drvinceknight/Lab-sheet-07-Symbolic-calculus.ipynb Secret

## Lab-sheet-07-Symbolic-calculus.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lab sheet 07: Symbolic calculus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q1: Limits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import sympy as sym  # Importing the library\n",
    "sym.init_printing()  # Pretty print"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOBAMAAADDIxFwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoklEQVQIHWNgEDIxZQCBPIGgAgaGdAaJCSDe\n+v+fGRhYFjAwJwA5DMnngILcBxjYP4J4B0AE/wEG3u8gxgEQIa/AwPsPxDjip8rAMN+AgRWonYFh\nI8P8CwzzFaA8oBENCJUMDEw/Qaawg0zhSWBg/svAvYGBBWQDVwID00eQ7UwJQB4TAwMHkG5nkApg\n/MbAqcDgBXSM8LujDAw9DAyTbBQZAHJpJhsPt5MdAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0.5$$"
      ],
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):  # Writing a simple function\n",
    "    return 1 / x\n",
    "f(2)  # Evaluating the function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAAqBAMAAAB1rqf/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiRInd\nuzLEnmxuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWNggAFGZRDLJOQzWICNGhSr03cP\nmPGk0//BgHR92HWYpczKTmBg3sBxjKeAgduA+zPjBAZmBvYLYNX8CWDq/QMgxWiQz8BmwLBe4D7D\nDGB4pJlFNDAAAN1MIGxCExJiAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{x}$$"
      ],
      "text/plain": [
       "1\n",
       "─\n",
       "x"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x  = sym.symbols('x')\n",
    "f(x)  # Evaluating the function of a symbolic variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, -sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, +sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGd\ngSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, 0)  # The default direction of approach is positive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGd\ngSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, 0, dir=\"+\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0A\nV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAm\nwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, 0, dir=\"-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A different function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def g(x):\n",
    "    return 1 / (x ** 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(g(x), x, sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(g(x), x, -sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(g(x), x, +sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGd\ngSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(g(x), x, 0, dir=\"+\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGd\ngSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(g(x), x, 0, dir=\"-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q2 Differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAqBAMAAADCE1/YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMlTvq5l2ZiJE\nibtxnypmAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAo0lEQVQoFWNgIAhYEzGViLd/xBRk4Bxughxl\nn6qw+HPwCgkZg4AKA/9/OPhAvmt5+xoxNd9mKMIUtGbw34AhasOw/wCGIANDvAAWQUssYrwNWARX\noIgxzqw+eZOB5wKPA5KwLMPiCUYMTqFxyBZdZLgvEM2Q//8/kkIGAQY9ZC6M/RPGQKJZviBxoMxA\nzg8ME9GEeb7yfeBegCbIWDlxxiqoGAAnPzT4q3C2QQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{1}{x^{2}}$$"
      ],
      "text/plain": [
       "-1 \n",
       "───\n",
       "  2\n",
       " x "
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(f(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnZUZrtE7yKJ\nmatywa77AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQUlEQVQoFWNgwAm4CyBS7BvQlXgFIER2wJiv\nYQwY7X8AxmJg0IYxhWAMLDT7BZggD5IFMDEYzQx3DuMCmBiU3uIAF9gMZzFoIpgMrulB3P0PWLSu\nhD4FicYCsWjczhigdWcRqhgbGJgZmB4wMLcy7AdZ1s7AwHiA3ZRZgYGhCKGKcyEDKwPHAwaOBQz8\nBUBhMwYGHgGej2xAHckMQsYgoAIUtm/dAFalwMAPtIVhJdAsBqYGIIvBH0RAgYz+X7CqBrgqBggL\nWRW3AEP9BJCNUFVAG4GmOIDIySACAtgCGPgKkFQBXc8mEM/AKoDieraFDBwbgAbBzAJ6f35CP8N2\noBl3YCYxMHAeDz/J8v6fyPt/bv3rgMJAa+RCRa+DvAvyGw6AiCFuAxxKgMLExTZDN8wEfCmHAZ4K\n4cph2pBppBQNAKW2Q7c+KCqVAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-sin(x)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(sym.cos(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All the below are the same way of calculating the second derivative:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAUBAMAAADVZMaCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(sym.diff(sym.cos(x), x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAUBAMAAADVZMaCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(sym.cos(x), x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAUBAMAAADVZMaCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(sym.cos(x), x, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiple variables:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = sym.symbols('y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAAAUBAMAAABRzuPpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nmOsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\sin{\\left (x \\right )} \\sin{\\left (y \\right )}$$"
      ],
      "text/plain": [
       "-sin(x)⋅sin(y)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(sym.cos(x) * sym.sin(y), x)  # Diff with respect to x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHcAAAAUBAMAAABFd79NAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nG/oDzE2JJftJp6QAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\cos{\\left (x \\right )} \\cos{\\left (y \\right )}$$"
      ],
      "text/plain": [
       "cos(x)⋅cos(y)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " sym.diff(sym.cos(x) * sym.sin(y), y)  # Diff with respect to y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g = sym.Function('g')  # We create g as a generic function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAvBAMAAACyMpGWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBc+T+DYRbNEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{d^{2}}{d x^{2}}  g{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "  2      \n",
       " d       \n",
       "───(g(x))\n",
       "  2      \n",
       "dx       "
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(g(x), x, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a more complicated function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAckAAAA/BAMAAAB5rS7yAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\frac{1}{\\left(e^{x} - \\cos{\\left (x \\right )}\\right)^{2}} \\left(- \\frac{2 \\left(e^{x} + \\sin{\\left (x \\right )}\\right)^{2}}{e^{x} - \\cos{\\left (x \\right )}} + e^{x} + \\cos{\\left (x \\right )}\\right)$$"
      ],
      "text/plain": [
       "                 2              \n",
       "    ⎛ x         ⎞               \n",
       "  2⋅⎝ℯ  + sin(x)⎠     x         \n",
       "- ──────────────── + ℯ  + cos(x)\n",
       "     x                          \n",
       "    ℯ  - cos(x)                 \n",
       "────────────────────────────────\n",
       "                      2         \n",
       "         ⎛ x         ⎞          \n",
       "         ⎝ℯ  - cos(x)⎠          "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(1 / (sym.cos(x) - sym.exp(x)), x, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q3 Integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAUBAMAAAAqxuNgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMqvNiRDvuyJ2RN1U\nmWaBK2/dAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABSklEQVQoFWNgQAccDyAifA0MDEImaLLtML4G\nAwNrEowDpS1g/GAgowjGgdB8F2B8lgkYktwNMEnWBAzJbpgcA4MVWDJkphcDwyLfl9OBEnOBOHJS\n1xSgmb4gScYdDOwTuL4y7OMEStQB3ejAt5PbgIHhMUiS24GB8SPTBgZtkIl7GBhYAli+8jQwMJwG\nScofYGD4wpsAkcwC6mRgLgCpkwdJnl/AwPCd4yNrDkgEKMnAwA+0ESoJ0vmXY8oUAZAI0FigMJi5\nGqST7QID1y+eBSBRsIN4AuYzcAaAHVTOwJjIwOzAZSXYAJIEun/9gXqGPiDzCoPQfw2GXhctBgb7\n/98uAEWAZgXNjLz3AMg0BGIIYD/A2rMTyEQEH8cGmBzDcSDrORCjBjxUOhrowBoQuwIqwACKMihg\ndZnpC5RnYIBHNlwVTAmQRkomAPJISik0ozTBAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\log{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "log(x)"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.integrate(f(x), x)  # An indefinite integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPBAMAAAAv0UM9AAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAMpndu3bvImbNiRBUq9OB\nhjcAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABESURBVAgdY2BgYBACYgYGExDBmgIiK6aAyAUgkqMA\nRG5lAJELQCSPAIjcxQAiz969++wqUIIBrIvhCYi55N0NEMXAAABbkhBrtxdTYQAAAABJRU5ErkJg\ngg==\n",
      "text/latex": [
       "$$4$$"
      ],
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.integrate(f(x), (x, sym.exp(1), sym.exp(5)))  # A definite integral"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verifying the fundamental theorem of calculus, for a particular function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAAqBAMAAAB1rqf/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiRInd\nuzLEnmxuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWNggAFGZRDLJOQzWICNGhSr03cP\nmPGk0//BgHR92HWYpczKTmBg3sBxjKeAgduA+zPjBAZmBvYLYNX8CWDq/QMgxWiQz8BmwLBe4D7D\nDGB4pJlFNDAAAN1MIGxCExJiAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{x}$$"
      ],
      "text/plain": [
       "1\n",
       "─\n",
       "x"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.integrate(sym.diff(f(x), x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "for a generic function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = sym.Function('g')\n",
    "sym.integrate(sym.diff(g(x), x), x) == g(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(sym.integrate(g(x), x), x) == g(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us integrate a more complicated function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWIAAAAqBAMAAAB7Dkz8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nKTsqxYO2/Uen1P4LuNaWoBQpVlkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\frac{1}{2} \\log{\\left (\\sin{\\left (x \\right )} - 1 \\right )} + \\frac{1}{2} \\log{\\left (\\sin{\\left (x \\right )} + 1 \\right )}$$"
      ],
      "text/plain": [
       "  log(sin(x) - 1)   log(sin(x) + 1)\n",
       "- ─────────────── + ───────────────\n",
       "         2                 2       "
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.integrate(f(sym.cos(x)), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q4 Plotting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline  \n",
    "# We need to tell Jupyter to display plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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XmDB4cLR7INI7FOISEz77LNo9EOkdCnGJCZ18LLmI5ynEJSYoxGWgUoiLiHiYQlxigmbi\nMlApxCUmKMRloFKIS0xQiMtApRCXmKAQl4FKIS4xQSEuA5VCXETEwxTiEhM0E5eBSiEuMUEhLgOV\nQlxigkJcBiqFuMQEhbgMVApxEREPU4hLTNBMXAYqhbiIiIcpxCUmaCYuA5VCXGKCQlwGqh4N8V27\ndvXk6mKens+uq6ysJDMzk2AwyMqVKy9brhDvWdo2e5ZzLre791WI92N6PrsmHA6zcOFCqqqqOHDg\nABs3buTgwYPt2nzpS1Hq3AClbbPH5Xb3jtqdIp5XXV1NIBBg+PDhJCQkUFhYSHl5ebs2N9zQ8nPL\nFjh7NgqdFOklg3pyZR9/DA0N7Wtml/8r29UadFzrrG1Hyzpre6V1fNHH6G5/r1b/8EN4992u9yku\n7sptu/IcX+156cr94zqZGlxq15Wfly6XbsfF/fPnpeuXhEIh0tLSIrdTU1Oprq5u99jx8S0/f/Ur\neOgh8Pvhtttg8uSW6wkJLZdBg/55Hdo/XmeXtn3uyvg6e+66Uu9oWWfPeVdfC59fZ1e2neZmOHmy\n8/5caX1d6Vd3ttVryZqu1L5IP7ubO93hzKznVuZcz61MRCSGmFm34r1HZ+I9+QdB5ErC4ZbLxYvw\nl7/s5plnlvPv/17JRx/BL39ZyqefOsaNW8rOnVBfDx99BPv2OW2j0l91e37eozNxQK8Q6XMXL14k\nIyOD7du3k5KSwsSJE9m4cSMjR44EYPToln9j//Y3hbj0W90Ocb2xKZ4XHx/Ps88+S35+PrfeeiuF\nhYWRAIeWfd8ffBDFDor0Is3EZcBbswb+9V/h7FnNxKXfis5M3Dl3n3PuP51zF51zOZ21u9qJGNKi\nqamJ/Px8MjIyKCgooLm5ucN28fHx5OTkMG7cOO65554+7mX/1tG2Nnlyy5E+AOfPn6ewsJBAIMBt\nt93GyUuHWEiHrvbaLSsrY9iwYeTk5JCTk8OaNWui0EtvKC4uxufzkZWV1Wkb59y/OeeOOOdqnXPZ\nXVqxmXX7AmQAAWAHkGMduHjxot1yyy12/PhxO3/+vI0dO9bq6uo6ahrzlixZYitXrjQzs9LSUlu6\ndGmH7b7yla/0Zbc8o7Nt7eJFs+uvNwPs+eeft0cffdTMzDZt2mT3339/lHvdf3Xltbtu3Tr7wQ9+\nEKUeessf//hH279/v40ZM6ajxQB3Aa+0Xp8E7LYu5PA1zcTN7JCZHeEK/wp05UQMaVFeXk5RUREA\nRUVFbNmypcN2pl0CHepsW4uLg08+aWnT9jm+77772L59exR73L919bWr7bFrbr/9dpKSkq7UZBaw\nHsDM9gBDnHO+q62319/Y7OhEjFAo1NsP60nvvvsuPl/L7yw5OZl3257p08ann37KxIkT+eY3v6k/\niG1caVu7dIJF2zbx8fEkJiZyVqdwdqirr90XX3yR7OxsZs+eTX19fV92caDxA6fa3A611q7oqseJ\nO+e2AW3/Gjha3sD8qZm9/AU7GfOmTZtGY2Nj5LaZ4ZzjmWeeuayt6+TUrhMnTpCSksLbb79NXl4e\nWVlZ3Hzzzb3W54Gg8zNdNYu8FjNnzuSBBx4gISGB3/72txQVFem/mz521RA3s2nX8gB+v7/dm0f1\n9fX4/Vf94zJgbdu2rdNlPp+PxsZGfD4fDQ0NDBs2rMN2KSkpANx8883k5uayf/9+hThd29ZSU1M5\ndeoUX/va17h48SLvv/8+Q4cO7euuekJXns+2uwceeeQRlixZ0mf9G4BCQFqb26mttSvqyd0pHc51\nJkyYwNGjRzlx4gTnz59n06ZNzJw5swcfduCYOXMm69atA1re9Z81a9Zlbc6dO8f58+cBeO+993j9\n9dcZNWpUX3az37rStnZpJj5jxgzKysoAeOGFF8jLy4tWd/u9rrx2G9p8WFJ5ebm2xauwfx4U0pH/\nAOYCOOcmA+fMrLGzxpettDsX4B5a9uF8DLwzffp0MzM7ffq0ffvb34687VpRUWHBYNDS09OtpKTk\nmt/lHajOnDljU6dOtWAwaNOmTbOmpiYzM9u7d6/NmzfPzMxef/11GzNmjGVnZ1tWVpatXbs2ij3u\nfzra1n72s59ZfPzLBtgnn3xi3/ve9yw9Pd0mTZpkb7/9dnQ73M919ny+/PLLZmb24x//2G699VbL\nzs62vLw8O3ToUDS726/NmTPHUlJSbPDgwZaWlmZr1qyx3/zmN7Zq1Sqzf2bqs8BR4E1ajvi7ag7r\nZB+JCTfcAB9+qJN9pN/SafciV/LZZ9HugUjvUIiLiHiYQlxExMMU4iIiHqYQl5igb7uXgUohLiLi\nYQpxEREPU4iLiHiYQlxEJAr27t3L2LFjOX/+PM65L7d+wc4X/twChbiISBSMHz+eWbNm8dOf/hRg\nJbDBzP72Rdej0+4lJlx3HXz6qU67l/7ls88+Y8KECbz55pu7gW9aNzZQzcRFRKLkvffe44MPPgD4\nCnBdd9ahEBcRiZL58+df+kKY3wO/6M46FOIiIlGwYcMGBg8eTGFhIbTsEx/vnMv9ouvRPnGJCdon\nLv2cPopWRCQWKcRFRDxMIS4xQR+AJQOVQlxExMMU4tLnVqxYQWpqKjk5OeTk5FBZWRlZVlJSQiAQ\nYOTIkWzdujVSr6ysJDMzk2AwyMqVKyP148ePM3nyZILBIHPmzOHChQt9OhaRaFOIS1Q8+eST1NTU\nUFNTw/Tp0wGoq6tj8+bN1NXVUVFRwYIFCzAzwuEwCxcupKqqigMHDrBx40YOHjwIwNKlS1m8eDGH\nDx8mMTGR1atXR3NYIn1OIS5R0dGhfuXl5RQWFjJo0CBGjBhBIBCgurqa6upqAoEAw4cPJyEhgcLC\nQsrLywHYsWMH9957LwBFRUW89NJLfToOkWhTiEtUPPfcc2RnZ/PII4/Q3NwMQCgUIi0tLdLG7/cT\nCoUuq6emphIKhThz5gxJSUnExcVF6qdPn+7bgYhEmUJcesW0adPIysqKXMaMGUNWVhYvv/wyCxYs\n4NixY9TW1pKcnMzixYu7/Tg6eUdi3aBod0AGpm3btnWp3bx585gxYwbQMvM+depUZFl9fT1+vx8z\n4+TJk5fVb7zxRs6dO0c4HCYuLi5Sv5Lly5dHrufm5pKbm9v1QYn0Qwpx6XMNDQ0kJycD8OKLLzJ6\n9GgAZs6cyYMPPsgTTzxBKBTi6NGjTJw4kXA4zNGjRzlx4gQpKSls2rSJTZs2AZCXl8cLL7zA/fff\nT1lZGbNmzbriY7cNcZGBQCEufW7JkiXU1tYSFxfHiBEjWLVqFQCjRo1i9uzZjBo1ioSEBJ5//nmc\nc8THx/Pss8+Sn59POBymuLiYzMxMAEpLSyksLOSpp55i3LhxFBcXR3NoIn1OH4AlMeFf/gU+/lgf\ngCX9lj4AS0QkFinERUQ8TCEuIuJhCnEREQ9TiIuIeJhCXETEwxTiEhN+8pNo90Ckd+g4cYkZzuk4\ncem3dJy4iEgsUoiLiHiYQlxExMMU4iIiHqYQFxHxMIW4iIiHKcRFRDxMIS4i4mEKcRERD1OIi4h4\nmEJcRMTDFOIiIh6mEBcR8TCFuIiIhynERUQ8TCEuIuJhCnEREQ9TiIuIeJhCXETEwxTiIiIephCX\nXvOHP/yB0aNHEx8fT01NTbtlJSUlBAIBRo4cydatWyP1yspKMjMzCQaDrFy5MlI/fvw4kydPJhgM\nMmfOHC5cuADA+fPnKSwsJBAIcNttt3Hy5Mm+GZxIP6EQl14zZswYXnrpJe6444529bq6OjZv3kxd\nXR0VFRUsWLAAMyMcDrNw4UKqqqo4cOAAGzdu5ODBgwAsXbqUxYsXc/jwYRITE1m9ejUAq1evZujQ\noRw5coTHH3+cJUuW9Pk4RaJJIS69JiMjg0AggJm1q5eXl1NYWMigQYMYMWIEgUCA6upqqqurCQQC\nDB8+nISEBAoLCykvLwdgx44d3HvvvQAUFRWxZcuWyLqKiooAuO+++9i+fXsfjlAk+hTi0udCoRBp\naWmR236/n1AodFk9NTWVUCjEmTNnSEpKIi4url398+uKj48nMTGRs2fP9uFoRKJrULQ7IN42bdo0\nGhsbI7fNDOccP//5z5kxY0aPPc7nZ/PX2k5koHDa6KW3Oed2AovNrKb19o8AM7OVrbcrgWWAA5ab\n2fTPt3PO/R3wmVnYOTcZWGZmd126r5ntcc7FA++Y2bBO+mHAijalXWa2q1cGLdJHNBOXvuLaXP8P\n4PfOuf8N+IF0oJqW3XvpzrnhwDtAYesFYAfwPeD/AkVAeZt1FQF7Wpfv6KwDZuY6WybiVZqJS69x\nzt0D/Ar4L8A5oNbM7mpd9mOgGPgM+KGZbW2tTwd+SUugrzaz0tb6zcAmIAnYD/w3M/vMOfclYAMw\nDjgDFJrZ8T4bpEiUKcRFRDxMR6eIiHiYQlxExMMU4iIiHqYQFxHxMIW4iIiHKcRFRDxMIS4i4mEK\ncRERD/v/EWDmG68tOqUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb0afa759e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fb0afa75908>"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.plot(f(x), (x, -1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb0a5616240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fb0a5616320>"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.plot(f(x), (x, -1, 1), ylim=(-10, 10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb0a55feb00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fb0a55fe080>"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.plot(f(x), (x, -1, 1), ylim=(-10, 10), xlabel=None, ylabel=\"$1/x$\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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evTuLFi2yVKmIHQpx8TT/4GbPnj2ZO3cugwYNqvL9DRs2MGvWLDZs2MCCBQt4\n4IEHcNX/IjFEIS6e5m+Jd+vWjS5dupwT0PPmzWPcuHEkJCSQk5NDly5dKCkpsVStSONTiIunXWia\n4a5du8jOzj57OzMzk127djVCZSLekGC7ABG/IUOGUFZWdva2aXU7bNz4JKdOjSQx0V5tIl7lqP9Q\nIonjOMuA/++67pozt4sA13Xd35+5vRCY7LruZzU81wWmBtxV7LpucfirFgkftcQlEjkBX88H3nIc\n51kgE+gM1Ngp7rquU9P9IpFMfeISERzHuclxnB1Af+B9x3EWALiu+xUwC/gK+AB4wNWflxJD1J0i\nIhLB1BIXEYlgCnERkQimEBcRiWAKcRGRCKYQFxGJYApxEZEIphAXEYlgCnERkQj2f0kQBoGbdHpQ\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb0a55456a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = sym.plot(f(x), (x, -1, 1), show=False)\n",
    "p2 = sym.plot(x, (x, -1, 1), show=False, line_color=\"red\")\n",
    "p1.extend(p2)\n",
    "p1.ylim = (-10, 10)\n",
    "p1.xlabel = None  # Try using strings here\n",
    "p1.ylabel = None\n",
    "p1.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Saving the file:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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SuzQSF08LtlS6dOlCZmYmbpnVP0uXLuWmm26iXr16ZGRkkJmZyZo1ayxVK1L7\nFOLiaReaobJv3z7atGlz7nZaWhr79u2rhcpEvEHtFPGMESNGUFBQcO6267qcOOHg8z0CjLZXmIiH\nOWX/PBXxMsdxVgEPuK67/uzt6YDruu5/nL29DJjhuu7HFTzXBWaF3JXvum5+9KsWiR6NxCUWhZ61\nfBX4i+M4fwTSgE5AhU1x13V1tlPqHPXEJSY4jjPOcZw9wADgdcdx3gRwXfczYCHwGfAGcJerPy8l\njqidIiISwzQSFxGJYQpxEZEYphAXEYlhCnERkRimEBcRiWEKcRGRGKYQFxGJYQpxEZEY9v/hD2LK\n+QAUngAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb0a55610f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1.save(\"my_plot.pdf\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q5 Worked example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Tgjs11b92pkiiKbjFdT0luDXiFqcouMV1PSW4NeIWpyi4xXU9Kbg14hYnKLjF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\nW0QkySi4RUSSzP8H2CkG9oQEg28AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb0a5489dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fb0a5489e10>"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return  -x ** 4 + 9 * x ** 2 + 4 * x - 12\n",
    "sym.plot(f(x), (x, -20, 20) , ylim=(-20, 20))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFQAAAAVBAMAAAAqQdQ7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\left\\{-2, 1, 3\\right\\}$$"
      ],
      "text/plain": [
       "{-2, 1, 3}"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solveset(f(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOEAAAAbBAMAAAB4lPU2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\njlJCCfwFZSWuYx7fF4kAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\left(x - 3\\right) \\left(x - 1\\right) \\left(x + 2\\right)^{2}$$"
      ],
      "text/plain": [
       "                        2\n",
       "-(x - 3)⋅(x - 1)⋅(x + 2) "
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0A\nV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAm\nwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0A\nV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAm\nwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, -sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOIAAAA/BAMAAAAPngkVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/AEJ2V+bHYm5mgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left\\{-2, 1 + \\frac{\\sqrt{6}}{2}, - \\frac{\\sqrt{6}}{2} + 1\\right\\}$$"
      ],
      "text/plain": [
       "⎧        √6    √6    ⎫\n",
       "⎨-2, 1 + ──, - ── + 1⎬\n",
       "⎩        2     2     ⎭"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "poi = sym.solveset(sym.diff(f(x), x), x)\n",
    "poi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-2 False\n",
      "1 + sqrt(6)/2 False\n",
      "-sqrt(6)/2 + 1 True\n"
     ]
    }
   ],
   "source": [
    "for point in list(poi):  # We convert the poi to a list\n",
    "    print(point, (sym.diff(f(x), x, 2).subs({x: point})) > 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q6 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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+2bkTLr3UdhWRRbMhpDwFsPhmwwaoV892FZFlwABYvhwOHbJdiUQCBbD4Jj8fzj/fdhWR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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb0a50d8860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fb0a50d8be0>"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return  -x ** 4 + 10 * x ** 3 - 29 * x ** 2 + 8 * x + 48 \n",
    "sym.plot(f(x), (x, -5, 5) , ylim=(-5, 50))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFQAAAAVBAMAAAAqQdQ7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZpkQ3Ynvq81UMrtE\ndiLw+n06AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABT0lEQVQoFY2TvUvDQBjGn/Sa1viRZHSSqoNr\n/wHRoYJLQfBjDjgIIgguHRtE6X/QwaXB1aWrIFgcOgcEV7u4Cw7aRXzf+0rEaHyG95577pe74+4C\nwFlGqa4fGRFbpSAB44SK81qIvuzfmny2AVRi6rkRFat+U1m/gUGo0+qQqE3q5FFx1tXowhQBj7Ke\nGI3I5FHgWaPzJ9iIaZTkL/6NEmI24M6UofZklixaj3kVI7MB+HurKhMpo/Kc+onBuLUocDCSAy4Y\nxTlwc0ytuGiRtpNvaO1QolcKrbf1PciQi5nVn2BuyoHYUWiPfPFegzeNep1O9yhVe5X3wN9LmVlr\nQ1Q/dUb+lysQH4Q4IQapdEBQiK6d9kK0msQ+rNxBOff+PSyaldf0JlxZmSt4Awy4XKRyLqLg53td\nVxzVzFViDi+55BXaTubGIw69//xbu8AXr5lO/mSMCkwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left\\{-1, 3, 4\\right\\}$$"
      ],
      "text/plain": [
       "{-1, 3, 4}"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solveset(f(x), x)  # Finding the roots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0A\nV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAm\nwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0A\nV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAm\nwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x), x, -sym.oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPEAAAA/BAMAAADDBzP+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMrtUdhCZiUSr72bd\nIs25ozBRAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAE9klEQVRYCe1ZTWgkVRCuyaRn8jOdzMmTkAbx\nnAgi6CVz8BTBzCEgijIDHmQPsqMechDMHDyIl52DLMYVEiIeBEnm5kmNAQ8uQeeylxU2QRBcWce4\nuhtX3MR69ar7vX79XnfvNI4XC6ZfVX1V9fV7Pd3VbwYApfSIOI5L/J8U0+qjSh+DttEOSSqnoTae\nsXwW8lzqhdqYxtW2JPL2mqmM/nkkJ6mBEjyKos9d0XO/SWT2ritC+qe76biBvm/YFrN8Tzon/rSA\nmuuGpmer/nF2DPBq1HjuroyXXYDVX1myuuNOvr4z9+Nuw/L20fGQ4WTz4cvr65049CKapXbcx1bp\nm1fW10nfG9CQwUyTOMDIp5pcQQ1f4repp0yhiRWqBQDlj4UVk1mMllc4H7OYxOQOeJe3ksyvAkzH\nigOt0EoAz1+4EwfQquBM5XnmY34Tc77/AQ+rSeYOwGtxgmofwP8iwJNNMlcBqnUKz8VMk+g4mAGm\nGnFmsULVucDKjNDbMjoX8yROYnbgZJ6QpaLjM6h97mYOZGAm8zsYV8PPGjiZxbVgmeqisg/gNZzM\n4loIyWKe3MWgp/HTcTJ7J6KQlJVjXOMGLjY4mVscmsV8bbVLkyjXncyTyMbir+GzUKzQY27mZzk2\nixlaAcx2AJ4DJ7O4c0PxbgN8hovdcTMfcGwmc+U+1JoAT25u/vWe9a6CVi/kxfHiAL7GR8jm5tYn\nDdtdBR53iszrDP5dMQkhO/ix3M+w0BYoy1FbrBDKTGC/q0phV8ycM2w1cRJCcCGtzMttgbJM3xEr\nhDIf2JnL+ZkX3miLSvD6+RW4sHezT4Z+uL6kWaXTJ8iqLt7rVz44u6JBUvU/YpcxZ0s/mtuVk0jU\nsDt+/tHuT3iZeZ6fsOEXT4uLlkfzpaixtU+Jg+8GhM5v04D9KCm/JF3oKTesbpiTBU3wcdMBi3Vy\nLQQ0UD9KxFgdLmZrMECSeaNDoRt9GqgfOXINd2Hmo2NR0fuU6sp+RGrmoTDzNL10TsgXQNmPMkkp\noDAzHIo6G12qJvsRqZmH4szXA4CJr4iI+xHq3q1DlG/rMK82C5GGeHU4vLU7HPZQlRGUDxDFKOUE\nobeGw6vD4U0RRIDwIcmHA2jJ7xf3I3JnHorPGd5dAv+gKZi4H2WSUkBx5urvWOiFY6azPUkYMobi\nzPTlmgobyG3woo2tQWWYBnNWWvJJUj6liosDGkQ/OqSVN3iSpsGclZZkntmmosu9qHZZnkRks1Jq\nxz0aM0FGmtn0NGaGljtUsLUd1cVdgE30ty2Be+oECTLSDowaLymboYtdcs2ofciaitG1lUC3YjpB\n8TRr05NJIfRrnWxtL9mPVQ0N2ieFRnyUUDwtpemFkPFOEq+pWbRP0mxNtUEpTS+E8jLTPkmj01QL\nlNL0Iigns9wnaXRKtUEpTS+CcjLLfZKi0zQblNL0Iigns9wnaXxKtUCq6akw1hSUj5n3SYk66LBB\nKU1PQfmYeZ9kY7ZBKU1PQfmYkVLskxxig3YcsehmKDez2Cc5xAaJTZhDGMrLLPZJjko2SDQ9h4RQ\nXmZHmQLu/5kLLN4Dp/53q30+oJMNf9t/4DMfPYF/WR/3Xzf4kyy/4/p/j37yo2VW/+C8raXRCoyc\n1drn1Fr63wkjEzgT+QUQl/1q3Rn0bwA31J9FFbG/Gpt4Z/KeIsJrY6MVRHKF/wG9arGnwnIFwQAA\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left\\{4, - \\frac{\\sqrt{41}}{4} + \\frac{7}{4}, \\frac{\\sqrt{41}}{4} + \\frac{7}{4}\\right\\}$$"
      ],
      "text/plain": [
       "⎧     √41   7  √41   7⎫\n",
       "⎨4, - ─── + ─, ─── + ─⎬\n",
       "⎩      4    4   4    4⎭"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "poi = sym.solveset(sym.diff(f(x), x), x)\n",
    "poi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4 False\n",
      "-sqrt(41)/4 + 7/4 False\n",
      "sqrt(41)/4 + 7/4 True\n"
     ]
    }
   ],
   "source": [
    "for point in list(poi):  # We convert the poi to a list\n",
    "    print(point, (sym.diff(f(x), x, 2).subs({x: point})) > 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x ** 2 + x\n",
    "\n",
    "def g(x):\n",
    "    return 1 / (x ** 2 + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a = sym.symbols('a')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAAsBAMAAACebLDoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzythP4kXfpNJzI1z2jaHmdjav/r97nNi4qV5AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{a^{2} + 1} \\left(a^{4} + a^{3} + a^{2} + a + 1\\right)$$"
      ],
      "text/plain": [
       " 4    3    2        \n",
       "a  + a  + a  + a + 1\n",
       "────────────────────\n",
       "        2           \n",
       "       a  + 1       "
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x) + g(x), x, a).factor() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAAsBAMAAACebLDoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzythP4kXfpNJzI1z2jaHmdjav/r97nNi4qV5AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{a^{2} + 1} \\left(a^{4} + a^{3} + a^{2} + a + 1\\right)$$"
      ],
      "text/plain": [
       " 4    3    2        \n",
       "a  + a  + a  + a + 1\n",
       "────────────────────\n",
       "        2           \n",
       "       a  + 1       "
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sym.limit(f(x), x, a) + sym.limit(g(x), x, a) ).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAuBAMAAAB0cDJ0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nETSRAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{a \\left(a + 1\\right)}{a^{2} + 1}$$"
      ],
      "text/plain": [
       "a⋅(a + 1)\n",
       "─────────\n",
       "   2     \n",
       "  a  + 1 "
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x) * g(x), x, a).factor() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAuBAMAAAB0cDJ0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nETSRAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{a \\left(a + 1\\right)}{a^{2} + 1}$$"
      ],
      "text/plain": [
       "a⋅(a + 1)\n",
       "─────────\n",
       "   2     \n",
       "  a  + 1 "
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sym.limit(f(x), x, a) * sym.limit(g(x), x, a) ).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJkAAAAcBAMAAACADwDiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACX0lEQVRIDa1UPYgTQRh9u0k2iclKagUv4CFy\nxRHlCoUrgiCWWvlT3VoIViaNlYUnWCwIx3liHzvPSkURCyWddkmhCIJcEAUVRKvD7nyzM7Mzk92s\nBBxI5v1883Z+dhaYv5WWjsw/aOaIY/g205vDeCNrn2DUyo4qZ6VCpTmQ9gss9LOF1UlWK1JqkXZ3\nUqQV9ictDG/TZja+q8hiKj5LkQUakUUOWNiF++UuebtaLrU1svuyrd6ynRSH54FgPaEmYyu1NXgf\nAd5vzYB622Ag7Ep2bVlMaCUhlbHUUO1UhxK+k13w+VdE9FIy8d+Qy1GCTkNZpF1J1FGkzLefPiqo\n0oCzQlhSPruDBhK5abW+MO/rirW9PQXdtNNUvcOL3BucMpDITZM7dtSpIAHctJsRcL1VE9u3zZ+C\nRG5afZ0SHvFnKoQylTYaIrgEf0DjFc9OQTI3LXxMCRtOhVCm0nhD/F00ujQuIIWlOL5zMY4nVOUp\nNMXk8dRUCIoTcfw8jvkMtuQUFsaoDNAbkjNNQzJ3bjKN46wKFuXMrdfBjjC4Ug3J3LSqXqlVIcZM\nrZRflt4YD8NWcgoaZtLkKfzg81RxEpVJ45nWuvXXJRrHkcJMmt8Rw/k66WJBRXPfEL6QwZfVQ2do\n3DCQTK/U//rnO0OGlMBH62JBRdNpy2sbm85dmHGzkkHynlT6CbH/dJrUbhvLvfUBd9K0lQTK9RqV\n6LLNAnGldXugQaZXX6TmuYzjCPWJRWd/Lfepif60qnNgY2iJ//6SX7Wqc+C9HK1ACtsFJrxBkZvj\nfcjRUsmPUvifwF+lb5DKtAqj6QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$a \\left(a + 1\\right) \\left(a^{2} + 1\\right)$$"
      ],
      "text/plain": [
       "          ⎛ 2    ⎞\n",
       "a⋅(a + 1)⋅⎝a  + 1⎠"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(f(x) / g(x), x, a).factor() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJkAAAAcBAMAAACADwDiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfcjRUsmPUvifwF+lb5DKtAqj6QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$a \\left(a + 1\\right) \\left(a^{2} + 1\\right)$$"
      ],
      "text/plain": [
       "          ⎛ 2    ⎞\n",
       "a⋅(a + 1)⋅⎝a  + 1⎠"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sym.limit(f(x), x, a) / sym.limit(g(x), x, a) ).factor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x ** 3 + 3 * x - 20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANAAAAAqBAMAAAAwg8g4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{h} \\left(3 h - x^{3} + \\left(h + x\\right)^{3}\\right)$$"
      ],
      "text/plain": [
       "       3          3\n",
       "3⋅h - x  + (h + x) \n",
       "───────────────────\n",
       "         h         "
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = sym.symbols('h')\n",
    "rhs = (f(x + h) - f(x)) / h\n",
    "rhs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAWBAMAAABppzwEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nqzIhoty3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQUlEQVQoFW2PP0vDQByG38QjTRNtzoKCW6kU\nxKmDDuLQDO4tSF1cMjkXFyepCOJoEFwcdHCqOtxWFzGDg0MGP4LfQAvFP0SIaZK7XuQyXN7fc8/d\nywHS11hn0qSKOp0LVFxiJq2MpFEVTV/7UfECI1FhVA1mT0VRra8J3hApC1p92QW0DtpevmN08sB/\nCyCfQGkM5yhH57jKEskL+y6+gfIAT0G2UXne2iwql779mxJeZMXxR1FBWgTYx8DwcGlfeg0vAh5q\nySFtZRd2zTg1g+yGySqU6glL6aqvMz0iVKFAH6TUurMx2xSCHoYv92H4moFbBo1iZgw4PaEkgRfN\nAy0XTpQq728qJWYTxWrCHBHWR4lNJX7LDXBGYXho967dFhanhijaA/lK8EV3A48Hwx1PoZS7276E\n5ciLZPYv21QCf4IwRNg4bvHgAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$3 x^{2} + 3$$"
      ],
      "text/plain": [
       "   2    \n",
       "3⋅x  + 3"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(rhs, h, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAWBAMAAABppzwEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nqzIhoty3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQUlEQVQoFW2PP0vDQByG38QjTRNtzoKCW6kU\nxKmDDuLQDO4tSF1cMjkXFyepCOJoEFwcdHCqOtxWFzGDg0MGP4LfQAvFP0SIaZK7XuQyXN7fc8/d\nywHS11hn0qSKOp0LVFxiJq2MpFEVTV/7UfECI1FhVA1mT0VRra8J3hApC1p92QW0DtpevmN08sB/\nCyCfQGkM5yhH57jKEskL+y6+gfIAT0G2UXne2iwql779mxJeZMXxR1FBWgTYx8DwcGlfeg0vAh5q\nySFtZRd2zTg1g+yGySqU6glL6aqvMz0iVKFAH6TUurMx2xSCHoYv92H4moFbBo1iZgw4PaEkgRfN\nAy0XTpQq728qJWYTxWrCHBHWR4lNJX7LDXBGYXho967dFhanhijaA/lK8EV3A48Hwx1PoZS7276E\n5ciLZPYv21QCf4IwRNg4bvHgAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$3 x^{2} + 3$$"
      ],
      "text/plain": [
       "   2    \n",
       "3⋅x  + 3"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(f(x), x)"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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}