SymPy-Exemplos.ipynb

{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# SymPy: Open Source Symbolic Mathematics\n",
      "\n",
      "This notebook uses the [SymPy](http://sympy.org) package to perform symbolic manipulations,\n",
      "and combined with numpy and matplotlib, also displays numerical visualizations of symbolically\n",
      "constructed expressions.\n",
      "\n",
      "We first load sympy printing extensions, as well as all of sympy:"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.display import display\n",
      "\n",
      "from sympy.interactive import printing\n",
      "printing.init_printing()\n",
      "\n",
      "# from __future__ import division\n",
      "# import sympy as sym\n",
      "from sympy import *\n",
      "x, y, z = symbols(\"x y z\")\n",
      "k, m, n = symbols(\"k m n\", integer=True)\n",
      "f, g, h = map(Function, 'fgh')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h2>Elementary operations</h2>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Rational(3,2)*pi + exp(I*x) / (x**2 + y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{3 \\pi}{2} + \\frac{e^{i x}}{x^{2} + y}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGUAAAAoCAYAAADnujR9AAAABHNCSVQICAgIfAhkiAAABN9JREFU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\nAElFTkSuQmCC\n",
       "prompt_number": 2,
       "text": [
        "        \u2148\u22c5x \n",
        "3\u22c5\u03c0    \u212f    \n",
        "\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\n",
        " 2     2    \n",
        "      x  + y"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "exp(I*x).subs(x,pi).evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-1.0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAASCAYAAADCKCelAAAABHNCSVQICAgIfAhkiAAAAVxJREFU\nSInt1b9LHEEYxvGP0UT00EQRIUiwUJEgsTCEkEZS+keYXv8AG/tgay0pDiSQIl2wsLEOpFBQm4CF\nIRhOm6Bco/ijmBH3zolkZY+cxKd5medhdr47M7svd0gtDXhmJ75iPOe8USxgF+fowxwqhdJl9Arf\n4mJ59Bg/MZ3x5rGFR8WgXek5VlAWdjUv7Hvsoy3j9eIEswXw/VFl+WG/40vC38Ta5eDB7ZkKUxdG\n8COR7eHl5aAZYAdjPUxkVXSjneaA7Y71OJFVY31Cc8Cexpq65w9jbaU5YA9uyEqxHlH7q3iBJX/f\nKDYwkxvtuirCrvYkshJ+S8Bu4k0Bi+dVFet4lsiGhU3Bv7kGI+io81bwWu2pDgkv8LmRMJ+EY+1M\nZJM4w2qd/1Q47ncZbxHbMu22TTHqxzIGMBa9XaG3f8DH6FWED2qnbv4vvBXa7oTQKHoxJf1Lu9f/\nqwtgsDsUgp/FvwAAAABJRU5ErkJggg==\n",
       "prompt_number": 3,
       "text": [
        "-1.00000000000000"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "e = x + 2*y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "srepr(e)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "\"Add(Symbol('x'), Mul(Integer(2), Symbol('y')))\""
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "exp(pi * sqrt(163)).evalf(50)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$262537412640768743.99999999999925007259719818568888$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfwAAAASCAYAAAC+a2xVAAAABHNCSVQICAgIfAhkiAAACXpJREFU\neJztnWusXUUVx3+nD27L7b28WrmFElpKQQSVS3gFpBGEEAIfSGN4JBUrJpZiFNA2AqFwpb1YMEGe\nqTxCDhS0AYESIBGNUGoitT54FCSIoGCNCEQU5S29fFizuXPmzuy9Zp+5OZ5m/slJ71l77bXWrD3r\nv+fMzN6FjIyMjIyMjG0eDef7ocD5wFRgFrAJuBj4m+fcfYAh4B3gPfPvEPCfGvaeA1YAjwBvAQeb\n874BPGvpHQ4sNr6mAtsDlwFPWjonAD80siK2rdbx3wDXe9oDsBSYDHwvcHx7YCPwmcDx1PnTxrev\nkb0EjADTgWXAPyyd2Lxo4psFXGB89iD5WQVsbsOvpr2afhDCXsAlwLvAh0Av8B3glay3Teil5hxN\nbWn1YmohdTs0+atTq1WcqL1uGi6J8au1p80fpOPETup9jIOAnwE7mu/TgA3Aq8BsR/dTwMvAEeb7\nAPBnpPF17I04n/eBrzs6g8ADwBRLthp4EzjQkn3bY8/+nIAfeyIXfChw/BCkw48EjqfOnza+HYAt\nwEJLdiHwNLCdJYvJiya+XYB1wExLNgcpoHk1/Wraq+0HPswBXgcWWLKFCMFNynpdr5eac7S1lboG\nU7dDm7/YWq3iRK1fLZdo/cbY0+QP0nJip/Ra8CCwtyMbRJKw1pJNMobOtWR7AK8B59SwBzIqvhG4\nF7gC2M8T3w/MuadaspOM7BpLttrEMxmYYMmPBK7z2C1wo7E15Mj3M21pIiPKUCdLnT9tfMMIEdgF\ntDPwAbDEkmnzoo1vGTIKdnEhcHkNvy5C7dX2Ax/WAf904piK/PpYlPW6Xi8152hrK3UNpm6HNn/a\n+LScqPWr5RKtX6090OUvNSd2Sq8F/0VGMJ9w5G8go7QCZyKjoB0ph9YewPoKWwBnAP8GjrNkpyEX\nfZUl891EpgE/RaY5fFhg2RoqiaFJuJOlzp82vj8C93vO2Qw8bH3X5kUb32qkSFx8C7iyhl8bZe3V\n9gMX2yEEvMlz7Dlk1iDrda8epOccbW2lrsGU7YjJX51abeLnxBi/Wi7R+I21tz5gw0ZqTuyUXgs2\nI+sScxz5K8jUaoGfA38IGalhD3RJ9+EK4H/AZyv0VgOHBY5NA242f7dzw0+dP018fUbmWwN/CLkx\nlsGXF218i43vHwE7GVkP8HvC62plfgvEXI8Cmn4wYOxt8BzbhPxCy3rdqwdpOUdbW+NRgynbEZM/\nbXw2mvg5McZvHS4J+Y21tz5gw0ZqTuyUXgt6kYtkYzdj6BHzvYFsVHgUmA+sBK5Cpm4Ga9gr8Biw\n3Nj6PnAPskGiDHOQDTFfq9A7EpmyCWEYmGv+bueGnzp/mvgOMDLfJsN7GN3A4YMvLzHx9TC6jvZ3\nZG3udsJr8mV+bcRcD9D3g4kIWfp+cWwxviZlva7Vg7Sco62t1DWYuh0x+dPGZ6OJnxNj/NbhkpDf\nWHtV+RsPTuyUXiVWITsri40K043RZ4GzLL3PIzsV94+0V+BPyJpIgYUIibudHmS9dgXwFLJGMcGj\nY+N3jB0pFzgQ2dlYoJ0bvg/t5q8qviOM7Lse32vMsV0DsfnyEhtfHzLdV2x4eaDEX5nfAjHXI7Yf\nANyCTInaT6cMIDuRRxidQs163annQ13O0dZW6hoMoR3urJs/TXxNwpwY4zeWS8r8xtiryt94cWKn\n9ILYG1lLWmnJdjXG3kU2X9jYQuu6jMZeAZesJwJvU74JaxIy1bIRuSg+fIHwVMwE4FZad9GmvOG3\nmz9NfIeVxLzWHNvNcyyUl9jrez5wE3Ai8CKjI8xPe2yX+YV61wN0/aDADGS9dbF17kpk6msE2e2a\n9bpXz0U7nKOtrdQ16EO73Fknf9r4moQ5McZvLJeU+Y2xV5W/8eLETul50YNMxbgL/pONoac952xE\n1p1801che2V4yXzKMN/E85PA8buRUaYPS4CjHVmqG36K/Gni28sjK3C/OdbnORbKS0x85yE7Zgv0\nIjvotxJ+Hj719ShQ1Q9s7IQ8F3wlQj57Ar9Fpu0aWa/r9Qq0yzna2kpdgy5ScWds/rTxNSnnRI3f\nOlxS5reOPRt2/saDEzul50UDWfy/NHD8VeCXHvmjSGJmOvIqexsC9rYgo6oCn2TsBoR+43MrstnL\nxmRkVDzssT0AXOuRp7jhp8ifNr5epO1Xe3QfRnb1uijLiza+hrF9gEfvLKPnPt6S6nrE9gMN/gr8\nIuttM3opOEdbW+NRgwVScWcIofxp44P4ZU7Xbx0uKfMbY0+bv5Sc2Ck9wL/uuQJZr7jYkp1h/R2a\nOu1BRjuvRdobxE/S0xkdZfUDjyNTQXMtnQ/Nvw1kKsbGoUgxuo+xAByL3DjWWZ9ihHSa+b7Ac54G\nKfKnje8tJC97uMaQacAnPPKyvGjjm4E8ovK8R+8G5CU4Uxx5iutRpx9UYQbylqq7st42o5eCc7S1\nNR41WCBFO0Ioy582vjpw/dbhkir7Wnva/KXkxE7pefEV/KNJe6fm6cgah72e0QD+BdxZw95djF1D\nKl4ysdx8n4I80/k88kKLAgcbvV97fHzZHFviOebDbNr/hZ86f5r4LkVet2lPy801ur43RlXlRRNf\nA3lE6HOe8/uQEbK7tJPiesT2g3m0tuNcZLQ+y5ItNfFOyXpdrwfpOAf0tZW6BiFtO2Lyp42vQJMw\nJ2r81uGSMr8x9rT5S8mJndIbg2OQEd3tzmct8GNLbwLyKMNSS3YKMsqZXcPeIcjaa/FihwZwG/Ar\nJ9DLkLcJ2UW1BtkpeZCnPcuQC/dVX2M9mGf0y6axik04vpdQpM6fNr6ZSMf7kiW7CniG1g1wBary\noo3vVOPDlvUjRfTFGn5dhNqr7QfzkanWhyzZRcBfGN2BO4jsyHV3Pme97tRLzTna2kpdg6nboc2f\nNj4bZZyo9RvLJVV+tfa0+UvNiZ3SayHNNwi/SWglrSOenZFNGL3IL64GcnFfqGnvKOCbyEaOfuTF\nE8OMXYdaBByPvGRlAHlt43JkJ6iLk5FR4LHIJpEQ+oH7kP/8YiYyRfM48krCdcijI2uA3Rl9BON1\nZBPHzcAdNdqryZ82PpDH2YaRPPQZ++cga2UuNHnRxjcfeZvTe0jxTUReQLK+pl/QtXcR1f1gX2SN\n7m7gbCObivxqmo7kaQdkuvQxJ4as151648E52tpKWYOp26HNnzY+LSfG+NVwidav1h7o+0FqTuyU\nXkZGRkZGRkZGRkZGRkZGRkZGRkZGRkbG/z8+AhxLZRB3IKHvAAAAAElFTkSuQmCC\n",
       "prompt_number": 6,
       "text": [
        "262537412640768743.99999999999925007259719818568888"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h2>Algebra<h2>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eq = ((x+y)**2 * (x+1))\n",
      "eq"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + 1\\right) \\left(x + y\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAAbCAYAAACtOKuoAAAABHNCSVQICAgIfAhkiAAAA+lJREFU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\n",
       "prompt_number": 7,
       "text": [
        "               2\n",
        "(x + 1)\u22c5(x + y) "
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand(eq)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 2 x^{2} y + x^{2} + x y^{2} + 2 x y + y^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQ0AAAAbCAYAAABm6to6AAAABHNCSVQICAgIfAhkiAAABP5JREFU\neJztnFuIV0Ucxz+7SFtpGQW1uJTrpQw0w0u26XYh8kWCksyHAiOMQgwKrKduDxotXR8qIbpN96AL\nEaUSkdpLGBVUEBiVFD3YBaIIKSPr4Xf+dDz+d3d+czlzTv/5gOyZmXN+8/v+ZpwzM2d2IZPJZBT0\nOT53ATATGAAuAgzwbhiXknAusByYDqwANgPvJ/XIj6wn40OUeP8MXFNcXwkcAI7zNZqIacBYKb0W\n0TOUxh1vsp6MD9HivQCYWlyvAf6ivYPGQuAQMKdIHw/8gwSrjWQ9GR9qifeLwG0hDdZMHzId6yzV\n5iNBWpTMIz+ynowPUeO9GLgFeBw4NoRBBcuA14HtwOfAk4Sbrj4HPBDIVhPIevyJ2d+aTpR4Xw98\njKyF6mAx8A5wQpGehmzS/AgMe9peD9yL+wZx08h6/InZ35pOsHiPAD8As4r0mcj0ZY2HzYXAFMt7\n3wbmVvIWFT687OHDpUiQAI6mng6h0a0lpJ6YftqSSk+s/haSGO0T9P/DUuA95HMrwCrgIEcGVoNR\nOPU78B1wciX/F+SrjgsXIgEaLP5dDpznaEuDIc7gFFqPIe1bNaWeGP0tNIaw7TNhvF1Gp4+Ap4Ab\nkV3WUWRU+srXU0v2AWfw39ebDn/i9gVnNvAWRy6vpjvYagJZT1hC97emM2m8q4PGEmAd8Dcycl0H\n3ICs54aAu4CvgedLzzwU0mMLRpDG2l/KmwGcAuyq3GurR9P4w8AmJLgvIF+POmwEVgOXKOxpiKEn\nBrZ+DmMXy5R6YvS3YdL1IZu6reM9G3gU6C/SBvgSORm2AplVbPJ0eDwMftOrMaShlpfyYunZChwF\n3AR8Vinbg26da7DX3Zb20fgZMpYaDGn7W2jdBns9QeveyuEjzCuFEYBTgfuBkzQGFRjcG3Eusu7c\nUsmPoWcUuKq43g68Wiqbihxy26CwZ9A1dhvax9bP0LHUYEjX32LoNtjpCV73rEr6e+BujQEPDG6N\nOAB8CDzYpSyGnsGiziHkTbO6VLYS2VGfr7BnsNfdlvax9TN0LDUY0vW3GLoNdnqC1F3e09hXup5X\nGN5p4YiGZ4Czu+SfhhygOdilbD1yDqRKH/A0sAO4s0t5DD2dde1a5G2zrVR2PrKb/kWX50Lobkv7\n2PrpGksNTexvPrp99USN+QZkd7h80nPOOPeGwKAf+bcAd1Ty1o1zb2g9O4A3Knm7kFODGgxub7w2\ntA/Y+RkqlhoM6ftbSN0GnR6vujubNscgJ7/OKtIrkU2SA6X7blU4FZtrkY2lzZX80eJnbD0zgb2l\n9AByXj/Wr2u3pX1c/Kw7li7E6G8pdXvV3VmerEJEfYJshpwO/Fq673ZkatQELgbuQ0bL8qffKcia\nDOLr+RY4sZQeQ07N7fawORFtaR8XP+uOpZZY/S2lbq+6O4PGbmSKswQ5az+C7AY/hqyT3gQ+COKu\nP68h372v7lLW2dGOredm4AngEeRtshTpJJ962JyItrSPi591x1JLrP6WUnfTY26Fob2//NOPbDA9\n6/CsoR26DfX46RNLDYZmxd1Xt8Fdj7ru/slvqYXfgD9SO2HJSxx+KOYyZKp3j4OttuiO5WfIWGpI\nHffQujV6UsW8p/kJeLi4ngF8Q/epa2ZyejWWKXV71/1/+TsLdXIFcA6yHzSINMCeCZ/IjEevxjKl\n7l6NeSaTyWQymUwmk8lkMpke518QvOZokU+z1gAAAABJRU5ErkJggg==\n",
       "prompt_number": 8,
       "text": [
        " 3      2      2      2            2\n",
        "x  + 2\u22c5x \u22c5y + x  + x\u22c5y  + 2\u22c5x\u22c5y + y "
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = 1/x + (x*sin(x) - 1)/x\n",
      "a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{x} \\left(x \\sin{\\left (x \\right )} - 1\\right) + \\frac{1}{x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKUAAAAZCAYAAABQI+7UAAAABHNCSVQICAgIfAhkiAAABIxJREFU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\n1xx4dj8YGGx6OnTo0KFDhw694z92gzEEahzUhQAAAABJRU5ErkJggg==\n",
       "prompt_number": 9,
       "text": [
        "x\u22c5sin(x) - 1   1\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\n",
        "     x         x"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "simplify(a)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sin{\\left (x \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACwAAAASCAYAAAAg9DzcAAAABHNCSVQICAgIfAhkiAAAAjFJREFU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\nRU5ErkJggg==\n",
       "prompt_number": 10,
       "text": [
        "sin(x)"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eq = Eq(x**3 + 2*x**2 + 4*x + 8, 0)\n",
      "eq"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 2 x^{2} + 4 x + 8 = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALIAAAAZCAYAAACVUXRFAAAABHNCSVQICAgIfAhkiAAABSNJREFU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\naRJDrJmEryoAAAAASUVORK5CYII=\n",
       "prompt_number": 11,
       "text": [
        " 3      2              \n",
        "x  + 2\u22c5x  + 4\u22c5x + 8 = 0"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve(eq, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}-2, & - 2 i, & 2 i\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "[-2, -2\u22c5\u2148, 2\u22c5\u2148]"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a, b = symbols('a b')\n",
      "Sum(6*n**2 + 2**n, (n, a, b))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sum_{n=a}^{b} \\left(2^{n} + 6 n^{2}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHgAAAA4CAYAAAA2PDy+AAAABHNCSVQICAgIfAhkiAAABvBJREFU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\nkTgo+R+8NYcrc8wvbwAAAABJRU5ErkJggg==\n",
       "prompt_number": 13,
       "text": [
        "  b              \n",
        " ___             \n",
        " \u2572               \n",
        "  \u2572   \u239b n      2\u239e\n",
        "  \u2571   \u239d2  + 6\u22c5n \u23a0\n",
        " \u2571               \n",
        " \u203e\u203e\u203e             \n",
        "n = a            "
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h2>Calculus</h2>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit((sin(x)-x)/x**3, x, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{1}{6}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAeCAYAAADZ7LXbAAAABHNCSVQICAgIfAhkiAAAAPZJREFU\nSInt1T8vBEEYx/GPP1GhcgnB3UanUJ1GQq3xCkhEobh4E5QqL8Ab0HgBKnpRXKNVSzQkEolmFTMr\nS0RCnk1c3LeZZ2cnv+/M7O4sA8YUztFuSrCPI5QompJUfCkZbdo6lPxNyVhQzg4OsIo5zOI6KPu/\nMVKrV3D6qe87+uj9VPJbyoCMASFiuyp2pX/JPaZxEpgN9nCc6wKvmIkUTOABnVrfUn3AeIBkTZp1\nB+vo4gJ3AdnvbEuv8Ua+nsQjFqoBEafwU25vcvuMF2xFSvrSSuoneinmUXzgEpu5bkmrm69uRn0n\nizjELZZxhqug7CEN8AajfR9caTPg4AAAAABJRU5ErkJggg==\n",
       "prompt_number": 14,
       "text": [
        "-1/6"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(1/cos(x)).series(x, 0, 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 + \\frac{x^{2}}{2} + \\frac{5 x^{4}}{24} + \\mathcal{O}\\left(x^{6}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOMAAAAcCAYAAACTSgiAAAAABHNCSVQICAgIfAhkiAAABxBJREFU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\n",
       "prompt_number": 15,
       "text": [
        "     2      4        \n",
        "    x    5\u22c5x     \u239b 6\u239e\n",
        "1 + \u2500\u2500 + \u2500\u2500\u2500\u2500 + O\u239dx \u23a0\n",
        "    2     24         "
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(cos(x**2)**2 / (1+x), x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{4 x \\cos{\\left (x^{2} \\right )}}{x + 1} \\sin{\\left (x^{2} \\right )} - \\frac{\\cos^{2}{\\left (x^{2} \\right )}}{\\left(x + 1\\right)^{2}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAM0AAAAoCAYAAABdNX5YAAAABHNCSVQICAgIfAhkiAAACHNJREFU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\n6AAAAABJRU5ErkJggg==\n",
       "prompt_number": 16,
       "text": [
        "         \u239b 2\u239e    \u239b 2\u239e      2\u239b 2\u239e\n",
        "  4\u22c5x\u22c5sin\u239dx \u23a0\u22c5cos\u239dx \u23a0   cos \u239dx \u23a0\n",
        "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "         x + 1                 2\n",
        "                        (x + 1) "
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(x**2 * cos(x), (x, 0, pi/2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-2 + \\frac{\\pi^{2}}{4}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAcCAYAAADhqahzAAAABHNCSVQICAgIfAhkiAAAAytJREFU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\nyJAhQ4acAvwHiTObAuvxk64AAAAASUVORK5CYII=\n",
       "prompt_number": 17,
       "text": [
        "      2\n",
        "     \u03c0 \n",
        "-2 + \u2500\u2500\n",
        "     4 "
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit(exp(-1/x**2),x,0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAASCAYAAABvqT8MAAAABHNCSVQICAgIfAhkiAAAANtJREFU\nKJHN0r0uhEEUxvGf9VWItTYahSgQPYV2SxfBBbiAbfSidSE60e4trASNjghZGiHbEFnFzpscb84W\nOqd5cp6Z/5nJzMMfa6rWb+MU9xhhBV0MMngJjzgI3jFuMJcBJ3jBTPDa+MJRBtzhIvGv0auaRtFF\nbOEhAZ6wWwfWi74nwBBNzEegWfRzAgCtCHwXHSXAbNHpCLwmG6taKPoRgUGZvjwBeKsDQ/SxlgCb\nuKqaRli4xJ7fcdkoQ86zu66Wow+Dd4ZbIRoxBs/oGEdkx/gz29iXP/d/qR8UhCWn2ReXCwAAAABJ\nRU5ErkJggg==\n",
       "prompt_number": 18,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(exp(-1/x**2),x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2}{x^{3}} e^{- \\frac{1}{x^{2}}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADMAAAAqCAYAAAAeeGN5AAAABHNCSVQICAgIfAhkiAAAAqVJREFU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\nAElFTkSuQmCC\n",
       "prompt_number": 19,
       "text": [
        "   -1 \n",
        "   \u2500\u2500\u2500\n",
        "     2\n",
        "    x \n",
        "2\u22c5\u212f   \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "   3  \n",
        "  x   "
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eqn = Eq(Derivative(f(x),x,x) + 9*f(x), 1)\n",
      "display(eqn)\n",
      "dsolve(eqn, f(x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$9 f{\\left (x \\right )} + \\frac{d^{2}}{d x^{2}}  f{\\left (x \\right )} = 1$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI8AAAAnCAYAAAA2LBsQAAAABHNCSVQICAgIfAhkiAAABV1JREFU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\nH8u6S9fBSsfeAAAAAElFTkSuQmCC\n",
       "text": [
        "           2          \n",
        "          d           \n",
        "9\u22c5f(x) + \u2500\u2500\u2500(f(x)) = 1\n",
        "           2          \n",
        "         dx           "
       ]
      },
      {
       "latex": [
        "$$f{\\left (x \\right )} = C_{1} \\sin{\\left (3 x \\right )} + C_{2} \\cos{\\left (3 x \\right )} + \\frac{1}{9}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQAAAAAeCAYAAADD2cA5AAAABHNCSVQICAgIfAhkiAAACDJJREFU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\nRK5CYII=\n",
       "prompt_number": 20,
       "text": [
        "f(x) = C\u2081\u22c5sin(3\u22c5x) + C\u2082\u22c5cos(3\u22c5x) + 1/9"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Illustrating Taylor series\n",
      "\n",
      "We will define a function to compute the Taylor series expansions of a symbolically defined expression at\n",
      "various orders and visualize all the approximations together with the original function"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "# You can change the default figure size to be a bit larger if you want,\n",
      "# uncomment the next line for that:\n",
      "#plt.rc('figure', figsize=(10, 6))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "def plot_taylor_approximations(func, x0=None, orders=(2, 4), xrange=(0,1), yrange=None, npts=200):\n",
      "    \"\"\"Plot the Taylor series approximations to a function at various orders.\n",
      "\n",
      "    Parameters\n",
      "    ----------\n",
      "    func : a sympy function\n",
      "    x0 : float\n",
      "      Origin of the Taylor series expansion.  If not given, x0=xrange[0].\n",
      "    orders : list\n",
      "      List of integers with the orders of Taylor series to show.  Default is (2, 4).\n",
      "    xrange : 2-tuple or array.\n",
      "      Either an (xmin, xmax) tuple indicating the x range for the plot (default is (0, 1)),\n",
      "      or the actual array of values to use.\n",
      "    yrange : 2-tuple\n",
      "      (ymin, ymax) tuple indicating the y range for the plot.  If not given,\n",
      "      the full range of values will be automatically used. \n",
      "    npts : int\n",
      "      Number of points to sample the x range with.  Default is 200.\n",
      "    \"\"\"\n",
      "    if not callable(func):\n",
      "        raise ValueError('func must be callable')\n",
      "    if isinstance(xrange, (list, tuple)):\n",
      "        x = np.linspace(float(xrange[0]), float(xrange[1]), npts)\n",
      "    else:\n",
      "        x = xrange\n",
      "    if x0 is None: x0 = x[0]\n",
      "    xs = Symbol('x')\n",
      "    # Make a numpy-callable form of the original function for plotting\n",
      "    fx = func(xs)\n",
      "    f = lambdify(xs, fx, modules=['numpy'])\n",
      "    # We could use latex(fx) instead of str(), but matploblib gets confused\n",
      "    # with some of the (valid) latex constructs sympy emits.  So we play it safe.\n",
      "    plt.plot(x, f(x), label=str(fx), lw=2)\n",
      "    # Build the Taylor approximations, plotting as we go\n",
      "    apps = {}\n",
      "    for order in orders:\n",
      "        app = fx.series(xs, x0, n=order).removeO() #aparentemente tem um erro aqui.\n",
      "        apps[order] = app\n",
      "        # Must be careful here: if the approximation is a constant, we can't\n",
      "        # blindly use lambdify as it won't do the right thing.  In that case, \n",
      "        # evaluate the number as a float and fill the y array with that value.\n",
      "        if isinstance(app, numbers.Number):\n",
      "            y = np.zeros_like(x)\n",
      "            y.fill(app.evalf())\n",
      "        else:\n",
      "            fa = lambdify(xs, app, modules=['numpy'])\n",
      "            y = fa(x)\n",
      "        tex = latex(app).replace('$', '')\n",
      "        plt.plot(x, y, label=r'$n=%s:\\, %s$' % (order, tex) )\n",
      "        \n",
      "    # Plot refinements\n",
      "    if yrange is not None:\n",
      "        plt.ylim(*yrange)\n",
      "    plt.grid()\n",
      "    plt.legend(loc='best').get_frame().set_alpha(0.8)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "With this function defined, we can now use it for any sympy function or expression"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_taylor_approximations(sin, 0, [2, 4, 6], (0, 2*pi), (-2,2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXoAAAEACAYAAAC9Gb03AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nDo4y9AqFQuHg/D9EdorxYwC9PwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb062fe6c>"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_taylor_approximations(cos, 0, [2, 4, 6], (0, 2*pi), (-2,2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXoAAAEACAYAAAC9Gb03AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb0bd22ec>"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This shows easily how a Taylor series is useless beyond its convergence radius, illustrated by \n",
      "a simple function that has singularities on the real axis:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# For an expression made from elementary functions, we must first make it into\n",
      "# a callable function, the simplest way is to use the Python lambda construct.\n",
      "plot_taylor_approximations(lambda x: 1/cos(x), 0, [2,4,6], (0, 2*pi), (-5,5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXAAAAD7CAYAAABzGc+QAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nQqFQSIoycIVCoZCU/wdDeJclXk9EegAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb0bc326c>"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = lambda x : (x**2)*sin(2*x)+1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f(x)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{2} \\sin{\\left (2 x \\right )} + 1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAAZCAYAAADQfBuCAAAABHNCSVQICAgIfAhkiAAABBtJREFU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\nowevUtEC9uCvkVaioqKioqLicOB/5oXqgmek4sEAAAAASUVORK5CYII=\n",
       "prompt_number": 28,
       "text": [
        " 2             \n",
        "x \u22c5sin(2\u22c5x) + 1"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_taylor_approximations(f, 0, [2,4,6], (0,2), (-3,3)) "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAEACAYAAABMEua6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzt3XlYVdX6wPHvYZZZwBFn0BRFUVGcwxxyTLtl6U0TG+3+\nbpZa2XAzLctsupV1u7fBKW+ZVqZlmpZgDpmVYileZwRFQSaZ5/X7YwtKAh4OB/YZ3s/znEf2GfZ+\nWW5f13n32msZlFIKIYQQNsFB7wCEEEKYjyR1IYSwIZLUhRDChkhSF0IIGyJJXQghbIgkdSGEsCF1\nSuoFBQVEREQQFhZGSEgITz31lLniEkIIYQJDXcep5+Xl4e7uTklJCYMGDeK1115j0KBB5opPCCFE\nLdS5/OLu7g5AUVERpaWl+Pn51TkoIYQQpqlzUi8rKyMsLIxmzZoxdOhQQkJCzBGXEEIIE9Q5qTs4\nOBAbG8vZs2f58ccfiYmJMUNYQgghTOFkrh35+PgwduxYfv31VyIjIyueDwwMJCkpyVyHEUIIuxAU\nFMSJEydq/bk6JfXU1FScnJzw9fUlPz+fbdu28dxzz1V6T1JSEjJnmPksWLCABQsW6B2G9SothZtv\nhogIFjg7S1uakZyb5mUwGEz6XJ2S+vnz55k+fTplZWWUlZUxbdo0hg0bVpddiuuIj4/XOwTrtnAh\nlJXBwoXE33ef3tHYFDk3LUOdknpoaCj79+83VyxC1K/Nm2HZMvj1V3AyW+VRCIsiZ7aViYqK0jsE\n65SQADNmwNq10Lw5IG1pbtKelqHONx9d9wAGg9TUhb7y8mDQILjrLpg7V+9ohDCKqblT5n6xMjJk\ntJaUgnvuga5dYc6cSi9JW5qXtKdlkPKLsG1LlsDJk/Djj2DiaAIhrImUX4Tt2rQJHngA9u2DwEC9\noxGiVkzNndJTF7bpyBHtwuiGDZLQhV2RmrqVkbqlETIzYcIEePll6N+/2rdJW5qXtKdlkKQubEtR\nEdx2G4werV0gFcLOSE1d2I7ykS5pabB+PTg66h2RECaTmroQixbBH3/Ajh2S0IXdkvKLlZG6ZTU+\n/hg++gi++QY8PIz6iLSleUl7WgbpqQvrFx2t3SkaHV0xBYAQ9kpq6sK6xcXB0KHw6adw0016RyOE\n2cg0AcL+nDkDo0bBa69JQhfiMknqVkbqlpelpMCIEVrZZdo0k3YhbWle0p6WQZK6sD6XLmk99MmT\n4ZFH9I5GCIsiNXVhXfLztYQeGgpLl8okXcJmmZo7JakL61FcDH/5C3h7a0MYHeSLprBdcqHUTtht\n3bKkBO6+W1tfdMUKsyR0u23LeiLtaRlknLqwfKWlMH26dvv/hg3g7Kx3REJYLCm/CMtWWgpRUXDh\nAmzcCI0a6R2REA1C5n4Rtqe0VJsTPSkJvv5aEroQRpCaupWxm7plaSncey+cPasldHd3sx/Cbtqy\ngUh7WgbpqQvLU1KiJfQzZ7Ql6eohoQthq6SmLixLYaF2U1FeHnz5pdEzLgpha2RIo7B+ubkwbpw2\nF/rGjZLQhTCBJHUrY7N1y4wMbS6X1q1hzRpwda33Q9psW+pE2tMySFIX+ktO1qbP7dsXPvwQnORS\njxCmqlNNPTExkbvvvpuUlBQMBgMPPPAAs2bNqnwAqamLmhw7BmPGaDMtzp8vc7kIcZkuc79cuHCB\nCxcuEBYWRk5ODr179+arr76iS5cudQ5M2IHdu+G22+DFF7XRLkKICrpcKG3evDlhYWEAeHp60qVL\nF5KSkuqyS3EdNlO3XLcObr0VVq7ULaHbTFtaCGlPy2C24mV8fDwHDhwgIiLCXLsUtkgpeOMNePNN\n2LoVLncKhBDmYZZx6jk5OURGRvKPf/yDiRMnVj6AlF9EueJibVGLnTvh22+1kS5WJjdXm4am/JGc\nDOnp2vPlj5wc7Vd1ctJGZ5Y/GjUCPz/t4e+v/dmsGXToAE2ayOUEUZluc78UFxdz2223MXXq1GsS\nermoqCjatWsHgK+vL2FhYURGRgJXvrLJto1vh4TApEnEFBXB4sVEXk7oFhPfVdv5+eDnF8nRo/D9\n9zGcOwfZ2ZGcOAEZGdr7IfLyn+bZ9vSMpEMH8PaOoVUruOWWSHr2hHPnYnB0tKz2ke362Y6JiWHF\nihUAFfnSFHXqqSulmD59Ov7+/vzzn/+s+gDSUzermJiYihPCauzfry1uMXUqLFyodVstQExMDN26\nRfLzz1qIv/8OBw/CiRNalagqrq7QvHnlh58feHpq90qVP1xctOlrSkq0P0tLtZtkMzK0GYTT07VH\nUhKcPAmZmVUfz8MDIiJg8GDt0b+/5c6aYJXnpgXTpae+e/duVq9eTffu3enZsycAixcvZtSoUXXZ\nrbAln34Ks2bBv/4FkybpGkppKcTFwU8/wZ498MMP2nxhf+bkBF26QEgIdOwIwcFXHk2b1k+ZJD1d\nS+6nTmkxHjigPc6ehe3btQdo/6lERsLo0TB2rBaTEFeTuV9E/Sguhqefhi++gPXroUePBg9BKTh0\nCL7/Xnvs2gVZWZXf4+YGffpojx49tEeXLlpP2xIkJ2tx79wJP/6oJfqr9egBd9yhPSTB2xZZo1RY\njoQEbVIuX19YtQoCAhr00N9/r/XCv/8eUlIqv962rVbC6N8fBgzQkqI1LaSUnAzffQebN2vXmq/+\nT6pfP2106J13gpeXfjEK85Ckbicsvm759ddw330wdy489li9Lw5dUgJ798I332iHjour/HrLltqU\nMsOGaTMRtGp15TWLb8vrKCyEbdtg7Vr46ivIztae9/CAKVO0gUbdujVcPNbenpZGVj4S+ioqgqee\ngs8/18otAwbU26EyMrTe6jffaD3W9PQrr3l5wU03wfDh2uOGG2x3qKCrqzap5bhx2kXYzz/Xps7Z\nuVP788MPtTaYPVurwdtqO4jKpKcu6u7oUW3ulqZNtTtE/f3NfohLl7Te6Gefab3TkpIrrwUHw/jx\nWnIbPNi6yin14ehRWLoUli/Xkj1o93jNnw8TJtT7lydhJlJ+EQ2vrEwb1bJggTZU8aGHzJoxcnK0\nkspnn2k98qIi7XlHRxg0SEvk48dDp05mO6RNyciADz7Qbt49f157rnt37a9qwgTpuVs6k3OnqmcN\ncAi7Eh0drXcImsREpYYPVyoiQqmjR82227w8pT7/XKlJk5Rq1EgpbQyLUgaDUpGRSr33nlIpKeY5\nlsW0ZT3Lz1dq6VKlAgOvtOegQUr99JN5j2Mv7dlQTM2d8kVM1I5SsHo19OoFN96ojberY1e5sFDr\nkU+dqlVwbr9dm+8rP18rzb/1ljZeOzoaZs7UbqkXxnNzg7//XRsHv3Sp1n67dmkjgO64QxsxJGyH\nlF+E8U6d0kosFy7AsmXQu7fJuyou1m6oWbNGu6566dKV18LDtRGRkyZBmzZXns8tLSW+oIDT+fmc\nLiggsbCQ9OJiMkpKKh6XSkooUYoypSgDXAwGPBwdcXd0xNfJiZYuLrR0dSXQxYXO7u6EeHjQwsUF\ngx3VIrKy4JVXtHnV8vO1O1SffRbmzLGc8flCauqiPpWUwD//CUuWwOOPa//6TbgaWVqq3UCzZo12\nT1Ja2pXXevTQxlffcQc0a1vCodxc/rj8+D0nhyN5eWSVltLW1ZX2jRrR3s2N1q6uBDg709jJicbO\nzvg5OeHt5ISzwYAD4GAwUFhWRm5pKbllZWQUF3O+qIikoiISCwr4X14ecXl5FCtFb09PBvn4MMjH\nh37e3njawepLZ89qI0/XrtW2b7gB3n8fhgzRNy6hkaRuJxp8LPAvv8ADD2jf2d97D4KCavXxsjLt\ntvw1a7QhdxcuXHmtc2e4Y7Ki919ySfHP4qcs7XGmoIDO7u509/Qk1MODUA8Punp40NzFBQcz9qjL\n2/JiURH7srPZdekSuy5d4mBODoN9fJgYEMB4f3+aN8B6qXratk0rzxw7pm3//e+weLE2n01tyDh1\n85Jx6sK8LlyAZ57Rblt89VW46y6jh0sopf1f8NlnWi/w6vlV2t9QyoD7svAclMlJtyzezM6iaZ4L\n/Z286eftzd8DAwn18MCpAcfdNXFxYay/P2MvD8XMKilhc3o6X6Wm8sSpU/T39ub+Fi0Y5++Psw2O\nBxwxQpvM7KWXtMc778CmTVqFTXK09ZGeuqissBDeflsrtcyYAf/4B/j4XPdjSmkzHH72mfY4ffry\nCw5lNBuSQ6c7MyjsmkkcWYS4u3NT48YM9PYmwtubJhZcyM0vLeXzixf54Px5jufnM7NlS2YFBtLY\nRgfDx8Zqf+2xsdr/4U88Ac8/L7V2PUj5RdSNUtoQlLlzteLqG28YNaolLk5L4mvWXPn6TtMCvEek\nEzA2jZQWmbT3cOMmX19uatyYG3198bHSevWR3FxeS0zkq9RUHmjZktmtWtHUBrNdcbG2bOwLL2jl\ns9694ZNP5H6AhiZJ3U7US93yxx+1GRUzMuC117R7ymtw4sSVRH7oEOBUBt0u0SgyHbfINEp9ihjX\nxI8xAf4Mb9yYZhaa+ExtyzMFBbySkMBnKSk83qYNj7ZqhasNlmV279aqbmfOaPPJfPCBNqdMdaSm\nbl5SUxe1t3+/lsyPHtVuM7zrrmoXsDh9Whs7/tln2scIKIS+aTjflY6hVwbtndyZ1NaPcU06E+7l\nhaMNDxFs6+bGu506MbtVK+aePMmH58/zRlAQ4xtwNsqGMHCgVlJ76CFtWvy//lW76P3aa1KOsWTS\nU7dHBw/CokVaV+yZZ+D++6v8V1qeyNeuhd8OKLghC/qn4TAwDaeWhfRz8iOqsx9jm/rZZBnCWN+l\npzPr+HF6enrybqdO+NtYvV0p+Pe/tVkfi4u1m5bWrYPAQL0js21SfhHXt2ePNrxh/35t6r6//U37\nXn2VU6e0f7Dr1sFvR0ogPB36p2Hol453mQvDGvnzt95+3Bjg3aAjVCxdfmkpT58+zbqUFD644QZG\n18OkZnrbt0+72zcxEVq00CZY69tX76hslyR1O1HruqVS2kDkl17SiqPz5kFUlHbv+GUnT2pjyNet\ng9/O50H/NO3RJZsWqT6MC/BnzgA/Ovs2Mvvvo6f6qAFHZ2QQ9b//cUtAAK8HBeFiY//xpaZqd/rG\nxGin0LJlV+rsUlM3L6mpi8pyc7U5Wt5+W5s5cd487d57JyfKyuC3X2DDBli/sYw4hyzolwazUjF4\nltIhxZ9JrVoxZ4gvTTzlFKmNoY0bczA8nKlHjjD84EHWde1qsReKTREQAFu3wsMPw3/+o9XZjxzR\nLskIyyA9dVsTHw/vvqtNpj1okFYIjYykqNhAdPTlRL69mAut07VE3icdhxQ3Qi75E9UpgJnDPPFw\nt92LnA2lTCmei49n5YULfNG1K328vfUOyayU0m5SevRRbdjjPfdoSd5KR6taJCm/2LPCQi1bL1um\n3co5Ywb83/+R7N5eWyFos2LTiWzyQtKhTzq0y8XtqC/9lT8P9PDnLze6ymiGevLlxYs8eOwYy2+4\ngXE2NjoGtDtPJ03SJgYbN04bHeXurndUtkGSup2oVLf8/Xctkf/3vxAaSmnUvexr9Rc2bW/Exj2F\n/OF2OYn3yoB0FwJO+zHSz4//G+RD/96Odr9IQkPVgPdlZXHLH3/wWlAQU5s3r/fjNbS9e2HsWEhP\nj6F//0i+/rpeFr+yO1JTtxfnzmm3+61ZA5mZZP0liu8e+5k1sW35btklckPOaYm8TxEOsY0JyfXj\n9nNBTBvlRocZegdvn/p6e7M9LIxRv/9OekkJs65e/doG9OunjY698UZtHPvgwbBlS+Vpk0XDkZ66\nNTh5Upt0fM0aSs8kcrLnJNa638m/i7pyrkkWhGVC5yw46Yn/KT+Geftxdz8vbrrRQCPbGrBi1c4U\nFDDi4EHua9GCJ2ww4507B6NGaXcZt2qlLWoSHKx3VNZLyi+2pLRUGxS8cSPFX26kJDmNPa0n8Gqz\nSWwN7IAKy9KSeLwHLnG+hOHLHV19uHWkEx066B28qMm5wkIGHzjAY61b8zcbvHsnI0NbN3b3bmjZ\nUkvsMmeMaSSpW7vEREq++4HMz7fhvvt7DjXpxH86TeLb9r24EOIOwTkQ74Hj1v8RFjiKCe19GDXE\niZ49ZcSBqfQaV30qP58bY2N5qX17ptlQjb28PXNytBr7jz9qNylFR2tzxInakZq6lSmJP8v5L/aQ\nt+VH3H6P4Wgzb/7bZSw7+00m4Z6ZlLoaIM4blxM+hB/yZmyBF6OHOJHTKZthw+QqlDXr0KgR33Xv\nzrCDB/F0dORWG1t01dNTm4Z/3DjtJqXISG3pwi5d9I7MPkhPvQFcSi0m4euDZH+3h4K4X8lwS+Zw\n25ZEd+zDoeB2pAW6o854Qpw3LTK86e/jzchujejfz0DXrtXOsSWs3P7sbEb9/jubQkNtbhw7QF6e\nVorZvh2aNdP+DAnROyrrIeUXnSmlLRZ0OvYSqdt/59Lvf5Can0C2VyYZrZzY37Yzh9u345KPO06n\n3Cg4HoB/hhehjTwZ0sadfr0ciYgAPz+9fxPRkL66eJG/Hz/Oz717E2iDy+bl5cGECfD999C0Kezc\nKTV2Y+mW1O+55x42bdpE06ZN+eOPP8wWmKVRCtLTtSv8584qkvefIz3uOGkZZ8jmIrk+eRS2KOVC\nS18OtelAhrcn3oklcMadvHPNaVrWmG6eHgwJakTvngbCwkxL4DK/hvlYSlsuSUhgbUoKO3v2xN2K\nv5ZV1575+Vpi37ZNG+a4axe0bt3w8Vkb3WrqM2bM4OGHH+buu++u664anFKQk6Otap+aqv2Zlgap\n54tJO3GWjLSzZBekkO+QSaF7HsX+JRQEGMgOcOV0SAuywt3xu+CLe5IPJDWiOKUpbqmtGRDvSZ82\nbnQLMRAyDNq1kxKKqN4TrVsTl5vL3UeOsK5rVww2dldYo0baiNwRI7Rx7CNHahdRbexSgsUwS/kl\nPj6e8ePHV9tTLylRKKXNEXH1n7V5rrQUioq0R2Gh9qju55wcyM6G3EsF5GZdIj83i8LCbAqKciku\nzaOYXIqccilxLaTMq5gSHyj0diDf24ksn0Zk+HiS5+aGf2oO3qkFuF1UOKQ6Y7jkgVOJP05ubWjj\n60+3li50DDIQFAQdOoCvb11bUtirwrIybjxwgDuaNmWOjXZjMzK0G5T++ENbIm/7drDBSwlmY9Gj\nX4bdvxwMgMPlAB0Ag0JV/Fz+UGAAZbj6Oe2hHKDMGZSToszZQGnFw4ESZwdKnR0odnak2NmR/Bau\n5HZwpdDZGfeCQjzyCmmUX4RrfjGueaW45JbhdAkMWa44ZHnhnOaOZ5kX/k6+NGrkj49PAO2budEq\n0EBgV+1GisBA7aq+EPXB1cGBNSEhROzfz0AfHyJsMNs1bgzffafdcfrbb3DLLbB5M3KDnJk1SE+9\nWfcOuPlqizE4uTrj2dwX33ZNUUqRGZ8MKHzbNQGlyDyTgkLh2zbg8vbFy9t+KFVCRnwyZZTi0caD\nUoq5lJBBKSW4tnWhmGJyEnIodCiksH0JRQ4FEH85kPaX/zxt5ds/Ac0tKB5r3i7/2VLiAUjvBi0n\nQsmbUJKjfzzW3p7WtH0aiL287QvswHJ76hcOnmyIw9gFS7m4ZwsstS1nHT/O2cI7+cLK6uu1ac/D\nh2HIEG3wwUMPwbvzsfsJ5v7M1L9721qWxQ5YYhKyVpbalq8GBXGmoIB/JyXpHUqt1KY9u3aFr78G\nV1d47z144436i8ve1DmpT5kyhQEDBnDs2DFat27N8uXLzRGXEHbL1cGB1V268Ozp05zOz9c7nHoz\nYACsWqX9/Pjj8MUX+sZjK+TmIytjqSUDa2TpbflKQgJb0tP5vkcPHKygNmFqe778Mjz1lLbmaUwM\nRESYPTSrZGrulPKLEBZqTqtW5JaW8h8rK8PU1rx5cN99UFCgTStw6pTeEVk36akLYcHicnMZcuAA\nv/buTTsbHvtXXKxNALZ1K3TurN2kZO/3fUhPXQgbFOLhweNt2nD/sWM23Tlydoa1a6FbN/jf/+Cu\nu7QbDkXtSVK3MjExMXqHYDOspS3ntmpFclERn1+8qHcoNapre/r4aOun+/lpU/fOn2+euOyNJHUh\nLJyTgwPvdOzI3JMnySkp0TucetWhA3z2GTg4wEsvwbp1ekdkfaSmLoSVmBoXR2s3NxbbwZqF//wn\nzJkD7u5afb17d70jangyn7oQNu58YSGhv/zC7l69uMHdXe9w6pVScPfdsHo1tG8Pv/wC/na24Jdc\nKLUT1lIHtgbW1pYtXF15um1bZh0/bpEdJXO2p8EA77+vzeZ4+jTceSfYeOXJbCSpC2FFHg4MJLGw\nkE1paXqHUu/K52Fv2hR++AEWLNA7Iusg5RchrMzG1FSePnWKg3364GgFd5rWVUwMDBumra/w7bcw\nerTeETUMKb8IYSfG+/vT2NmZjy9c0DuUBhEZCS+8oP08bRokJuoajsWTpG5lrK0ObMmstS0NBgNL\nOnRgfnw8+RZ0h059tueTT8KoUdpyk3fcoa10JqomSV0IKzTAx4feXl68c+6c3qE0CAcH+PhjbRWy\nvXu1CcBE1aSmLoSV+l9uLoNjYznWty+NnZ31DqdB7NmjrXNaUgJffgm33qp3RPVHaupC2JnOHh7c\nGhDAK3ZUZB4wAJYs0X6eMUMb7igqk6RuZay1DmyJbKEtn2nblveTkkgrLtY7lAZrz9mzYeJEuHRJ\nm/hLxq9XJkldCCvW1s2NWwMCeOvsWb1DaTAGA3z4IQQGalMILFqkd0SWRWrqQli5k/n5RPz2Gyci\nIvC1k9o6QHS0Nn7dYIAdO2DQIL0jMi+pqQthp4IaNWKcvz9L7WQkTLmhQ7VVk8rKYOpUyMzUOyLL\nIEndythCHdhS2FJbPt22LW+fO0eWjgVmPdpz4UIID4czZ+Chh7SJwOydJHUhbEAnd3dGNm7Mv+ys\nt+7iAp98Ah4esGaNNpbd3klNXQgbEZeby9DYWOL79aORo6Pe4TSo5cvhnnvA0xNiYyEoSO+I6k5q\n6kLYuRAPD/p4efHf5GS9Q2lwUVEwaRLk5Gg/W9DsCQ1OkrqVsaU6sN5ssS3ntG7NG2fP6vLtWM/2\nNBjgvfegeXPYtQvefFO3UHQnSV0IGzLU1xcXg4Hv0tP1DqXB+ftr49cBnnkGDh/WNx69SE1dCBuz\n6sIFVicns7VHD71D0cV998FHH2mrJv30E1jr0H2pqQshAJjctCmHcnP5PSdH71B08cYb0LYt/PYb\nvPSS3tE0PEnqVsYW68B6sdW2dHFw4O+BgfyzgacOsJT29PbWRsOANoXAb7/pG09Dq3NS37JlC507\nd6Zjx44sKZ8+TQihqwdbtuSr1FQuFBbqHYouhg6FRx7RJvu6+24oKNA7ooZTp5p6aWkpN9xwA99/\n/z2BgYH06dOHTz/9lC5dulw5gNTUhdDFg0eP0trVlX+0a6d3KLrIy4OePeHYMW3lpMWL9Y6odnSp\nqe/bt4/g4GDatWuHs7MzkydPZsOGDXXZpRDCTGa2bMn7589TaqedKnd3WLFCG+746quwf7/eETWM\nOiX1c+fO0bp164rtVq1acc7OblNuaJZSt7QFtt6WPb28aOHiwua0tAY5niW2Z//+MGuWdjPSvfeC\nBUw7X++c6vJhg8Fg1PuioqJod/kroK+vL2FhYURGRgJXTgTZNm47NjbWouKRbcveHhofz6LYWMbd\ne69FxKPH9s03w8aNkcTGwsyZMUybZlnxlW/HxMSwYsUKgIp8aYo61dT37t3LggUL2LJlCwCLFy/G\nwcGBefPmXTmA1NSF0E1eaSmtf/qJ33r3pl2jRnqHo5sffoDhw7UJwA4cgJAQvSO6Pl1q6uHh4Rw/\nfpz4+HiKior47LPPuOWWW+qySyGEGbk7OjKtWTM+OH9e71B0NWyYdlNSUZFWhrHluWHqlNSdnJx4\n5513uPnmmwkJCeHOO++sNPJFmF/51zVRd/bSlg+2bMlH589TVFZWr8ex9PZ89VVo2RL27oWlS/WO\npv7UeZz66NGjOXr0KCdOnOCpp54yR0xCCDPq4uFBFw8PvkpN1TsUXfn6apN+gTY3zKlT+sZTX2Tu\nFyHswKfJySy/cMFu54O52pQp2oIaN98MmzdrQx4tkcz9IoSo1sSAAH7NzuasPd1aWY0339R67d99\nB2vX6h2N+UlStzKWXre0JvbUlo0cHZnUpAkf1+MCGtbSns2aQfmMJo8+ansLVktSF8JOTG/enJUX\nLkg5FG0kzIABcOECPP203tGYl9TUhbATSilu2LePVZ0708/HR+9wdHfokDY3TGkp7NkD/frpHVFl\nUlMXQtTIYDBovXU7XMO0Kt26wWOPgVLw4IO2M4WAJHUrYy11S2tgj205rVkz1qakUFAPd99YY3s+\n+yy0bw+//w5vvaV3NOYhSV0IO9LGzY2enp5sbKBJviyduzu8+67283PPwZkz+sZjDrrV1G+66Say\nsrLq89BC4O3tzfbt2/UOw6KsvnCBT1JS+LZ7d71DsRh33qkNbxw7Fr7+2jLGrptaU9ctqYeHh/Pr\nr7/W56GFkPOsCrmlpQTu2cOxiAiaurjoHY5FOH8eOneGrCz4/HO47Ta9I5ILpXYjOztb7xBshjXW\ngM3Bw9GRsf7+fHHxoln3a83t2aLFlZWRZs3Skru1kqQuhB2a3LQpa1JS9A7Dojz4IPTtC0lJ8MIL\nekdjOknqVsbLy0vvEGxG+UIF9miknx9/5OaaddoAa29PR0ftoqnBoE0lEBend0SmkaRuRRISEvDy\n8qpVnW3gwIEcPHiw3mKKiIggzlrPfjvm6uDAxIAA1pm5BGPtwsPh/vuhpAQeflgbw25tJKmbYMeO\nHaxcufKa51euXMmOHTsqti9evMiUKVMIDAzE19eXQYMGsW/fvms+N2PGDKOO26ZNG5KSkoxeRvDr\nr7/Gx8eHHpdn5lu5ciXh4eH4+PjQunVr5s2bR+mfxiv/+Xe4nscee4z58+cb/X5jHTp0iJtvvpkm\nTZrg4FA/p6k114DNwdwlGFtpz5deAj8/2L5du2hqbSSp18KGDRv44IMPKrbXr1/P+++/z/vvv89X\nX30FaLf+IVezAAAe+0lEQVRiv//++2zYsIGcnBz69u3L/v37ycjIYPr06YwdO5a8vDzy8vKYOXMm\n6enpgNYLf+ihh8wa77///W+mTZtWsZ2fn89bb71FWloaP//8Mz/88AOvvfYaQLW/w/WMHz+e6Oho\nko28S9HYBO3i4sLkyZP56KOPjHq/qL2bfH05XVDA6fx8vUOxKP7+WmIHmDMHcnL0jafWVD2r7hC9\ne/eu70Ob5MSJE8rPz0/t379fKaXUuXPnVEBAgNqxY4dSSqlly5ap/v37q7CwMPXaa6+pkpISVVxc\nrF599VUVFham+vfvr5YvX17t/r29vSv2fejQIXXHHXeooKAgFRUVpRISEpRSSv3888+qd+/eytvb\nWzVr1kzNmTNHKaXU6dOnlcFgUKWlpUoppW688Ub17LPPqoEDByovLy81cuRIlZqaqpRSqrCwUDVq\n1EidO3eu2ljeeOMNNX78eKWUqvZ32L17twoICFCJiYlKKaViY2NV48aN1dGjRyv2M2LECLVy5Uqj\n2tdgMBj1vnLHjx+v9WeuZqnnmaV46OhRtTg+Xu8wLE5JiVK9eikFSj31lD4xmJqeLbKnbjCY71Fb\nQUFBLFmyhKlTp5Kfn8+MGTOYMWMGQ4YMqahlGwwGDAYDDg4OKKVQSlX0QGvqicbGxlJUVERwcDCg\n9YjLx6KW7wvgkUceYfbs2Vy6dIlTp05xxx13VLvPTz/9lBUrVpCSkkJRUVFFz/v48eM4ODjQsmXL\naj+7Y8cOunXrVhFLVb/DgAEDePDBB5k+fTr5+flMnTqVRYsW0alTp4r3dOnSpV7r9qL+3CmjYKpU\nftEU4LXX4NgxfeOpDYtM6nq77777CA4Opm/fviQnJ/Piiy8CWvmluLiYF198kUceeYR27dqxbNky\nli9fTvv27Xn00UdZtGgRhYWFFaWMcllZWUybNo0FCxbg5eVFXl4eS5cu5V//+hdDhgzhueeeY/Hl\ngbIuLi4cP36c1NRU3N3diYiIqDJOg8HAjBkzCA4Oxs3NjTvuuIPY2FgAMjMzaxwps2zZMvbv389j\njz0GUOPvsGDBAi5dukTfvn1p3bo1f/vb3yrty8vLi0wjJqVWFnbVyVZqwHUxyMeHi8XFHMnNrfO+\nbK09+/WDe+7RJvqaNcuKLpqa6ZtCtao7hKV/Ld64caMyGAzqww8/vOa1mJgYtWLFimueX7FihYqJ\nibnm+by8PDVkyBD1wAMPVHmsqKioStvHjx9XU6ZMUQEBAapPnz7qm2++UUpdW36JjIxUH330UcXn\nli9frgYNGqSUUurw4cPKw8OjyuOtX79eNWvWTB06dMjo3+Htt99WBoNBff/999e89vDDD6u5c+dW\neaydO3cqX1/fiofBYKi0vXv37io/d3Vb1Ff5JTo62uT92pJZx46pF06frvN+bLE9k5OV8vXVyjDr\n1zfssU1Nz5LUq5Cdna06dOig7r//fhUYGKjS09NN3ldBQYEaOXKkmjp1qkmf//zzz5Wbm5vKy8ur\nVVIvLCxU7u7uKikpqdL+Nm/erJo0aaJ++eUXo2M4e/asCggIUPfee6/q3r27KiwsrPT68OHD1apV\nq4zal9TULU90errqVYvzwd4sXaol9bZtlcrNbbjjmprUpfxShUceeYS+ffvy/vvvM3bsWGbOnGnS\nfoqLi7n99ttxd3dnxYoVRn9u9erVXLw8ftjHx6eifl8VVc13QhcXF4YPH17pK/H27du56667+PLL\nLwkPDzcqFqUUUVFR3HfffXz44Ye0aNGCZ599tuL1goIC9u/fz4gRI4z87YxXUFBAUVERAIWFhRQW\nFpr9GEIrwZwpKCBB1i+t0syZ0L27NoPjyy/rHY0RzPpfSxWqO4Sl9qC++uor1apVK5WRkaGUUion\nJ0cFBwerTz75pNb7iomJUQaDQXl4eChPT8+Kx65du2r83NSpU1XTpk2Vp6en6tatm9qwYYNSSiu/\nODg4VNtTX7FihRo8eHDF9qZNm9To0aMrtocOHaqcnZ0rxTJmzJgaY3nzzTdVWFiYKi4uVkoplZSU\npJo0aVLxO6xdu1bddtttRreJg4ODUe8r/1ZiMBiUg4ODMhgMqn379kYfp5yUX4wTdeSIeuvyCCdT\n2XJ77typ9dZdXZUyQ6XKKKamZ0nqViYrK6tW7x84cKCKjY2tp2iUioiIUIcPH663/deVJHXjbLh4\nUUUeOFCnfdh6e06ZoiX2SZMa5nimJnWZelfYNDnPjJNfWkrzPXs4GRFBgEzHW6XERLjhBsjPhx07\nYMiQ+j2eTL0rhDBZI0dHhjduzNeyIlK1WreGefO0nx99VFuw2hJJUrcyMp+6+djauOq6ujUggK9S\nU03+vD205+OPa8n9wAFYvlzvaKomSV0IAcBYf3+iMzPJtdQuqAVwd4clS7Sfn3nGMhfTMDmpr1u3\njq5du+Lo6Mj+/fvNGZOogcynbj7WPv+3uTV2dqaftzdbLk8yV1v20p6TJ8PAgZCSAosW6R3NtUxO\n6qGhoaxfv54h9X21QAjRYCYGBLBe5livUfkiGqD9efy4vvH8mclJvXPnzpUmdRINQ2rq5mMPNeDa\nGufvz5b0dEpNGHVhT+0ZHg5RUdq8MJenT7IYUlMXQlRo4+ZGS1dX9llisdjCvPQSeHrCxo3w/fd6\nR3OFU00vjhgxggsXLlzz/EsvvcT48eONPkhUVBTt2rUDwNfXl7CwsIrXynue5bVi2a55u/w5S4nH\n0rcLCgqIiYmpqPeW9yYjIyOJjIystP3n1+11u1tSEpv8/env41Orz9tbe7ZoAVOmxPDBB/Doo5HE\nxsKuXabvLyYmpmI6kfJ8aYo633w0dOhQXn/9dXr16lX1AeTmIz755BPOnz/Pvn37uPXWW5k8eXKt\n9xEbG8vq1asr5ksXxrGn88xcdmZmMuvECQ4YOT+QPSsogJAQOH1am3/9T7NS14muNx/V802pVu3E\niROkpaUxd+5c3n33XR566CFOnz5dq3288cYbPP/886SlpUlN3YzsqQZcG/29vUkoKOBcLSdQs8f2\ndHPTFtEAeO45uHRJ33igDkl9/fr1tG7dmr179zJ27FhGjx5tzrhsxuHDh3nllVcACAgIIDg4mN9+\n+61W+5gzZw4TJkyoj/CEuIaTgwMj/fzYLHeXGuXWW2HwYEhNvbK2qZ5qrKnX5NZbb+XWW281ZyxW\n5dSpU5UWof6zfv36MWHCBMaMGcPmzZsB7RvN+fPnK5azKxcaGsrKlSurLWGVfxZknLo52cu4alOM\n8fNjfWoq99WwHOKf2Wt7GgzwxhvQp482xPGhh6AOJfE6Mzmp1zfDQhMWGK2Ceq72paH9+/ezd+9e\nkpKSCA8Pp7S0lE2bNrFs2bKK93To0KFi+bmaODs7V6wDumnTJsLDwytdKAZ44YUXrjs81GDKgqtC\nmGiUnx8PHz9OYVkZrjWsuys04eFw113w3//CU0/Bp5/qF4vFJnVTkrG5pKSk0LlzZ7Zt28aiRYtQ\nSvHEE0/UaZ+ZmZmsWLGC1atXX/PaxIkTr/v58p761SNfRN1cPSpGVNbExYUuHh7szMxkuJ+fUZ+x\n9/Z86SX44gtYs0ab8KuapYXrncUmdT2NGjWKp556imnTpgHw008/0adPn0rvMbb8AlpCfvnll/nw\nww/x9PTkzJkztG3btlYxSU9dNLQxfn58m55udFK3d23awOzZsHgxzJkDu3ZppZmGJvOpV6Nfv358\n9913+Pj4MHPmTCZNmkRxcTGjRo2q9b7efvttBg4cSGBgIAkJCeTn53PjjTdWvL5+/XpGjhyJh4dH\ntftYsWIFO3bsYLmlTg1noSz9PLNk+7OzmRIXx1G9upxWKCsLOnbU5oVZtw5uv930fcl86maUl5eH\nr68vPj4+AHh4eJCSkoKfCT2WXbt2MXv2bPr06UPLli3p37//NRdKn3/+eU6ePFntPt555x2WLVtG\nTEwMCxcuJEvu9hMNoKenJ1mlpZzKz9c7FKvh7Q0LF2o/z5sHeiyrKz11KyM19dqp6Tyz9xqwMaYd\nOcIgHx8eNGIUjLSnpqQEevSAuDh4/XWtFGMK6akLIcxuROPGbDNxKl575eQEr76q/fzCC9DQw/0l\nqVsZ6aWbj/Qqr29448Zsz8w0atZGac8rRo+G4cMhM1NL7A1JkroQolotXV1p6eLCbzI9Ra0YDNr0\nAQaDNifMsWMNd2xJ6lZG5n4xH3ucq8QUI/z82JaRcd33SXtW1qMH3HOPVmMvX7C6IUhSF0LUSOrq\npnvhBfDwgK++gh07GuaYktStjNTUzUdqwMa50deX33JyyCkpqfF90p7XatECym9GnzsXysrq/5iS\n1IUQNfJwdKS3pyc/WsK8slZo7lxo2RJ++w0++6z+jydJ3cpITd18pAZsPGPq6tKeVfPwuDIC5umn\n6/+GJEnqQojrkrp63UyfDl27Qnw8/Otf9XssSepWxpia+pkzZ1i3bh2LFy+u9YIc9kRqwMbr7eVF\nUlERSTV0M6U9q+foCJfXyuGFF8CIwUQmk6Rug3bv3o2/vz8dO3bkWEMOkBU2y9Fg4CZfX76vz2xk\n40aPhqFDtYT+8sv1dxxJ6g1o3759Ri2s8WeffPIJr7/+OnfeeadRszT+9a9/pX379vz666/cdttt\npoRaSWxsLI899lid92NppAZcOzc1bkxMZma1r0t71sxguNJbf+stOHOmfo4jSb2BlJWVMX/+fIqL\ni2v1uT8vXD1nzhyjFq5u3749EydOZMGCBdd977vvvlvta1cvei3sW6Svb41JXVxfeDhMmaJdLH32\n2fo5hiT1BrJu3TqGDx9e61nXTFm4et68ecTFxeHq6srRo0eve4zU1NRqX7PlRa+lBlw7XdzdySkt\n5UxBQZWvS3sa58UXwdkZVq+GAwfMv39Z+chEtVn56OLFizg6OtKkSRNyc3OveW9NC08bs3D1n02c\nOJETJ05w+PBhnn/++dr8WlWq59mZhZUwGAxE+vqyIzOTu5s31zscq9W+Pfz97/DPf2rTB2zdat79\nS1KvgjkXngb48ssveeCBB1i1alWVr9e08PSfF64OCwurWLh648aNODo6snPnTkJDQ9myZQvPPPMM\n/fv3B+CWW24x+neuia0upSfzf9depK8v0dUkdWlP4z3zDCxbBtu2aUl95Ejz7VuSehXMufD03r17\niYiIqHHCe2MWni5fuLr820FCQgIhISEEBwczf/58nnzySXx8fGjTps1193XkyJFK/8Hs2rWLgqu+\nUg8ePJgxY8ZUbEtPXZQb6uvLq4mJeodh9fz9tcT+xBPaY9gwbdijOVhuUjdX79CEhGTOhad/+eUX\n8vLy+O6779i9ezf5+fls3LixVr3oqxeu9vX1rbRwdXJyMl5eXvj6+jJu3Dij9telS5dK3zIWLlzI\nc889V+37bbWnLr3K2uvs7k5eaSnx+fm0a9So0mvSnrXz8MPwzjtw8CD8979w993m2a/lJnWde4fR\n0dE8+eSTAKxatYr777+fLVu2VCw8bWz55eGHH674ecGCBRgMhmsS+vUWnl66dCmTJk2ioKCAffv2\nkZ+fT35+PgUFBRw4cIAhQ4YA8M033xid2GtDeuqiXEVd/dKla5K6qB03N1i0SEvm//gHTJoE5mhS\nGf1SBXMuPF1u7dq1bNy4kY0bN7Ju3bpKr9W08HR1C1dv3bqVTZs2oZSioKCA9evX07RpU5Pjq44t\nL3ot46pNU93QRmnP2rvrLm3e9cREePttM+1U1bPqDtG7d+/6PrRNysrKMvs+lyxZYvZ9WoqazrPo\n6OiGC8SGxOXkqHY//XTN89Keptm6VSlQyttbqYsXrzxvanqWnrqVqY/51E29CGztpAZsmqvr6leT\n9jTNiBHa6JesLG0Me12ZnNQff/xxunTpQo8ePfjLX/7CJZlrWQi7cHVdXZjHkiVX1jM9dapu+zI5\nqY8cOZLDhw9z8OBBOnXqZNKcJqL2ZD5185EasOmqqqtLe5ouLAymTYPiYm2oY12YnNRHjBiBg4P2\n8YiICM6ePVu3SIQQVuPGy3eWCvN54QVwdYU1a+CXX0zfj1lq6suWLat0s4qoP7JGqflIDdh0nd3d\nySwp4fxV86tLe9ZNmzbwyCPaz48/bvp+akzqI0aMIDQ09JrH119/XfGeF198ERcXF/7617+aHoUQ\nwqo4GAwM9PFht9TVzeqpp8DPD3bsMH0fNd58tG3btho/vGLFCr799lt++OGHGt8XFRVFu3btAPD1\n9a2YuwSu1IjLe6CyXfN2cnIy7u7uFhOPpW8XFBRUmpOkvO4bGRlZqQZc1euyXfP2QG9v1mzdSkCr\nVtKeZtiOiYlhxYoVBAVBeno7TGbq2MrNmzerkJAQdfHqgZVVqO4QMk7dNPUxTt2WyTj1+rMzI0OF\n//prxba0p3kUFCh1002mj1M3KGXaPeAdO3akqKio4i7L/v37868qVlStbiKr8PBwfv31V1MOLYTR\n5DyrPwWlpfjv3k3ygAF4OlnujCPWqqZJAGti8t/E8ePHTf2oEMIGuDk6Eubpyb7sbG5q3FjvcMRl\nckeplZFx6uYj46rrbqCPD7suXyyV9rQMktTtxJdffsncuXOJj4/XOxRhQwbJCBiLI4UwK2PqOPVm\nzZrx+uuvmzka6ybjqutugLc3044coVQpaU8LIT11GzJ37txqX7vhhhs4dOgQ0dHRDRiR5syZM6xb\nt47Fixdfd9FsYV0CXFxo6erKHzk5eociLpOkbmWqq6mfPHmS2NjYaj/3448/0q1bN/bs2VNfoVVr\n9+7d+Pv707FjR44dO9bgx6+O1IDNY6C3N7suXZL2tBBSfrERZ86cqVifNCkpiSNHjvDDDz/QrFkz\nQkJCaNeuHXv37mXw4MENHttf//pXTp8+zdatW3n++ecb/Piifg3y8eG7jAy66R2IAKSn3iBycnKY\nP38+H3zwAa+//nqdloerqqa+d+9e+vbtW+m5YcOGkZaWxsyZM7npppvo1asX/fr1q1j6ri4++eQT\nXn/9de68807WrFlj1Gfat2/PxIkTWbBgQZ2PDxAbG8tjjz1Wp31IDdg8ykfASHtaBknqDWDWrFnc\ne++93H///SxbtoyEhASz7j8+Pp4ffviBhIQEoqOj8fT0JDk5maZNm1JYWEhubm6t9/nuu+9W+fyJ\nEydIS0tj7ty5vPvuuzz00EOcPn26xn3NmzePuLg4XF1dOXr0aJ2OD/DGG2/w/PPPk5aWZtS+RP0K\nbtSIorIyEgoK9A5FIEm93p06dYqkpCTatm0LwNatWyt+NkVVNfXJkycTGhpKYWEh+fn5LFq0iJiY\nGDw8PNi+fTve3t61Pk5qamqVzx8+fJhXXnkFgICAAIKDg6978XPixImcOHGCLVu2GF1+qe74AHPm\nzGHChAlG7acmUgM2D4PBQD9vbz7avFnvUARSUzfZqVOn+OCDD6p9vV+/fkyYMIHt27fj6+vLxx9/\nTGZmJl5eXkRFRVV6b2hoKCtXrqRXr14mx9OhQwd27doFUK/TII8ZM4bNl//xKqU4f/48wcHBNX6m\nf//+ANxyyy1mi6MuJSxhfv28vdmfl6d3GAJJ6lXav38/e/fuJSkpifDwcEpLS9m0aRPLli2reE+H\nDh2MWu0pOTmZQ4cOVdSeBw8ezMCBA+nYsWPFe1544QU6depkVGx6z6fu7OxMt27aJbFNmzYRHh5e\nMevmxo0bcXR0ZOfOnYSGhrJlyxaeeeYZOnfubPY4DAZDnfchNWDzifD25psuXfQOQ2DBSd1gpq/G\nyoR/uCkpKXTu3Jlt27axaNEilFImL87s7e1NaGhoxXabNm3YunVrpaQ+ceLEGvfxyiuvkP+nRX7L\nTZ8+nQ4dOpgUG2jJsbS0lCNHjrBq1aqK53ft2kXBVTXSwYMHV/oGkJmZyYoVK1i9ejUACQkJhISE\nEBwczPz583nyySfx8fGpGJFzPbU9vvTULUsfLy8O5uRQVFaGi4NUdfVksUndlGRsLqNGjeKpp55i\n2rRpAPz000/06dOn0nuMLb907dqVnTt3Vjzv4OBAWVlZreK5+j+U7Ozsa3rrtd1fVbp06VLpm8fC\nhQt57rnnqnyvUoqXX36ZDz/8EE9PT86cOVNxnSA5ORkvLy98fX0ZN25cvRwfzNNTv3qedVE3Xk5O\nNImL4/ewMMJNuIYjzMdik7reoqOjefLJJwFYtWoV999/P1u2bGHUqFGA8eWXgQMH8vTTT1dsnzx5\n8pphfevXr2fkyJF4eHiYLf74+Hj27dvHyZMnGTlyJL179zbbvpcuXcqkSZMoKChg37595Ofnk5+f\nT0FBAQcOHKgYNvnNN9/UKrHXhvTULU+Iuzs/Z2dLUteZJPUq5OXl4evri4+PDwAeHh6kpKQQFBRU\n6325urqyYMEC5s+fT1lZGf/3f/93zX6ef/55goKC6N69+3X3Z2xNfc+ePTRv3hwHBweOHTtmtqS+\na9cuZs+eXZFUDQYDCQkJfPHFF2RnZ9OiRQsKCgpYv349gYGBZjnmn73zzjusXbuWxMREFi5cyOzZ\ns00a4SO9dPO6dfhwdly6xP/V09+7MJIZFuqoUXWHkJWP6t+pU6fUvHnzVGFhYa0/u2TJknqIqOGP\nL+dZw/k9O1t13LtX7zBshqnpWa5oWJnazKdel7s4Tb0wbC4NcXwZp25eKb/8woWiItKKi/UOxa5J\nUrdRptzFKURdOBoMhHt5sS8rS+9Q7JokdStjbE3dlLs47Y3U1M0rMjKSCG9vfpakriu5UGqj6uMu\nTiGup5+3N/9OStI7DLsmPXUrI2uUmo/U1M0rJiaGCC8vfs7KkiGnOpKkLoQwm+aurng5OnK8mjug\nRf2TpG5l9J77xZZITd28ytuzn7c3e6WurhtJ6kIIs4rw9pYRMDqSpG5lpKZuPlJTN6/y9uzt5cVv\nshC1biSpCyHMqqenJ7/n5FBihonmRO1JUrcyUlM3H6mpm1d5e3o7OdHGzY04WTRDF7qNU/f29iY8\nPFyvwws7YcpEX6Luent68mt2Nt09PfUOxe4YlIkDSp999lk2btyIwWDA39+fFStW0Lp162sPYDDI\nmFUzkjnAzUfa0ryubs83ExM5np/Pu0au6CWuZWruNLn88sQTT3Dw4EFiY2OZOHEiCxcuNHVXohZi\nY2P1DsFmSFua19XtGe7lxa9yUV8XJif1q2u7OTk5BAQEmCUgUbPMzEy9Q7AZ0pbmdXV7hnl6cig3\nl2K5WNrg6lRTf+aZZ/j4449xd3dn79695opJCGHlPJ2caOfmxuHcXMLk4n6DqrGnPmLECEJDQ695\nfP311wC8+OKLJCQkEBUVxezZsxskYHsXHx+vdwg2Q9rSvP7cnlKC0YfJF0qvlpCQwJgxYzh06NA1\nrwUHB3Py5Mm6HkIIIexKUFAQJ06cqPXnTC6/HD9+nI4dOwKwYcMGevbsWeX7TAlKCCGEaUzuqd9+\n++0cPXoUR0dHgoKCeO+992jatKm54xNCCFELZim/CCGEsAxmmyZgy5YtdO7cmY4dO7JkyZIq3zNr\n1iw6duxIjx49OHDggLkObXOu15YxMTH4+PjQs2dPevbsyaJFi3SI0jrcc889NGvWjNDQ0GrfI+el\n8a7XnnJu1k5iYiJDhw6la9eudOvWjbfffrvK99XqHFVmUFJSooKCgtTp06dVUVGR6tGjh4qLi6v0\nnk2bNqnRo0crpZTau3evioiIMMehbY4xbRkdHa3Gjx+vU4TW5ccff1T79+9X3bp1q/J1OS9r53rt\nKedm7Zw/f14dOHBAKaVUdna26tSpU51zp1l66vv27SM4OJh27drh7OzM5MmT2bBhQ6X3bNy4kenT\npwMQERFBZmYmycnJ5ji8TTGmLQGZesFIgwcPpnHjxtW+Ludl7VyvPUHOzdpo3rw5YWFhAHh6etKl\nSxeS/rTGa23PUbMk9XPnzlWa96VVq1acO3fuuu85e/asOQ5vU4xpS4PBwJ49e+jRowdjxowhLi6u\nocO0GXJempecm6aLj4/nwIEDREREVHq+tueoWWZpNBgMRr3vz/+DG/s5e2JMm/Tq1YvExETc3d3Z\nvHkzEydO5NixYw0QnW2S89J85Nw0TU5ODrfffjtvvfUWnlXMbFmbc9QsPfXAwEASExMrthMTE2nV\nqlWN7zl79iyBgYHmOLxNMaYtvby8cHd3B2D06NEUFxeTnp7eoHHaCjkvzUvOzdorLi7mtttuY+rU\nqUycOPGa12t7jpolqYeHh3P8+HHi4+MpKiris88+45Zbbqn0nltuuYVVq1YBsHfvXnx9fWnWrJk5\nDm9TjGnL5OTkiv+59+3bh1IKPz8/PcK1enJempecm7WjlOLee+8lJCSERx99tMr31PYcNUv5xcnJ\niXfeeYebb76Z0tJS7r33Xrp06cJ//vMfAB588EHGjBnDt99+S3BwMB4eHixfvtwch7Y5xrTl559/\nznvvvYeTkxPu7u6sWbNG56gt15QpU9ixYwepqam0bt2ahQsXUlxcDMh5aYrrtaecm7Wze/duVq9e\nTffu3Svuyn/ppZdISEgATDtH5eYjIYSwIbJGqRBC2BBJ6kIIYUMkqQshhA2RpC6EEDZEkroQQtgQ\nSepCCGFDJKkLIYQNkaQuhBA25P8ByRUuDWZFhf4AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb094a80c>"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{2} \\sin{\\left (2 x \\right )} + 1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAAZCAYAAADQfBuCAAAABHNCSVQICAgIfAhkiAAABBtJREFU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\nowevUtEC9uCvkVaioqKioqLicOB/5oXqgmek4sEAAAAASUVORK5CYII=\n",
       "prompt_number": 31,
       "text": [
        " 2             \n",
        "x \u22c5sin(2\u22c5x) + 1"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f(x).series(x,1,4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 + \\sin{\\left (2 \\right )} + x \\left(2 \\cos{\\left (2 \\right )} + 2 \\sin{\\left (2 \\right )}\\right) + x^{2} \\left(4 \\cos{\\left (2 \\right )} - \\sin{\\left (2 \\right )}\\right) + x^{3} \\left(- 4 \\sin{\\left (2 \\right )} + \\frac{2}{3} \\cos{\\left (2 \\right )}\\right) + \\mathcal{O}\\left(x^{4}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAtAAAAAcCAYAAABI4oh6AAAABHNCSVQICAgIfAhkiAAADwVJREFU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\nGIZhGIZhGIZhGIZhGIZhGIZhdGP+D+v/dxNW6AvBAAAAAElFTkSuQmCC\n",
       "prompt_number": 42,
       "text": [
        "                                        2                        3 \u239b          \n",
        "1 + sin(2) + x\u22c5(2\u22c5cos(2) + 2\u22c5sin(2)) + x \u22c5(4\u22c5cos(2) - sin(2)) + x \u22c5\u239c-4\u22c5sin(2) \n",
        "                                                                   \u239d          \n",
        "\n",
        "  2\u22c5cos(2)\u239e    \u239b 4\u239e\n",
        "+ \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + O\u239dx \u23a0\n",
        "     3    \u23a0        "
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(f(x),x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2 x^{2} \\cos{\\left (2 x \\right )} + 2 x \\sin{\\left (2 x \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAL4AAAAZCAYAAACPbZTLAAAABHNCSVQICAgIfAhkiAAABWxJREFU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\nRTCPx+PxeDwej8fj8STgP3eU4jeMwCUjAAAAAElFTkSuQmCC\n",
       "prompt_number": 51,
       "text": [
        "   2                        \n",
        "2\u22c5x \u22c5cos(2\u22c5x) + 2\u22c5x\u22c5sin(2\u22c5x)"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}