{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "toc": "true"
   },
   "source": [
    "# Table of Contents\n",
    " <p><div class=\"lev1 toc-item\"><a href=\"#1\" data-toc-modified-id=\"1-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>1</a></div><div class=\"lev2 toc-item\"><a href=\"#1(b)\" data-toc-modified-id=\"1(b)-11\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>1(b)</a></div><div class=\"lev1 toc-item\"><a href=\"#2\" data-toc-modified-id=\"2-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>2</a></div><div class=\"lev2 toc-item\"><a href=\"#2(b)\" data-toc-modified-id=\"2(b)-21\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>2(b)</a></div><div class=\"lev1 toc-item\"><a href=\"#3(a)\" data-toc-modified-id=\"3(a)-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>3(a)</a></div><div class=\"lev2 toc-item\"><a href=\"#3(b)\" data-toc-modified-id=\"3(b)-31\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>3(b)</a></div><div class=\"lev1 toc-item\"><a href=\"#4\" data-toc-modified-id=\"4-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>4</a></div><div class=\"lev2 toc-item\"><a href=\"#3(b)\" data-toc-modified-id=\"3(b)-41\"><span class=\"toc-item-num\">4.1&nbsp;&nbsp;</span>3(b)</a></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAZBAMAAADqPuxnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnZmzRC73UTvIomZ\nVKu7zOipAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACE0lEQVQ4EaWUPUgcQRiG37m7vTs3p5ySWheD\nTRLCYQpBLQ6RlGEr4bCxCEIsgoWksDrShVQJpAkWC0FFsDgOBdGg1wfCEUggafJHQEKK04AYf3L5\nvpnd2b9j4XCK3Xfeeebl+2Z/gG5G/Z3dDR5jC6XCXMzsxijU0q1u+A7sFSsAKsUOqRHrtTsfj/g8\nTU0pM8qIFz5szrm6z/FNrVQLMUacawIZb59R9k2tzEuWicykhutaeaJSNP+wTmKw7NGoaOWJnJN6\nyTrGmP0eAnNRy0xVS1eI+r5DMsyIG6PltX+4vjJwx6LFVAMwhn89ofqvkYTx9k0Ts8M7YI7mPMLM\nDLCI+8BjyzyjxWwJmMWYtQSkyzTfaIpXYhlZS3Jk0AgzT21YeAQ8BPiAck1gGwf2Z6CXu5lG71Gu\nBnEkOd4eZXInN8EBD4BjWuypAjbuMSda1MEFiSEKPZcc21HG2Go3OGDeD1BKBuQ59EsVOJEcb1cB\nPrOK9GkggFuggpnLUwu6ggvJsa1a8Jn3wI9AAB/Qh1QLI+4hTtNZ9DSQP5WcCggzn2wscP1uC/Ts\n8pfZlkk5hRrxGyXsiufI1CSnAsLMs9GJ6mD752D728e/X2XdxveR23cJzRTpYvy+5WBz7xDM0ZwH\n9RZilOtd/dd03bNi90Qm8UNxoxKZrONSHT9ntZbI6J+FptzAwC2Zif6uAhu1DDD/Af9rnHH4sKkJ\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sin{\\left (x \\right )} \\cos^{3}{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "          3   \n",
       "sin(x)⋅cos (x)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "x = symbols('x')\n",
    "f = sin(x)*cos(x)**3\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "fs = f.series(x,0,15)\n",
    "f0 = fs.removeO()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeEAAAAwBAMAAADUcd8DAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nolO+TKr4tFxsFIHMzpJEYeT+qHb8H5eTbdJVUDjeAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{8194 x^{13}}{6081075} - \\frac{82 x^{11}}{6237} + \\frac{257 x^{9}}{2835} - \\frac{26 x^{7}}{63} + \\frac{17 x^{5}}{15} - \\frac{5 x^{3}}{3} + x$$"
      ],
      "text/plain": [
       "      13       11        9       7       5      3    \n",
       "8194⋅x     82⋅x     257⋅x    26⋅x    17⋅x    5⋅x     \n",
       "──────── - ────── + ────── - ───── + ───── - ──── + x\n",
       "6081075     6237     2835      63      15     3      "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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/7hhKk1A7r36fzrp9uVbHUR7gyYLRDjhQ43VG9bJG3VZEJotIqoikZmdnNyjo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\nKeUWLRhKKaXcogVDKaWUW/4fT9kMvhdotP8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f65e9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x10f4a2780>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(f,f0,(x,0,pi/2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g = integrate(f,(x,0,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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iweDH3U52/vgIShva2XdMdVcN1aK0MGKDPHlibR5NDrQaf0QlDlssrnpmwyFa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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fe98828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x10fe8f1d0>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(f,g,(x,0,2*pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFsAAAAyBAMAAADB47o4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiRDdIrtUdpkyzUSr\nZu9x33h7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACXUlEQVRIDY1UPWgUQRh9u7d3e8kaE38ggkXG\nNkQIKAgKuiAINiZNOsFVVOxc0iSN5BACMRZJLUIWQcQqsbJM0IBgkwMbG/EgvZ5EFDFhnbnbmflm\nnM3dwO33/u5Ydr93gDxjee/TlmFgUsM+UDDdR0hHaqnGfaD66uEhb/ExDVQoceCT+EjVcUoc+DzW\nYiLfJtgFL2Fqg+inCXbDh4zoLwh2wwtUzihxYa9B1IgSomv4WUMgTCgzsD+f4gaLRiLyZqrMiFDi\n1VpYwLP790jkkLd0rZ5hBt/znPzGc4ItyI6m+G1pjyxO6WY88IdyjrctTukywhblHE9Kbm2ekG+i\nuiXt7tTlsDZP2IvYbJrxoVRya/OEPDG6Eku7O3U5rM3r2stmGvSxG5vHc0cy/68Vp+UwNo/nKs2q\nutXia6QcxuYJO5p7UqTEuC4upBzG5gnPPK8EFeXw57bfjsLcPDPKWW1fSBn/nMGn9KK1ecIzztnL\nnHbKcRxj7K61eUZUkGSFAV7CEcNLfi3O+DlxXgum/2nbnH1NgKIc9pqKsH2mRvij7Yj/rakdFXw4\nAzrluBO2MetKGNogvyNRjuhXtR00DctFPN4UUQ7/w+z8F1fA1GoHpBympViwNLNaEH8fuhwqYIIJ\neGopd6HLYaYUewd8k2SH6XJIzZq7wI68m4UNWg4r2KXrTMfXUloOZ5yLV1jhDE+TcpSlox/SGWw9\nlbB8VrakV/+ZSVg+l5Q1dNA7HjZUPNjTWIkWeIAgllKeSFQ2BxoIY2neYhKVzavHTr1X3gmFysB6\nnu8p741CPcE/Ddif4WwEVjAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sqrt{x^{2} - \\frac{y^{2}}{2}}$$"
      ],
      "text/plain": [
       "     _________\n",
       "    ╱       2 \n",
       "   ╱   2   y  \n",
       "  ╱   x  - ── \n",
       "╲╱         2  "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "y = symbols('y')\n",
    "x = symbols('x',positive = True)\n",
    "f = sqrt(x**2-Rational(1/2)*y**2)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAAAwBAMAAAA1E18+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nl1jBf4V84pvfNMnHAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2} x^{2}}{4} + \\frac{\\sqrt{2} \\pi}{8} x^{2}$$"
      ],
      "text/plain": [
       "    2         2\n",
       "√2⋅x    √2⋅π⋅x \n",
       "───── + ───────\n",
       "  4        8   "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "int1 = integrate(f,(y,0,x))\n",
    "int1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAAAwBAMAAAD3D9n/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwxFLZvFjISmRhackw9p7emvIK/xl+AYJocTzEw3F4AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2}}{12} + \\frac{\\sqrt{2} \\pi}{24}$$"
      ],
      "text/plain": [
       "√2   √2⋅π\n",
       "── + ────\n",
       "12    24 "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(int1, (x,0,1))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAABLCAMAAABjltjdAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n/q1fsgj7w5JhyOqkioq7X7IMA/sCnk8k8xXXnZYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 1 & 1\\\\1 & -1 & -3\\\\0 & 1 & 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  1   1 ⎤\n",
       "⎢         ⎥\n",
       "⎢1  -1  -3⎥\n",
       "⎢         ⎥\n",
       "⎣0  1   2 ⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A=Matrix([[1,1,1],[1,-1,-3],[0,1,2]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAIpm7MhCriUTv3c12VLge\nopIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABOSURBVAgdY2BUMnZgYAhjYH/BwJDKwDCTgWEWA0Oe\nA8O+ABAJBOsCgATHcxCTKwFEKoEIHgUQeYmBUYCBRYGBR4BBqrwoi4Fh37t3rxgAK5QOlzv7snYA\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.rank()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x1=Matrix([1,1,0])\n",
    "x2=Matrix([1,0,-1])\n",
    "x3=Matrix([0,-1,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACsAAABXCAMAAACqY0xIAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nCPgBf+InUaNaxekAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡√2⎤\n",
       "⎢──⎥\n",
       "⎢2 ⎥\n",
       "⎢  ⎥\n",
       "⎢√2⎥\n",
       "⎢──⎥\n",
       "⎢2 ⎥\n",
       "⎢  ⎥\n",
       "⎣0 ⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y1=x1\n",
    "a1=y1/y1.norm()\n",
    "a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y2=x2-(x2.T*a1)[0]*a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAABLCAMAAAAxteWWAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRO/NZondIrtsRPdGkQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAQ1JREFUSA3tl8kS\ngyAMQNlrZdPm//+1LAMVJlpmetKaC2F4BJKwEgpJGNkVnwlCKHARRO6SxMZ2BhGl+9Snxf6OSq0+\n9rKGWlWMMxhDgxVxo6eJAGcaDLPtKkDXQIuU2o2ePwKKO+v6Iwd3y4TzS/mlZD+XKGp1bHRmAF3T\n+Sm7PYNahYTO0B6jGKqAx6FnEM0MMHQBFxmai4rjaLI6gh5M4DE96yhJyW7Jzq3XhNwF6xp72JFg\n5RTwkRQQHxOru0WARSBgLlxTHRkmhMy1dbLUbvQ/I7B9bBxFoHtsHKFhSW0fG2dFlfFVTN5ZV3Cr\nbPhato+NL9mqvYJyaXT4C6HiB0GIeRuZVk9fCCHIGxvsGDM7PBEXAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1\\\\0\\\\-1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1 ⎤\n",
       "⎢  ⎥\n",
       "⎢0 ⎥\n",
       "⎢  ⎥\n",
       "⎣-1⎦"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAABNCAMAAABtw5/KAAAAQlBMVEX///8AAAAAAAAAAAAAAAAA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\nAjalHe98F03OAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{1}{2}\\\\\\frac{1}{2}\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1/2⎤\n",
       "⎢   ⎥\n",
       "⎢1/2⎥\n",
       "⎢   ⎥\n",
       "⎣ 0 ⎦"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x2a = (x2.T*a1)[0]*a1\n",
    "x2a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC0AAABNCAMAAAAFMB3yAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nOh1pDd+3LRwNP5f0egAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{1}{2}\\\\- \\frac{1}{2}\\\\-1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1/2 ⎤\n",
       "⎢    ⎥\n",
       "⎢-1/2⎥\n",
       "⎢    ⎥\n",
       "⎣ -1 ⎦"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y2 = x2-x2a\n",
    "y2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAABYCAMAAAB8m391AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nWktfzwcoAHbtB8AAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\sqrt{6}}{6}\\\\- \\frac{\\sqrt{6}}{6}\\\\- \\frac{\\sqrt{6}}{3}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ √6 ⎤\n",
       "⎢ ── ⎥\n",
       "⎢ 6  ⎥\n",
       "⎢    ⎥\n",
       "⎢-√6 ⎥\n",
       "⎢────⎥\n",
       "⎢ 6  ⎥\n",
       "⎢    ⎥\n",
       "⎢-√6 ⎥\n",
       "⎢────⎥\n",
       "⎣ 3  ⎦"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2=y2/y2.norm()\n",
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x3a1 = (x3.T*a1)[0]*a1\n",
    "x3a1\n",
    "x3a2 = (x3.T*a2)[0]*a2\n",
    "x3a2\n",
    "y3=x3-x3a1-x3a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC0AAABNCAMAAAAFMB3yAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{2}{3}\\\\- \\frac{2}{3}\\\\\\frac{2}{3}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2/3 ⎤\n",
       "⎢    ⎥\n",
       "⎢-2/3⎥\n",
       "⎢    ⎥\n",
       "⎣2/3 ⎦"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAABYCAMAAAB8m391AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\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": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a3 = y3/y3.norm()\n",
    "a3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAZBAMAAADDORPgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iEJmJZjLNVO+r\nRHY8nXFNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAeElEQVQYGWMQUjJgAAJeJTUGBRADBGzQmCxh\nZQ1QUc4FLMpQZjUDQyaUKcPAsN4Bou0PA8P9BWAmy1cg8wGYyfyNgaE/AcIEikKZSAoYgNrWQ7Qx\nCDIw1DtADANaEckAYTItYNkBZbLkvYE5B+RaqCiFTCVYOOgAALX2IMEQ20hWAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[0]"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a3.T*a1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "\n",
    "a, x = symbols('a x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAvBAMAAABJZWRJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABO0lEQVQ4EWNggIDcvQJQFlaKzYF5AlYJqCCH\nA88XvPIBLF/xyTMwcH3EL8+hgF8+F1Oa6yFCjP0Cgg1lCV9GsjKMIQZDARNCnmfWqo045IWUXdUU\n+P7//4BdnjGBvYkDe9iBzWcTYPvI5YChFyQAlmdk4NyAVRYqz8DAr4Bf/vwBfPJcAvoMTAJYVYDt\nj1+wn8EdqzTEfcJKQtcLsMrz2v00xypBmSCXAn79g0z+PwIA3c02c+aMlzNnGgCZ/GAJTL8MMvcD\nHchyd/cBhDvh7oMLCzOwIWVuRgeoWriwIQPDO4R+OAsu/IKBwf4AXBjOgAvPF8Aqjyzcjz0Fw4RZ\nv8FNRWbAhTlAEYIJ4MJ3MeVAIjBhpgtY5eHCaxlYHLCogAnzXGBgwiIPF/ZIy7XAoh0uPP///x9Y\n5MHCAEGjX/wk9GLMAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{x^{2}}{2} + \\frac{1}{2}$$"
      ],
      "text/plain": [
       " 2    \n",
       "x    1\n",
       "── + ─\n",
       "2    2"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C1 = Rational(1/2)*x**2 + Rational(1/2)\n",
    "C1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAwBAMAAADnb5cjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAu0lEQVQoFWNggIDcvQJQFgObA/MEGJvDgecL\nnB3A8hXGZmDg+ohgcygg2LkIJvsFBDuMIQbG4Zm1aiOQLaTsqqbA9///BwYGxgT2Jg6onWwCbB+5\nHCBqGRk4N8B0MTDwKyDY5w/A2FwC+gxMAhBe/IL9DO5QCWEloesFMEU0pv8jABVsOotkRjeCzfQa\nwXafjWAnINjsDgi2EAOCnYBgcx1AsMUYEOzFxsb/jBCGItnF8BshnP3fHMEBswDEfDA5IHsTfwAA\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{x^{2}}{4}$$"
      ],
      "text/plain": [
       " 2\n",
       "x \n",
       "──\n",
       "4 "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C2 = Rational(1/4)*x**2\n",
    "C2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHEAAAAwBAMAAADUVIgqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJr9TlElYh6N/N0OjEA1TCPkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{a^{2}}{4} + \\frac{a}{4} + \\frac{7}{12}$$"
      ],
      "text/plain": [
       " 2         \n",
       "a    a   7 \n",
       "── + ─ + ──\n",
       "4    4   12"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = expand(integrate(C1-C2,(x,a,a+1)))\n",
    "S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAqBAMAAAC9wk0pAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMlTvq5l2ZiK7\niUTiBfEGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfUlEQVQoFWNgwA3CBJDlGCvqUfgMDPMHN789\nX30BsgcGhi1kDAIqDAz/QeADDR2xo+cIsumMBQz3HyAJsAswsCUg8dkMGNi/IPGZPqLygVLMH5Hk\ngcz3F1D55qhcjgZU/gxULmsDw25kEQ8GhtlIfG7bu+0GSHwmYHAZMAAAbgMkLgI1UYUAAAAASUVO\nRK5CYII=\n",
      "text/latex": [
       "$$- \\frac{1}{2}$$"
      ],
      "text/plain": [
       "-1/2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S0 = solve(diff(S,a),a)[0]\n",
    "S0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAArBAMAAABlSd54AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABF0lEQVQoFbWQPUvDUBSGn0SbNNaW2B+gAT9H\nBd2lg3M2l4KZOpUYEHTTzC4KxdmAoGsglC4O6uJoJ+eugoNSg3bQ67mx4A8Qz/DyvPece849F+rz\n63DtN/pQCjiKOVfvgOXjnNJru8JOgvVGIASV/JfFTuWEzwtFhpOIDebiwlxqnUi0ml6hr1rvpHGK\n/SFY9ZhxUsxcuAG3pjRLwbg6WEoMjycZXFFKJdT3F6Xkv0NGjePlr6Nkm+b9yk+XC/Cw+tpMd6gO\nMGRHmM2wVyklgkQZ5aFvDwRrbga7X4eCNBG2VVdzpPnxeBTLt20K1wIezmAbYcdl8lOyrdaovSO1\nob5AR5+zVvCQcpfajeY9FbK1PH5bkRT5Br5STo08aKlvAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{25}{48}$$"
      ],
      "text/plain": [
       "25\n",
       "──\n",
       "48"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S.subs({a:S0})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(C1-1,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEECKZu6vv\nVGbHkCAIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7UlEQVQoFWMQMmEgBB6rMYTB1XDMXOUA58AY\nnbOWMADVIFRVM7B+g0nCaPYNDPEJKKq2MzCcgcnCaO4CBp4DKKoOMTDsd4BJQ2keBQbuLyiq7Asw\nVTF/R1cF1K5fgGYWkMv0HcUsBga2r5iKGPID0FQxbcCiShsSEsVKIAAK3JlYFHFOgKiCSzECBTCA\nJFAEOVQZbjJwNKCr4p3A0IGiimUCAyOGKk8GBgkUVU2hkZsY6g1QTONQDZ2tgKLK/v//TwxM6iiq\nmP///4+qCiJ9A0UVhIPierBQADGq2AWIUcWHRRHQ9UIq2MRRxITUALzvNFwqWCFtAAAAAElFTkSu\nQmCC\n",
      "text/latex": [
       "$$\\left [ -2, \\quad 2\\right ]$$"
      ],
      "text/plain": [
       "[-2, 2]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(C2-1,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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8uundvPcKQHguu8PJjoZethzu4eWKJjoHLYQE+nHO7GSWF6dw5qwkQoPk0T9P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\nAAghhI/6PwdoXu7Q4Cl1AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1122dbf98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x10bcd9c18>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#from sympy.plotting import plot\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "plot(C1,C2,1,(x,0,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](plot1.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAvBAMAAACyMpGWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{a^{3}}{6} + \\frac{a^{2}}{2}$$"
      ],
      "text/plain": [
       " 3    2\n",
       "a    a \n",
       "── + ──\n",
       "6    2 "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Su = expand(integrate(C1-1,(x,1,a+1)))\n",
    "Su"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALYAAAAwBAMAAABON53yAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- \\frac{a^{3}}{6} - \\frac{a^{2}}{4} + \\frac{a}{4} + \\frac{7}{12}$$"
      ],
      "text/plain": [
       "   3    2         \n",
       "  a    a    a   7 \n",
       "- ── - ── + ─ + ──\n",
       "  6    4    4   12"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S-Su"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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bAzl4opDbX15KQZHGXQsVBY6I1Hv926Tw7C39WbvvBN97dQUlGgInJBQ4IiLA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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1157557b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x115755ef0>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(S-Su,(a,0,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAN8AAAA/BAMAAACSkwCdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\left [ - \\frac{1}{2} + \\frac{\\sqrt{3}}{2}, \\quad - \\frac{\\sqrt{3}}{2} - \\frac{1}{2}\\right ]$$"
      ],
      "text/plain": [
       "⎡  1   √3    √3   1⎤\n",
       "⎢- ─ + ──, - ── - ─⎥\n",
       "⎣  2   2     2    2⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(diff(S-Su,a),a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "\n",
    "a, x = symbols('a x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAvBAMAAABJZWRJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABO0lEQVQ4EWNggIDcvQJQFlaKzYF5AlYJqCCH\nA88XvPIBLF/xyTMwcH3EL8+hgF8+F1Oa6yFCjP0Cgg1lCV9GsjKMIQZDARNCnmfWqo045IWUXdUU\n+P7//4BdnjGBvYkDe9iBzWcTYPvI5YChFyQAlmdk4NyAVRYqz8DAr4Bf/vwBfPJcAvoMTAJYVYDt\nj1+wn8EdqzTEfcJKQtcLsMrz2v00xypBmSCXAn79g0z+PwIA3c02c+aMlzNnGgCZ/GAJTL8MMvcD\nHchyd/cBhDvh7oMLCzOwIWVuRgeoWriwIQPDO4R+OAsu/IKBwf4AXBjOgAvPF8Aqjyzcjz0Fw4RZ\nv8FNRWbAhTlAEYIJ4MJ3MeVAIjBhpgtY5eHCaxlYHLCogAnzXGBgwiIPF/ZIy7XAoh0uPP///x9Y\n5MHCAEGjX/wk9GLMAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{x^{2}}{2} + \\frac{1}{2}$$"
      ],
      "text/plain": [
       " 2    \n",
       "x    1\n",
       "── + ─\n",
       "2    2"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C1 = Rational(1/2)*x**2 + Rational(1/2)\n",
    "C1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAAVBAMAAADLKsFqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVQYGWNgwAKMNglgEWVgnsCTgE2cYwLr\nT6ziFzj/YhNnYGD/iF2c4wEDg5CJKYakEVAknUFiAliiCu44FgUGBpYFDMwJDAwMjIfOw8UvM1xl\n4D4As+U9TJw1vXwjA/8BBt7vQPUMDHBxrv//PzDIKzDw/oOLy72Z+/gBmDffgIH1M0yc0YCljyMB\nIq6AJM4swPyRHeI6FHMYGdg2gFUzgOxlR7KX/wFUnHsDAwskNMDuiQ+AigP9xZQAZgPF2QXeM/BB\nvdHOIBXA+A0oAxS/X7CfYQpUg/C7owwMPQwMOutzHGTeyak4QMWxUwBL5z5LNxvIbwAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$0.1 x^{2}$$"
      ],
      "text/plain": [
       "     2\n",
       "0.1⋅x "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C2 = 0.1*x**2\n",
    "C2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATIAAAAWBAMAAACxog1iAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$0.4 a^{2} + 0.4 a + 0.633333333333333$$"
      ],
      "text/plain": [
       "     2                            \n",
       "0.4⋅a  + 0.4⋅a + 0.633333333333333"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = expand(integrate(C1-C2,(x,a,a+1)))\n",
    "S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAOBAMAAABA5yhLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMpmJZlQi77ur\nRHZNUE1LAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuklEQVQYGWNggALGsDIHEJPpIUsEVAjEa2C0\nBPG4/v9PANEQUMXAkARiMeW8hIqAqD0MDP0CQJoHSYyB4Q8Dw/0GdFHGr0DRByDR8qcOQAoCOL8x\nMPgnANncApx/oWIMDJxAtWBRoMhmBgYhYxBQgZsAFI2fAFcMtK2/Acg7xMCQfwEuupuBoR7kskVA\ntSAaAoC+CAKxAhgYjKBCQIq5gXEJg7wCw1QG9pUIUca8Nw4MzAcYOMOOCjAAAKpqJ8okT/2ZAAAA\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$-0.5$$"
      ],
      "text/plain": [
       "-0.500000000000000"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S0 = solve(diff(S,a),a)[0]\n",
    "S0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAOBAMAAABN1ZpJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVI8TNwFkJ6YaWGjtG6W/ATTOIqceE7CXAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0.533333333333333$$"
      ],
      "text/plain": [
       "0.533333333333333"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S.subs({a:S0})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "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": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(C1-1,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAPBAMAAACRq9klAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nlORyewlQZmaxL+hSvl6QKuYr/7H9AUIMDcbuSSteAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$3.16227766016838$$"
      ],
      "text/plain": [
       "3.16227766016838"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s1 = solve(C2-1,x)\n",
    "s1[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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j3Uj8C38R2AZkjsC2ROQE/WNXK7cvX0+vN8Cdl8/iIwtLtWRjnIjoxB3GmFLgYuDXkdyO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\nACIicer/AxkYSasF5Jq3AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115e75438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x115e7c908>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#from sympy.plotting import plot\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "plot(C1,C2,1,(x,0,4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAvBAMAAACyMpGWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{a^{3}}{6} + \\frac{a^{2}}{2}$$"
      ],
      "text/plain": [
       " 3    2\n",
       "a    a \n",
       "── + ──\n",
       "6    2 "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Su = expand(integrate(C1-1,(x,1,a+1)))\n",
    "Su"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXMAAAAvBAMAAADjrzeZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZiXbvRGa7\nVKtFbb1tAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEP0lEQVRoBdWYS4wURRjH/73DbLPzYAhEHqee\njJx8EKJZHwRh4kGOOxf1YOKMHHZ9bMgEV3aNHCYeQDjAYryIEOZkwok5aYwHNh5MNCh91ESd1fjA\nmCC7BEVwHb6u7urv63K36QNZrTr8u+r7/7v6l6K6pxYg2dzD3ycL9oxewaQ9sEnSCVxIFmwaWbvq\ncN6xaZ0TrG92EkOrBvZuGDR9q1Y6hqWd3uzFI7s6ozi63i7igDY/8eJJ7D3yk33kONcYumohNiE7\n72F4jq4WtuEF5DrZuN3JF7IFVyk1Mgevnu1Zr+GHbMFVSnlt9DM+ahRjjYzRVYl5Pj4pZyPaiWZ3\nVZgyPmSoU9ztZsyi/7/6+DuXp54/lhX9sazB/za36Y0pE8BtGRXOPGU4auhMfFhXHXdmCvnpQ74p\neHb6CwjhjCiqjHCWe1Ky9iA2hq/A1/Eu+SCZAOIMdplWMB7yndHg6tyHfXgG5Vum5Fvo91iEzUXV\nE04wYXorzsKtUsT5+XeNXmgX6uFNX4UXnQHWng4roZY74fUj4EDQK7VwESd8XDNl7RIqO1iEzUXV\nE044dZqW5lFeUIETGv3Jb7+JuhE6Z557QM6l0b9E+GKfb5B7russmlI4Ba/KImwuqp5w5IOW71fm\nUVhSVox+djCIshE6Z9rLov8DnPfpls/C22jDINg1UmjD0JiFbVFUtnDC6VZUr4XCDeUG6Op4ydkI\nPc4UG4TOmWjVnT8IvUN3Le493KXLxva/xXmUaiwiw0XVEw7dkdqabaz5UyUCdON4GaHHmS0gdM5E\n6Hm6f4xwncUeztCH5BGayJD8q3QsYhE2F1VPOKnYgdlsCXTzeKnRdaZN6CKj0WnVFfpgPV4ibPcU\nzWsIXu9SkYVtUVR27Gx4OGjbgEHcrtIccRme3DCJ4+WBWu3xWm07pXWmXCd0nXFrtXvfrtXmabX1\nhrkGXOhS/l3iN2XkJNVYhM1F1RMO3ZHW6BUsx6+pebyMVl1nNoPQRSZaddBr2vfpIQ8Reu9p4JJv\nSL6B0hIL2Oai6gknDTr0SnMoxh9H83gZoevMd1eu/HVZZDT6RUIOFvrXYNVp11zyDaksEDoL2Oai\n6gnnzujFWQxXVYxeU/N4GaFzBqdlRqPTT9K+YIox2uuUwJ6GIfQvlbvOIjJcVD3hKKZ0eQJb6/Qj\nAhC6ebyM0BFn8LfMaPScT/9L4lWxru2cwcco3zCl2EO/wyJsLqqecNKplbv5rR+B3cD+s9t75vFS\no+sMDg1+ExmN7swcryM3DxzcX0dhfLJrCt4f/xxCOCOKKiOcBPv0kXpifMeBRl8hqNFXsO9ieUM3\n2tWZ57wnPek00v27534K+lRY2QrXrcQOoEuz1qJXts28bCm8dz9Ggp9rC5u3gOHgDGRhq+xA8aaF\n3IScq6IY/NVlYVs3a+2qr6G9buv38RdstfQLA3eczojWtNujf/ahm9j7yQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{a^{3}}{6} - 0.1 a^{2} + 0.4 a + 0.633333333333333$$"
      ],
      "text/plain": [
       "   3                                     \n",
       "  a         2                            \n",
       "- ── - 0.1⋅a  + 0.4⋅a + 0.633333333333333\n",
       "  6                                      "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S-Su"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x115e7c9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x115ea4208>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(S-Su,(a,0,4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAAUBAMAAABynCdQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3acAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ -1.11651513899117, \\quad 0.716515138991168\\right ]$$"
      ],
      "text/plain": [
       "[-1.11651513899117, 0.716515138991168]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(diff(S-Su,a),a)"
   ]
  }
 ],
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  "toc": {
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