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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Quiz 2 : Batch 2"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Find the value of $\\pi$ upto $5$ decimal places by doing the following steps.\n",
      "\n",
      "1. Write a **function** to compute arctan of a number $x$, $-1 < x < 1$ using the formula\n",
      "    \\begin{equation}\n",
      "        \\arctan(x) = x - \\frac{x^3}{3} + \\frac{x^5}{5} - \\dotsb\n",
      "    \\end{equation}\n",
      "    The function should have $x$ and a positive integer $N$ as input. $N$ is the number of terms of the series to compute. Call this function \n",
      "    *compute_arctan*.\n",
      "    \n",
      "1. Then use this function (with $N = 20$) to compute the value of pi using the following formula :\n",
      "    \\begin{equation}\n",
      "        \\pi = 4 * \\arctan\\left(\\frac{1}{2}\\right) + 4 * \\arctan\\left(\\frac{1}{3}\\right).\n",
      "    \\end{equation}\n",
      "    \n",
      "1. Print the answer to 5 decimal places."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "####Grading scheme.\n",
      "* Total points : 10 marks\n",
      "* -5 if any kind of loop (while or for) is not used to compute the sum\n",
      "* -2 if the logic is correct but the code is not running due to syntax error\n",
      "* -1 is not upto the specification of 5 decimal spaces."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def compute_arctan(x, N) :\n",
      "    t = float(x)\n",
      "    s = 0.0\n",
      "    d = 1.0\n",
      "    for i in range(N) :\n",
      "        s += t/d\n",
      "        t *= - x*x\n",
      "        d += 2\n",
      "    return s"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "val_of_pi = 4 * compute_arctan(1.0/2, 20) + 4 * compute_arctan(1.0/3, 20)\n",
      "print \"The value of pi is %7.5f.\" % val_of_pi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The value of pi is 3.14159.\n"
       ]
      }
     ],
     "prompt_number": 2
    }
   ],
   "metadata": {}
  }
 ]
}