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 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Solutions to Quiz 06"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Imports\n",
      "import math"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Note to the T.A.s** : Only the output of this file should match with the student's answers. Note that they are free to use calculators to do the problems. This code is just so that I don't have to do the computations myself."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Solution to Problem 1 for each batch\n",
      "def solve1(f, b, e, l) :\n",
      "    \"\"\"Solves problem 1\n",
      "    f : the function\n",
      "    b : integrate from\n",
      "    e : integrate to\n",
      "    l : list of number of divisions of the interval to apply\n",
      "        trapezoidal rule.\n",
      "    \"\"\"\n",
      "    for d in l :\n",
      "        print \"Computing using %d divisions.\\n__________________________\" \\\n",
      "        % d\n",
      "        h = float(e - b)/d\n",
      "        integral = 0\n",
      "        for i in range(d) :\n",
      "            print \"Computing trapezoid %d...\" % (i + 1)\n",
      "            base1 = b + i * h\n",
      "            base2 = b + (i+1) * h\n",
      "            f1 = f(base1)\n",
      "            f2 = f(base2)\n",
      "            print \"Corners of the trapezoid :\", (base1, 0), (base2, 0),\\\n",
      "            (base2, f2), (base1, f1),\n",
      "            a = 0.5 * (f2 + f1) * h\n",
      "            print \"\\tArea of the trapezoid :\",  a\n",
      "            integral += a\n",
      "        print \"Integral with %d divisions = %g.\" % (d, integral)\n",
      "        print \"-\"*40\n",
      "        print\n",
      "    return None"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Solution to problem 2 for each batch\n",
      "def solve2(f, x, lh) :\n",
      "    \"\"\" Solves problem 2\n",
      "    f : Function to differentiate\n",
      "    x : point at which we want the derivative\n",
      "    lh : lists of deviations, h to consider\n",
      "    \"\"\"\n",
      "    \n",
      "    for h in lh :\n",
      "        fxph = f(x + h) # f (of) x plus h\n",
      "        fxmh = f(x - h) # f (of) x minus h\n",
      "        dfwh = float(fxph - fxmh)/(2*h) # derivative (of) f with h\n",
      "        print \"h = %6g, f(x + h) = %15g, f(x - h) = %15g\" % (h, fxph, fxmh)\n",
      "        print \"Approximate derivative = (f(x + h) - f(x - h))/(2h) = %g\" % dfwh\n",
      "        print \"-\"*40\n",
      "        print\n",
      "    return None"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Common variables\n",
      "l = [2, 3, 4]\n",
      "lh = [.5, .001]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Solutions for batch 1"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \"Solution to problem 1\"\n",
      "print \"_____________________\"\n",
      "solve1(lambda x : x*x, 0, 1, l)\n",
      "print\n",
      "print \"Solution to problem 2\"\n",
      "print \"_____________________\"\n",
      "solve2(lambda x : x**4, 1, lh)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Solution to problem 1\n",
        "_____________________\n",
        "Computing using 2 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (0.0, 0) (0.5, 0) (0.5, 0.25) (0.0, 0.0) \tArea of the trapezoid : 0.0625\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (0.5, 0) (1.0, 0) (1.0, 1.0) (0.5, 0.25) \tArea of the trapezoid : 0.3125\n",
        "Integral with 2 divisions = 0.375.\n",
        "----------------------------------------\n",
        "\n",
        "Computing using 3 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (0.0, 0) (0.3333333333333333, 0) (0.3333333333333333, 0.1111111111111111) (0.0, 0.0) \tArea of the trapezoid : 0.0185185185185\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (0.3333333333333333, 0) (0.6666666666666666, 0) (0.6666666666666666, 0.4444444444444444) (0.3333333333333333, 0.1111111111111111) \tArea of the trapezoid : 0.0925925925926\n",
        "Computing trapezoid 3...\n",
        "Corners of the trapezoid : (0.6666666666666666, 0) (1.0, 0) (1.0, 1.0) (0.6666666666666666, 0.4444444444444444) \tArea of the trapezoid : 0.240740740741\n",
        "Integral with 3 divisions = 0.351852.\n",
        "----------------------------------------\n",
        "\n",
        "Computing using 4 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (0.0, 0) (0.25, 0) (0.25, 0.0625) (0.0, 0.0) \tArea of the trapezoid : 0.0078125\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (0.25, 0) (0.5, 0) (0.5, 0.25) (0.25, 0.0625) \tArea of the trapezoid : 0.0390625\n",
        "Computing trapezoid 3...\n",
        "Corners of the trapezoid : (0.5, 0) (0.75, 0) (0.75, 0.5625) (0.5, 0.25) \tArea of the trapezoid : 0.1015625\n",
        "Computing trapezoid 4...\n",
        "Corners of the trapezoid : (0.75, 0) (1.0, 0) (1.0, 1.0) (0.75, 0.5625) \tArea of the trapezoid : 0.1953125\n",
        "Integral with 4 divisions = 0.34375.\n",
        "----------------------------------------\n",
        "\n",
        "\n",
        "Solution to problem 2\n",
        "_____________________\n",
        "h =    0.5, f(x + h) =          5.0625, f(x - h) =          0.0625\n",
        "Approximate derivative = (f(x + h) - f(x - h))/(2h) = 5\n",
        "----------------------------------------\n",
        "\n",
        "h =  0.001, f(x + h) =         1.00401, f(x - h) =        0.996006\n",
        "Approximate derivative = (f(x + h) - f(x - h))/(2h) = 4\n",
        "----------------------------------------\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Solutions for batch 2"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \"Solution to problem 1\"\n",
      "print \"_____________________\"\n",
      "solve1(lambda x : 1.0/x, 1, 2, l)\n",
      "print\n",
      "print \"Solution to problem 2\"\n",
      "print \"_____________________\"\n",
      "solve2(lambda x : x*x*(x-1), 1, lh)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Solution to problem 1\n",
        "_____________________\n",
        "Computing using 2 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (1.0, 0) (1.5, 0) (1.5, 0.6666666666666666) (1.0, 1.0) \tArea of the trapezoid : 0.416666666667\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (1.5, 0) (2.0, 0) (2.0, 0.5) (1.5, 0.6666666666666666) \tArea of the trapezoid : 0.291666666667\n",
        "Integral with 2 divisions = 0.708333.\n",
        "----------------------------------------\n",
        "\n",
        "Computing using 3 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (1.0, 0) (1.3333333333333333, 0) (1.3333333333333333, 0.75) (1.0, 1.0) \tArea of the trapezoid : 0.291666666667\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (1.3333333333333333, 0) (1.6666666666666665, 0) (1.6666666666666665, 0.6000000000000001) (1.3333333333333333, 0.75) \tArea of the trapezoid : 0.225\n",
        "Computing trapezoid 3...\n",
        "Corners of the trapezoid : (1.6666666666666665, 0) (2.0, 0) (2.0, 0.5) (1.6666666666666665, 0.6000000000000001) \tArea of the trapezoid : 0.183333333333\n",
        "Integral with 3 divisions = 0.7.\n",
        "----------------------------------------\n",
        "\n",
        "Computing using 4 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (1.0, 0) (1.25, 0) (1.25, 0.8) (1.0, 1.0) \tArea of the trapezoid : 0.225\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (1.25, 0) (1.5, 0) (1.5, 0.6666666666666666) (1.25, 0.8) \tArea of the trapezoid : 0.183333333333\n",
        "Computing trapezoid 3...\n",
        "Corners of the trapezoid : (1.5, 0) (1.75, 0) (1.75, 0.5714285714285714) (1.5, 0.6666666666666666) \tArea of the trapezoid : 0.154761904762\n",
        "Computing trapezoid 4...\n",
        "Corners of the trapezoid : (1.75, 0) (2.0, 0) (2.0, 0.5) (1.75, 0.5714285714285714) \tArea of the trapezoid : 0.133928571429\n",
        "Integral with 4 divisions = 0.697024.\n",
        "----------------------------------------\n",
        "\n",
        "\n",
        "Solution to problem 2\n",
        "_____________________\n",
        "h =    0.5, f(x + h) =           1.125, f(x - h) =          -0.125\n",
        "Approximate derivative = (f(x + h) - f(x - h))/(2h) = 1.25\n",
        "----------------------------------------\n",
        "\n",
        "h =  0.001, f(x + h) =        0.001002, f(x - h) =    -0.000998001\n",
        "Approximate derivative = (f(x + h) - f(x - h))/(2h) = 1\n",
        "----------------------------------------\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Solutions for batch 3"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \"Solution to problem 1\"\n",
      "print \"_____________________\"\n",
      "solve1(lambda x : 3*x*x + 2*x, 0, 1, l)\n",
      "print\n",
      "print \"Solution to problem 2\"\n",
      "print \"_____________________\"\n",
      "solve2(lambda x : math.sin(x), math.pi/3.0, lh)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Solution to problem 1\n",
        "_____________________\n",
        "Computing using 2 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (0.0, 0) (0.5, 0) (0.5, 1.75) (0.0, 0.0) \tArea of the trapezoid : 0.4375\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (0.5, 0) (1.0, 0) (1.0, 5.0) (0.5, 1.75) \tArea of the trapezoid : 1.6875\n",
        "Integral with 2 divisions = 2.125.\n",
        "----------------------------------------\n",
        "\n",
        "Computing using 3 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (0.0, 0) (0.3333333333333333, 0) (0.3333333333333333, 1.0) (0.0, 0.0) \tArea of the trapezoid : 0.166666666667\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (0.3333333333333333, 0) (0.6666666666666666, 0) (0.6666666666666666, 2.6666666666666665) (0.3333333333333333, 1.0) \tArea of the trapezoid : 0.611111111111\n",
        "Computing trapezoid 3...\n",
        "Corners of the trapezoid : (0.6666666666666666, 0) (1.0, 0) (1.0, 5.0) (0.6666666666666666, 2.6666666666666665) \tArea of the trapezoid : 1.27777777778\n",
        "Integral with 3 divisions = 2.05556.\n",
        "----------------------------------------\n",
        "\n",
        "Computing using 4 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (0.0, 0) (0.25, 0) (0.25, 0.6875) (0.0, 0.0) \tArea of the trapezoid : 0.0859375\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (0.25, 0) (0.5, 0) (0.5, 1.75) (0.25, 0.6875) \tArea of the trapezoid : 0.3046875\n",
        "Computing trapezoid 3...\n",
        "Corners of the trapezoid : (0.5, 0) (0.75, 0) (0.75, 3.1875) (0.5, 1.75) \tArea of the trapezoid : 0.6171875\n",
        "Computing trapezoid 4...\n",
        "Corners of the trapezoid : (0.75, 0) (1.0, 0) (1.0, 5.0) (0.75, 3.1875) \tArea of the trapezoid : 1.0234375\n",
        "Integral with 4 divisions = 2.03125.\n",
        "----------------------------------------\n",
        "\n",
        "\n",
        "Solution to problem 2\n",
        "_____________________\n",
        "h =    0.5, f(x + h) =        0.999722, f(x - h) =        0.520296\n",
        "Approximate derivative = (f(x + h) - f(x - h))/(2h) = 0.479426\n",
        "----------------------------------------\n",
        "\n",
        "h =  0.001, f(x + h) =        0.866525, f(x - h) =        0.865525\n",
        "Approximate derivative = (f(x + h) - f(x - h))/(2h) = 0.5\n",
        "----------------------------------------\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Solutions for batch 4"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \"Solution to problem 1\"\n",
      "print \"_____________________\"\n",
      "solve1(lambda x : 1/(x*x), 1, 2, l)\n",
      "print\n",
      "print \"Solution to problem 2\"\n",
      "print \"_____________________\"\n",
      "solve2(lambda x : math.log(x), 1, lh)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Solution to problem 1\n",
        "_____________________\n",
        "Computing using 2 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (1.0, 0) (1.5, 0) (1.5, 0.4444444444444444) (1.0, 1.0) \tArea of the trapezoid : 0.361111111111\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (1.5, 0) (2.0, 0) (2.0, 0.25) (1.5, 0.4444444444444444) \tArea of the trapezoid : 0.173611111111\n",
        "Integral with 2 divisions = 0.534722.\n",
        "----------------------------------------\n",
        "\n",
        "Computing using 3 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (1.0, 0) (1.3333333333333333, 0) (1.3333333333333333, 0.5625) (1.0, 1.0) \tArea of the trapezoid : 0.260416666667\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (1.3333333333333333, 0) (1.6666666666666665, 0) (1.6666666666666665, 0.3600000000000001) (1.3333333333333333, 0.5625) \tArea of the trapezoid : 0.15375\n",
        "Computing trapezoid 3...\n",
        "Corners of the trapezoid : (1.6666666666666665, 0) (2.0, 0) (2.0, 0.25) (1.6666666666666665, 0.3600000000000001) \tArea of the trapezoid : 0.101666666667\n",
        "Integral with 3 divisions = 0.515833.\n",
        "----------------------------------------\n",
        "\n",
        "Computing using 4 divisions.\n",
        "__________________________\n",
        "Computing trapezoid 1...\n",
        "Corners of the trapezoid : (1.0, 0) (1.25, 0) (1.25, 0.64) (1.0, 1.0) \tArea of the trapezoid : 0.205\n",
        "Computing trapezoid 2...\n",
        "Corners of the trapezoid : (1.25, 0) (1.5, 0) (1.5, 0.4444444444444444) (1.25, 0.64) \tArea of the trapezoid : 0.135555555556\n",
        "Computing trapezoid 3...\n",
        "Corners of the trapezoid : (1.5, 0) (1.75, 0) (1.75, 0.32653061224489793) (1.5, 0.4444444444444444) \tArea of the trapezoid : 0.0963718820862\n",
        "Computing trapezoid 4...\n",
        "Corners of the trapezoid : (1.75, 0) (2.0, 0) (2.0, 0.25) (1.75, 0.32653061224489793) \tArea of the trapezoid : 0.0720663265306\n",
        "Integral with 4 divisions = 0.508994.\n",
        "----------------------------------------\n",
        "\n",
        "\n",
        "Solution to problem 2\n",
        "_____________________\n",
        "h =    0.5, f(x + h) =        0.405465, f(x - h) =       -0.693147\n",
        "Approximate derivative = (f(x + h) - f(x - h))/(2h) = 1.09861\n",
        "----------------------------------------\n",
        "\n",
        "h =  0.001, f(x + h) =       0.0009995, f(x - h) =      -0.0010005\n",
        "Approximate derivative = (f(x + h) - f(x - h))/(2h) = 1\n",
        "----------------------------------------\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 8
    }
   ],
   "metadata": {}
  }
 ]
}