5A2uZyVNWvsKydnzABGjJAaDsuBx1UgbNKEruG3bpUdiWb++Ye+7du3abfAsmX6WS+pixGPoksXeyX7d9+lO5+Mueyvv6iOp4luocfG0m3z27cp+SxYoJ+kA+gs8QDABx8AAwfSpdabb3KLZOYGVittGHz1VdmRaCIhga4coqKoC8zatfqb1tJd4rFYgG++oTo+d+8CHTpwl1Lmgvh4usYwSQcJIegP95491HJq40bgscdkR/Uo3SUegOq7rl4NPPkkEB1Nl2CuNIhnJrZ0Kd3RGjhQdiSa+Pxz+pa9vel36KmnZEeUNV0mHoBus2/cSK2S9+8H+vfnO13MSUoHiY4dJS7R1U5EhL3v1cyZtEhbr3SbeABqYf3DDzRCXrGCsjljDtu3jzpImKDY15Ej1FgBoL2Qb78tN57c6DrxALR7VtnNPnYssG6d3HjMzqM6iZqkg8T16zQXevcu/b5Mny47otzpah1PTkJCaNLZ15cuvWrUkB2Ruel+Hc+NG7QT8tNP6VapQSUl0UrknTuBKlWowqcnlFbS/YhHMWMG1YKNjwc6daLt/YxlKzycbpHqZamuSkaOpKRTtChdDXhC0gE8KPEod7oqVADOnKGSGikpsqNiupSaSr20Dd5BYsECugqwWIDlyz3rKsBjEg9A6xEiImhP108/0Y5bxh6hdJAw8Erl33+3z5l//DHduPMkHjPHk9GyZfZyuRERQOfOUsMxJV3P8bz2GiWev/7S1z4BN7l+HahfH7h8mV77P/zgeY1QPSxc0rMn7eUCgD59qJYPYwCog8SGDYbtIJGcTKVjLl8GqlWjmuWelnQAD008AG0mbdYMiIujlc337smOiOnC/Pm0MalnT9mRqOK//wW2bweKFKGRjt4Gm47y2MSjTDaXKUNN5gcP5pXNppexg4Sn/kbmYP16e/WG8HCgZk2p4bjEYxMPADz+OK1o9vam/Snz58uOiEm1cSN1kDDgSuWoKKB3b3r+zjtA165y43GVR04uP+x//wNGj6YR9u+/A2ZqiimLLieXW7emhV779smOxK0SE4Hnn6fFgY0bU13y/PllR+Uajx7xKN57D3jlFdrB/sYb1KiMmczp08C2bYYc7YwbZ1+RvHKl5ycdwCCJx8uLZvefeILucL3zjuyImOaUDhJvvCE7Erf66Sfgyy/peXg4LaA1AkMkHoAWqC5fTndQw8Np7oe5ny43iT54YMgOElev0nIRgP6YGqmAoiHmeDL66CPaF1isGJUKKF9edkTGpKs5niVLaOb11CmqpWIAQtD0wU8/0Zzl/v20Yt8oDDPiUUycSBNwcXH014L3c5mA1UqlLwySdAD6ln76iZperlhhrKQDGDDx5MsHLF5MC6x27rTX8mEGpXSQMNCk8qlTwPvv0/OpUz1r86ejDJd4APrDpxRD+vBD4MQJufEwFYWFAWXLet4uyWykpAB9+1KZ6JdeAoYPlx2ROgyZeACq7d2mDbX66NePL7kMKT6eVo4GBxvjHjOo7tS+fVTw7ttvPXMfliMM+m3R3a1582iSef9+vuQypGXLaGgQHCw7Erc4c4baDAN0C93IN0YMm3gAIDAQ+OILej5hAnD+vNRwmDsZrIOEEDRNlZBAl1gDBsiOSF2GTjwAtcVp1owKYQ8bxhtJDWPfPmoza5BiX0uWUP2yggXtVQWNzPCJx8uLLrny5wc2baJ2rswAlA4Sem4e5aDbt+13sSZONNSqgGwZPvEAQPXq9v/Y99+naQHmwW7cAL7/nmqhGGD29eOPqapg9erAqFGyo9GG5/+vOejDD2kv17lztJudebDwcLpm7ttXdiQuO3YMmDWLns+YARQoIDUczZgm8RQtCkybRs8//xy4dk1uPJ5K+l6tjB0kHntMTgxuNGYMLfXo3JmWf5iF4fZq5UQIoFEjIDKSWrzOni07Is8lba/Wtm1Ud2fPHqBpU+3Oq4Ldu4HmzWm1/bFjwJNPyo5IO6YZ8QB0p0ApHTl3Lq2bYB7GagVq1QKaNJEdiUuEoNEOQItdzZR0AJMlHgBo2RJo25aq9U+ZIjsa5pTLl6nwsAE6SGzZQgtbCxc2Z3840yUeAJg0iR6XLAEuXJAaCnOG0kFCaarmwSZPpschQ2irmdmYMvE0agS8+CKNepTqbkznkpMN00Hit99oiqpAASrba0amTDwAMHYsPc6bB9y8KTcW5oANG+hSywArlZXlHH36AAEBcmORxbSJ58UXgbp1aTFheLjsaFiurFaq8FavnuxIXHLhAuVQwDyLBbNi2sRjsdAtdYDucJlnUYEHOnPGMB0kFi6kpUitWtFKZbMybeIBqAd1oULUmWLvXtnRaCs0NBQNGzaEr68vSpcujc6dO+OkXpvQz5lDvV26dZMdiUtSUijxAMbffZ4bUyeeYsWA//yHni9eLDcWre3atQshISHYv38/tm3bhuTkZLRu3Rp3796VHVpmGTtIeHjh4V27gIsXKYd26SI7GsmEyW3bJgQgRMmSQiQmyo5GntjYWAFA7Nq1y6HPt9lsAoCw2WzqBrZkCf0HnTyp7nk0EBxM30pwsOxI5MsnO/HJ1rIl9eS6cYMut5o3lx2RHDabDQBQsmTJTO9PTaWyNzt2JOHoUYGoKC/cuGGBzZYKAGjYkHrYly9Pq28bN6adDEWKuCkwq5XuBHj40t7UVPuksqf3PXcL2ZlPD3r1or9Eo0fLjkSO1NRU0bFjR9GsWbP09x05IsTIkUIEBNDP5tE3GvHQY+aPFSwoRJcuQkRECJGU5EJgf/1FB1y92uXvUbY//6RvpWhRIR48kB2NfKYf8QC053DJEuCXX2RHIsewYcPw999/Y8+ePfj9d+rVvWOH/eNFigDNmqWgVq1kVK6ciscfT4XFchddutAOhvv3gago4OhRmseIjgbWrKG38uVpzdSgQYC3t5OBWa20rNcALTR//pkemzenXlmmJzvz6cGVK/a/1jdvyo5GW8OGDRPlypUTf/8dJQYPFsJioZ+Dt7cQr78uxLp1Wf+Fzm6OJzVViEOHhPjgAyFKlbL/XJ95Roi//3YisLg4Gh5MmODS96cXrVvTz+Grr2RHog+ceNJUrUovjE2bZEeijdTUVBESEiICAgLE5s1RokoVe5Lo00eIixdz/npHJpfv3xdi5kwh/PzouAUKCLFggYMBWq1CeHnlHogHSE4Wolgx+hkcOiQ7Gn3gxJNGmef55BPZkWhj6NChws/PT8yfHylKlkwRgBABAcli82bHJiCcuat15YoQHTrYE9v06bl8QWqqEHXqCNGpk0Ox6N2JE/R9Fy5MSYgJYep1PBnVr0+Phw9LDUMzVqsVNlsxDBxYHjdvegH4EzExZXD16gq3n6tMGZoLGj2a/v3uu7lsU9m/n26lGWClMkBdlgGgdu08zHMZFCeeNEp/ar0u3nW3pCSB5567CKA06tUDbt16BkJcR1+V6hhbLNQHXNmc+/bbVHUvS1YrUKmSITpIAPYW2jVryo1DTzjxpFFaipw5Y459W3PmUGuqYsXo7lPx4uqf02KhOjStW9OdsCFDsvhZ//sv8N13hukgAdgrXVapIjcOPTHG/6wbKM0oExJoMaGRxcdTSxWARiFBQdqd28uLeoIXLEg1aZRFdemUDhL9+mkXlMouXaLHChXkxqEnnHjSFChgb1oQEyM3FrV9/z31capSher9aq1cOWD4cHr+zTcZPmCwDhIK5fVkxkqD2eHEk4HyWv/3X7lxqG3lSnrs3586rMoQHEyPP/9MSRAAreA8c8YQxb4yUkbQBsqlLuPEk4Eyz5G2bcmQkpLoEgegXk6yVKlCzSJSUuzxpHeQ8PC2NRkJYX89aTGP5ik48WSgbGy8c0duHGr65x+a2PXzA6pVkxtL48b0eOAADNVBIqPERCoXDVBTSUY48WRQsCA9JiTIjUNNFy/SY+XKebtp5M5OosqdxOhoGKqDREYZX0u8R8uOE08GynxHUpLcONSkzF/ldb4hJCQEx48fR2RkpMuxKDHcjE3rINGzp8d3kHiYMtoBqGMoI5x4smCgkf4jlHUzelgio/ycG13faJgOEjkx8uvKWTp4+elHSgo96uGXUi2FC9OjHuaxlBg6XrJSszNl34qBZNwioby+GCeeTJRLLFm3mLVQpgw96mGtUkwMUBlnUC92q2H2ZT0s42spMVFeHHrDiSeDBw/o0cNriudIWbZ//rz9+5XlxAlgEObiQeESwBtvyA1GJcoNC8DYNy2cxYknA6XBgpETT9mygL8/DfuPHJEby/GDD9AfC3Ctned3kMiOl5f9bpbeGnjIxIkng/h4evT1lRuHmiwW+/qZ9IV7Ely+DDS4sBql8C/8PxwsLxANKK8nPcyr6QUnngzi4ujRYHd0H6F00ti2TV4Mv/wCDIUVv/u+iKJPe3YHidworycjr4h3FieeDG7doscSJeTGobZ27ehx+3b7KE9rhxb9jabYi3Ots55Uvnv3LmbMmIEOHTqgT58++PXXXzWO0H2UrRK3b8uMQl94SVOaxET7NbjRE0/NmrRd4uRJICIC6N1b2/PfugVU3xmGGJTFU2Me7SBx7949NG/eHH/++Wf6+5YsWQKr1YrBgz3vskx5Pd28KTcOPeERTxrlRWGxGH8zn8Vi35mQYwlSlUQsjkeP1CVY/9hA1G3w6NqF8PDwTEkHAIQQGDt2LO7du6dVmG7j70+PRq964AxOPGmU0gX+/sZeQKjo3ZsS0I4d9gp5Wrn65XIUxj14DQ7OcjXvjoxNvTK4ffs2Dh06pHJ07leqFD1y4rEzwa+YY5TEo7xIjK58eaBtW3qeqRhXLlzdJPrH7wKvXLTiR0sHdBkRmOXnlC5dOtuvz+ljeqW8pmJj5cahJ5x40igvCjMVa3rnHXqcN88+sZ4bVzeJrh27H/XwF069ODTbJD9w4EB4Z9GOoVWrVqiqbGn3IMprKr3gGePEo1BeFGZKPG3bUsuVO3eA2bPVP98//wDVd1pxFpXQ9svW2X5e/fr1sXTpUpRR9ncAaN26NVascH/rHS0ogzQe8dhx4klz7Ro9euBIPs8sFuqTDgAzZqi/wG3WpH/xBr7DnpqDUbN2zi+97t274+LFizh48CDOnz+PLVu2eORlFsCJJyuceNIoL4rHH5cbh9a6daP9W//+C8yapd55Tp4EinwfDgsE6nzpWAeJ/Pnzo379+qjg4e0ZlMTDl1p2nHjSKCMesyWefPmASZPo+dSp6t15GT8uFYNEGPY/0RX1W5voehb219StW7xDXcGJJ40y4vHQ0bxL3nwTqFuXtoyEhrr/+H/8AdyO2I6qOIPyocYsf5GTEiXs1Qd51EM48aQx410thZeXPeF8/TVw7pz7ji0E8MEHtC/rUvGaCHrLOB0kHOXlxbfUH8aJJ43yl8iMIx6A7nC1akU1Y0aNct9xV60Czuy+jE5Yh8KjjNVBwhnKHzROPIQTD+i6W9k5bMYRD0D5YNYsKtW5bh2taHbV/fvA6NHAAHyL1AIFUXJEL9cP6qGU15XR22M7ihMP7BOqXl7G3yCakxo17PXW33nH9YnQqVOBmOhkDPWeC6+3jNdBwhm8iDAzTjywJ56SJc2xTysnH39M8xHHjgFffJH345w4QfNGHbARZVMuI98wY3eQyI2yUZR3qBOT/5oR5cVQsqTcOPTA39+ecD79lCoFOksIYMQIKp4/oZQVwqAdJJyhvLZ4oyjhxAN74jHzZVZGvXoBTZoA9+5R8welFxfg2CbRZcuArVuB6vnP4ukbW2ExaAcJZyiJx9E9cUbHiQf2ynBGr8PjKIsFCAsDChQANmwAVq+2fyy3TaI3bgAjR9LzbxvNoWxu0A4SzlD+qHHiIZx4YC//aeK5z0fUrg2MHUvPR4xwvETq2LF0OfFMzQd47p8FQN++hu0g4Qyl4LusUrN6w4kH9s2RRu4ukRfjxgGVKwNXrlDyyc3mzcC339LzZV1+gOXffwEPLFWqBu40kRknHthfDEWKyI1DbwoWBBYupEuvhQuBn3/O/nPv3gUGDaLnw4cD1bZbaUVitWraBKtzymuLEw/hxAOaRAU48WTl+eftBcPeftveAuhh48cDly4BFSsCn711BPjtN8O2Jc4LpWf9/fty49ALTjywvxh4KiJrkyYBTzwBnD5N+64e9vPPwFdf0fOZM4GCC63UsrRTJ03j1DPlteWBtepVwYkH9p7WSqtZllmJEsDSpfR87lwgY4urhAQgOJieh4QAHVvEA0uWAAMHAvkf7SBhVspri/unE048sG8NKFBAbhx61qIF5RIA6N/fvudo9Gjg/Hka4ISGAli+nP6sK9mIAbC/trgeD+HEA1phC9hrprCsTZ5MczhnztjX6oSH0+Sz1Qr4FhX0pEMHIDDrDhJmpSSelJTMCzLNihMP6MUAcOLJTenSwNq19HzDBvv7P/wwbTrn99+Bv/7iSeUsZGyakZoqLw69cOhXTQiBeAOvfFKuuxMSsr9rY3YJCQlISEiAvz/Qpk0hbNlCM/JFi8ahf/+0n9tXXwEVKgCNG/MP8iEZb6PfumXsy3pfX19Ycqm7ZBEi94FfXFwc/Pz83BYYY8y4bDYbiuWyDcChxJPdiKdhw4ZON3Zz9mvi4uIQGBiI6OjoXL+ZvJ6nRw9g0yZg+nSaOFXjHIA230tevya3z1dGPEIArVoVwZkzdwEEAojGb78VQ62dXwMTJ1I7iRzasWrxvTj7NVr8v9y+TYNBgGryODPi0eJ15s6fsSMjHocutSwWS5bfgLe3t1P/UXn9GgAoVqyYU1/nzHmUW50FCji3X0uP30tev8bRz1+0iCaXixb1Srt8KIbQyUWx/nQ4LN26AZUquTUuLb9Gzf8X5QYGQMsTsmiU6pbzZOTM96Pl7zLg4uRySEiIJl+TF86cR5lUTk5W7xyu0OLn7MjnR0bat0UEB9N94fz5Be7/uB2W06cdmlTW6jWjxf+NM+fI+Npyttic3r4XV74mndA5m80mAAibzabaOXr2FAIQYto01U4hhNDme1FLcrIQzz1HP6dOnYS4cCFaABAzZ14Rq9FFHLXUFCf+SZUdZp5o8f8SHU0/u3z5VDtFOk94nen+drqPjw8mTpwIHxWXFSuHVntxlxbfi1pmzgT27aNd1rNmAYUK0ffwnxdi0dmyDt+IoRgw0OL0qFEPtPh/0XJ1vEe8zmRnPj0YOpT+Gk2YIDsSfTpyhP5SA0JYrfS+9L+q48aJlEKFRUCR2wIQ4uOP5caqV8eO0c/P3192JPqg+xGPFnjncPZSU6nMRXIy0LFjFuV1wsPh9VZPfBZGyy2mTgWiorSPU++UzaG8EZlw4oG9HMbdu3Lj0KNZs6jHVqFCtD7wkbukV64AQ4eiZ0+geXNK3r162VeDM6IkHuWPnNlx4gEnnuxER9N2CIA6TwQFZfFJzzwD1K+fXizM15dK8cyfr2mouqesXC5aVG4cesGJB1wPNzvDh9Nf6qZN7Y3+0ikN1gcMSH9XUBC1xAGo9vK1a9rE6QmU1xaX1yWceGBfNKi0MWbAypW0ITRfPuo48cglVng4PXbpkundb79NLbRu3+a9ohkpW9e4oQDxyMSTkJCAevXqwWKx4PDhwy4fT2lro1XrkfPnz2PAgAEICgpCoUKFULlyZUycOBGJOinWcucO8N579Hz8eKBWrYc+ISGBin0Bj8yW5stHl1z58gEREcCWLerHm1ehoaFo2LAhfH19Ubp0aXTu3BknT55U5VzKa0ur3m2hoaGwWCwYqdQv0RmPTDyjR49GQECA246ndbO1EydOIDU1FXPmzMGxY8cwffp0hIWF4UNlQkWyiROBmBi6dBo3LvPHZs+ejdGVKuXYi7duXXud5mHDgAcPVAzWBbt27UJISAj279+Pbdu2ITk5Ga1bt8ZdFSb7tGwaGRkZiblz56JOnTrqnyyvZN/Pd9aPP/4oqlevLo4dOyYAiEOHDrl8zBMnaI1FsWKux5dXn3/+uQgKCpIXQJrDh4Xw9qafx48/ZvNJTZsK2wsv5Lg61mYTIiCAjvPRR+rF606xsbECgNi1a5fbjz1wIP0sPvnE7YfOJD4+XlStWlVs27ZNNG/eXIwYMULdE+aRR414rl27huDgYCxZsgSF3XhfUtlMHReXeTOflmw2G0pKbt4uBPDuu3QrvGtXoF27LD7pSFoHiQyTylkpVoxWOwPA//4HXLzo/njdzZY2yafG/4PSM93f3+2HziQkJATt27fHSy+9pO6JXOQxiUcIgb59+2LIkCFo0KCBW49dooR9455SS1hLZ8+exaxZszDkkVtH2oqIoDU7Pj7AtGnZfFJYGFCmDNC+fa7H69KF1vY8eEC1mfVMCIFRo0ahWbNmqPXIpJbrrl+nxxwqhrhs5cqVOHjwIEJDQ9U7iZtITzyTJk2CxWLJ8e3AgQOYNWsW4uLiMO7hSQc38PKy/yVSXiB54ej3klFMTAzatm2Lbt26YaBSTV2CBw+AUaPo+fvv22vHZHLnjlMdJCwWYMYM+vmuWgXs2uXWkN1q2LBh+Pvvv7FixQpVjq+8rkqXVuXwiI6OxogRI7B06VIULFhQnZO4kUOFwNR048YN3MhlmFGxYkV0794dGzZsyFRgKCUlBd7e3ujZsycWLVrkUhy1agHHjgHbtgF5HaU6+r0oL4yYmBi0bNkSjRo1Qnh4OLycrZfgRtOnU+J54gng1KlsVtjOnUv3yKOiEFe8OPz8/ByqNjd0KA2UGjcG9u7N4ta8ZO+88w7Wrl2L3bt3IyjLVZKu8/enCeajR4GaNd1//LVr1+K1116Dd4ZCPykpKbBYLPDy8kJCQkKmj0knd4rJcRcuXBBHjhxJf9uyZYsAIFavXi2io6NdPn7LljT5t3y5G4J1wKVLl0TVqlVF9+7dRXJysjYnzYbNRpsXASHmzcvmk1JThahXT4iOHdO+xvHSCzExQhQuTMePiHBf3K5KTU0VISEhIiAgQJw6dUq18yQm0vcOCHH9ujrniIuLy/T7ceTIEdGgQQPx1ltviSNHjqhzUhd4TF+F8uXLZ/p30bS155UrV0a5cuVcPr4yBNZitW1MTAxatGiB8uXLY9q0abie4fquTJky6gfwkP/9jyY/q1UD+vbN5pN+/x04fBiYMsXp45ctS+1wpkyh2/MdOuijo0dISAiWL1+OdevWwdfXF1evXgUA+Pn5oZAbd3Mqg2AvL/Vup/v6+j4yN1WkSBH4+/urMmflKulzPHqhJJ7YWPXPtXXrVpw5cwbbt29HuXLlULZs2fQ3rd26ZW8/PGVKDgnBaqWFPW3a5Ok8o0fTeqkTJ4Dvv89brO5mtVphs9nQokWLTP8Hq1atcut5lD9mpUo5V/LU0GQPufTi009pKDxggOxItPXJJ/R916olREpKNp/0779C+PgIMXVq+rvyUuVOOVft2jmcy4C2bLF/34zwiCeNlpdaenH3rn208+GHOdQCDg+nwjz9+rl0vmHDaJPkkSPAjz+6dCiPooyiH3tMbhx6woknjZJ4XLmd7mkWLaK5naAgoFu3bD4pNZVuSXXt6vK94BIl7BtHP//cpUN5FCXxPP643Dj0hBNPGuWvkVkSj0hrcw7QxG+2czs7dgAOdpBwxPDhNM/x66+0fMEMlNcUj3jsOPGkUVaUmiXx/P47rSkpVAjo3TuHT7RaaeFJs2YAaJNojRo10LBhwzyd94knqIQqAMybl6dDeBzlrhYnHjtOPGmUlcvx8ep3m9ADZb1lt272siCPiImhojxDhqSv+gsJCcHx48ed7jqZkbJAe/lyeXvjtKQkHrX3aXkSTjxpihe3r6jNoeKDISQlAd99R89zHO18+y1t3OrVy63nb9OGpouuXwd+/tmth9Yl5fUkeQ+wrnDiSePtDfhRowTcvi01FNX9+iv9Mjz2GNCiRTaflJxMWyR69LD/YNwkXz6aqwZoQGV0yuuJE48dJ54MtK5EKMvWrfT4yis5LGjbtAm4dEm1+qUdOtCjnisUuouSeNycvz0aJ54MlL2ORi/6vns3PbZsmcMnWa3As88CTz+tSgzNmtG6oQsXqJuFkSn1ljnx2HHiyUBpPaK0IjGilBTg0CF63qRJNp907hwNRVSs1u7rCyiVOR+qFGI43NrmUZx4MlBKQSjN14zo7FmqvVO4MFC5cjafNGcOXXe+8YaqsSiJ5+hRVU8jVVIS0vvJcxdRO048GSg97o18O/3CBXoMCspmi0RCArBgAW1TV7ntpZL4PKEsal5lfC15QH0uzXDiyUApqmfkxKMs38+2+sYPP9DCEw3KsCoxGHl/XMbXkgNFG02DE08GEgsAakZZsFegQDafYLXSrHO1aqrHoowwlUsRI8pY31NvlRdl0kE5Jv2QWwRWG8qerCxHdUeOAHv22FcXqkyJwcg1ajImm9RUeXHojQn+xjtO+UXIdjRgAMoO6bRie5nNmUPXP507axLLlSv0KKHoomYyXl6ZYXuIozjxZHD/Pj0aeRKwShV6PH2a5pHT3bkDLF6cawcJVzeJZqTczcr27poBZHwt6bWjqgyceDIww3qLihVpq0RiIvDHHxk+sHw5VQYLDs7x692xSRSgy449e+h5o0YuHUrX8uWzJx+jL0x1BieeDMywmc9iAdq2pefppYWV4jzt2wMPFdVXy549dKlVrBjw3HOanFIapcC70TcfO4MTTwbKrWY1uz3qQY8e9Lh4cVr9oT/+oA4SKq5Uftj06fT4+uvGvrQFzFdkzhGceNLEx9v31DzxhNxY1Na6NVC/Pn3PkyaBRjsVK9IHNLBrF+1K9/KirqVGp7yeLl+WG4eecOJJc/48PRYvTvuIjMzLy94bfcU3N5GyYhUweLAm97Vv3rTXAAoOBmrUUP2U0ilXr1FRcuPQE048aU6fpscnn5Qbh1ZataKuD32wCCmJKThYr7/q54yPpzv1Fy/SnSyzFHyvWpUeT52SG4eecOJJc+QIPT71lNw4tDTtfwLvFgrDD3gdLd4orWo1wGvXgJdfpiJkxYrRzoxcWq4bhjKqM/JmWGdx4klz8CA91qsnNQxN+fy2HeXvn8L+ekMRH09TPB98QHfV3UUISjJ16lCB+RIlgF9+AerWdd859E55TZ044d6frSfjxAP65di7l54beU0JAJw/fx4DBgxAUFAQIlq3xqn8+VG03U7065cCIWjup0YNYP5818qDCEEFx9q1ozKnsbFA7dr0c27QwH3fjycoWxYoV47WLv3+u+xo9IETD6i/040bdFtXpYJ7unHixAmkpqYifMoUdLZY8KBfP8z/djZKlhyDDRuAChVoDiY4mH5ZBg8Gtm2zr+rOiRB0OTF5MlCrFtC8OdUTK1AA+O9/6a599erqf4969Pzz9Lhjh9w4dEN2D2U9mD6delu3aSM7Eg198okQhQsLcfu2+Pzzz0VQUJAQQoi7d4WYNk2IihXpZ6K8WSxC1KghRNeuQoSECPHee9Q7PSTEJnr3FqJZMyH8/TN/TeHCQgwaJMSpU5K/Vx1YsIB+Jo0by45EH3h3OoCffqLHl16SG4dmHuogYbPZUDJtuXbhwsB771F30R07aHXzjz8C168LHD9uwfHjmQ81e3bmfxcuTCOdrl2BLl1y6NllMi++SI9//EFto83eY8sihBmKQWQvLs6+d+n4cZPc1Vq/HujUCThwAGeLF8fTTz+NL774AgOVTnsPEQJ4990Z+OqrnwFUBlAKgA+AMRg+3IaAgGKoUIFuG9eubezd/a6oU4funi5alEs/MzOQPeSSbdEiGgJXqyZEaqrsaPJu4sSJAkCOb5GRkfTJbdsK0bChuHz5sqhSpYoYMGBArsd/8OCBsNls6W/R0dECgLDZbCp/Z8YxYQK91tq3lx2JfKYf8bz0Et3enTQJmDhRdjR5d+PGDdxQeuVmo2LFiigYEwNUqYJbX3yBxmFhaNSoEcLDw+HlZPnFuLg4+KVdphUzy4IcF504QSPqfPmopY+R6xDlxtSJJyoKqFSJdmyfO0fblQxv7FikhoWhXqlSqNmwIZYuXQrvPGyV4MSTN02aAPv2AZ99BoweLTsaeUx9O33+fHp88UWTJJ2EBKTMm4fFAB6rUAHTpk3D9evXcfXqVVzNsiQhc7f+aTtT5s83dylU0454HjygNSuxscDq1VSewfCWLwd69kR1ACez+LAzLwUe8eTNnTu0Wz0uDti82V4byWxMO+JZtIiSTmAg8OqrsqPRSFoHiRNCQGTxxtRXtKh91GOWTbJZMWXiSU62/6e/955J+h0dPUpl/zQs9sWyNmoUTTDv2GHeLRSmTDyrVtFkcqlSVNvcFMLCqMVEp06yIzG9wEDgrbfo+ZQpcmORxXSJJykpreoegHffBYoUkRqONjJ2kODVfbowdizdTV2/3pyjHtMlnvnzgTNngNKlgXfekR2NRlasoHoMgwbJjoSlqVYN6NOHno8da45mkhmZ6q7WnTvUV+raNWDmTJMkHiGAZ56hWykbNrjtsHxXy3XR0fR6TEyk/XDt2smOSDumGvF89hklncqVqdyDKfzxB3DokNsmld3Z0M/sAgOB4cPp+fvvm6vTqGlGPGfPUoGrxEQTrdsBgL59qa3DmTNuLebOIx73uHWL6nzfuEFF2N57T3ZE2jDNiOfddynpvPwylWswhZs36RaeRh0kmPNKlACmTqXnn3ySTU97AzJF4lm7lqY38ucHvvqK7iaYwqJFQEqKfcUa06W+fYGGDWk186hRsqPRhuETj80GvP02PX//fZPU2wFoUjksjK4pS5eWHQ3Lgbc3LSr38qIbkJs2yY5IfYZPPOPHU4/uqlWBjz6SHY2GduygRk68UtkjPPMMTQcAQEgI3YE1MkMnnh077KU5rVagUCG58WjKaqXZdKXKONO9SZNo4/KFC9RmyMgMm3ji44EBA+j5kCH2mremcOUKTWwNGWKiCS3PV7QosGABPQ8LowJ1RmXYxPPBB1Toq0IFE+4C/vZb2hph+sK+nqdVK/vVcf/+NEdpRIZMPOvXA3Pm0POFCwFfX7nxaColhTpIvPkm4OcnOxqWB59/DgQFUX+zkBDZ0ajDcIknNpaa0QG0GKtlS7nxaG7TJlqLz5PKHqtoUWDZMrrLtWwZ8P33siNyP0MlntRUurqIjaVOlp9+KjsiCaxWWhTyzDOyI2EueO454MMP6XlwMHD+vNRw3M5QiWfGDGqZW6gQsHIltSQ2lXPn6AfAox1DmDCBisPbbFS/JzlZdkTuY5jEExlJ5QUA4IsvgJo15cYjxdy5NK/zn/+odgreJKqd/PmBpUuBYsWA337z7PZLDzPEJtHbt4H69Wk4+vrrdE1survICQlAuXJAz5409FMZbxLVznff2f+WGKVAvMePeISg247nz1OPrPnzTZh0AGDNGtriPGSI7EiYm73xhv3quXdv4PJlufG4g8cnnmnTgIgIGpauWgUULy47IkmsVqBFC6B6ddmRMBV8+SVQrx5w/TrQtavn1+7x6MSzZw8wbhw9nzkTaNBAbjzSHD0K/PorTyobWMGCwA8/0B/W/fuBMWNkR+Qaj008MTE0n5OSQmvlTFNRMCtz5lAHic6dZUfCVFSpEi2IBYDp02knu6fyyMSTmEjXvbGxQJ06wLx5Jp3XAewdJAYM4A4SJtC5c+b1PUePSg0nzzwy8YwcSbcX/fyojKkpWtRkZ8UK2hHLHSRM45NPaNPz3buUiG7dkh2R8zwu8SxYQPOoFgstJ69aVXZEEglBP4z27Wk3LDMFb29aIFuxItUS79mTphw8iUclnv377fOnH39Mv2+mFhnp1g4SzHOUKkV3cwsVorU9//2v7Iic4zGJ5/JlGlYmJgKvvUaVBU3PaqWRTps2siNhEtSrRxVQACoY70mTzR6ReO7do2Rz7RpQuzbNpXp5ROQqunmTxtvcQcLU3nzTvqRkwAAaBHsC3f/6CkEtvyMjgZIlaXhZtKjsqHRg8WLuIMEAAP/3f8ArrwD379NVwZUrsiPKne4Tz+TJNITMl492BVSuLDsiHVA6SHTpQut3NMSbRPVHmWyuUYPWt3XuTFcJeqbrTaJr1tg7flqtvA0p3fbtdD91506geXMpIfAmUf05cwZo1Iiuwrt3B5Yv1+/6Nt2OeA4coBokAPWX5qSTgdJB4oUXZEfCdKRKFZqKyJePRkCffCI7ouzpMvFcugR06kTXrG3bUn0dloY7SLAcvPAC/V0CqF3OypVSw8mW7hLPnTtAx450rVqjBu04z5dPdlQ6onSQ6NVLdiRMpwYOpK65ALVH3rtXajhZ0lXiUTZ8Hj5MXXc3baLqayxNxg4Spq3/wRwxdSrw6qtUH65TJ6qKqye6SjzvvQds3EglANatoyXhLIMff+QOEswh3t40ufz001Qfrn17fe3p0k3imTkT+Oorer54MdC4sdx4dMlqpaJD3EGCOaBIEWDDBqqIe+IELcJNTJQdFdFF4lm/3t6w/rPPgG7d5MajS1FRwE8/8WiHOSUggKYsfH2BXbuolIYeFtBITzyRkbTmIDWVfihGb1afZ3Pn0oRX9+5uO2RCQgLq1asHi8WCw4cPu+24TF/q1KGC8d7edDUxaZLsiCQnnnPn6A6Wctt89my+Q5ylhAS6m9WnD1C4sNsOO3r0aAQEBLjteEy/2ralxe4Are9RKhnKIi3x3LwJtGtHGz/r1KHb5vnzy4pG59asoSrfblxFuXnzZmzduhXTpk1z2zGZvg0caO89FxwMbNsmLxYpief+fbrFd+oUEBhI9UT4tnkOlA4STz3llsNdu3YNwcHBWLJkCQo7OIJKSEhAXFxcpjfmeaZMoR0BKSm0HenQITlxaJ54lP7me/ZQ6dJNm2gCjGXj2DG3dpAQQqBv374YMmQIGjjRliM0NBR+fn7pb4GBgW6Jh2nLYqHecy1aUMXc9u2Bixe1j0PTxCME3b1avZouq9ato/o6LAdhYQ51kJg0aRIsFkuObwcOHMCsWbMQFxeHcUoRFweNGzcONpst/S06OtqFb4rJ5ONDu25q1aIdOG3b0tSHljTdnf7FF/al3MuX0wJcloM7d4AnngCGDaP6IDm4ceMGbty4kePnVKxYEd27d8eGDRtgyTCLn5KSAm9vb/Ts2ROLFi1yKDTene75oqOBJk1ob+TzzwNbt9LiXS1olniWL6ei1AB1/3zvPS3O6uHmz6fuEVFRbivmfvHixUzzMzExMWjTpg1Wr16NRo0aoVy5cg4dhxOPMRw9CjRtCsTFUXkn5ba76oQGfv5ZiPz5hQCEGDlSiNRULc7q4VJThXj6aSHat1f1NFFRUQKAOHTokFNfZ7PZBABhs9nUCYxpZscOIQoUoN/PYcO0+f1UfY7n0CFaqp2URCuSv/iC1+o4JDISOHiQVyoz1bVoASxZQr+XX39NG0zVpuql1rlzdA157Rp9cz/9RBNbzAH9+1OlwbNndVnMnS+1jGfmTGDECHq+cCGV1FCLaiOe2FiaLVcWCK5dy0nHYbduUQWnQYN0mXSYMQ0fDoweTc8HDqSlLmpRJfHcuUNV70+fpjnRzZtpzQ5z0KJFQHIy9SthTENTp9I6u5QUmhrZv1+d87g98SQm0uz4n39St8OtW3mBoFMkdpBgTFlg2K4d7TBo3x44ftz953Fr4klNpevCbduoFsimTcCTT7rzDCawcydw8iRPKjNp8ucHvv/e3rGiTRta6+NObks8QgAjR2bugfXss+46uolYrbQniztIMImUgUO1apR0Xn4Z+Pdf9x3fbYknNBSYNYuGauHhQOvW7jqyiVy5Qv1JuIME0wF/f2DLFnsFww4dgLt33XNstySeuXOB8ePp+YwZ9hXKzEkLFtA4t3dv2ZFkizuJmkuFCpR8SpSgiebXX6c1ea5yeR3PmjU0+52aSsnn009dD8qUUlKASpWAl16iol86x+t4zGXfPnpp3rtHA4vFiwEvF4YtLo14tm+njZ5K2dL/+z9XjmZyP/5I9Ql4Upnp0HPPUVUJb29g2TKaz3VlyJLnxHPgAFVqSEwEunalOVGelnCB0kHCiRo5jGmpXTsa6QA0n+vK1U2eE48QNB3RsiXt8+AFti7gDhLMQ/ToQfO4gGsdfl2a4zlxgsrF+PrmPQAGYNw4GvFcvkz3MT0Az/GY24EDrg3OXepKXr26K1/NAGTuIOEhSYcxV2cEpPfVMr2ICLd3kGBM7zjxyGa1As2bu62DBGOewKVLLeaiY8eA3bupBAZjJsIjHpnmzAFKl6YSjYyZCCceWe7epbo7AwYABQrIjoYxTXHikWXFCuqoNmiQ7Eicwnu1mDto2leLpRGC7keWLQts3Cg7mjzhdTzMFTy5LMOBA9RBwkOTDmOu4kstGaxWqjfQtq3sSBiTghOP1riDBGOceDS3eDFVUuIOEszEOPFoiTtIMAaAE4+2du6kLf1c/oKZHCceLYWF0Zb+5s1lR8KYVHw7XUurVsmOgDFd4AWELE+EEIiPj4evry8sXPOWOYkTD2NMczzHwxjTHCcexpjmOPEwxjTHiYcxpjlOPIwxzXHiYYxpjhMPY0xz/w+vTaHbJK4hnAAAAABJRU5ErkJggg==", "text/plain": [ "Graphics object consisting of 3 graphics primitives" ] }, "execution_count": 12, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var ('x, y')\n", "implicit_plot(y^4-4*y^2-x^4+9*x^2==0,(x,-5,5),\n", "(y,-5,5), axes=True, frame=False)+plot(TL,xmin=-5,xmax=5,ymin=-5,color='red')+point([(1,1)],color='black',size=25) " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "## Question 2\n", "\n", "[2 points] Consider the curve defined by $ x^3+y^3=9xy$\n", "\n", "### Part a\n", "\n", "Calculate the derivative $\\frac{dy}{dx}$." ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[diff(y(x), x) == -(x^2 - 3*y(x))/(y(x)^2 - 3*x)]" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "y=function ('y')(x)\n", "solve (derivative(x^3+y^3==9*x*y,x), derivative(y,x))" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\frac{\\partial}{\\partial x}y\\left(x\\right) = -\\frac{x^{2} - 3 \\, y\\left(x\\right)}{y\\left(x\\right)^{2} - 3 \\, x}\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\frac{\\partial}{\\partial x}y\\left(x\\right) = -\\frac{x^{2} - 3 \\, y\\left(x\\right)}{y\\left(x\\right)^{2} - 3 \\, x}\\right]$" ], "text/plain": [ "[diff(y(x), x) == -(x^2 - 3*y(x))/(y(x)^2 - 3*x)]" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "show(_)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part b\n", "\n", "Calculate the slope $m$ at the point $(2,4)$.\n", "\n", "[Check: The slope should be 4/5]" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "4/5" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" } ], "source": [ "-(2^2-(3*4))/(4^2-(3*2))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part c\n", "\n", "Calculate the tangent line at the given point $(x_0,y_0): \\quad TL(x)=y_0+m\\cdot(x-x_0)$." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "4/5*x + 12/5" ] }, "execution_count": 16, "metadata": { }, "output_type": "execute_result" } ], "source": [ "TL (x)=4+ (4/5)*(x-2)\n", "TL (x)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part d\n", "\n", "Graph the original equation and the tangent line on the same window.\n" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "execution_count": 18, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var ('y')\n", "implicit_plot (x^3+y^3==9*x*y, (x, -5,5),\n", "(y, -5,5) ,axes=true, frame=false)+plot (TL (x) , xmin=-5, xmax=5, ymin=-5, ymax=5, color= 'red' ) +point ( ( ) , color= 'black' , size=25)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "## Question 3\n", "\n", "[2 points] Consider the curve defined by $\\displaystyle (x^2+y^2-1)^3=x^2y^3$\n", "\n", "### Part a\n", "\n", "Calculate the derivative $\\frac{dy}{dx}$." ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[diff(y(x), x) == -2/3*(3*x^5 + 3*x*y(x)^4 - x*y(x)^3 - 6*x^3 + 6*(x^3 - x)*y(x)^2 + 3*x)/(2*y(x)^5 - x^2*y(x)^2 + 4*(x^2 - 1)*y(x)^3 + 2*(x^4 - 2*x^2 + 1)*y(x))]" ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "y=function ('y')(x)\n", "solve (derivative((x^2+y^2-1)^3==x^2*y^3,x),derivative (y,x))" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\frac{\\partial}{\\partial x}y\\left(x\\right) = -\\frac{2 \\, {\\left(3 \\, x^{5} + 3 \\, x y\\left(x\\right)^{4} - x y\\left(x\\right)^{3} - 6 \\, x^{3} + 6 \\, {\\left(x^{3} - x\\right)} y\\left(x\\right)^{2} + 3 \\, x\\right)}}{3 \\, {\\left(2 \\, y\\left(x\\right)^{5} - x^{2} y\\left(x\\right)^{2} + 4 \\, {\\left(x^{2} - 1\\right)} y\\left(x\\right)^{3} + 2 \\, {\\left(x^{4} - 2 \\, x^{2} + 1\\right)} y\\left(x\\right)\\right)}}\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\frac{\\partial}{\\partial x}y\\left(x\\right) = -\\frac{2 \\, {\\left(3 \\, x^{5} + 3 \\, x y\\left(x\\right)^{4} - x y\\left(x\\right)^{3} - 6 \\, x^{3} + 6 \\, {\\left(x^{3} - x\\right)} y\\left(x\\right)^{2} + 3 \\, x\\right)}}{3 \\, {\\left(2 \\, y\\left(x\\right)^{5} - x^{2} y\\left(x\\right)^{2} + 4 \\, {\\left(x^{2} - 1\\right)} y\\left(x\\right)^{3} + 2 \\, {\\left(x^{4} - 2 \\, x^{2} + 1\\right)} y\\left(x\\right)\\right)}}\\right]$" ], "text/plain": [ "[diff(y(x), x) == -2/3*(3*x^5 + 3*x*y(x)^4 - x*y(x)^3 - 6*x^3 + 6*(x^3 - x)*y(x)^2 + 3*x)/(2*y(x)^5 - x^2*y(x)^2 + 4*(x^2 - 1)*y(x)^3 + 2*(x^4 - 2*x^2 + 1)*y(x))]" ] }, "execution_count": 20, "metadata": { }, "output_type": "execute_result" } ], "source": [ "show(_)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part b\n", "\n", "Calculate the slope $m$ at the point $(1,1)$.\n", "\n", "[Check: The slope should be -4/3]" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-4/3" ] }, "execution_count": 21, "metadata": { }, "output_type": "execute_result" } ], "source": [ "-2* (3*1^5+ (3*1*1^4) -(1*1^3)-6*1^3+6*(1^3-1) *1^2+3*1) / (3* (2*1^5-(1^2*1^2)+4*(1^2-1)*1^3+2*(1^4-2*1^2+1) *1))" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[diff(y(x), x) == 1/2*(3*x^2 - 4)/(3*y(x) + 1)]" ] }, "execution_count": 23, "metadata": { }, "output_type": "execute_result" } ], "source": [ "y=function ('y')(x)\n", "solve(derivative ((x^3-3*y^2) ==4*x+2*y), derivative (y, x))" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.487088178015674" ] }, "execution_count": 24, "metadata": { }, "output_type": "execute_result" } ], "source": [ "1/2* (3*4^2-4)/((3*3.68053* (4)+1))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part c\n", "\n", "Calculate the tangent line at the given point $(x_0,y_0): \\quad TL(x)=y_0+m\\cdot(x-x_0)$." ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-4/3*x + 7/3" ] }, "execution_count": 25, "metadata": { }, "output_type": "execute_result" } ], "source": [ "TL(x)=1+-4/3*(x-1)\n", "TL(x)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part d\n", "\n", "Graph the original equation and the tangent line on the same window.\n" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 2 graphics primitives" ] }, "execution_count": 36, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var ('y')\n", "implicit_plot((x^2+y^2-1)^3 == x^2*y^3, (x, -5, 5), (y, -5, 5), axes=True, frame=False) + plot(TL, xmin=-5, xmax=5, ymin=-5, ymax=5, color='red') + point((), color='black', size=25)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "## Question 4\n", "\n", "[3 points] Consider the curves defined by $y^2=x^3$ and $2x^2+3y^2=5$.\n", "\n", "### Part a\n", "Find $\\frac{dy}{dx}$ for the first curve." ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[diff(y(x), x) == 3/2*x^2/y(x)]" ] }, "execution_count": 37, "metadata": { }, "output_type": "execute_result" } ], "source": [ "y=function ('y')(x)\n", "solve(derivative(y^2==x^3,x),derivative(y,x))" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\frac{\\partial}{\\partial x}y\\left(x\\right) = \\frac{3 \\, x^{2}}{2 \\, y\\left(x\\right)}\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\frac{\\partial}{\\partial x}y\\left(x\\right) = \\frac{3 \\, x^{2}}{2 \\, y\\left(x\\right)}\\right]$" ], "text/plain": [ "[diff(y(x), x) == 3/2*x^2/y(x)]" ] }, "execution_count": 38, "metadata": { }, "output_type": "execute_result" } ], "source": [ "show(_)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part b\n", "\n", "Find the tangent line to the first curve at $(1,1)$." ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "3/2" ] }, "execution_count": 39, "metadata": { }, "output_type": "execute_result" } ], "source": [ "3*1^2/(2*1)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part c\n", "\n", "Find $\\frac{dy}{dx}$ for the second curve." ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[diff(y(x), x) == -2/3*x/y(x)]" ] }, "execution_count": 40, "metadata": { }, "output_type": "execute_result" } ], "source": [ "y=function ('y')(x)\n", "solve(derivative(2*x^2+3*y^2==5,x),derivative(y,x))" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\frac{\\partial}{\\partial x}y\\left(x\\right) = -\\frac{2 \\, x}{3 \\, y\\left(x\\right)}\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\frac{\\partial}{\\partial x}y\\left(x\\right) = -\\frac{2 \\, x}{3 \\, y\\left(x\\right)}\\right]$" ], "text/plain": [ "[diff(y(x), x) == -2/3*x/y(x)]" ] }, "execution_count": 41, "metadata": { }, "output_type": "execute_result" } ], "source": [ "show(_)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part d\n", "Find the tangent line to the second curve at $(1,1)$.\n", "\n", "[Caution: Make sure you give this tangent line a different name than the tangent line in Part b, like TL2(x).]" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-2/3" ] }, "execution_count": 42, "metadata": { }, "output_type": "execute_result" } ], "source": [ "-2*1/(3*1)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-2/3*x + 5/3" ] }, "execution_count": 43, "metadata": { }, "output_type": "execute_result" } ], "source": [ "TL2(x)=1+(-2/3)*(x-1)\n", "TL2(x)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "### Part e\n", "Graph the two curves and the two tangent lines on the same axes (use red for the tangent lines).\n", "\n", "[Notice that the two tangent lines are perpendicular; their slopes should be 3/2 and -2/3.]" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 5 graphics primitives" ] }, "execution_count": 51, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('y')\n", "implicit_plot(y^2==x^3,(x, -5,5),(y,-5,5),axes=True, frame=False)+implicit_plot(2*x^2+3*y^2==5,(x, -5,5),(y, -5,5), axes=True, frame=False) + plot(TL, xmin=-5, xmax=5, ymin=-5, ymax=5, color='red')+ point((1, 1), color='black',size=25) + plot(TL2, xmin=-5, xmax=5, ymin=-5, ymax=5, color='red')" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": false }, "source": [ "## Question 5\n", "\n", "[1 point] Graph the curve given by $\\displaystyle\\frac{1}{x^2} - \\frac{1}{y^2} + \\frac{1}{z^2}=0$ with $-5 < x < 5$, $-5 < y < 5$, and $-5 < z < 5$." ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": "\n\n", "text/plain": [ "Graphics3d Object" ] }, "execution_count": 52, "metadata": { }, "output_type": "execute_result" } ], "source": [ "var('y,z')\n", "implicit_plot3d((1/x^2)-(1/y^2)+(1/z^2)==0, (x, -5,5), (y,-5,5), (z,-5,5))" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "argv": [ "sage-10.0", "--python", "-m", "sage.repl.ipython_kernel", "--matplotlib=inline", "-f", "{connection_file}" ], "display_name": "SageMath 10.0", "env": { }, "language": "sagemath", "metadata": { "cocalc": { "description": "Open-source mathematical software system", "priority": 1, "url": "https://www.sagemath.org/" } }, "name": "sage-10.0", "resource_dir": "/ext/jupyter/kernels/sage-10.0" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.1" } }, "nbformat": 4, "nbformat_minor": 4 }