{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Python Exam\n",
    "\n",
    "This Python exam covers the following topics:\n",
    "\n",
    "* Problem 1: Variable types and formats\n",
    "* Problem 2: Debugging\n",
    "* Problem 3: Functions, Conditionals, and Loops\n",
    "* Problem 4: Solve Linear Equations\n",
    "* Problem 5: Solve Nonlinear Equations\n",
    "* Problem 6: Integration\n",
    "* Problem 7: Differentiate\n",
    "* Problem 8: Interpolation\n",
    "* Problem 9: Polynomial Regression\n",
    "* Problem 10: General Nonlinear Regression\n",
    "* Problem 11: Solve Differential Equation\n",
    "* Problem 12: Optimization\n",
    "\n",
    "A template for each problem is provided with instructions on what is needed. In some cases, the problem is to debug a section of code to produce a specific result. In other cases, the problem is to import a module and produce a result. This exam allows open access to any online resources, prior homework solutions, or other files."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Current conda install:\n",
      "\n",
      "               platform : win-64\n",
      "          conda version : 4.3.30\n",
      "       conda is private : False\n",
      "      conda-env version : 4.3.30\n",
      "    conda-build version : 3.0.27\n",
      "         python version : 3.6.3.final.0\n",
      "       requests version : 2.18.4\n",
      "       root environment : C:\\Anaconda3  (writable)\n",
      "    default environment : C:\\Anaconda3\n",
      "       envs directories : C:\\Anaconda3\\envs\n",
      "                          C:\\Users\\John Hedengren\\AppData\\Local\\conda\\conda\\envs\n",
      "                          C:\\Users\\John Hedengren\\.conda\\envs\n",
      "          package cache : C:\\Anaconda3\\pkgs\n",
      "                          C:\\Users\\John Hedengren\\AppData\\Local\\conda\\conda\\pkgs\n",
      "           channel URLs : https://repo.continuum.io/pkgs/main/win-64\n",
      "                          https://repo.continuum.io/pkgs/main/noarch\n",
      "                          https://repo.continuum.io/pkgs/free/win-64\n",
      "                          https://repo.continuum.io/pkgs/free/noarch\n",
      "                          https://repo.continuum.io/pkgs/r/win-64\n",
      "                          https://repo.continuum.io/pkgs/r/noarch\n",
      "                          https://repo.continuum.io/pkgs/pro/win-64\n",
      "                          https://repo.continuum.io/pkgs/pro/noarch\n",
      "                          https://repo.continuum.io/pkgs/msys2/win-64\n",
      "                          https://repo.continuum.io/pkgs/msys2/noarch\n",
      "            config file : C:\\Users\\John Hedengren\\.condarc\n",
      "             netrc file : None\n",
      "           offline mode : False\n",
      "             user-agent : conda/4.3.30 requests/2.18.4 CPython/3.6.3 Windows/10 Windows/10.0.17134    \n",
      "          administrator : False\n"
     ]
    }
   ],
   "source": [
    "!conda info"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: gekko in c:\\anaconda3\\lib\\site-packages (0.0.4rc3)\n",
      "Requirement already satisfied: numpy>=1.8 in c:\\anaconda3\\lib\\site-packages (from gekko) (1.13.3)\n",
      "Requirement already satisfied: flask in c:\\anaconda3\\lib\\site-packages (from gekko) (0.12.2)\n",
      "Requirement already satisfied: Werkzeug>=0.7 in c:\\anaconda3\\lib\\site-packages (from flask->gekko) (0.12.2)\n",
      "Requirement already satisfied: Jinja2>=2.4 in c:\\anaconda3\\lib\\site-packages (from flask->gekko) (2.9.6)\n",
      "Requirement already satisfied: itsdangerous>=0.21 in c:\\anaconda3\\lib\\site-packages (from flask->gekko) (0.24)\n",
      "Requirement already satisfied: click>=2.0 in c:\\anaconda3\\lib\\site-packages (from flask->gekko) (6.7)\n",
      "Requirement already satisfied: MarkupSafe>=0.23 in c:\\anaconda3\\lib\\site-packages (from Jinja2>=2.4->flask->gekko) (1.0)\n"
     ]
    }
   ],
   "source": [
    "# install new packages with \"!pip install module-name\"\n",
    "!pip install gekko"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Hello, World!\n",
      "Hello, BYU Student!\n"
     ]
    }
   ],
   "source": [
    "print('Hello, World!')\n",
    "\n",
    "myName = 'BYU Student'  # input('Name: ')\n",
    "print('Hello, ' + myName + '!')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 1: Variable types and formats\n",
    "\n",
    "#### Part A\n",
    "Add **x=5** and **y='4'** together to produce a string (54) in addition to the example code of below of adding them as integers (9). In addition to printing the results, show the type of variable with **type()** to confirm that the result is a string.\n",
    "\n",
    "Hint: An integer is converted to a string with **str()** and a string is converted to an integer with **int()**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5\n",
      "<class 'int'>\n",
      "23\n",
      "<class 'str'>\n"
     ]
    }
   ],
   "source": [
    "x = 2\n",
    "y = '3'\n",
    "\n",
    "# adding two integers together (5)\n",
    "z = x + int(y)\n",
    "# show the result of adding 2+3\n",
    "print(z)\n",
    "# show that z is an integer\n",
    "print(type(z))\n",
    "\n",
    "# add strings\n",
    "z = str(x)+y\n",
    "print(z)\n",
    "print(type(z))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Part B\n",
    "\n",
    "Show the result of $y = exp(0.0)$ and $z = 1.0$ to 20 decimal places. Is there a difference in the results?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "NumPy version:\n",
      "1.13.3\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "print('NumPy version:')\n",
    "print(np.__version__)\n",
    "\n",
    "y = np.exp(0.0)\n",
    "z = 1.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y = 1.00000000000000000000\n",
      "z = 1.00000000000000000000\n",
      "Another way to format numbers\n",
      "z = 1.00000000000000000000\n",
      "Difference\n",
      "0.00000000000000000000e+00\n"
     ]
    }
   ],
   "source": [
    "#### Solution\n",
    "print('y = {:.20f}'.format(y))\n",
    "print('z = {:.20f}'.format(z))\n",
    "\n",
    "print('Another way to format numbers')\n",
    "print('z = %.20f' %z)\n",
    "\n",
    "# difference\n",
    "print('Difference')\n",
    "print('{:.20e}'.format(y-z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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N7tOunURl1FVzekAyQT/5RAIdtMSZKR5goXu58d13ErGrbaVQQKasittVxsYC\nzZopleA5jRpJ5EGAfPetMAxbAFQjoipEFAagJ4AM0UVEVBfAJxCjcCrd9uJElM/xuBSApgD2WqDJ\nfcqVkxRkheER+fMDgwb5tKilb6lRQxKkli5VrcSv3HefGIc2bVQr8ZDu3YGJE1WrQL16EvF84IBq\nJR4QEiLrMytWSJKs5nhtGJg5FcBQAKsB7AMwn5n3ENE4InJGGb0JoBCABZnCUqsDiCeiHQC+BfAa\nM/vfMHz3nUyjbeDcv3hRZg1a5ooRyQrsQw+pVuJXQkNlpKtl0bzERMlat4FvPMqxwqjtuCIyUn7A\nWk8dBWINyxhERERwfHy8dQfs2hX46Sf5kYSozfk7e1Z6jb/4ouZ9cYOELVuA2bOB0aM1rcD88cfA\nkCHSf8EGzanr1ZPiAz/9pFqJB1y9KqHuY8eKb9GGENFWx5pujpjM56tXpVBRZKRyowBIKPSBA3Kh\n0ZbERPmbBgGzZ8u1VcvZAiDD8zvusIVRADSPXwgPl3wGmxoFd1B/JVTNunUSRG2j8qaVK2ucCQoA\n48aJ3zoAfK05wSzX1datpSKIdpw9K2XTo7KNEvc70dHyd12mtgaCd6SmSuEsjTGGIS7Op72dPSEt\nDXj4YeDdd1Ur8ZCoKPlhBICvNSd27ZJkYRtdV93j/HkZ3dqouNNddwG33qrxOsO1a5Lw9sorqpV4\nhTEMCQnSrSpfPtVK/iEkBNi/X3lYuee0aiXdvwI82S0uTmZ2nTqpVuIhFSuKL6x+fdVK/oFIZg3r\n1gHnzqlW4wFhYWLdtLVsgjEMa9fa8gocEwNs3w78/rtqJR6QL5/0C46L82sJZ3+zdKmEr998s2ol\nHpCcLKMPGxIdLQGCK1fmvq8tiYqSxfxff1WtxGOC2zA4I7Kc/TVthPa5YlFRQFKSpnG3uXP0qNSd\ns9HSlHt8840sONvQ3deo0b99LbSksyNKX9sfb7AbhoYNgfHjVavIkqpVgZo1Nf5uRUZKGfPatVUr\n8QnOxVFt1xeWLhV3X9OmqpXcQEiIJAs6C/ZqR4UKEnersTspeA3Dr79KEHrRoqqVZEtkJLBhg6b1\nYwoU0NTH4hpxcbaK8nSPtDT5ADZbW0vP2bMSsr1hg2olHjJ+vNYL0MFrGJxDcRv7AiIjxUW/fLlq\nJR6yd6+sNQRYZzdmmQg98ohqJR6yeTNw8qStpzv58wOTJwM//6xaiYfcf7+tIh3dJY9qAcqIi5OO\n7RUq5L6vIurVA265RaT26aNajQcUKyYriE2aSKRGgEAEvPlm7vvZlqVLgTx5xGjblHz5gGPHbLn8\n5zpbt0oEyYABqpW4TXDOGP4yVuvYAAAgAElEQVT8U9IrbTxbAMTX2rmzJBFfvapajQfccousJAZY\n2Or+/ZoHW40eLQbb5h3HnEZBw6o9wuzZ0ppOw2S34DQMADBypBap6336AM8+K3kzWhIVJeE7R4/m\nvq8GJCcDERHA8OGqlXhBsWKSrm1z0tJkEXrsWNVKPCQqSn64GpaHCU7DcPPNwKuvAtWrq1aSK02a\nAP/3f9IhTEucszKtaxz8CxEwZYrGLTznzAEmTNBiGO4sXbZokVodHtOkidTQ1zA6KfgMw8WLEiSt\n0RD88mVpDJWWplqJB9x5p8zMFLWMtJqwMKBHD1n/0ZL33hMXhybFuCIjgd9+k5t2hIZKWvzy5bYo\n6e8OwWcYVq+WiAGN6vouWSLfL20jNObPB3r2VK3Ca5iB998H/vhDtRIPOX5cIpJsHI2UGe1zxaKi\n5IujWRa0JYaBiNoR0W9EdJCIRmXxej4imud4fTMRVU732mjH9t+I6H4r9ORIXBxQogRw770+P5VV\ndOwIrFmj8SgVkJnasWOqVXjF1q3AsGHA+vWqlXiI051n86CL9FSsKN12tTUM7dtLBYC771atxC28\nNgxEFArgIwDtAdQA0IuIamTabQCAv5n5NgATALzueG8NSCvQuwC0A/Cx43i+ITVVfDIdO0q4niYU\nLSprhXnzqlbiIczS9vOZZ1Qr8Yq4OPF7d+igWomHxMVJ6VLNQocjI4GNGyX1Qjvy5tUyhduKGUMD\nAAeZOYGZrwGYCyDzkCQSgLNS3UIArYiIHNvnMnMyM/8O4KDjeL5hwwbg77+1GjE5OXZMAqkSElQr\n8QAisWwrVmjna01PXBzwn/9o2pObWW5dumizvuAkMlKkf/WVaiUesm2bDIy2blWtxGWsMAzlAKSP\nRUx0bMtyH0eP6HMASrr4XutYvVqCo+/3vcfKalJTgTfeABYvVq3EQyIjpY7yd9+pVuIRCQnSf0HD\nMYVAJGGTr72mWonb1K4NVKqksTupYkVZPdfoA1hhGLIafmSOhctuH1feKwcgGkRE8UQUn5SU5KZE\nB+PHi9UuWNCz9yukUiX5gWj03cpImzZS50DTD6BBBZWccWZIajZbAERy586yznb5smo1HlCqlBQr\n1Oi7b4VhSASQvq5EeQDHs9uHiPIAKArgjIvvBQAw82RmjmDmiNKlS3umNDRUpnSaEhkpwVSe2kWl\nFCggxiEuTosY+szExUm126pVVSvxgJQUGbVqOFtwMmAA8MknWto1ITIS2LlTmwYrVhiGLQCqEVEV\nIgqDLCZnzmZaBsCZEtQVwDfMzI7tPR1RS1UAVAOga1Cmz4mMlFwGbX2tL7wAfPmlahVu89dfsjyl\n7WxhwwYZTWhZClaoXRvo21cmnVqiWaKn14bBsWYwFMBqAPsAzGfmPUQ0jogcUcj4DEBJIjoI4L8A\nRjneuwfAfAB7AawCMISZda5C41Pq1pWafxrNSDNSt678wjUb9i1fLgZZo/D/jMTFydpa27aqlXjF\n8ePAxx9rWqfqttukjkrNmqqVuASxhtP6iIgIjo+PVy1DCUOHAlOnyii2QAHVajzghx8k83zcONVK\nXObtt4HPPpPq4ZrZNHHbVakicfQaztbSM2+e5Elu3gw08F3sYkBDRFuZOSK3/YIv81lzIiOBK1ek\nVbWWbNwIvPSSVunDTz+tqVEAxK/9xx8aT3f+pWNH4MABzY3CoUPSp8TmGMOgGc2bS8Kbtu4kzXyt\nzpJaWhoFQApGvvWW1FTRnIIFxSOjLcxAs2ZSFdPmGMOgGWFhQP/+GnfNvP12KayniWV77DGpnqKh\nx1W46SaZ8pQpo1qJJRw4IO4kzUoPCURioDVosGIMg4a8846kZGhLZKQkup09q1pJrtx7r5TA0HLG\ncOKEVFK9eFG1EsvIn1/WGjSsZC1ERgKXLgHffKNaSY4Yw6ApaWlAYqJqFR4SGQmULQscPKhaSa48\n/LA0PNOSxYuBhx7SvnhhesqXl2KSmkw4b6RlS6BQIdt/AGMYNKVvX1lv0NLF0aiRLIhG5BocoZRt\n24AzZ1Sr8IKlSyV3QeP8hayIjJTOvCdOqFbiAfnyScXVFSts/eM1hkFT+vWTfDEtm/cQyS0tzbZB\n6cxA9+5A796qlXjI2bNSH1zbrLzscQZYaRK/cCOvvw5s325r/6QxDJrSurX0gw71XZFy37JrF3DL\nLVIAx4bs3SuRhdpeV1eulMqL2n6A7KlZU6qHL1miWomHVKli+46GxjBozJEjwMyZqlV4SLVqsghn\n01+30wWsbZTnzz9LJFLDhqqVWA4REB0t67fnzqlW4yFLl8oClk0xhkFj5s2TpvQa5Yr9S3i4+Frj\n4mzpTlq6VK6pt9yiWomHTJgA7N6t8ZQyZ6KjpTbgihWqlXjIkSPA9OkSf2tDjGHQGKev1eYBDtkT\nHS1tuTZtUq0kA8eOAVu2BIAXxtMqxBrQqJGkaNh0wpk7zi+XTX+8xjBoTLVqUkVc25juBx6Q1oc2\n+3U7FzW1rSIxciQwaJBqFT4lJAQYMUI66mlJpUpAnTq2/fEaw6A5UVHA998Dp0+rVuIBRYtKhTqb\nXYHj4sTo3nmnaiUecP06MGOGtLANcEaMAJ54QrUKL4iOlgYrf/6pWskNGMOgOVFRci1Yvly1Eg95\n4glJL7YJ587JomZUlK2jCbNn0yZxz8XEqFbiFy5cALQttBwTI53dTp1SreQGjGHQnHr1gHLlbDsj\ndY34eGDdOtUqAEgwz/XrGq8vLF4sBbU6dFCtxC889ph4JG0Yv5A7NWtKE6VatVQruYE8qgUYvCMk\nRC5i06ZJP1wtezQMHy5Dv+3bVStBmzYysy9RQrUSD2CW9ZrWrYEiRVSr8QtPPw0MHqzp7M7J2bNi\nzG304/VqxkBEJYhoDREdcNwXz2KfOkS0kYj2ENFOIuqR7rXpRPQ7EW133Op4oydYiYrSvEdDdDSw\nY4dt+uGWLq1plGdysrgn+vdXrcRv1K0rC9Ahuvo+fv1VvnCLFqlWkgFv/5yjAKxj5moA1jmeZ+Yy\ngL7MfBeAdgDeJaJi6V5/hpnrOG7qh4wa0ry51KTTMp8BEMMAKI9OWr1aZgxHjyqV4Tnh4dJ7oUsX\n1Ur8yo4d0uLAxqWHsuf22yXudvFi1Uoy4K1hiAQww/F4BoAbwkuYeT8zH3A8Pg7gFIDADbBWQFiY\n5MtoG6FRpYr0glZsGC5dklm9tq0LfvpJymAEGfHx0hRw507VSjwgJEQGRqtWyRfQJnhrGG5i5hMA\n4LjP8SdFRA0AhAE4lG7zeIeLaQIR5fNST9CSx7FapOUiHCA/jp07lfYOiImRxLZ8On4L9++XCJdP\nPlGtxO906iTXV5ulw7hOTIw07lm1SrWSf8jVMBDRWiLancXNrbgNIioL4HMADzOzsyboaAB3AqgP\noASAkTm8fxARxRNRfFJSkjunDgrS0uS68OyzqpV4yFNPSZhloUJKTn/mzL9tPLXEeVXs3FmtDgWU\nKSPffW0Nw3/+I0X1bPQBcjUMzNyamWtmcYsDcNJxwXde+LMMyCWiIgCWAxjLzJvSHfsEC8kApgHI\nts03M09m5ghmjigdwKn+nhISIukANWuqVuIhRYuKj1wRY8YAVatqPONavBioXx+oUEG1EiVERcmE\nMyFBtRIPyJNHkhKfe061kn/w1pW0DECs43EsgBsKfxBRGIAlAGYy84JMrzmNCkHWJ3Z7qSeoef11\nWxdszJ2vv5bEDD+XzLx+XQZrTZpoGo109KgkYDgX8YMQ50fXNp+nQwdbNVTy1jC8BqANER0A0Mbx\nHEQUQUSfOvbpDqAZgH5ZhKXOIqJdAHYBKAXgZS/1BD3nzwO//KJahYcULCht0/xcMvOnn8SLpW0w\nz5dfyn2QZDtnhU3iF7zjyy+BTz/NfT8/QKxhjFdERATHa5sH71s6dZJqywkJGib9pKVJnetmzYD5\n8/122mHDZM02KQkoXNhvp7WO1FSZMTRpolqJUl54ARg3Tlp+3nSTajUe0LOn1GM5ccJnU1ci2srM\nufbU1TUtxJAN0dHA4cPA1q2qlXiAM4175UqJ0vADaWninm/XTlOjAIiPOsiNAiATJmaN3UkxMTI6\n+fFH1UqMYQg0oqLkOrFwoWolHhIdLSGrfkrj/vlnIDFRYzfS4sVSZvTKFdVKlHP33ZKgqN1M2Un7\n9hIrbYNkN2MYAowSJYBWrYAFCzTNBG3ZEujRw2/FihYtkpYQ2rbwnDpVRgEKI7rsApHEL2jbiqJw\nYaBtWzEMin+8xjAEIF27yhqDDWrSuU9YGDB3rl9cI8xiGFq3BooVy31/23HhArBmjbggtB0mW8/1\n6+Km15KYGPkNKO7RYAxDABIVJWtXCxbkvq9tOXLE5/1wd++Wun3aupFWrJCsvCCORsqKli2BXr1U\nq/CQPn3ke1+2rFIZpux2AFKqlPw4FiwAxo/XcDB5/brkM7RpA8ye7bPT3H03sGeP9LPQkgULJPym\ncWPVSmzFE09oXG3VGY107Zr4OBX9eHX98xlyoVs34OBBTQuLhYbKIvSyZdJkwofUqCFJ11pSsqSU\n2NYyK893dO2q+STqu++Am29W+uM1hiFAcUYnbdyoWomH9Ogh1SZ9lOy2dy/Qu7emJRScfPIJ8Mor\nqlXYkoQEYPp01So85K67JFN1zhxlEoxhCFBKl5b1q0cfVa3EQ5o3l+po8+b55PAHDkj/BRs1zXIP\nbZtv+IdZs6Q8zLFjqpV4QKlS4kadO1dZdJIxDAFMyZKqFXhBnjziE1i1SjqTWUxkpBjOm2+2/NC+\n5/Rp4LbbgDfeUK3EtnTvLvfa5vP06iXGf9Om3Pf1AcYwBDDXr0sV5pd1rUA1erQslFjcICE5WQZi\n2rrmFy+WMhht2qhWYlvuuENqJ/mxsoq1REXJ916RO8kYhgAmNFTi8wsWVK3EQ8qX90nRm9dekwuH\ntsnCc+ZIS8g6pkV6TnTvLgUStWzVWqQI8MEHshCmAGMYApyZM4Hhw1Wr8IKffpJCRufPW3I4ZvE/\nly8P5M9vySH9y4kTwPr1UnBNuzhk/9Ktm9xr60565BGgQbYtanyKMQxBQGqq5muVq1cDcTe0+vCI\nLVtk4VnRQMx7nLVOevZUrcT2VKsG1K2rsTsJADZv9mkuT3YYwxAEdOki9bm0rJ3UqJF0JbMoOmnW\nLHHdapvt/PDDYiSrV1etRAu6d5f1W20HRh9+CAwZ4ve+s8YwBAHt2wP79mlaOykkRH7dX38N/P23\nV4dKTZUIwI4dNU5qK1w4KPs6e0pARCedPSuzZj/ilWEgohJEtIaIDjjui2ez3/V03duWpdtehYg2\nO94/z9EG1GAx3btLXa6ZM1Ur8ZAePYCUFK/bc61dC5w6pbEb6fPPgbff1nTqp4aqVYH77pNcSS1p\n00bizv0cneTtjGEUgHXMXA3AOsfzrLjCzHUct/TDndcBTHC8/28AA7zUY8iCEiWkrPTs2XJ91Y6I\nCCmR4WUJ1Fmz5BDt21uky9+89ZaUgzWLzm6xbh3wf/+nWoWH5M0r+TxxcX61bt4ahkgAMxyPZwCI\ncvWNREQAWgJwTvLcer/BPfr2ldHy11+rVuIBRBK770UBnEuXZMLRrZvlaRH+Ye9eqZ1jFp3dxmlH\nz5xRq8NjevWSL+3evX47pbeG4SZmPgEAjvsy2ewXTkTxRLSJiJwX/5IAzjJzquN5IgBd61zannbt\nJNNeW3cSICGrBw969NavvhLjoK0bae7cf9dbDG4zYIC0+NDSC/ef/0iafv36fjtlrmW3iWgtgKwK\nB4xx4zwVmfk4EVUF8A0R7QKQVWB6tv82IhoEYBAAVKxY0Y1TGwBZY+jVC5g8WdaytGxM07y5LL5+\n/73bb+3aVfqs33uvD3T5GmYxDC1aaFrDQz3R0VLJ/fp1qbaiFSEh8gNm9tsHyHXGwMytmblmFrc4\nACeJqCwAOO5PZXOM4477BADrAdQF8BeAYkTk/JTlARzPQcdkZo5g5ojSpUu78RENTvr2lXIQ2kZo\n9OwJbNgA7N/v9ltDQ2URUss6/efOAZUrazzdUU/HjsDjj2toFJz8+adku/tpyu/tz2QZgFjH41gA\nN2QhEVFxIsrneFwKQFMAe5mZAXwLoGtO7zdYR716wJ13Srl3LenbV67wbtZT/vxzYMQITRfeAZne\nff215DAYPOb8eWDKFJ+3+PANN90kMwY/RSd5axheA9CGiA4AaON4DiKKIKJPHftUBxBPRDsghuA1\nZnauoowE8F8iOghZc/jMSz2GHCASo6DtOkPZshJSNGOGJCW4yN694n3Km9eH2nxFaqqmtaPtx7Zt\nwKBBXkc9q4FIZswnT/ol2Y1Yw9WYiIgIjo+PVy1Da5g1jXpcskSik9avlzUHF7l+XdNqqsuWiYN8\nwwZZPTV4TFqaVCuvWlVyWrQjJUV8YV78cIloKzNH5Lafjh5Xg5e8954EOGg4JgA6dJCwTReNgjP0\nW0ujAACffipdl/wYkRKohIQAsbEShKBliQw/9oA2hiEIuflm6R548aJqJR4QFgbcfbdLuzID99yj\ncXXZY8eA5ctlbUFLP5j9iI2V74W27lQ/YQxDENKjh7jpCxdWrcRDkpOBhx4CJk7McbctWySAyUU7\nYj+mTRP/x8CBqpUEDJUrAy1bSvxCWppqNfbFGIYgZscO4OpV1So8IF8+SXSbODFHf9i0aUB4uKaV\nVJnFerdsCdx6q2o1AUW/fkBCAvDDD6qV2BdjGIKUDRukAdhXX6lW4iH9+wO7dgFbt2b58vnzEqba\ns6emlVSJZIH9vfdUKwk4YmJktjxtmmol9sUYhiClSROJ/tTW19qzp0wHpk7N8uXPP5eF5yFD/KzL\nSsqVA2rWVK0i4ChYUCqLLFig6TqbHzCGIUgJDZVE2pUrJalSO4oWlToXs2ff0LyZGfjoIwnkicg1\nMM+GJCVJvsYvv6hWErD07y89oE5lWavBYAxDEDNwoORPffKJaiUe8thjwODBNyyUfPedNCZ6/HFF\nurxlxgxg1SpNy8DqQZMmkstQtapqJfbEJLgFOQ88IBmhR45IJGgg0L27/OiPHQPy51etxk2YpW1n\nyZLAjz+qVhPwnDghOWPBUn7NJLgZXOLJJyXLXtuG6devy+g6MRGArCusWyeuAu2MAiBRAb/9Bjzy\niGolAc+ZM0ClSmZ9PyuMYQhy2rYF7rhDfhwaTh6B48dl2jNlCgBZWDx8GBg9Wq0sj/n0U6BIEeko\nZPApJUoAH35o0kSywhiGICckRGYN8fHAxo2q1XhAhQpi3T79FGlXr4FZQhFLllQtzEMaNgRGjRIL\nZ/A5gwZJ0pshI8YwGNC3ryzCaVk/BpCaF8ePY8mIH1GrlqyXaMuQIRpPd/Rk82bg0UdNJnR6jGEw\noFAh4MAB6fCmJW3bArVqoeCXc1GlMqOcjg1imaVybKbQW4PvOXRIIvNWrlStxD4Yw2AAIC6ltDQx\nENpBBDz7LNrxSiz78IielVTXrpWU3NmzVSsJOrp1E4/kW2+pVmIfjGEw/MPQoUDjxnoOWtff3BNn\ntx6SMBMdGT9eMp1N+06/kzcv8NRTUoHERMELXnVAJaISAOYBqAzgMIDuzPx3pn3uAzAh3aY7AfRk\n5qVENB1AcwDnHK/1Y+btnmhJSUlBYmIirmpZFc5awsPDUb58eeR1s1Tzww8D//mPfn1xL14EImNC\nER0diumfJANnz0orRF344QfJynv3XZPUpoiBA4EXXwTefttv3TNtjVcJbkT0BoAzzPwaEY0CUJyZ\nR+awfwkABwGUZ+bLDsPwFTO71Z4+qwS333//HYULF0bJkiVBWrYmswZmxunTp3HhwgVUqVJFtRy/\nMHmyJED/+AOjyaO1gGrVgMWLVctynfbtpRjg4cNAgQKq1QQtzzwDTJggaw66Tjxzw18JbpEAZjge\nzwAQlcv+XQGsZGbL23FfvXo16I0CABARSpYs6fHM6dIl4NVXgZ9+sliYj0hNlVFenTpA4yYEREUB\nS5dKkpgOXLoktZGGDzdGQTFPPinLVSbhzXvDcBMznwAAx32ZXPbvCSDzRG08Ee0koglElO08mogG\nEVE8EcUnJSVlt48b0gMXb/4OISFyodVlIW7mTGnG88ILjq6HTzwhtT3eflu1NNcoWFA6Co0YoVpJ\n0FOhgjSxmjJFvJHBTK6GgYjWEtHuLG6R7pyIiMoCuBvA6nSbR0PWHOoDKAEgWzcUM09m5ghmjiit\nSWGTgQMHYu/evapluEX+/JL0ExcH/P67ajU5k5wsfuEGDYDOnR0by5SRxZIZM+xfNvb4canLQGRa\nd9qEp5+WNavJk1UrUUuuhoGZWzNzzSxucQBOOi74zgt/TkVsuwNYwswp6Y59goVkANMANPDu49iL\nTz/9FDVq1FAtw22GDJHr1IsvqlaSM5MnSzLbyy9n6pH+9NNASor9VxFHjgRq1ACuXVOtxOCgbl3p\n5RHsZTK8dSUtAxDreBwLIC6HfXshkxspnVEhyPrEbi/1KOPSpUvo0KEDateujZo1a2LevHlo0aIF\nnIvkhQoVwpgxY1C7dm00atQIJ0+eBAAkJSWhS5cuqF+/PurXr48fbVBRs1w5YNgwcdPYtSXApUsS\n4dmiBdC6daYXb7tNhD/1lApprnHokBiu3r0Dp6xtgNC7t9RRCma8NQyvAWhDRAcAtHE8BxFFENGn\nzp2IqDKACgC+y/T+WUS0C8AuAKUAvOylnn9p0eLG28cfy2uXL2f9+vTp8vpff934Wi6sWrUKt9xy\nC3bs2IHdu3ejXbt2GV6/dOkSGjVqhB07dqBZs2aY4ij6NmzYMAwfPhxbtmzBokWLMNAmQ5XRo+XH\n8cwz9iyu98EHUhV2/PhMswUntWvLC3atc/D66xIX/PTTqpUYsmDVKuDBB6V4bzDilWFg5tPM3IqZ\nqznuzzi2xzPzwHT7HWbmcsyclun9LZn5bodrqjcza9to7+6778batWsxcuRIbNiwAUUzNRoOCwtD\nx44dAQD16tXD4cOHAQBr167F0KFDUadOHXTu3Bnnz5/HhQsX/C3/BooVA55/XkpYr1qlWs2NNGsG\n/O9/0nAlWyZPBu66SwYCdiIxUQYh/ftLf1WD7bhwQfqUHDumWokaNEtlcoP167N/rUCBnF8vVSrn\n17Pg9ttvx9atW7FixQqMHj0abdu2zfB63rx5/4kWCg0NRWpqKgAgLS0NGzduRH4bNg8YPBh4/30J\nmGnTxl6Jb02a5GIUAKkn/uuvEn/70kt+0eUSK1bINOzZZ1UrMWRD165AZGTwevlMSQyLOH78OAoU\nKIDevXtjxIgR2LZtm0vva9u2LT788MN/nm/f7lHit08ICxOPx9699pk1JCVJvLlLAUfNmwMPPQS8\n8Ya98hoGDZKQL1Pv2bYQyff/4kWpbRhsGMNgEbt27UKDBg1Qp04djB8/HmPHjnXpfe+//z7i4+NR\nq1Yt1KhRA5MmTfKxUveIjpZkN4cXTDnff+9mnPlbb0kM7pAh9lgscVYpLF9erQ6DS7z+OtCliySm\nq+bwYTFU/iBgej7v27cP1atXV6TIfvji73HxopToVs3p02424vnoI5lm/PwzUK+ez3TlysKFkkG1\nZg3QsqU6HQaXOXcOuP126Vfy44+SAKqClBTp4ZQvnwzUPM1hNT2fDZYyezZQsaLaxThnwp3b3dke\nfVTCV1UahdOnZdZSt66snBu0oGhR8URu2iTh26p4/335Co8c6blRcAdjGAwu0bixtAtQtQC9bh1w\n663SUMVtQkOBWrXksSMazO8MHy5Zzp99Zq9VfEOu9Okj3/+RI2UGoYLBgyWQLSq3anQWYQyDwSWq\nVJE+9SqqWZ88KWvId9zhZbuCWbMk+c3FwADLWLFC0mlHj5b8CoNWhIQAH34ogQ8vvODfcycnS3+U\nQoWA2Njc97cKYxgMbrF/v4Ty+SvVIi1NelKfOwfMny815zymQwfxQz32mH8zl44eFTfWmDH+O6fB\nUu65R0btH3wA7PZjfYYXXpBz+zu1yRgGg1v88YeE7/Xt65+k4jfeAL7+Wkoh3323lwcrVkyqrv78\ns0x//MXgwdJx3jTh0ZqXX5Y1hyee8F9CfatWEhVVuLB/zufEGAaDW7RpI9fWpUt9X2Tvxx+BsWMl\nkOeRRyw66EMPSYmTUaNkJO9LNmyQPxQAPRtRG9JTsiTw2muS++pm/qvHtG4tBsnfGMNgE+Lj4/Hk\nk0+qluESw4YB/foB48YBCxb45hxnzgC9ekknrcmTLYzEIAImTpTysfv2WXTQLLh0Sf5II0aIo9gQ\nEAwcKOGivo42fuYZGRSpyiYw4RE2ISIiAhERuYYX2wIiYNIkSSbu1086adapY93xmaWlwp9/yo+w\nSBHrjg0AuPNOiX11LlikplobKXTpkizEJCTI0NK4kAIGIolQAmRGe/Cg9YvCa9ZIXuaQIf4JTc0K\nM2OwkJkzZ6JWrVqoXbs2+vTpgz/++AOtWrVCrVq10KpVKxw5cgQAsGDBAtSsWRO1a9dGM0dM+/r1\n6/8psvfCCy+gf//+aNGiBapWrYr333//n3N88cUX/2RYDx48GNcVlX/Ml0/aKhcvLjVlTuXUicNN\nTp2SMhxvvgn4zFY6jcIXX0jm0Jkz1hz3zBnxt339taRoN29uzXENtuOdd2QNzMp2GvPmAZ06AdWr\nS9a1MphZu1u9evU4M3v37s3wvHnz3G9vvplx/2nT5HFS0o375sbu3bv59ttv56SkJGZmPn36NHfs\n2JGnT5/OzMyfffYZR0ZGMjNzzZo1OTExkZmZ//77b2Zm/vbbb7lDhw7MzPz8889z48aN+erVq5yU\nlMQlSpTga9eu8d69e7ljx4587do1ZmZ+7LHHeMaMGVnqyfz38BVbtjCHhzP/5z/MycneHSstjfny\nZXl84YI89zmrVzOHhTHXr8989qz3x5s+XY63aJH3xzLYmosXmU+dsuZYaWnMr7zCDDDfey/zX39Z\nc9zMAIhnF66xZsZgEcXjdcwAAAi7SURBVN988w26du2KUqVKAQBKlCiBjRs34sEHHwQA9OnTBz/8\n8AMAoGnTpujXrx+mTJmS7Yi/Q4cOyJcvH0qVKoUyZcrg5MmTWLduHbZu3Yr69eujTp06WLduHRIS\nEvzzAbMhIkJytjZsABwfzyOYJYEuNlYeFyrkp2l027ZSquKXX4AHHvC8GI2jWi5iY2XtIibGOo0G\nW1KwIFC6tMwYBg3yvKlVSooEV/zvf7KutmaNB9n9FuOVY5WIugF4AUB1AA2YOT6b/doBeA9AKIBP\nmdnZ0KcKgLmQfs/bAPRhZksmZu5GDaTf34Oq22Dmf8pqZ4fz9UmTJmHz5s1Yvnw56tSpk2VF1Xzp\n/NLOMt3MjNjYWLz66qvuifMxDz4oderuu0+eb9okYfvutDEmkkoRSlofd+ok3dR69JDHq1e7V295\n2zage3fxA9SrJ4V1DEHDmTPylYmLk7Jc0dGuB6GdOwd06ybGYMwYCehQVY8pPd5K2A0gBsD32e1A\nRKEAPgLQHkANAL2IyNkI+XUAE5i5GoC/AQzwUo8yWrVqhfnz5+P06dMAgDNnzqBJkyaYO3cuAGDW\nrFm49957AQCHDh1Cw4YNMW7cOJQqVQpHXQybbNWqFRYuXIhTDof+mTNn8Mcff/jg07hPdLRc3E+e\nFAMxcmTu70lJkajRZcvk+fDhwNChihbcunaVYjgtWoh1SkuTfIecwkISEyVkqkUL+TD+DjY32IKb\nb5YlpaJF5SJfrZrUNnJl8tmvH/Dtt8DUqRKWagejAHg5Y2DmfQByGyk3AHCQmRMc+84FEElE+wC0\nBPCgY78ZkNnHRG80qeKuu+7CmDFj0Lx5c4SGhqJu3bp4//330b9/f7z55psoXbo0pk2bBgB45pln\ncODAATAzWrVqhdq1a+O77zJ3Pb2RGjVq4OWXX0bbtm2RlpaGvHnz4qOPPkKlSpV8/fFcpkwZCWF1\nliY6dEjqx6Wmysjq9Gm5P3NGejzEx4sR6dxZrW4AkuPgZN06WUS+6y6JUezdWy78zplcjx6Sig3I\nh12+3JTSDmLuuEM8iHFxkuczbJh0QBw8WBLibrlFBjyJiTL4GTZMBlBvvCHPW7VS/Qky4cpCRG43\nAOsBRGTzWleI+8j5vA+ADyE9ng+m214BwG5XzufK4nOwY5e/R6dOsqCW+RYSwlyhAvOCBaoVZsP5\n88xTpjA3bCiCw8KYixZlvnJFXp82TaIXduzw0yq5QSc2bmTu2lW+56GhzM89J9svXGCuXl1dbAJc\nXHzOdcZARGsB3JzFS2OYOc4F25PVdIJz2J6djkEABgFAxYoVXTitwQ5MnCiD7WLFgBIl/r0VKWKf\naXOWFC4sM4WBA6U4zsyZIvjKFSA8XHwABkM2NGokM+eEBHET1XA4zwsVklBsu5OrYWDm1l6eIxEy\nG3BSHsBxAH8BKEZEeZg5Nd327HRMBjAZkEY9Xmoy+Ily5WRdVmtq1pQ5v8HgJlWrqilp4S3+GLNt\nAVCNiKoQURiAngCWOaY130JcTQAQC8CVGYjBYDAYfIhXhoGIookoEUBjAMuJaLVj+y1EtAIAHLOB\noQBWA9gHYD4z73EcYiSA/xLRQQAlAXzmjR7WsE2pLzB/B4PB4A3eRiUtAbAki+3HATyQ7vkKACuy\n2C8BErXkNeHh4Th9+jRKliyZaz5BIMPMOH36NMLDw1VLMRgMmhIwRfTKly+PxMREJCUlqZainPDw\ncJQ3oZMGg8FDAsYw5M2bF1WqVFEtw2AwGLTHzgGDBoPBYFCAMQwGg8FgyIAxDAaDwWDIAOkY2khE\nSQA8rR5XCpJcF0yYzxwcmM8c+Hj7eSsxc+ncdtLSMHgDEcUzsx49NC3CfObgwHzmwMdfn9e4kgwG\ng8GQAWMYDAaDwZCBYDQMk1ULUID5zMGB+cyBj18+b9CtMRgMBoMhZ4JxxmAwGAyGHAgqw0BE7Yjo\nNyI6SESjVOvxJURUgYi+JaJ9RLSHiIap1uQviCiUiH4hoq9Ua/EHRFSMiBYS0a+O/3dj1Zp8DREN\nd3yvdxPRHCIKuKqRRDSViE4R0e5020oQ0RoiOuC4L+6LcweNYSCiUAAfAWgPoAaAXkRUQ60qn5IK\n4Glmrg6gEYAhAf550zMMUuI9WHgPwCpmvhNAbQT4ZyeicgCehLQTrgkgFNLnJdCYDqBdpm2jAKxj\n5moA1jmeW07QGAZIee+DzJzAzNcAzAUQqViTz2DmE8y8zfH4AuRiUU6tKt9DROUBdADwqWot/oCI\nigBoBkcvE2a+xsxn1aryC3kA5CeiPAAKIIfuj7rCzN8DOJNpcySAGY7HMwBE+eLcwWQYygE4mu55\nIoLgQgkARFQZQF0Am9Uq8QvvAngWQJpqIX6iKoAkANMc7rNPiaigalG+hJmPAXgLwBEAJwCcY+av\n1aryGzcx8wlABn8AyvjiJMFkGLLq3hPwIVlEVAjAIgBPMfN51Xp8CRF1BHCKmbeq1uJH8gC4B8BE\nZq4L4BJ85F6wCw6/eiSAKgBuAVCQiHqrVRVYBJNhSARQId3z8gjA6Wd6iCgvxCjMYubFqvX4gaYA\nOhPRYYirsCURfaFWks9JBJDIzM7Z4EKIoQhkWgP4nZmTmDkFwGIATRRr8hcniagsADjuT/niJMFk\nGLYAqEZEVYgoDLJYtUyxJp9B0t/0MwD7mPkd1Xr8ATOPZubyzFwZ8v/9hpkDeiTJzH8COEpEdzg2\ntQKwV6Ekf3AEQCMiKuD4nrdCgC+4p2MZgFjH41gAcb44ScB0cMsNZk4loqEAVkOiGKYy8x7FsnxJ\nUwB9AOwiou2Obf9z9N82BBZPAJjlGPAkAHhYsR6fwsybiWghgG2Q6LtfEIAZ0EQ0B0ALAKWIKBHA\n8wBeAzCfiAZADGQ3n5zbZD4bDAaDIT3B5EoyGAwGgwsYw2AwGAyGDBjDYDAYDIYMGMNgMBgMhgwY\nw2AwGAyGDBjDYDAYDIYMGMNgMBgMhgwYw2AwGAyGDPw/x/1Q28CeuyQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x207be3aadd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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ONRaRt0Xkq8jP3f24dugTgojkAY8AJwDtgLNFpF2wUfmuBLhGVQ8GDgcuy4HP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\n99YAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x207bf3903c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "x = np.linspace(0,10)\n",
    "y = np.sin(x)\n",
    "z = np.cos(x)\n",
    "\n",
    "plt.figure(1)\n",
    "plt.plot(x,y,'r--',label='sine')\n",
    "plt.plot(x,z,'b-.',label='cosine')\n",
    "plt.legend(loc='best')\n",
    "\n",
    "plt.figure(2)\n",
    "plt.subplot(2,1,1)\n",
    "plt.plot(x,y,'r--',linewidth=5)\n",
    "plt.legend(['sin(x)'],loc='best')\n",
    "plt.ylabel('sin(x)')\n",
    "plt.subplot(2,1,2)\n",
    "plt.plot(x,z,'bo',markersize=5)\n",
    "plt.legend(['cosine(x)'],loc='best')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('cos(x)')\n",
    "plt.savefig('myPlot.png')\n",
    "plt.savefig('myPlot.eps')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2: Read / Write Files"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Writing to a text file \n",
      "\n",
      "Second lineThird line?\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'cos(x)')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hIk7bSU6O874yIamh41CGq2rVYuGquldEzghSTCaSZGUhWVl8UrCbrOc/AyA2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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x207bb3349b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# write messages or data to file\n",
    "fid = open('msg.txt','w')\n",
    "fid.write('Writing to a text file \\n')\n",
    "fid.write('Second line')\n",
    "fid.write('Third line?')\n",
    "fid.close()\n",
    "\n",
    "# read one line at a time\n",
    "fid = open('msg.txt','r')\n",
    "text = fid.readline()\n",
    "print(text)\n",
    "text = fid.readline()\n",
    "print(text)\n",
    "fid.close()\n",
    "\n",
    "x = np.linspace(0,2*np.pi,20)\n",
    "y = np.cos(x)\n",
    "\n",
    "#--- stack rows together from (x,y) to produce data \n",
    "#--- then transpose data to convert rows into columns\n",
    "data = np.vstack((x,y))\n",
    "data = data.T\n",
    "\n",
    "#--- save data to data.txt\n",
    "np.savetxt('data.txt',data)\n",
    "#--- load text file\n",
    "z = np.loadtxt('data.txt')\n",
    "#print(z)\n",
    "\n",
    "#--- create plot of y = sin(x)\n",
    "plt.plot(x,y,linewidth=3)\n",
    "plt.plot(z[:,0],z[:,1],'ro')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('cos(x)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3: Functions, Conditionals, and Loops"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,0,'y')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ySzGr8otZvrmYFZuL2RNr12/cyOiV2pJRvVIY1KUNg7u2YWBaa53YFZG40jfM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\nBCIiIacgEBEJuf8HDCjQmyIAb48AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x207be3aa3c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x):\n",
    "    y = x**2\n",
    "    if y>=20:\n",
    "        y = np.nan # nan = not a number\n",
    "    return y\n",
    "\n",
    "x = np.linspace(0,5,100)\n",
    "y = np.empty_like(x)\n",
    "\n",
    "# calculate with loop\n",
    "i = 0\n",
    "for xi in x:\n",
    "    y[i] = f(xi)\n",
    "    i = i + 1\n",
    "\n",
    "# plot result\n",
    "plt.plot(x,y)\n",
    "# optional to make plot look better\n",
    "plt.xlim([0,5])\n",
    "plt.ylim([0,30])\n",
    "plt.title('Clipped Function')\n",
    "plt.ylabel('x')\n",
    "plt.xlabel('y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 4: Matrices, Solve Linear Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A matrix\n",
      "[[3 2]\n",
      " [1 2]]\n",
      "b vector\n",
      "[1 0]\n",
      "[ 0.5  -0.25]\n",
      "[ 0.5  -0.25]\n",
      "{(1/2, -1/4)}\n",
      "{(1/2, -1/4)}\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{(1/2, -1/4)}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.array([[3,2],[1,2]])\n",
    "print('A matrix')\n",
    "print(A)\n",
    "\n",
    "# b in matrix form\n",
    "b = np.array([1,0])\n",
    "print('b vector')\n",
    "print(b)\n",
    "\n",
    "#### Solution 1\n",
    "x = np.dot(np.linalg.inv(A),b)\n",
    "print(x)\n",
    "\n",
    "#### Solution 2\n",
    "x = np.linalg.solve(A,b)\n",
    "print(x)\n",
    "\n",
    "#### Solution 3\n",
    "import sympy as sym\n",
    "from IPython.display import display\n",
    "x, y = sym.symbols('x y')\n",
    "ans = sym.linsolve([3*x + 2*y - 1, x + 2*y], (x, y))\n",
    "# 3 ways to display answer (ans)\n",
    "print(ans)\n",
    "sym.pprint(ans)\n",
    "display(ans)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 5: Solve Nonlinear Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.89442719  0.4472136 ]\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAV8AAAA/BAMAAAC1GNZcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nOuvOjAQ4VV8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left\\{\\left ( - \\frac{2 \\sqrt{5}}{5}, \\quad \\frac{\\sqrt{5}}{5}\\right ), \\left ( \\frac{2 \\sqrt{5}}{5}, \\quad - \\frac{\\sqrt{5}}{5}\\right )\\right\\}$$"
      ],
      "text/plain": [
       "⎧⎛-2⋅√5   √5⎞  ⎛2⋅√5  -√5 ⎞⎫\n",
       "⎨⎜──────, ──⎟, ⎜────, ────⎟⎬\n",
       "⎩⎝  5     5 ⎠  ⎝ 5     5  ⎠⎭"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.optimize import fsolve\n",
    "import sympy as sym\n",
    "sym.init_printing()\n",
    "\n",
    "#### Solution with fsolve\n",
    "def f(z):\n",
    "    x = z[0]\n",
    "    y = z[1]\n",
    "    return [x + 2*y, x**2+y**2-1]\n",
    "\n",
    "zguess = [0,1]\n",
    "z = fsolve(f,zguess)\n",
    "print(z)\n",
    "\n",
    "#### Solution with Sympy\n",
    "ans = sym.nonlinsolve([x + 2*y, x**2+y**2-1], [x,y])\n",
    "display(ans)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 6: Integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SciPy Solution\n",
      "2.666666666666667\n",
      "2.6666666666666665\n",
      "SymPy Solution\n",
      "x**3/3\n",
      " 3\n",
      "x \n",
      "──\n",
      "3 \n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAvBAMAAAAV7ydtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1klEQVQoFWNggICkywJQFgNvAe8GODuB/QOM\nzcCAEGdgkDsAF2eaBGeiqmH5BZOQO8DyHcbmcGB6CGQLKbuqKfAkJTswMDAmsDdxTIDIswmwfeQC\nioEAIwMn3CUMDPwKYEEwcf4AjM0loM/AJADhxS/Yz+AOlRBWErpeAFNEY/o/AlBsk1vaLpgZLBcY\n1sO8wPSLgb8BKsH6lEF+AkwRA0INMMDa4MIsGRfhbAaGvAAEhw8U+iDA4sDADYsh/o8INt8GBo4v\nECUM7AUM6xWgbIbYu1tBTABGZDQhWVh3aAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{x^{3}}{3}$$"
      ],
      "text/plain": [
       " 3\n",
       "x \n",
       "──\n",
       "3 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.integrate import quad\n",
    "import sympy as sym\n",
    "\n",
    "#### Solution (scipy)\n",
    "print('SciPy Solution')\n",
    "def f(x):\n",
    "    return x**2\n",
    "print(quad(f,0,2)[0])\n",
    "print(8/3)\n",
    "\n",
    "#### Solution (sympy)\n",
    "print('SymPy Solution')\n",
    "x = sym.Symbol('x')\n",
    "f = sym.integrate(x**2,x)\n",
    "print(f)\n",
    "sym.pprint(f)\n",
    "display(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 7: Differentiate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11.0\n",
      "10.0000008274\n",
      "10.00000082740371\n",
      "10.0000000000000\n",
      "   2              \n",
      "3⋅x  + 4.0⋅x + 3.0\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJMAAAAWBAMAAAA2trNNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$3 x^{2} + 4.0 x + 3.0$$"
      ],
      "text/plain": [
       "   2              \n",
       "3⋅x  + 4.0⋅x + 3.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.misc import derivative\n",
    "import sympy as sym\n",
    "\n",
    "# numeric method 1\n",
    "def f(x):\n",
    "    return x**3 + 2.0*x**2 + 3.0*x + 1.0\n",
    "print(derivative(f,1))\n",
    "print(derivative(f,1,dx=1e-10))\n",
    "der = (f(1.0+1e-10) - f(1.0-1e-10))/2.e-10\n",
    "print(der)\n",
    "#help(derivative)\n",
    "\n",
    "# analytic solution with sympy\n",
    "x = sym.Symbol('x')\n",
    "f=sym.diff(x**3 + 2.0*x**2 + 3.0*x + 1.0,x)\n",
    "print(f.subs(x,1))\n",
    "sym.pprint(f)\n",
    "display(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 8: Interpolation\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.5\n",
      "0.5\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x207c1cfb160>]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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vM7f9XMqdCXA2pC5VCp5/HvLnh/LlnSdkywa5crkbWiSNaFpGMpxj547RdUZX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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x207c1c8fc50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.interpolate import interp1d\n",
    "x = np.array([0,1,2,3,4,5])\n",
    "y = np.array([0.1,0.2,0.3,0.5,1.0,0.9])\n",
    "plt.plot(x,y,'bo')\n",
    "\n",
    "xp = np.linspace(0,5,100)\n",
    "y1 = interp1d(x,y,kind='linear')\n",
    "print(y1(3))\n",
    "plt.plot(xp,y1(xp),'g-')\n",
    "\n",
    "y2 = interp1d(x,y,kind='quadratic')\n",
    "print(y2(3))\n",
    "plt.plot(xp,y2(xp),'r--')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 9: Linear and Polynomial Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.04166667 -0.2         0.33690476  0.07857143]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'y')"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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27eDkSXPf3R2++gqefjpfp/Gv7M8fg/4AoLxneTt0rGSSzWtCiKtcvmwSjn70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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x207bf3ad588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.array([0,1,2,3,4,5])\n",
    "y = np.array([0.1,0.2,0.3,0.5,0.8,2.0])\n",
    "\n",
    "#### Solution\n",
    "# linear\n",
    "p1 = np.polyfit(x,y,1)\n",
    "# quadratic\n",
    "p2 = np.polyfit(x,y,2)\n",
    "# cubic\n",
    "p3 = np.polyfit(x,y,3)\n",
    "print(p3)\n",
    "\n",
    "# plot fit\n",
    "plt.plot(x,y,'ko',markersize=10)\n",
    "xp = np.linspace(0,5,100)\n",
    "plt.plot(xp,np.polyval(p1,xp),'b--',linewidth=2)\n",
    "plt.plot(xp,np.polyval(p2,xp),'r--',linewidth=3)\n",
    "plt.plot(xp,np.polyval(p3,xp),'g:',linewidth=2)\n",
    "plt.legend(['Data','Linear','Quadratic','Cubic'],loc='best')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 10: Nonlinear Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a = [ 0.58628748]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x207c1cbd9b0>]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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1ttuOgsL9ePLzwXywdl9msh8fsQ+17X9IaSmgB5xEpAEK/iisXRumNZg7F+bM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\n0WHkSloAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x207c1ce9e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.optimize import curve_fit\n",
    "\n",
    "x = np.array([0,1,2,3,4,5])\n",
    "y = np.array([0.1,0.2,0.3,0.5,0.8,2.0])\n",
    "\n",
    "## Solution\n",
    "def f(x,a):\n",
    "    return 0.1 * np.exp(a*x)\n",
    "\n",
    "aopt, pcov = curve_fit(f,x,y)\n",
    "print('a = '+str(aopt))\n",
    "\n",
    "plt.plot(x,y,'bo')\n",
    "xp = np.linspace(0,5)\n",
    "plt.plot(xp,f(xp,aopt),'r-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 11: Solve Differential Equation\n",
    "\n",
    "Solve the following differential equation with initial condition $y(0) = 5$:\n",
    "\n",
    "$ k \\, \\frac{dy}{dt} = -t \\, y$\n",
    "\n",
    "where $k=10$. The solution of $y(t)$ should be reported from an initial time $0$ to final time $20$. Create of plot of the result for $y(t)$ versus $t$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'y')"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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tlt4Y8RIbI3wcKHHO/bKFaXof3ZZgZuOJLvu9HufqZGZZR68T3ei3sslkfwau\nt6gJwIGjHzUToMU1Lz+WVyONX0M3ALOameZvwCVmlh0bwrgkdptnzGwq8EPgSudcZQvTtOY5j3eu\nxtt8rm5hfq1573rhImC1c25bc3d6vbxO0A3+vMa82DLt5Q/RvUrWEN3i/2+x2+4j+iYAyCQ6RLAO\nWAQMSkCmyUQ/an0MLIv9TANuA26LTXMnsIro3gkfAuckINeg2PyWx+Z9dHk1zmXAw7HluQIoTNDz\n2JFogXdtdFvClxfRPzg7gVqia1Q3Ed3mMwdYG7vsHpu2EHis0WNvjL3O1gHfSECudUTHdI++xo7u\njdYXmH2i59zjXE/HXjsfEy2yPk1zxX4/7r3rZa7Y7b8/+ppqNG0il1dL3eDLa0zftBURCYhUG9IR\nEZFTpMIXEQkIFb6ISECo8EVEAkKFLyISECp8CSwz62Zm34xd72tmf/A7k4iXtFumBFbs2CavO+dG\n+hxFJCHS/A4g4qOfAoNjx0lfC5zhnBtpZl8nevTCMDCS6LFh0oHrgGpgmnNun5kNJvqltVygErjF\nObc68f8NkdbRkI4E2d1ED808BviXJveNBGYQPa7K/UClc24ssAC4PjbNo8BdzrmzgO8DjyQktcgp\n0hq+SPPmuujxyyvM7ADwWuz2FcCZsaMfngO83Oh0CxmJjynSeip8keZVN7re0Oj3BqLvmxCwP/bp\nQCQlaEhHgqyC6Gnn2sxFj2m+0cz+GY6dG3h0PMOJxJsKXwLLObcXmB878fXPT+GfmAncZGZHj7QY\nt1P2iXhBu2WKiASE1vBFRAJChS8iEhAqfBGRgFDhi4gEhApfRCQgVPgiIgGhwhcRCQgVvohIQPx/\nrveUmtyplEcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x207c20484a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.integrate import odeint\n",
    "\n",
    "def dydt(y,t,k):\n",
    "    return -t*y/k\n",
    "\n",
    "k = 10.0\n",
    "t = np.linspace(0,20)\n",
    "y0 = 5.0\n",
    "y = odeint(dydt,y0,t,args=(k,))\n",
    "plt.plot(t,y)\n",
    "plt.xlabel('time')\n",
    "plt.ylabel('y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 12: Nonlinear Optimization\n",
    "\n",
    "Solve the following nonlinear optimization problem:\n",
    "\n",
    "$\\min x_1 x_4 \\left(x_1 + x_2 + x_3\\right) + x_3$\n",
    "\n",
    "$\\mathrm{s.t.} \\quad x_1 x_2 x_3 x_4 \\ge 25$\n",
    "\n",
    "$x_1^2 + x_2^2 + x_3^2 + x_4^2 = 40$\n",
    "\n",
    "$1\\le x_1, x_2, x_3, x_4 \\le 5$\n",
    "\n",
    "with initial conditions:\n",
    "\n",
    "$x_0 = (1,5,5,1)$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial Objective: 16.0\n",
      "Final Objective: 17.0140172456\n",
      "Solution\n",
      "x1 = 1.0\n",
      "x2 = 4.74299609688\n",
      "x3 = 3.82115462341\n",
      "x4 = 1.37940764508\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.optimize import minimize\n",
    "\n",
    "def objective(x):\n",
    "    return x[0]*x[3]*(x[0]+x[1]+x[2])+x[2]\n",
    "\n",
    "def constraint1(x):\n",
    "    return x[0]*x[1]*x[2]*x[3]-25.0\n",
    "\n",
    "def constraint2(x):\n",
    "    sum_eq = 40.0\n",
    "    for i in range(4):\n",
    "        sum_eq = sum_eq - x[i]**2\n",
    "    return sum_eq\n",
    "\n",
    "# initial guesses\n",
    "n = 4\n",
    "x0 = np.zeros(n)\n",
    "x0[0] = 1.0\n",
    "x0[1] = 5.0\n",
    "x0[2] = 5.0\n",
    "x0[3] = 1.0\n",
    "\n",
    "# show initial objective\n",
    "print('Initial Objective: ' + str(objective(x0)))\n",
    "\n",
    "# optimize\n",
    "b = (1.0,5.0)\n",
    "bnds = (b, b, b, b)\n",
    "con1 = {'type': 'ineq', 'fun': constraint1} \n",
    "con2 = {'type': 'eq', 'fun': constraint2}\n",
    "cons = ([con1,con2])\n",
    "solution = minimize(objective,x0,method='SLSQP',\\\n",
    "                    bounds=bnds,constraints=cons)\n",
    "x = solution.x\n",
    "\n",
    "# show final objective\n",
    "print('Final Objective: ' + str(objective(x)))\n",
    "\n",
    "# print solution\n",
    "print('Solution')\n",
    "print('x1 = ' + str(x[0]))\n",
    "print('x2 = ' + str(x[1]))\n",
    "print('x3 = ' + str(x[2]))\n",
    "print('x4 = ' + str(x[3]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Use Python GEKKO\n",
    "\n",
    "See [GEKKO documentation](http://gekko.readthedocs.io/en/latest/overview.html) and [example problems](https://apmonitor.com/wiki/index.php/Main/GekkoPythonOptimization). GEKKO is a Python package developed in the [BYU PRISM Lab](https://apm.byu.edu/prism) by [Logan Beal](https://www.linkedin.com/in/logan-beal-2461ba57). Logan is supported by the National Science Foundation, [Grant #1547110](https://nsf.gov/awardsearch/showAward?AWD_ID=1547110), EAGER: Cyber-Manufacturing with Multi-echelon Control and Scheduling."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Final Objective: 17.0140171\n",
      "Solution\n",
      "x1: [1.0]\n",
      "x2: [4.743]\n",
      "x3: [3.82115]\n",
      "x4: [1.379408]\n"
     ]
    }
   ],
   "source": [
    "from gekko import GEKKO\n",
    "\n",
    "# Initialize Model\n",
    "m = GEKKO()\n",
    "\n",
    "# initialize variables\n",
    "x1,x2,x3,x4 = [m.Var(lb=1, ub=5) for i in range(4)]\n",
    "\n",
    "# initial values\n",
    "x1.value = 1\n",
    "x2.value = 5\n",
    "x3.value = 5\n",
    "x4.value = 1\n",
    "\n",
    "# Equations\n",
    "m.Equation(x1*x2*x3*x4>=25)\n",
    "m.Equation(x1**2+x2**2+x3**2+x4**2==40)\n",
    "\n",
    "# Objective\n",
    "m.Obj(x1*x4*(x1+x2+x3)+x3)\n",
    "\n",
    "# Solve\n",
    "m.solve(disp=False)\n",
    "\n",
    "# show final objective\n",
    "print('Final Objective: ' + str(m.options.objfcnval))\n",
    "\n",
    "# print solution\n",
    "print('Solution')\n",
    "print('x1: ' + str(x1.value))\n",
    "print('x2: ' + str(x2.value))\n",
    "print('x3: ' + str(x3.value))\n",
    "print('x4: ' + str(x4.value))"
   ]
  }
 ],
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