{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Lab 8 Taylor Series"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.9999987317275388\n"
     ]
    }
   ],
   "source": [
    "from math import factorial\n",
    "\n",
    "approx = 1\n",
    "x = 3.14\n",
    "for i in range(1,20):\n",
    "    term = ((-1)**i)  * ((x**(i*2)) / (factorial(i*2)))\n",
    "    approx = approx + term\n",
    "print(approx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.9999987317275395"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import math\n",
    "math.cos(3.14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Volume in drive C is Windows\n",
      " Volume Serial Number is B899-AB94\n",
      "\n",
      " Directory of C:\\Users\\student\\Desktop\n",
      "\n",
      "11/20/2019  09:58 AM    <DIR>          .\n",
      "11/20/2019  09:58 AM    <DIR>          ..\n",
      "11/20/2019  09:35 AM    <DIR>          .ipynb_checkpoints\n",
      "11/20/2019  09:57 AM    <DIR>          __pycache__\n",
      "11/20/2019  08:56 AM             1,457 Kaltura Capture.lnk\n",
      "05/13/2016  12:33 PM               210 MSDS Online.url\n",
      "11/20/2019  09:56 AM               352 myfuncs.py\n",
      "08/01/2018  03:14 PM    <DIR>          Safety Information\n",
      "11/20/2019  08:49 AM             1,490 Week10_notes_numerical_methods.ipynb\n",
      "11/20/2019  09:58 AM             4,418 Week9_Notes_Taylor_Series_Lab_Intro.ipynb\n",
      "               5 File(s)          7,927 bytes\n",
      "               5 Dir(s)  321,499,787,264 bytes free\n"
     ]
    }
   ],
   "source": [
    "%ls"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### In a file called myfuncs.py\n",
    "\n",
    "```\n",
    "# myfuncs.py\n",
    "\"\"\"\n",
    "A python that contains a function that calculates the approximation of the cosine function using a taylor series\n",
    "\"\"\"\n",
    "\n",
    "from math import factorial\n",
    "\n",
    "def approx_cos(x,n_terms):\n",
    "    approx = 1\n",
    "    for i in range(1,n_terms+1):\n",
    "        term = ((-1)**i)  * ((x**(i*2)) / (factorial(i*2)))\n",
    "        approx = approx + term\n",
    "    \n",
    "    return approx\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "import myfuncs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.9999987317275388"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from myfuncs import approx_cos\n",
    "\n",
    "approx_cos(3.14,19)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "after 1 terms, the difference between cos(x) and aprox func is 2.9298012682724606\n",
      "after 2 terms, the difference between cos(x) and aprox func is 1.1206867383942072\n",
      "after 3 terms, the difference between cos(x) and aprox func is 0.2105196466234821\n",
      "after 4 terms, the difference between cos(x) and aprox func is 0.023858254692953773\n",
      "after 5 terms, the difference between cos(x) and aprox func is 0.0018181048161521884\n",
      "after 6 terms, the difference between cos(x) and aprox func is 9.976362487795765e-05\n",
      "after 7 terms, the difference between cos(x) and aprox func is 4.134263480115585e-06\n",
      "after 8 terms, the difference between cos(x) and aprox func is 1.3403493681618528e-07\n",
      "after 9 terms, the difference between cos(x) and aprox func is 3.493543787591591e-09\n",
      "after 10 terms, the difference between cos(x) and aprox func is 7.481359975969326e-11\n",
      "after 11 terms, the difference between cos(x) and aprox func is 1.3391510123028638e-12\n",
      "after 12 terms, the difference between cos(x) and aprox func is 2.1094237467877974e-14\n",
      "after 13 terms, the difference between cos(x) and aprox func is 4.440892098500626e-16\n",
      "after 14 terms, the difference between cos(x) and aprox func is 6.661338147750939e-16\n",
      "after 15 terms, the difference between cos(x) and aprox func is 6.661338147750939e-16\n",
      "after 16 terms, the difference between cos(x) and aprox func is 6.661338147750939e-16\n",
      "after 17 terms, the difference between cos(x) and aprox func is 6.661338147750939e-16\n",
      "after 18 terms, the difference between cos(x) and aprox func is 6.661338147750939e-16\n",
      "after 19 terms, the difference between cos(x) and aprox func is 6.661338147750939e-16\n"
     ]
    }
   ],
   "source": [
    "for n in range(1,20):\n",
    "    approx = approx_cos(3.14,n)\n",
    "    exact = math.cos(3.14)\n",
    "    print(f'after {n} terms, the difference between cos(x) and aprox func is {abs(exact-approx)}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}