{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Polynomial Approximation with Derivatives"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "from math import factorial"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## A Brief Intro to `sympy`"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Here we import `sympy`, a package for symbolic computation with Python."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "import sympy as sp\n",
        "sp.init_printing()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Next, we make a (symbolic) variable $x$ from which we can then build more complicated expressions:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEHarIkSJZt3NVLsymT3i\nYlMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABASURBVAgdY2AQUnZVU2BgTGBv4pjAwCbA9pDLgYGR\ngXMDAwjwKYCpcweAFJeAHgOTAEPcgn0M7gwMwkpC1wsYAB0ECeuXDPmiAAAAAElFTkSuQmCC\n",
            "text/latex": [
              "$$x$$"
            ],
            "text/plain": [
              "x"
            ]
          },
          "execution_count": 9,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "x = sp.Symbol(\"x\")\n",
        "x"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Build up an expression with $x$. Assign it to `expr`. Observe that this expression isn't evaluated--the result of this computation is some Python data that represents the computation:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHcAAAAcBAMAAACpJD0gAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
            "text/latex": [
              "$$\\sin^{2}{\\left (\\sqrt{x} + 2 \\right )}$$"
            ],
            "text/plain": [
              "   2        \n",
              "sin (\u221ax + 2)"
            ]
          },
          "execution_count": 10,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "expr = sp.sin(sp.sqrt(x)+2)**2\n",
        "expr"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Next, take a derivative, using `.diff(x)`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQcAAAAwBAMAAADujaI1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiRN2J\nuzLrRHL3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEgElEQVRYCe1XTYgcRRT+Znt+tmd2OoOi4Ckr\nERY9SB9EEFcdBEPMZUfIMdEhIesKQoZEbETEwUMIGHBICOLFGTx4ECGLCioIjkhMDIHMKeDFrAG9\nya5sVjFkN76q6qp61b09sof0KmzBdL33vfdVffW6th8L/HdGYc/2a3nswOr2i0B5R0T8FnYqoa/j\nTiX+P5UIBlpr1lzOCmyJOvZO/K638I/dEWNF+2auNo3pGoaK/r9Tx4o4oRcuDrSVnC8lgdg3VOzP\nyIChlp79a19WEkptHXpfG6m5FqYgAViq39k0gcAMKkVutC2nbzZ4wYLWOnfoZ6DcsgCzLLXUYLA2\nx1EpZ3moE4EL2vQWtcVmr4mrA/grDLKmoeJJCxprLNVkKeNB7bunuV/BxRAT08BXOsmZDRVOEQtt\nmTWW6qyDwDRY9zSxiIlFFDeAQy5JeZbqFjEWMY4qFvhwSS1Dz8lpbZp7LIFYRGVVijisk/hsqVNd\njscixlF/O/Wyf70ZHH9xfq9gljoxX5zGm3/+4x+UH4sgp0a1Oh+S4R280MDs/GUECycFy1DFlWDU\nWASnks2HN0INlSZq19DvUaA+iqPlLjCLd9tfKt+KOD+km7xE6JlG8HDwNertp4BF8g0VCw7VijBU\nyuajugdl1JuoT2NXlwKkB/iEfjX6fY+r4QM007AiviCvP6BHhMJGbYhg46MQbfIltSoiJIlRrYiY\nKr+m9kGEb6/1pIgOdrXJrbXoMzBDxjz9QrxFT+BiFH0WRa9Ju0QJ6Deo4GtkLJNxu7b+iggJKg53\niN+21KkoenMmipoUgqEKxxmzb69JESMlQhzn7LGBKjDwp841lTgqEFGJqojtJmPdu3hH7C+o/uO3\nVBEZ1VTCUCnZGX6I9wbidTAReGaEyaFIC+jPUQ0totDCB1SAHqvE2llM0tbqdXjrwGlBsVQtwlLV\nkvZJGcUuE1EcUaz0t/rG31tewRGVq0X8ClDt5V9HBBQmWqjeotg+ypJUvNTD5+QwqhbBqGpN86T/\ng+o9KoOuRHmaQv6aPE31j/qK31WpsQj/mysHFoF3BHimix+DR1EZ3hPiJ/IlFf2WKCKnxiI4VdDp\npep+X720cDmYu/n03M1frj9Egar8Ys6F4jTe3iNHT4lsGrGICvFIhPxies+92sO5K/vx6cnTA0pR\n1OKqaJScGovgVLEisvs9HhHh5e9aYmIjFhEjr7OINSU1uH2fRaSlX4eCLTW73+OgyK3M0Gmc8QT3\nPFkujkhbUnH8RCLg9RjAqKUGw13zhnAL4huQPSabm8YkFbuTRXRyGdXtkE5WUep7w8GSTm0piUhf\nUSv84Kk8RnX6vZvodVx/M09+CNKBrVFFh8wcyXuVTvRHaUwiW6JOdXmrTaxYGCSAlFsKU5ACtkSl\nK8G6dMaKdxumfs9a7d3eLb2+7ve6S6czckB0v2etNodd3S1sv7et1s3Iw9P9nrXaPLZN7KH6PW+1\niYQ8XNXveavNY9fEHqrfJ8C83XS/z1uB2C/V77dDxPh+n5Oi8f0+BxH/ALfsRAdkarRDAAAAAElF\nTkSuQmCC\n",
            "text/latex": [
              "$$\\frac{1}{\\sqrt{x}} \\sin{\\left (\\sqrt{x} + 2 \\right )} \\cos{\\left (\\sqrt{x} + 2 \\right )}$$"
            ],
            "text/plain": [
              "sin(\u221ax + 2)\u22c5cos(\u221ax + 2)\n",
              "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
              "           \u221ax          "
            ]
          },
          "execution_count": 11,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "expr.diff(x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Take 10 derivatives:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": 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TSnvIaAQeDjxhiOGQwmYpK1n8v+EX3vCL7Us64pP3/PUwvTp5fePNeKdHVlZgKltp6eFp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\n",
            "text/latex": [
              "$$\\frac{1}{512} \\left(- \\frac{256}{x^{5}} \\sin^{2}{\\left (\\sqrt{x} + 2 \\right )} + \\frac{256}{x^{5}} \\cos^{2}{\\left (\\sqrt{x} + 2 \\right )} + \\frac{63360}{x^{6}} \\sin^{2}{\\left (\\sqrt{x} + 2 \\right )} - \\frac{63360}{x^{6}} \\cos^{2}{\\left (\\sqrt{x} + 2 \\right )} - \\frac{2162160}{x^{7}} \\sin^{2}{\\left (\\sqrt{x} + 2 \\right )} + \\frac{2162160}{x^{7}} \\cos^{2}{\\left (\\sqrt{x} + 2 \\right )} + \\frac{18918900}{x^{8}} \\sin^{2}{\\left (\\sqrt{x} + 2 \\right )} - \\frac{18918900}{x^{8}} \\cos^{2}{\\left (\\sqrt{x} + 2 \\right )} - \\frac{34459425}{x^{9}} \\sin^{2}{\\left (\\sqrt{x} + 2 \\right )} + \\frac{34459425}{x^{9}} \\cos^{2}{\\left (\\sqrt{x} + 2 \\right )} - \\frac{11520}{x^{\\frac{11}{2}}} \\sin{\\left (\\sqrt{x} + 2 \\right )} \\cos{\\left (\\sqrt{x} + 2 \\right )} + \\frac{887040}{x^{\\frac{13}{2}}} \\sin{\\left (\\sqrt{x} + 2 \\right )} \\cos{\\left (\\sqrt{x} + 2 \\right )} - \\frac{15135120}{x^{\\frac{15}{2}}} \\sin{\\left (\\sqrt{x} + 2 \\right )} \\cos{\\left (\\sqrt{x} + 2 \\right )} + \\frac{64864800}{x^{\\frac{17}{2}}} \\sin{\\left (\\sqrt{x} + 2 \\right )} \\cos{\\left (\\sqrt{x} + 2 \\right )} - \\frac{34459425}{x^{\\frac{19}{2}}} \\sin{\\left (\\sqrt{x} + 2 \\right )} \\cos{\\left (\\sqrt{x} + 2 \\right )}\\right)$$"
            ],
            "text/plain": [
              "         2                  2                    2                    2       \n",
              "  256\u22c5sin (\u221ax + 2)   256\u22c5cos (\u221ax + 2)   63360\u22c5sin (\u221ax + 2)   63360\u22c5cos (\u221ax + 2\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\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
              "          5                  5                   6                    6       \n",
              "         x                  x                   x                    x        \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\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\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",
              "                                                                              \n",
              "\n",
              "               2                      2                       2               \n",
              ")   2162160\u22c5sin (\u221ax + 2)   2162160\u22c5cos (\u221ax + 2)   18918900\u22c5sin (\u221ax + 2)   1891\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\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\n",
              "              7                      7                       8                \n",
              "             x                      x                       x                 \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\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\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",
              "                                                                              \n",
              "\n",
              "        2                       2                       2                     \n",
              "8900\u22c5cos (\u221ax + 2)   34459425\u22c5sin (\u221ax + 2)   34459425\u22c5cos (\u221ax + 2)   11520\u22c5sin(\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\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\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
              "       8                       9                       9                      \n",
              "      x                       x                       x                       \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\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\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",
              "                                      512                                     \n",
              "\n",
              "                                                                              \n",
              "\u221ax + 2)\u22c5cos(\u221ax + 2)   887040\u22c5sin(\u221ax + 2)\u22c5cos(\u221ax + 2)   15135120\u22c5sin(\u221ax + 2)\u22c5co\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\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
              "   11/2                            13/2                              15/2     \n",
              "  x                               x                                 x         \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\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\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",
              "                                                                              \n",
              "\n",
              "                                                                              \n",
              "s(\u221ax + 2)   64864800\u22c5sin(\u221ax + 2)\u22c5cos(\u221ax + 2)   34459425\u22c5sin(\u221ax + 2)\u22c5cos(\u221ax + 2\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\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
              "                          17/2                               19/2             \n",
              "                         x                                  x                 \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\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\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",
              "                                                                              \n",
              "\n",
              " \n",
              ")\n",
              "\u2500\n",
              " \n",
              " \n",
              "\u2500\n",
              " "
            ]
          },
          "execution_count": 12,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "expr.diff(x, 10)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Use `.subs(x, ...)` and `.evalf()` to evaluate your expression for $x=1$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 13,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAPBAMAAABpSyLSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
            "text/latex": [
              "$$0.019914856674817$$"
            ],
            "text/plain": [
              "0.0199148566748170"
            ]
          },
          "execution_count": 13,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "expr.subs(x, 1).evalf()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Polynomial Approximation"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Here are some functions to play with:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 17,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "#f = sp.exp(x)\n",
        "f = sp.sqrt(1-x**2)\n",
        "#f = 1/(20*x-10)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Plot `f`:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 18,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "def plot_sympy(my_f, my_pts, **kwargs):\n",
        "    f_values = np.array([my_f.subs(x, pt) for pt in my_pts])\n",
        "    plt.plot(pts, f_values, **kwargs)\n",
        "    plt.ylim([-1.3, 1.3])"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": 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y6dOfJgAApGnChLhvYObM0q+76msCK1dK73uftHq1tOdbBqQEgDTMmBFdR5cufeu9A3Vd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\nAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f88ff94a240>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "pts = np.linspace(-1, 1, 1000)\n",
        "plot_sympy(f, pts)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The [Taylor polynomial](https://en.wikipedia.org/wiki/Taylor%27s_theorem) uses derivatives to *automatically* get close to the function--the closer the higher you increase the degree.\n",
        "\n",
        "Write out the degree 4 Taylor polynomial about 0, call it `taylor4`:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 21,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAvBAMAAAA4HQjXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ny7zVhIA/8a6Aecp4gw8AAAAASUVORK5CYII=\n",
            "text/latex": [
              "$$- \\frac{x^{4}}{8} - \\frac{x^{2}}{2} + 1$$"
            ],
            "text/plain": [
              "   4    2    \n",
              "  x    x     \n",
              "- \u2500\u2500 - \u2500\u2500 + 1\n",
              "  8    2     "
            ]
          },
          "execution_count": 21,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "taylor4 = (\n",
        "    f.subs(x, 0)\n",
        "    + f.diff(x).subs(x, 0) * x\n",
        "    + f.diff(x, 2).subs(x, 0)/2 * x**2\n",
        "    + f.diff(x, 3).subs(x, 0)/6 * x**3\n",
        "    + f.diff(x, 4).subs(x, 0)/24 * x**4\n",
        ")\n",
        "taylor4"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 22,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f88ff238e10>"
            ]
          },
          "execution_count": 22,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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mzRoAli5dSpUqVcjJyTn9/vHjx9OsWbM8+540aRKNGjWifPnytGvXjnXr1p1+\nrUaNGrz22ms0adKE0qVLn+5zzJgxJCQk0L59+0v4qUWkuFAIBOjDDz+kevXqTJ06laNHj/LYY4/R\nuXNnNm3axN69e2nWrBm//e1vAbj66qtJTEzkq6++Ov3+jz/+OM9hmw0bNtC7d2/eeust9u3bx803\n30zXrl3Jzs4+3WbMmDF88cUXHD58mIiICI4ePcrgwYN54403dBCYiBRIsQkBs+DcLtaZC93+/ftT\nsmRJoqOj+dOf/sTKlSs5duwY4F9zGDVqFAAHDx5k2rRp9O7d+xf9jR07li5dutCuXTsiIyN57LHH\nOHXqFAsWLDjd5ve//z1JSUnExsYC8Kc//Ynf/e53JCcnX/wHERFPKRLnDiqIcPnhm5OTwx//+EfG\njRvH/v37MTPMjP3791OmTBn69OlDgwYNOHnyJGPHjuXGG2+kUqVKv+hn586dpKSknH5sZlSrVo0d\nO3acfu6yyy47fX/FihXMmDGDFStWXNoPKCLFSrEJgVA6c4+e0aNHM3nyZGbOnEn16tU5cuQICQkJ\np9cUkpKSaNmyJRMmTOCjjz7iwQcfzLPPpKQkVq9e/bPntm3b9rMF/5nTnT17Nunp6VSvXh3nHMeP\nH8fn87FmzRqWLl0azI8rIsVIsRkOCqXKlSuzefNmAI4dO0ZsbCwJCQmcOHGCp59++he7fd511128\n9tprrF69mh49euTZZ8+ePZk6dSqzZs0iOzub119/nRIlStCyZcs82w8cOJBNmzaxYsUKVq5cyf33\n30+XLl2YPn16cD+siBQrCoEgePrpp3nxxRcpX748hw4dIiUlheTkZBo1akSrVq1+0b5Hjx6kp6dz\n2223ERcXl2efV1xxBR999BGDBg2iYsWKTJ06lcmTJxMV5V95OztYSpQoQaVKlU7fSpcuTYkSJShf\nvnzwP7CIFBs6lXSI1K5dmxEjRtCuXbuQ1VAc56uIF+lU0kXM+PHjiYiICGkAiIhAgCFgZr82s9Vm\n5jOzvI948rfrZGbrzGyDmT0ZyDSLurZt2/LQQw8xbNiwUJciIhLYcJCZ1QVygL8DjznnluXRJgLY\nALQHdgJLgF7OuXV5tPXMcFA40HwVKR4CGQ4KaBdR59z63ALONfFrgY3OufTctmOAbsAvQkBERApX\nYWwTSAa2nfF4e+5zIiISYuddEzCzr4DKZz4FOOAZ59zkAkwjr7WEfMcghgwZcvp+amoqqampBZiE\niIh3pKXR8i4gAAAE+0lEQVSlkZaWFpS+grKLqJnNAh7NZ5tAC2CIc65T7uOnAOecezWPttomUIg0\nX0WKh3DZRTS/ApYAtc0sxcxigF7ApCBOV0RELlKgu4h2N7NtQAtgipl9kft8VTObAuCc8wGDgOnA\n98AY59zawMoOLxs2bKBZs2bEx8czdOjQUJcjIlJgOmI4CAYMGEB8fDxvvPFGqEu5IOE+X0WkYMJl\nOMiz0tPTadiwYajLEBG5YFoTCFD79u2ZPXs20dHRREdHs2zZMmrXrh3qsgoknOeriBRcIGsCCoEg\naNu2LXfddRf33HNPqEu5IOE+X0WkYEJ2xHA4secDuDbkGdxgLRRFxDuKTQho4S0icuG0YVhExMMU\nAiIiHqYQCIJzn0RVRCR8ae8gD9N8FSkedLCYiIhcFIWAiIiHKQRERDxMISAi4mEKARERD1MIiIh4\nWJE4bURKSor2xb8EUlJSQl2CiIRYkThOQERE8qfjBCRPaWlpoS6hWNH8DC7Nz/CgECjG9E8WXJqf\nwaX5GR4UAiIiHqYQEBHxsLDbMBzqGkREiqJicY1hEREpXBoOEhHxMIWAiIiHhTQEzOzXZrbazHxm\n1uwc7TqZ2Toz22BmTxZmjUWJmSWY2XQzW29m08wsPp92PjNbZmbLzezzwq4z3J3v+2ZmMWY2xsw2\nmtlCM6seijqLggLMy35mtjf3+7jMzO4JRZ1FgZmNNLM9ZrbqHG3eyv1erjCzqwrSb6jXBL4DegCz\n82tgZhHAUKAj0BC408zqFU55Rc5TwAznXF1gJvB0Pu1OOOeaOeeaOue6F1554a+A37d7gYPOuTrA\nm8BrhVtl0XAB/7tjcr+PzZxz7xVqkUXL+/jnZZ7M7GagVu73ciAwvCCdhjQEnHPrnXMbgXNt1b4W\n2OicS3fOZQFjgG6FUmDR0w34IPf+B0B+C3idiCl/Bfm+nTmfxwHtC7G+oqSg/7v6PhaAc24ecOgc\nTboBH+a2XQzEm1nl8/Ub6jWBgkgGtp3xeHvuc/JLlZxzewCcc7uBivm0izWzb8xsgZkpUH+uIN+3\n022ccz7gsJmVL5zyipSC/u/eljt8MdbMLiuc0oqls+f3DgqwrLzkZxE1s6+AM9PIAAc845ybXJAu\n8njOs/u1nmN+PnsB3VR3zu02sxrATDNb5ZzbEsw6i7CCfN/ObmN5tJGCzctJwGjnXJaZDcS/hqU1\nq4tzUcvKSx4CzrkOAXaxHThzw9tlwM4A+yyyzjU/czcaVXbO7TGzKsDefPrYnft3i5mlAU0BhYBf\nQb5v24BqwE4ziwTKOufOtZruVeedl2fNt3eBVwuhruJqO/7v5X8UaFkZTsNB+Y0LLgFqm1mKmcUA\nvfD/epBfmgT0z73fD5h4dgMzK5c7HzGzRKAVsKawCiwCCvJ9m4x//gL8Bv9GePml887L3B8r/9EN\nfRfPx8h/WTkJ6AtgZi2Aw/8ZHj4n51zIbvg3XG4DTgG7gC9yn68KTDmjXSdgPbAReCqUNYfzDSgP\nzMidV18B5XKfbw6MyL3fElgFLAdWAv1DXXe43fL6vgHPA11y78cCY3NfXwRcHuqaw/VWgHn5ErA6\n9/v4NXBFqGsO1xswGv8v+wxgK3A3/r2A7jujzVDgh9z/7WYF6VenjRAR8bBwGg4SEZFCphAQEfEw\nhYCIiIcpBEREPEwhICLiYQoBEREPUwiIiHiYQkBExMP+P5UbIUfQ7Mr4AAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f88ff3a5f98>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "plot_sympy(taylor4, pts, label=\"taylor4\")\n",
        "plot_sympy(f, pts, label=\"f\")\n",
        "plt.legend(loc=\"best\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now write code to do this for any order `n`. Put the result in `taylor`:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 23,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "n = 20"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 26,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "taylor = 0\n",
        "for i in range(n):\n",
        "    taylor += f.diff(x, i).subs(x, 0)/factorial(i) * x**i"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 25,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
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            "text/latex": [
              "$$- \\frac{715 x^{18}}{65536} - \\frac{429 x^{16}}{32768} - \\frac{33 x^{14}}{2048} - \\frac{21 x^{12}}{1024} - \\frac{7 x^{10}}{256} - \\frac{5 x^{8}}{128} - \\frac{x^{6}}{16} - \\frac{x^{4}}{8} - \\frac{x^{2}}{2} + 1$$"
            ],
            "text/plain": [
              "       18        16       14       12      10      8    6    4    2    \n",
              "  715\u22c5x     429\u22c5x     33\u22c5x     21\u22c5x     7\u22c5x     5\u22c5x    x    x    x     \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\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500 - \u2500\u2500 - \u2500\u2500 - \u2500\u2500 + 1\n",
              "   65536     32768     2048     1024     256    128    16   8    2     "
            ]
          },
          "execution_count": 25,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "taylor"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 158,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f7cf1d64198>"
            ]
          },
          "execution_count": 158,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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gFdAB2AzMB250zq04qk0/4ALn3H0n6atYtwR278umxaN3syNhHlNv+ZjLGtcvtnmJiL89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\niIj4mEJARMTHFAIiIj6mEBAR8bH/B59DK25ALgQtAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f7cf2bc3b38>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "plot_sympy(taylor, pts, label=\"taylor\")\n",
        "plot_sympy(f, pts, label=\"f\")\n",
        "plt.legend(loc=\"best\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Behavior of the Error (Part II)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Let's repeat the setup, for clarity:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 166,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "f = 1/(20*x-10)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 167,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "n = 4\n",
        "\n",
        "taylor = 0\n",
        "for i in range(n):\n",
        "    taylor += f.diff(x, i).subs(x, 0)/factorial(i) * x**i\n",
        "    \n",
        "error = taylor - f"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 174,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f7cf1ba3f98>"
            ]
          },
          "execution_count": 174,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f7cf2bbea20>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "plot_sympy(error, pts, label=\"error\")\n",
        "plt.legend(loc=\"best\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "* What's the error at the expansion center?\n",
        "* How does the error depend on the distance to the expansion center?\n",
        "\n",
        "Let's look back at this Taylor expansion:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 169,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMwAAAAwBAMAAACvXIijAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMpm7du8iZolU\nq0RaI+fpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACu0lEQVRYCe2YP2gTURzHv9c0TY+r6eEqhYCi\ng0OlDpJFg+Dk0FZsB4twLt3EVCgWxEF0cJEGxDlVEasOjYIUEfEEFQShVYfaQqmbo/EfIrWe7+Vy\nl+t7v99LUJzMLfe77+e93yf3cknuAhCbM/maSOPIGR2O678pjmHUNP0h3phwy2wMc6axeQyWTLx1\nZjyb/Si/b72VYaR10QAFWnDN3Eg7hyJ8uhhV8d4eGS/EB/vi6g+KlBdP0hetD86vCDvx64kSft9z\nXmVTXpyU/bisF+ISvxRlN6Oi+b7v+CdlkH3dCxPxzpRnFIgLwFohDLOVbL1SBxHHGVXjdHj1YXlM\nae9xvxtrDr56oWFCEEaa5rbQWGeWP8zj8NllatoBN8TTQaDhkGgxoGqsotBsw43iADFYRtl1A2Yn\nqhoHQnMPT9yXjCa1aMDsRFXzTGpc7GQkwAgMmJ2oaKyK1ADfOU2m9mFhMTdR0fSsrq5dLqLzK6d5\nDrtkwNxERSO6d3t4mqlighSJb6JMiccs0TW9XvZnV9WeITVXHj14y2OeaBrn1Ma1pYnJO6QF/UHw\nxWIxS9LvPq/QDdtpewXaK9BsBbbuldsO9Aabt6qYaG2XbKAgypYwP6fZq2jz/24FdrmHfMNJ85gl\n1tjdgtZxOljXskTAY5LIG+gO38onOoTl7qWSliUCHlOkdgM9C9xPdAjLRS3ZFPCYJPIX+iT1OESO\nbph4TBKp2QA++o0OYbVy66gaJY95TBKhsb4JTTHZQ9YnzM+VPCaJ0NjikhqsqBqga4+eJRIeE0Rq\nxNlQmvSPRFO95DFBuEXbkks8YuoOHtNEaOQlsOArrbpzSAvEbTymidSMA3Ou0jANpHJKljjkMU2k\nRnw8jyRa1Ep7CLMlNWwc85gmUpPyiT8zrj4ebnTVKx5TpHYDbc2fK+iN/k3yGyAkBSIpbF2kAAAA\nAElFTkSuQmCC\n",
            "text/latex": [
              "$$- \\frac{4 x^{3}}{5} - \\frac{2 x^{2}}{5} - \\frac{x}{5} - \\frac{1}{10}$$"
            ],
            "text/plain": [
              "     3      2         \n",
              "  4\u22c5x    2\u22c5x    x   1 \n",
              "- \u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500 - \u2500 - \u2500\u2500\n",
              "   5      5     5   10"
            ]
          },
          "execution_count": 169,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "taylor"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "* To get an idea of what the error might look like, compute the Taylor polynomial up, store it in `next_taylor`.\n",
        "* Compare that to Taylor."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 190,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAvBAMAAABwAtUSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaJmUSrVGYizbvd\n7zLJaKAlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHklEQVQoFWNgQAbcyBwQu14AVYTRHk2ASR5N\n4DlYQEjZFabRACzgwMBWABHhmAASYG9g4FwAERBiAAkwb2BgPAARKHHx92BgYP0twNwAEWBgmA+y\nJf+HBQODkNpUZQUGjv3pQAHm/wsZGA04DvEsgKori/kbwCzA/JF9AkSAI4FB/yIjA9cGqDwD7wQG\nvu8MDPwKMAF5ICODgeH9A5gAUAXDTnYBfQYmkJVAwLqQgUMhvmA/wwwIn4HhmbMrg5iSkEcDTGCQ\n0P/RAC2dxaXI545iPu///wYoAlxmmih8BjZULgMWgQ6VByiKuAU4v6IIADl70AX8gTGGAFkMDPYB\nCC4Dwy0GBn9oVEKEHRgYTiErYJjCwHETRYDTOQWqAwDOp0Al2CEtBwAAAABJRU5ErkJggg==\n",
            "text/latex": [
              "$$\\frac{8 x^{4}}{5}$$"
            ],
            "text/plain": [
              "   4\n",
              "8\u22c5x \n",
              "\u2500\u2500\u2500\u2500\n",
              " 5  "
            ]
          },
          "execution_count": 190,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "next_taylor = 0\n",
        "for i in range(n+1):\n",
        "    next_taylor += f.diff(x, i).subs(x, 0)/factorial(i) * x**i\n",
        "    \n",
        "taylor-next_taylor"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "How does that difference compare to the actual error?"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 175,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f7cf00b54e0>"
            ]
          },
          "execution_count": 175,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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9QR43blyJj+WM84R1QF2lVGTuVUCDgMWX7PMNcE/u4/7AT04oVwghhIMcrgnk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\nEMLGJAkIIYSN/X+MDCQ2aOpvjwAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f7cf0326dd8>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "plot_sympy(taylor-next_taylor, pts, label=\"diff to next\")\n",
        "plot_sympy(error, pts, label=\"error\")\n",
        "plt.legend(loc=\"best\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "* Far away from the expansion center, it doesn't look like they have much to do with each other.\n",
        "* To get a better idea of what happens close to the center, use a log-log plot:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 189,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.text.Text at 0x7f7ceb075e80>"
            ]
          },
          "execution_count": 189,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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ylM9kKZ/xCPW/tEJ6lywD+q039gVwetmjS9Bnn8EZZ8DChX5vrd12Cx2RiEjy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\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f7ceb152cc0>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "pos_pts = 10**np.linspace(-3, 0.5)\n",
        "err_values = [abs(error.subs(x, pt)) for pt in pos_pts]\n",
        "plt.loglog(pos_pts, err_values)\n",
        "plt.grid()\n",
        "plt.xlabel(\"$x$\")\n",
        "plt.ylabel(\"Error\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "* What is the behavior of the error as we get very close to the expansion center?\n",
        "* Is there any predictive power in what we just observed?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Predicting Taylor Error (Part III)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 46,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAccAAAAwBAMAAACWMqyUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXzBi47DEj1q9vTs9Tf4/TP4Pj2kV/Mn0+6sAAAAASUVORK5CYII=\n",
            "text/latex": [
              "$$\\frac{\\sqrt{2}}{128} \\left(x - 12\\right)^{3} - \\frac{\\sqrt{2}}{32} \\left(x - 12\\right)^{2} + \\frac{\\sqrt{2}}{4} \\left(x - 12\\right) + \\sqrt{2}$$"
            ],
            "text/plain": [
              "           3              2                   \n",
              "\u221a2\u22c5(x - 12)    \u221a2\u22c5(x - 12)    \u221a2\u22c5(x - 12)     \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u221a2\n",
              "    128             32             4          "
            ]
          },
          "execution_count": 46,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "f = sp.sqrt(x-10)\n",
        "t = sum([f.diff(x, i).subs(x, 12)/factorial(i) * (x-12)**i for i in range(4)])\n",
        "t"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 47,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nvqMBk+pHKsYAAAAASUVORK5CYII=\n",
            "text/latex": [
              "$$1.58113883008419$$"
            ],
            "text/plain": [
              "1.58113883008419"
            ]
          },
          "execution_count": 47,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "f.subs(x, 12.5)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 48,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJYAAAAPBAMAAAD0RitaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nPBuGOzAxcKMs9yQco7l0QUwN/p/jDycI5Cea2vLTAAAAAElFTkSuQmCC\n",
            "text/latex": [
              "$$1.5813227821457$$"
            ],
            "text/plain": [
              "1.58132278214570"
            ]
          },
          "execution_count": 48,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "t.subs(x, 12.5).evalf()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 59,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMoAAAAPBAMAAABXbk2cAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
            "text/latex": [
              "$$0.000183952061507453$$"
            ],
            "text/plain": [
              "0.000183952061507453"
            ]
          },
          "execution_count": 59,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "error1 = f.subs(x, 12.5) - t.subs(x, 12.5).evalf()\n",
        "abs(error1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now predict the error at $12.25$:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 60,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAANgAAAAVBAMAAADWeD20AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
            "text/latex": [
              "$$1.14970038442158 \\cdot 10^{-5}$$"
            ],
            "text/plain": [
              "1.14970038442158e-5"
            ]
          },
          "execution_count": 60,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "abs(error1) * (0.25/0.5)**4"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 61,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAANgAAAAVBAMAAADWeD20AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nvP4DlF4+EdGgVBYAAAAASUVORK5CYII=\n",
            "text/latex": [
              "$$1.24076489040892 \\cdot 10^{-5}$$"
            ],
            "text/plain": [
              "1.24076489040892e-5"
            ]
          },
          "execution_count": 61,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "error2 = f.subs(x, 12.25) - t.subs(x, 12.25).evalf()\n",
        "abs(error2)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": []
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.5.2+"
    },
    "widgets": {
      "state": {},
      "version": "1.1.2"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}