Linear_and_higher_order_approximation.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Linear and higher order approximation.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyMoPuoTL+/IIK8j8kOVp2EO",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/awangberg/Calculus-Projects/blob/main/Linear_and_higher_order_approximation.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PKnVymVY9ZSI"
      },
      "source": [
        "# Approximating functions with derivatives\n",
        "\n",
        "Functions can be approximated in several different ways.\n",
        "\n",
        "A very simple way is to just choose some inputs that are spread out:\n",
        "\n",
        " $$  x = 1 \\hspace{1cm} x = 3 \\hspace{1cm} x = 5 \\hspace{1cm} x = 7 \\hspace{1cm} x = 9 $$\n",
        "\n",
        "and then use the outputs that are associated with them:\n",
        "\n",
        "In the case of $y = \\sqrt{x}$, the outputs would be\n",
        "\n",
        "$$ y = \\sqrt{1} \\hspace{1cm} y = \\sqrt{3} \\hspace{1cm} y = \\sqrt{5} \\hspace{1cm} y = \\sqrt{7} \\hspace{1cm} y = \\sqrt{9}$$\n",
        "\n",
        "In this scenario, we'd choose the best output by just finding the a nearby input.  So, to approximate the function when $x = 8.2$, we'd estimate that the output is \n",
        "\n",
        "$$ \\sqrt{8.2} \\approx \\sqrt{9} = 3 $$\n",
        "\n",
        "In this activity, we'll explore two better ways to approximate functions.\n",
        "\n",
        "Three quick notes:\n",
        "\n",
        "  1.  We'll be doing computations in Desmos.  It may be helpful to copy the Desmos URL into a new window and do the work there.\n",
        "  \n",
        "  2.  You'll save screenshots into this worksheet.  On a mac, hold **Shift**, **Command**, and **3** together.  On a PC, hold the **Windows Logo Key**, **Shift**, and **S** key together.  Then, copy the image to your clipboard, and paste it into a new cell.\n",
        "\n",
        "  3.  Be sure to save your worksheet back out to your github account.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Estimating Functions on Segments\n",
        "\n",
        "In this part of the activity, we'll appoximate the function\n",
        "\n",
        "$$ y = \\sqrt{x}$$\n",
        "\n",
        "by using the best point, line, and then parabola on small segments.\n",
        "\n",
        "All of the code is written within Desmos.  You'll be asked to provide a screenshot showing:\n",
        "\n",
        "  1.  The approximation just with points (already provided).\n",
        "  2.  The approximation with just lines, using a first derivative\n",
        "  3.  The approximation with quadratics, using a second derivative\n",
        "  4.  [Optional] A cubic approximation for the entire segment $0 \\le x \\le 10$"
      ],
      "metadata": {
        "id": "xjQHe1tDzV9a"
      }
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Oz0RjRHO9Ly_"
      },
      "source": [
        "DesmosURL = \"https://www.desmos.com/calculator/ai65zhhipu\"\n",
        "#  Uncomment these lines to display here:\n",
        "#from IPython.display import IFrame\n",
        "#IFrame(src=DesmosURL, width=1200, height=575)"
      ],
      "execution_count": 5,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "#  Approximation with a points:"
      ],
      "metadata": {
        "id": "qVg_HK4W0YSG"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here >>![image.png](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAABKMAAAJMCAYAAAAmOGV8AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAHQCSURBVHhe7d0JgF51eS/+XzKZLbNkYychAqKIbAECEnetS9vb5d7WVqvU5Qre9tZ621q7t7eLtb0t1ta6RWX5i1tFa7XV1rrjUg0SUBBQFAIEFMg2S2af/PMczsGXycxkZvLOzHvOfD72dN73zARySH7v8n2f5/ktO3BQanCXX355dmzZsiVde+21+dlq2bdvX9q7d2/auHFjfqZaduzYkVavXp1WrVqVn6meKl/jtm3b0vbt29Nll12Wn6kea7D8lsI1bt26NW3atClt3rw5P1MdVV+DweNM+VmD5eYay6/Ka7BQ5cdSa3B+RbQTx/j4eHaMjY2l0dHR7Lj//vvTcccdl/9kffT19aWPf/zj2XvFtra29NznPjc985nPTF/60peyf9eGDRvShz70ofSNb3wj/xUPW79+fXr5y1+eluf3AQAAAGBaDzzwQHrf+96XBVEdHR3pxS9+cXrqU5+aPvzhD2fHu971rnT99denF73oRen5z39+/qseTRgFAAAAwLSi8uqOO+5IV1xxRbrtttuyKqdf/dVfTSeeeGJ697vfnVVFhaGhoayr7V/+5V/Ss571rHTJJZdk1VNheHg4DQ4OCqMAAAAAmFq0/kW101VXXZVVRp111lnpV37lV7J2wAiiIpya6LrrrktXX311etzjHpeNfDn22GMfqaoSRgEAAAAwqZGRkfTZz342feQjH0n79+9PT3/607Nqp1tvvTW9/e1vTzt37sx/8lC33HJL9jPNzc3pVa96VTr11FOznxdGAQAAAHCIqHz69Kc/nT7xiU+kZcuWpV/8xV9MP/uzP5s+97nPpWuuuSYLpw4nwqcrr7wy3XvvvVmF1JOe9CRhFAAAAACHioqmLVu2pCc84QlZkHT22WdnIdQnP/nJ/CdmZvfu3em9731v+spXvpJe8IIXCKMAAAAAOFTMijr66KPTa17zmvT4xz8+3X777ekb3/hG/t3ZicHm8Wv37NkjjAIAAABgcrEDXrTjRZi0fPmRx0jR7ieMAgAAAGDOTj/99PTnf/7n6fLLL8+OX//1X8+/MzlhFAAAAABzFhVTMV+q9piOMAoAAACAI3LgwIE0OjqaHXF7OsIoAAAAgAqJMCiGj4+NjeVnGoswCgAAAKCkIniqDZ9i4Hh/f3/q6elJu3fvzn+qsQijAAAAAEqiCJ5qw6e+vr60d+/e9NBDD6Vdu3al3t7eNDAwUPfKqFNOOSU96UlPSmecccajjrVr12a75BViZtTEn4njvPPOS6tXrxZGAQAAADSq2qqnOAYHB7Pwac+ePenBBx98JHyK8/UOnyZas2ZNetnLXpbtlld7vOQlL0krV67MfyqlDRs2HPIzcfz4j/94WrFihTAKAAAAoFHUhk8xDDwqnCJsivDpgQceyL5GGDU0NJT93EIaGRlJ+/fvn/SI32shqrUm+5miWksYBQAAALBIasOnCHsisCnmPUXlU7TfxQyoCHjiZ6tAGAUAAACwACJMmhg+RcVQhE/RbleET3EuvleV8GkiYRQAAADAPCiCpyJ8Kna627dvXzZsPMKnuB3hU22bW73UDhVvJMIoAAAAgDqYKnyqHTYeVVDzsdNdiPBp+fLl2dHU1JTtateIhFEAAAAAczBV+BStdrU73c3XsPEifIrgKY62trbU0dGRVq9enY466qi0du3a/CcbizAKAAAAYAamC58ieIqvET4NDg7Oe/i0YsWK1N7enrq6utKaNWvSMccck33t7OxMra2t2c81KmEUAAAAwCRmWvm0EOFTtNxF+NTd3Z1VPB199NFZBVRUQrW0tDTsfKjJCKMAAAAADlrM8CnCpNrwKQKmlStXZuHTunXrHgmf4lwEU4cLn4p/XiMSRgEAAABL0kwGjs9n+BTBU234FFVOq1atyuY9xRG3I3yKlryZKMKnRg2hCsIoAAAAYEkowqcInqYLn+Zj4HgRPhXznmKuUzFsPKqeovopqqCiFS9+Zq5qA6nidlx3IxFGAQAAAJU0MXyKkGmytruFCp9iuHgRPsXcpxg+Hjvgxc/VUxFEFb+HkZGR/DuNQRgFAAAAVMJUlU9F+LR79+4FC58iZIrwKXa4K8KnYqe7+Nl6K/6Ztf/suB3tf6Ojo/mZxiCMAgAAAEqp0Sqfou2uCJ/ia4RPC7nTXW0gFb+3OGLm1P79+xuqVU8YBQAAAJTC4cKnxax8Kna6W8jwqVD774vbtUcEZPH77evry39i8QmjAAAAgIbUyOHTYlQ+HU7x+yiCqKI6Kn6v8d+op6enISqkhFEAAADAvBgYGUx7B/alXft3pwf7H0oP9j2U3Y/zkxE+HbkiiCqOuKb4Pcc1xH+zaF2M/6YDAwPZTK3FOIRRAAAAQN31DfWnT9362fSWL74rveGzf5/++HN/nX7vY3+e3vyFrenfb/lM2jfwcJWO8Kk+it9jEUIV11ccRSAV7YTx3zLmSO3bt2/Gx3e+8520c+fOSb8320MYBQAAANTV7v496UPb/yW96yvvSf915/Xp/p4fZOd/2PNA+vpdN6Qr/uuadMVXr0n37r5P+DRPJgZScd1xxAyp2Nnv2GOPTccff3w67rjjsiPuT3fccMMNWZA02fdmewijAAAAgLqJiqi//exb0kdu/Hh+5tGKaqhP3/aF9H8+/Pvp3gfvEz7VSXEdRRBVHBMDqfjvEUdxu/bcVMfEX3skhzAKAACAxhdDlx0NfxwYH08fu+mT6eZ7b0nLDp5aFufHD54fG8+O8dGx7Guci2NweCC9Z/sH09j46KT/vNkcEcPEsXzZstS0fHlqbW1JnR0dac3q1enoo47Kvsb9lubm7Ocm+2dU4Siurfa/x8Qj/vvUHisiZDrMEb8uuz1JuDTb4+Dfi/jdNrbLL788O7Zs2ZKuvfba/Gy1RKlblCVu3LgxP1MtO3bsyPpSV61alZ+pnipf47Zt29L27dvTZZddlp+pHmuw/JbCNW7dujVt2rQpbd68OT9THVVfg8HjTPlZg+XmGstt9OC1vW/rO9LpTzg9nXnmWfnZ6rnvvp2pq6v74NGVnymLH8UKIyOj6W8+/ea0c9992f0icsi+RECyPItKDvEbT/6VtKqtO783c7X/tOaW5rRiRXNW6dQcgdOyyf9d8yXaC3t7e9IJJ5yYn1lctXFP3H7kzyL7/4/+fph4f6KPfPjD6fTHPS6d85SnpObVa/Kzc7NsbGxs+n9bA3jjG9+YHRdffHH60Ic+lJ+tFi9Qy6/qYdSNN96YLr300vxM9ViD5bcUrvGd73xnOvfcc70RLimPM+VnDZabayyxg2+Qd/zjm9O9n/5UFjDEUVUxwDtaqRY6RJmriWFHGD8wnnoGe7Pbs9HR0pFWLG/K781M0YJW3F5s8d8gWg2jla1RTAx8Dhc4TScGnsf6W3v+BemU174uLTuC61y2e/fuhg+j3vKWt2RHPPFfffXV+dlqiSeOOE466aT8TLXcfffd2YvTKr9ArfI1fuMb30jf+ta30ste9rL8TPVYg+W3FK7xqquuSmeddVY6//zz8zPVUfU1GDzOlJ81WG6usbwOjIyk77/6V9P+/v4lEUZFqBKBVKOZLHiaLNgYGx9L/SP783sz19bcllqWT/9nOzF4aoQAqlYEUfHfpJHCqMJUwc9swqkijIrjpNf/VWpety7/zuwt27lz58z/zYvkHe94R3r729+eLrjgguwTqSrq6enJjvXr1+dnquXee+9N3d3d2VFVVb7GqIq6+eab00te8pL8TPVYg+W3FK7xmmuuSWeeeWZWmVE1VV+DweNM+VmD5eYay+3Bv3p92r1jR2ptaUmtra352eoZGR1NTcub0vIpWtkWShYSHDjw6HauGYYWURm1Z3DfrEKO0N3amVqaWiJhylvvisAp+9JwwdNkxscPZGFc84oV+ZnGNLs/mR+JNsRogWxftSod/Tu/n5avXJl/Z/bMjGoQ8QmG0v1yq3qbnplR5WYNVoN5NeXmcab8rMFyc43l1vPNm9Ln3/XOdOzRx2Rb0VfVvp59qa21NbW2tuVnFsLDs4SKYzyOsbEsWBkfH5t1qBS+uOO/sh31DqfIlyKAe9rJW1JXW2c2TPvhnd8ar7rocIaGBtPg0FBa1V3N58KbbroxHXPscelxz3lOWnPRk/KzcyOMahBeoJafMKrcrMHy80a43LxJLD9rsNyswWqo+jVWeQ0WFuqxtAie4ojWwOHh4TQyMpJ9jftH6sob3p++vONr+b2HFf++2hbEqHaKY+O6Deny//4Xqa253FVv1uDMNV4jKgAAAFA3EQLFPKMieIrZPxGcPPTQQ9kR7Z0DAwN1CaLC8057Vupq7czvPdrDVU/Ls7lK8TVmRf3M2T9R+iCK2RFGAQAAQIXUhk9R8RThUwROu3btysKnCKLqGT4VosopAqYNq09ILzr359LRXUelFU0rHgmg4iiqoeJYu3J1etppW9JzT39m/k9gqRBGAQAAQIlNDJ8iaIrwaffu3enBBx/MwqcIpEZHR/NfUR9F+BRVTitWrEhtbW2ps7MzrVmzJv3M+T+Rfu95v5FOPmpjWrNydepo7UjNTc1pZcvKtKq9O52w6rj06mdcln796dUdBcLUhFEAAABQIrXhUwRMg4OD2U5ne/bsySqfYm5RhE8RTNXTdOHT0UcfnX2N+7HjWvzs6ceeln7nOb+eLn3yL6fnn/7stOm4s9JPPvG56RVbLkl/9OOvTRectCn/J7PUCKMAAACggU0Mn4aGhlJfX18WPkXlU3zt7+/P5kHFz9bLxPCptbV12vBpMiesOj497bFb0s+d+d/Syze9KP3yRb+YnnXaU9KJq07If4KlSBgFAAAADaY2fIqQKcKmqHiKyqdov4swar7DpwiZOjo6sh3+jjrqqLR27drDhk8wE8IoAAAAWGS11U9R+VS7410MHo82vDgfP1NPRQBVhE8rV65Mq1atysKndevWpa6urqwiKn4G6sXfJgAAAFhgteFTUfkUgVNUPUX4FAPIYxbUfIVPUf1UW/kUwVMc3d3d2Swo4RPzyd8uAAAAmGcTw6fayqc4ivApvl9Pk4VPReVTHEX4FN+HhSKMAgAAgDqrDZ9iV7sInyJwiqqnCJ8iiBoYGJjX8Km5uTlru4vAKaqeivCpvb1d+MSiEkYBAADAEZoYPkXQVLTdxY53ET5FIBUDyetpYvgUQVMETjFsPHa8iyqoCKRiJhQ0CmEUAAAAzFJt+BQBU7TYRfi0Z8+erPIpdr6LOVARTNVTbfgUAVO02MWQ8TVr1mSVTzH/KcKnCKagUQmjAAAA4DAmhk+xs11fX18WOkXlU4RQET7FPKj42XqZGD7FznadnZ1Z6BSVTxFCxRyomAcVPwtlIIwCAACASdSGT8WOdxE+ReVTtN9FGBWhVD3Dp1AbPtXueBeVT9F+F2FUhFLCJ8pKGAUAAAAHRahUW/kU4VPMeoqh43FEG16cj5CqnmqrnyJ8ija7Yse7GDwebXgRPsXPQBX4mwwAAMCSVNt6V1Q+TQyfYhZUfL+eJoZPReVTtNxF5VMMII9ZUMInqsrfbAAAAJaEieFT7G4X4VO03cXR09OzIOFTbeVTHEX4FN+HpUAYBQAAQCXVhk+1O97FvKcInyKIGhgYmNfwKXa1a29vzwKnaLmL8CmCqDgnfGKpEkYBAABQGZPNfZq4493IyEj+0/UxMXyKKqeY8xQtd7HjXbTgRTVUDCQHhFEAAACU2HRzn6ICqhg6Hj9XL7XhUwRMET7FDncx8ykqn+JrzIGKYAo4lDAKAACA0qgNn6LCKdrsYtZThE/zNfdpYvgUO9sVQ8ej8inCpwijYh5U/CwwPWEUAAAADS3CpyKAiiqnvr6+rOUuwqdowYtB5NGWV0+ThU/F0PFov4s2vDgvfILZE0YBAADQUIrqp9q5T1EBFcFTtN5FGBUtefPRejcxfIqh40X4FO148TPAkbGKAAAAWFQTW++i0qmY+xRHzH2K8/Ez9VLbehftdbXhUxxF+BTfB+pLGAUAAMCCqg2fovopZjxF4BRVT7HrXQRRUQk1X3OfYrB4e3t76u7uzoKnaL2L23FO+ATzTxgFAADAvIsAqgifita7aLuL8CnmP8X9qH6qp4lzn2p3vIvh4ytXrsy+BywsYRQAAAB1V1v9FPOdImwqWu+iAioqoSKUWoi5T8XQcTveQWMQRgEAAHDEJs59ija7np6eLHyKXe/idrTjzWfrXVQ6TTb3KX4GaBxWJAAAAHNSO/cpqpxil7touYvwKVrwYhB5fK+ealvvImiKwCmqno4++ugsiDL3CRqfMAoAAIAZKaqfImCqbb2L8Cla7yKMivP1br2LcClCqGLXu5j3FOFTzH+K+1EVBZSHMAoAAIBJTWy9i0qnYu5TBFAx9yla7+Jn6qW29a4In6LiKeY9xdeohIp5UOY+QXkJowAAAMjUhk9R/RRBUwROUfUU4VMEUTELaj7nPkWbXXd3dzbzKQaPx21zn6BarGYAAIAlLAKoInyKFrtotYt5TxE+xfynaMWLqqh6tt7Vzn2KKqeoeoqWuwifogUvBpHH94BqEkYBAAAsIbXVTxE+Retd7HQX4VO030UYFcPI56P1LgKmaL0rdr2L8CmGj0cYFee13sHSIIwCAACosIlzn6ZqvZuvuU+1rXcRPGm9A6x8AACAiqltvYsqp6la7+qptvUugiatd8BUhFEAAAAlN1nrXVQ8RfgUFVBF61095z5F9VPt3KfY9S5CpwifIoTSegdMRRgFAABQQkX4VOx6F4FTEUDF1zg3n613UelU7HoXR1dXVxZKab0DDsejBAAAQAlMV/0UrXdxv56td7XhU9F6F4FTzH06+uijswHkMQsqvg8wG8IoAACABjWx+ql28HjsgFfv6qfa8CmqnKLVLlrvInyK1rtoxYuqKIAjIYwCAABoEEX1U4RPUf0Ug8bnc/B4Uf0U4VPMd4rWu6h4irlPUQEVYVSEUuY+AfUkjAIAAFgkta13RfVTVDwV1U9RCRWDx+tV/TRx7lO02cXcpwieIoCK29GOZ+4TMJ88wgAAACygw1U/xeyn+F69TJz7FNVO0XIX4VO04EU1VHwPYKEIowAAAObRxOqngYGBrPpp165d2TGf1U9F612EThE+RQgVYVSc13oHLBZhFAAAQJ1FAFWET1H91NfXl1U/Pfjgg9nXqH6K79fLxOqnYte7ovUu5j5pvQMahUcjAACAI1Rb/RQDxidWP0UYFdVP8XP1MHHweOxyF6FTVD7Z9Q5odMIoAACAOaitfoqgKQKnmPkUs5+i+ikCqXpVP9W23kUANXHweFRCqX4CysIjFQAAwAxMrH6KVruoforwKXa/izAqWvLms/qpmP1k8DhQZsIoAACAKUxX/bRv376s+mk+Bo9Hi11t9dO6detUPwGV4VEMAACorAiThkaHUt9Qf+ob7k8DIwNpdHw0/+6hiuqnCJ+K6qcInRaq+il2ulP9BFSdMAoAAKikCJ7u3H13+sqdX0+fvP3T6d/v+Gz63HeuS9/ceUvas39v/lPpkda7CKAGBwcPqX6Kc/Nd/RSVT3FEGBXVT/FzAFUljAIAACrnwb6H0mdu/2L680/+TXrjZ96aPnrzJ9Jnvv/F9Lbrrkx/8m9/la7+2vvTvXt2ZlVOtdVPEUJFGBVVUfWsfioGj0fQNFn1U3wfYKkQRgEAAJWyq393+vvPvyO940tXpYf6duVnfyRCpk/f9oX0uo/+afr2jtuyIeTzWf0UYVNR/RTzn1Q/AUudMAoAAKiMgZHB9M4vvyfddO/N+ZmHw6diFlTtsXf/3vTWr12ReoZ685+cu4nVTzFsfM2aNVn106pVq7J2PNVPAA9bdvBBuT61p/Po8ssvz44tW7aka6+9Nj9bLVEWvHfv3rRx48b8TLXs2LEjK0GOJ+KqqvI1btu2LW3fvj1ddtll+ZnqsQbLbylc49atW9OmTZvS5s2b8zPVUfU1GDzOlNtoX2/6/664Mp111pnp3HPPzc9WR09P78FjX1q/fn1+ppzirU3Mg/rbz7zlRy12EUTF/+LrwWOyneh+7LHPSD9+2rPzezPzcFHTsuzrsmXLs+HjLS3NB4/W1NS0OJ/5V+XPcSrvec816ayzz06bn/704g+gcqr8WOq5vvzq+VpUGNUgvEAtvypfozCq/KzBahBGlZvHmfIaGxhIP/z4v6QbPvvZdPTRR6Vjjjk2/051xNykoaGhrJqnnB4OmkZHx9Jdu+5O33nwjvz8jxTveibLMI7uPCo98ZjT83szE1VOD7fiLT/4tengP3fxw5Hy/zlO75ZbbklHn3hiOveVl6bW40/Iz1ZLlR9LPdeXnzCqgrxALb8qX6MwqvyswWoQRpWbx5ny2vv1r6XvXf432ZDrmP8TR9XEW4JoWytbG1nxVia+FrejTW94bDi7PVPLly1PXa2d+b3JFWFTfG3UWU9l/XOcqWINHvusZ6eTf+3X87PVUuXHUs/15VfP16JmRgEAwDTGhobSsklau1gcRfBUO/upCKLCwe/mt2Zusl9ThE5R/VR7NGoQtZSMH1yTQLmpjGoQPi0tvypfo8qo8rMGq0FlVLl5nCmvoR/+MN355jelnXd8L3V0dqSuzukraMpodGwsjY6OprbW1vzM4srCpjx0Ovj/4v8eCZwOFzbFMPIH+h7K781Md1tXOr772KxCKjwcRGU3S6XR/hzr7Qc/+GHq6OpMp13y0rTuaU/Pz1ZLlR9LPdeXnza9CvICtfyqfI3CqPKzBqtBGFVuHmfKbXxwMP1/b3trOuOMM9I555yTn62Onp6e1Nvbm0488cT8zMKKtyTZMR4h1HgWqMT8ozii8mk2bn/ojvSe7f+URsfH8jM/Ev+O5csfnTJF8PS805+d/sc5/y2taF6Rny2nxf5znG/vf//70xlnn5MufPbshs2XSZUfSz3Xl58wqoK8QC2/Kl+jMKr8rMFqEEaVm8eZ8rMG66sIoIpWu2L49sjIyKwDqFqDo0PpTV95R7pj1/fzMw8r/n3FPKWiDa+rrTP94fN/O51x3OOy82VW9ceZKq/BQpUfSz3Xl18916DmdwAAYEFEGDSWt5INDg5mVTx79uxJDz30UFbVE2HUkQRRoW1Fa/rZJ/xEOqpjbXa/CJ2Ko3b2U7Tn/cxZP16JIAqgTIRRAADAvCgqnyKAioqn2A0tKgcifIoQqr+/PztfD0XQtGLFinT2iWekSy9+adq4bkPqbOtMK5pWPGr4eEtTSzq6c1362XN+Mv3i+f8jOwfAwhFGAQAAdVNb/RStd319fY9UP0UQFRVRR1r9VIiAKdruIoBqb29PXV1dae3atemoo45KT338xen1P/VH6TlPeGZ6wvGPSyd0H5eO6TgqbVy7IW1+zHnpfz315ekFm34m/ycBsJCEUQAAwBEpqp8igBoYGMja73bt2pUdEUZFKBUh1ZGqrX5qaWlJK1euzGbrRPgUc3Y6Ojqy7xXWdqxOr7z4JemvfvpP0uue+er0vy54abr8f/xF+t3nvCZduPH8/KcAWGjCKAAAYFaK9rui+ina7WJob1Q/xddox4twqh4igCqqn9ra2rLqpzVr1mQBVHd3d3YuAqrDObrjqHR813GpdUVLfgaAxSKMAgAADqu2+mni8PG4XY/h46G2+qm5uflR1U8RQkX1U5wHoLyEUQAAwCEmVj9F+918Dh8vqp9aW1uz6qdou4sAKoKomVY/AVAOHtEBAIDMZMPHo+0uZj/Vc/h4Uf0UAVRR/RQtdxE+xQDyqH6KmVDxcwBUjzAKAACWsNr2u5j1NJ/Dx4vqpwiaOjs7H1X9FLvhqX4CWBo82gMAwBJS234Xc56i3S6qnnbv3p19refw8YnVT9F+t27duuyIMCpa8lQ/ASw9wigAAKi4ovopZjwVw8cjfIrqp7hdz+qnYvh4VD9Fu11t9VMEUhFOAbC0CaMAAKBipho+HuFTMXw8vlcPtdVP0WoX1U8x9ykCqLit+gmAiYRRAABQAROHj0fFUwwfj93v5mP4eLHzXVQ7RdVTtN5FFVRUQ8X3AGAqwigAACip2va7qH6qHT4e1U8RStVDbfVTW1tbVvG0Zs2aLICKXfDiXPwMAMyEZwwAACiRIoCa2H4XVVD1Gj4+cfZTtN8V1U8RQkX1UwRTADAXwigAAGhgE3e/6+vry+Y+RQBV7/a7qH4q2u+i+ina7iKAiiBK9RMA9eLZBAAAGkzt/KeJu99FGFWP3e9qq5/iiOqnaLmLweMxgDyqn6IqyvBxAOpNGAUAAA2gtv0u2u16enqy4eP13P2utvopgqbOzs5Hqp+iEioCKdVPAMw3zzQAALAIJrbfRQAVc5+K9ruYB1WP9rva2U+x812EThE+xRFhlOonABaaMAoAABZIbftdVEBF+10x/ykqoSKUqmf7Xcx+ina72tlPEUhFdRQALJZlB5/wjuzZbgG86U1vSn/3d3+XLr744vSBD3wgP1st8elXHCeddFJ+plruvvvu7MVPHFVV5Wu8/vrr04033phe+cpX5meqxxosv6Vwje9617vSueeemy644IL8THVUfQ0GjzPlN9c1WFRAxdeRkZEscIogqh673hUigCqOIoSKiqeZ7nhnDVZD1a+xys+DhSo/llqD5VfPNbjsnnvuafgwauvWrekd73hHOv/887PbVRSfisWnYSeeeGJ+plp27tyZDcSMsvCqqvI1RhB1yy23pBe/+MX5meqxBstvKVzje9/73vTEJz4xexFQNVVfg8HjTPnNZg0W4VN8jQAqwqf4Wo+2u0JRARVfI3SKIwKouVQ9WYPVUPVrrPLzYKHKj6XWYPnVcw0uO/gfq+HDqDe/+c3pH/7hH9JFF12UrrnmmvxstVQ9QVWVUW7f+MY30k033ZRe8YpX5Geqxxosv6VwjVdccUU655xzsg9nqqbqazB4nCm/6dZgbfVTMYQ8KqDqMXS8Vm0AFcFTcRzp0HFrsBqqfo1Vfh4sVPmx1Bosv3quwWUHnzAbPoy6/PLLs2PLli3p2muvzc9WS/yFjYGVGzduzM9Uy44dO7JZBVV+gVrla9y2bVvavn17uuyyy/Iz1WMNlt9SuMaoDt60aVPavHlzfqY6qr4Gg8eZ8pu4BmsDqNr2u3oGUBE6FQFUVDxF8BQteFEFFefrxRqshqpfY5WfBwtVfiy1BsuvnmvQAHMAAJihCJ+K4eOx+120Yzz00EPZEPK4X48gKkKmCJ6K2U/RrrNmzZpsAHncjkCqnkEUACw0YRQAAEyhqH6KkCmOgYGB7FPv2P0uPgGP+/WYA1UEUFHx1N7enoVOET6tXbs22w1vpoPIAaAMhFEAAFAjAqiY+xThU7TexUDaqHzq7+/Pqp+iKqoeky5qW+9WrlyZDS2OACpadOL+XAaRA0AZCKMAAFjyoropAqiY/RTVThFARfXT7t27sxAqzh9pAFXMfira7yJwiuApAqiYDxMVUUc6iBwAysCzHQAAS05t+10x/6lov4uvcT/CqSM1cf5TZ2fnIwFUVELFOfOfAFhqhFEAACwJE9vv+vr6HgmgYhB5nKv3/Ke2trZs/lPMfoojwqhoywOApUwYBQBAZRUBVLTZDQ4OZu130XoXR4RR9Z7/FAFU7fyn2AUv7kdlFADwMGEUAACVUhtA1c5/iiHk0X4XlVFHaqr5T0cddZT5TwBwGJ4hAQAovdoAKgKnaLurnf9Ur/a72gDK/CcAmBthFAAApVTsgFcMIN+3b18WQMXXqIiq9/ynODo6Osx/AoAjJIwCAKA0JtsBL+Y/RQAVM6HqEUBNNf8pbsdAcvOfAODICKMAAGhokwVQ9dwBr7b9LiqdzH8CgPnlWRUAgIYS85+KACrCpv7+/kcFUPXYAa9ovzP/CQAWnjAKAIBFNzGA6uvreySAit3w6h1ARbtdV1dXWrNmzaPmPwmgAGD+CaMAAFgUxQ54RQAVodOePXuyGVARRkUAdaSKACrmP0W7XQRQUf0UIVS048V5AGBhCaMAAFgwRQA1MjKSDRyPACrCpziiHS/OH6mpBpBHG17cj+8BAItHGAUAwLyaKoCKKqgYSB6VUUdq4gDyGDweAZQB5ADQeDwrAwBQd7UB1MDAQBZAxfynIoCK7x2p2gCqo6PjkQAqKqFiJpQACgAak2doAADqojaAisApdr6LACoGkcf9GFB+JGL+UxFAxW53EUAVO+DFLCg74AFAOQijAACYs2IHvDgicNq3b18WQMXXqIiqRwBV7IAXYVPsehfDx4sAyg54AFA+wigAAGalCKBit7sigIr2u2jFi5lQ9QqgYgB5tNtF6LR27drsiDDKDngAUG7CKAAADqs2gIpd76L1LiqgohVvaGho3gKoqIKKgeRRGQUAVIMwCgCASRUBVIRNtQFUVEBFKBUzoo5EzH8qAqgInGLweLTfFQFUfA8AqB5hFAAAmWIAeRFA9fX1zUsAVeyAF4FTsQNefG1vb7cDHgAsAZ7tAQCWsMkCqJj/FAFU3I4A6kjVBlCxA14RQEUlVLTkCaAAYGnxzA8AsMRMFUDt3r07a8cbGRnJf3JuYv5TEUDFDngRQK1evfqRHfDinB3wAGDpEkYBACwBCx1Axa53tQFUVEUJoACAIIwCAKiohQigigHkRQAVO+DFEbcjgAIAmEgYBQBQIRFA1e6CN58BVMx7iqqnCJ9iDlQMJI/KKACA6QijAABKbr4DqGi/KwKo2PEuBo9H+92aNWuyACq+BwAwU8IoAICSmqoFL27XI4AqdsCLAKrYAS/mQMV9O+ABAHPlVQQAQIlMVQFV7wAqKp6KACq+RkueAAoAqAevKAAAGtxkAdSuXbvq1oI3WQAVrXgxlNwOeABAvQmjAAAa0FQB1HxWQAmgAICFIIwCAJgn4wfGU89gb3qob1fa2XN/enD/rrRn/960f3h//hOPJoACAJYCYRQAwDzoG+pPt//wjvSJWz6V3rPtn9Jbv3JF+sDN/5yu3f6x9Jnbv5ju2/eD7OcEUADAUiOMAgCos70D+9J/fPuz6fLPvCW9d9u16bO3fzHdtfvu9O0Hbk8f+9Yn09YvX53edt0V6db7bxdAAQBLjjAKAKCOxsbH07/d/Kl01dfel37Y+0B+9kcOHDiQHTfcfVP6rY/8Ubr3oZ11CaCampoEUABAKQijAADq6L5992dteLWKACpa8ooj7oeP3/6pNDAykN2eDQEUAFBWyw6+EHr4lVADu/zyy7Njy5Yt6dprr83PVsu+ffvS3r1708aNG/Mz1bJjx460evXq7IVyVVX5Grdt25a2b9+eLrvssvxM9ViD5VflaxwfGUmjB/9+vve916Qzn/jEdPbZ5+TfqY7e3t7U09OTTjzxxPxMecVMqE9++z+z20UIFa+2Hn7JdSAtW3boZ4EvO++F6QnHPC6/N7Vl8b9lDx8RQsXRKMHTzp07syCsq6srP1M973v/+9KZ525Km5/xjPxMdVT9eTC4xvLbunVr2rTp4BrcvDk/Uz1Vfj1jDZZfPdegMKpBVP0vrTfC5SaMKj9rsNx6v31LevBTn0q33/yttHbduqz6pWqGh4ezo7OzMz9TNj8KnK6/e3vqHerLz9coXnJNEhyd1H1iWr/6hPxerUf/bFPTw9VQy5c3NUQAVSvmXRUBWVV99zvfSetOOy2d/+rXpKaVK/Oz1eBNYjV4I1x+wqhyswZnThjVIKr+l9Yb4XITRpWfNVhud7zh9WnfjdvT/v37U3Nzc3ZUTbwcida1CFrKpngpFV9jXlTvUG92fzaam5rTyub2/N6PFIFTUQ3VyMbGxrLWwUb/fR6JYg2e8Sd/mrrPOTc/Ww3eJFaDN8LlJ4wqN2tw5syMAoBGd2A8v0EjKQK02hlQRTB1JIrgKYKd4qhywFM6EZjW4c8ZAJYylVENouoJqqqMclMZVX7WYLnt+uLn0wOf/ES69557U3dXV+pe1Z1/pzpGR8eyHeXa29vyM41jfPzh0OnAgTh+VAk1VSBx976dk7fppeLnHx0sxb31a09I61auS5E5lTV4GhgYzKqGVqwoX3XbTGVr8KQN6fw/+OO0orQtpZNTsVANqjLKT2VUuVmDMyeMahBV/0vrjXC5CaPKzxosv9GenvSeq65KZ551Vjr3nOoNMO+JAeYH1+H69evzM4tr/ODLo6LiqZhnNXLwmInYHe9z37suv/cj8ZIr/rc8BpjneVMETy0rWtJrnvGqdM6JZz58sqTuvffe1H1w/UVgWlXXvOc96cyDr0er+EbYm8Rq8Ea4/IRR5WYNzpwwqkFU/S+tN8LlJowqP2uwGqr8IrwR1mC8JKoNoIaGhrKvcX82vr9nR3rzV995SHVUFkYdPGIuVtGKF8468Ynpj57/2tTe3HhVYbNhDZabN4nVUPVrFEaVmzVYfvVcg2ZGAQBLVoRDMXg7WgQHBgZST09P2rVrV/ZicnBwcNZBVDhlzcb0tJO3pI6Wh3dbK4KnyWZAnbDquPRL5/986YMoAIDZEEYBAEvKVAFUfJIZ9+cSQBUiaIrKp+c87hnpx05/Rjp+9XGpvaXtUQFU0/Km1NXWmU496uT0qqe8PJ15wunZeQCApUIYBQBUXm0AFRVPvb29affu3XUJoCJkigAqhne3t7en7u7u9NgTT0mveurL0isufnG6+OQL02nHnJrWrzohre8+IZsN9bwznp3+5Cdel87bcHb+TwEAWDqEUQBAZUXIFAFUzH/q6+tLe/bsyY79+/dn4dRc1QZQK1euzAKodevWZTM+IpCKSqjw5FMuSr/17P+d/uKn/iD9ypZXpP+1+WXpT3/yd9NLL3xhWrNydfYzAABLjTAKAKiUCKBGR0cfCaCi+imqoPr7+7PzczVZBdRkAdRkVja3p8es2ZDWta/JzwAALF3CKACg9IoAKna/KyqgYg5U3I7KqLmqDaDa2tpSV1dXWrt2bbbT0eECKAAAJucVFABQSrUBVFQ9RQVUPQKoECHTihUrUmtra+rs7Exr1qzJjmjJi3AKAIC5E0YBAKVRO4g85j7t27cvC6BiIHmEUvH9uZosgIoqqI6Ojuw8AAD1IYwCABraxACqp6cnC6Dia8yFqkcA1dLSkoVO0X4XAVSEUdGaBwBA/QmjAICGUxtADQwMPBJARSVU3I8WvbmqDaCi7S4GkMcg8pgHFediThQAAPNHGAUANITaAGpwcDBrvYtd8GIW1JEGUMUg8gibip3wogIqvkZbngAKAGDhCKMAgEUVAVQMIo+Wu2InvDiiJS++N1e1O+EVFVARQMXX2BnPTngAAIvDqzAAYMEVO+FFBVSxE1604MXtOD9XtQFUUQEVLXgCKACAxuEVGQCwIKINr6iAitApqp+iDS+qoeoVQEXgFLOfogIqhpFHICWAAgBoLF6dAQDzZrKd8CKAinlQce5IFIPIY+ZT7H63Zs2a7IiWvAinAABoTMIoAKDuigAqqqAieCp2wou2vAio5qoIoKICqgigogqqo6MjOw8AQOMTRgEAdVHMgSra8GIOVFRBRUXUkeyEV1sBFaFTtN9FABVhVLTmAQBQLsIoAGDOijlQw8PDWehUBFAxB+pI2vBqA6janfBiHlRLS0v+UwAAlJEwCgCYldo5UAMDA4/MgYqvEUrNtQ0vBpFHABVhU7ETXgRQ8TVCqfg+AADlJ4wCAGaktg0vKp8igIpKqJgDNdc2vIk74RUBVFRCxX0BFABA9QijAIApTQyg9uzZk4VQMRMqqqPmKtrwigAqWu9iEHkEUXE/vgcAQHV5tQcAPMrENrzYBa/ec6Bi+HgMIo8QKmZCCaAAAJYOr/wAgEwRQEXbXcx/2rVrVxZERVXUkcyBija8mANVO4jcTngAAEuXMAoAlrDJ2vDiiIqouc6BCsUcKIPIAQCYSBgFAEtMVDlFABU73+3fvz8bQl7MgYrzc1U7ByoqnyKAila8CKS04QEAUPDKEACWgNo5UBFARRteBFDxNUKpI2nDK+ZAdXR0ZOFThFBxO6qjAABgImEUAFRYtNpFABVteL29vVkAFXOgYi7UXNvwijlQUQUVc6CKNrzYFc8cKAAADmfBwqj4NPb+++9PN954Y7ruuuvStm3b0ne/+92sJQAAqJ+Jc6CKNryoiIrn47mqbcOL4GndunXZQPK4bw4UAAAztSBhVHwie9NNN6W//du/Ta9+9avTL//yL6fLLrssve51r0sf+9jHshfJAMDcFW14xRyoqH6KACrCqHgenqsIoKINr5gDFW14a9asySqizIECAGAuFuRV5O23355+8zd/M/3bv/1b2rJlS/qzP/uz9OIXvzgLof7kT/4kffzjH89ePAMAs1PMgYq2u9o5UFEVdSRzoKINr6WlJQudovop2vAijNKGBwDAkZr3MCpeIH/4wx9O3//+99Mb3/jG9IY3vCFdcskl6bWvfW1697vfnR7/+Menf/iHf0g7d+7MfwUAMJ2JbXh79uzJjoGBgTnPgQrFHKjY/a6YAxVfYzg5AADUy7yHUVH9dP3116enPe1p6SlPecqjSvpPOeWU9LznPS/7/n333ZefBQAmiiqnIoSqbcOL2Ytxbq5q50BF5VMEUNGKF4GUNjwAAObDglRGnX322enpT3969iK3VrQWxAvr2P5Z2T8AHCqeJ4s2vDgihKpHG17MgYqKp3gOjvApQqi4HdVRAAAwn+Y9jDrhhBPS61//+nTppZce8glrtOb953/+Z7rwwgvTsccem58FgKUtQqaiDS8qn6LKONrwYr7iXNvwJs6BKtrwYlc8HwgBALCQFq3+PloLrrnmmmyXvZ//+Z9P69evz78DAEtTMYw82vCi+ikCqN7e3uzcXNW24UXwFAFUDCSP+xFQAQDAQlt28MXu3Gr8Zyhe6MansPFCuHjR+8Mf/jC9733vS+9617vS85///PSHf/iHWYtAfAI82QvuN7/5zdlx0UUXpfe85z352WqJNx3RerFhw4b8TLXcc8892Zuf+CS+qqp8jTfccEMWHL/85S/Pz1SPNVh+Zb3GYhZUBFFFK17cnswHP/jBdMYZZ6SzzjorPzO5CKDiKCqhIniKtrxGVvU1GDzOlN+VV16ZzjnnnHTeeeflZ6rDGqyGql9jlddgocqPpdZg+dVzDS777Gc/O29hVIRPETKddNJJ2ddw9913p6uvvjqrivrpn/7p9LrXvS4dc8wx2R/aXXfdlVVMTfRP//RP6UMf+lD2AvxP//RP87PVErshRStGVdsVI4CMWSQT54ZVSZWv8bbbbkvf/e5300/91E/lZ6rHGiy/sl1jBFARRMWHMNF+F8fhfPrTn04nn3xyOvXUU/MzPxLPuRFAxdf4AKj4IKgsqr4Gg8eZ8vv4xz+eTjvttHT66afnZ6rDGqyGql9jlddgocqPpdZg+dVzDS772Mc+Nm9hVHwK+4QnPCFLBeMF96233pr+/u//Pn3pS19KL37xi9OrX/3qLKSKF+I7duzI3uxOtiPQRz/60fTP//zP2T/rd3/3d/Oz1RItGfEXN4K5KnrggQeyB9SYU1JVVb7GWJvf+973skrGqrIGy68M11hUQMURz3fx/DebIeSf//zn08aNG7NAqlBUQMUR4VM89xaVyGVS9TUYPM6U37//+79nYXC8EK8aa7Aaqn6NVV6DhSo/llqD5VfPNbjs4IvgeW3TC/GiO1p8/uzP/izdcsst6bWvfW265JJLZrzALr/88uzYsmVLuvbaa/Oz1RKlfDGgNt5kVFGEjRE8RslpVVX5Grdt25a2b9+eLrvssvxM9ViD5deo1xhPs0UAFe3oU7Wkz0RUFZ955plp06ZNjwRQsSNeHGWqgppM1ddg8DhTflu3bs3W3+bNm/Mz1WENVkPVr7HKa7BQ5cdSa7D86rkG532AebwIj4qKv/zLv8z6X//6r/86vfKVr6z0p2YALG1FABWhU3xCFq3o0YYen5TNNYgq2vCi8imeQ2OWRAwjj09Pyx5EAQCwtMx7GBW7AL397W9Pn/rUp9ILX/jCrKQrqqO++c1vPuqInwOAMos2vAibYgh5PK9FABWfkMX9uRYiFwFUDCGPIz4pjcNueAAAlNW8h1ExJ+ojH/lI9glxtBdceumlWWXUxCMGJANA2UTIFPOfov0uBlZGafaePXuyiqh47puLCJmKFryogopy/TVr1mRDySOcAgCAMpv3V7TRmvCYxzwm6ymMF9Lxonqyo729Pf8VAND4ija8gYGB7LkuAqiohpprG16IoCla7qLqKdrw4nkzvkYIBQAAVTHvYdRznvOcbPefwx0xkBUAGlm04UUVVLTdxfynCKCiEirux/fmomjDiyqomP9UVEHFhzSqoAAAqCKvcgFgGsUw8uHh4az1LmZARQgVLXkRTM1F0YYXFU9RHRwzoAwjBwBgqRBGAcAkJhtGHu14MRvqSIaRF214XV1dWQAVbXhRFWUYOQAAS4UwCgBytVVQE4eRz7UNL0Kmog2vo6PjkTa8qIjShgcAwFLkVTAAS95kVVBHOow82vCiCipmPxXDyKMaShseAABLnTAKgCVpYhVUMQvqSKqgimHk0YYX858igIpKqLivCgoAAB7mlTEAS0oETTF4PGY/1VZBRSg1F5MNI48QKlryIpgCAAAeTRgFQOVNtSPekVZBGUYOAACzJ4wCoLImVkFFAHUkO+IVbXiGkQMAwNx55QxA5UQVVAwfHxgYeFQVVJyfi6INrxhGHlVQhpEDAMDcCKMAqISodCqqoPr6+rIAKoKouVZBRatdUQVVDCOPeVDRlqcNDwAA5k4YBUCpFVVQg4ODjwwkj93xIpiai6iCioqnqIKK8CmqoKIlL84DAABHThgFQOnUVkFF8HSks6Amq4KKeVCGkQMAQP0JowAojRhIXsyCKqqgoiWv3lVQEUwBAADzQxgFQEObWAW1d+/e7IhA6kiroCJ4UgUFAAALSxgFQEMqqqBqZ0HF1zg3F7VVUMWOeNGSpwoKAAAWljAKgIYRlU4xkHx4eDirgord8GIeVL2roOyIBwAAi0cYBcCiiyqoohWvtgoqQqm5UAUFAACNSxgFwKIpqqD279//SBVU3I5warZUQQEAQDkIowBYULUDyWMnvAigenp6svtzacVbvnx5VgUVoVNXV5cqKAAAaHDCKAAWRDGQPOY/RfgUrXgxFyqqo2ZrsiqoaMezIx4AADQ+YRQA82biQPKogtq7d2+2Q96RVEHFLKiiCiq+qoICAIDyEEYBUHdTDSSPc7M11SyoCKRUQQEAQPkIowCom6IKKlrxjnQguSooAACoJmEUAEekdiB5tOJFG14EUXMZSD6xCioqoFRBAQBAtQijAJiTYiB5zH8qWvFid7y5tOJNVQUV5wAAgGoRRgEwK0UrXrTfFa140ZY3l4HkTU1NqaWlJa1cuTKtWrVKFRQAACwBwigADqu2FS+qoCKA6unpyUKp2aptxevs7MwGknd3d2ehFAAAUH3CKACmVLTiFQPJoxUvKqKiOmq2alvxInyKVryYCxXVUQAAwNIhjALgUaIKKsKmqIKKGVBRBRVDyecykDxM1orX1tamFQ8AAJYoYRQAmaIKqmjFO5KB5FrxAACAqQijACpuYGQw7RvqTQ/0PZT2DfSk4dFHz3kqBpJHK17MgSpa8eZSBaUVDwAAOBxhFEBFDY4Mpe89dFf6wne/nD5753Xpw9/61/SJW/4zffn7X0v37r0vjYw+XAXV39+fteHFTKi4Pxda8QAAgJkSRgFU0A96HkjXfe+r6R+/8M70li++K33qjs+lL9351fS+669Nl3/mLemKr7w3ffWObWnnD+/TigcAACwoYRRAxezZvze97/oPp3/4/DvSHQ9+Pz/78GDymAsVX//rzm3prz7zpvTlHV9Po+OzC6K04gEAAEdCGAVQIeMHxtMnv/3p9LnvfDE/83AIVRtExVH44Lc+mu7YdWd+b3pa8QAAgHpYdvBNyewn1C6wyy+/PDu2bNmSrr322vxstcSslpjZsnHjxvxMtezYsSN74xpvYKuqyte47QtfSLfc8I30whe+MD9TPbF7XAzvPvHEE/Mz5bR/eCD98b++Ie0b2Jfdzx7gDz7Mj42Np8iMJguOTjvq1PSK834pvzfBwR9fvmx5Vg3V2taahU+NWgG1c+fOrFKrq6srP1M9H/jAB9MTL7wwbX7yk/Mz1VH158Hgub78tm7dmjZt2pQ2b96cn6kOa7Aaqn6NVV6DhSo/llqD5VfPNSiMahBV/0srjCqv8ZGRtO0f35weuu3b6fTHn56frZ6R4eE0dPCI2UfldSD1DPSm/7rr+sifHq04MUkY1dLUnC5cf17kTj+S3VmWli+PuVDN2WyoRhezr1pbWlJzhWdW3Xr7d9IxWy5OF7705fmZ6vACtfyEUeVmDVZD1a9RGFVu1mD5CaMqqOp/aYVR5TX0wA/T9v91WRoZGcnas6oqHgqjja2sc4/i9x/H8Ohw2j8ykJ+dua7WzqwCqqicmvi1DMbGxrIKriq3De7fvz+1HHNMuujdV+VnqsML1PITRpWbNVgNVb9GYVS5WYPlV881aGYUML2x8ZTsjtawihCtCKMO3sq/Mzvxq4ogp/ha5VCnzOLPGQAAykxlVIOoeoKqMqrcvvbn/zft+85300kbT8rPVM/o6GhW/RU7xDWyLHAaG09j4+NxJ2X/q3kYHxobTrc/eEd+r0b8yIRsKa99Sm3NremJx5+ehVBlNjAwkO3yV4aWwrm6++570urNm9OFr35NfqY6fFpafiqjys0arIaqX6PKqHKzBsuvnmtQGNUgqv6XVhhVbtu2bUu3XHddetEvvSg/Uz0PDzDvTSeeeEJ+prFE9VO0oY2MjqahwaE0PDKcf+fR+ob60z98dWvqGerNzzwsfn1ttVN2O+Kog/939olPTK95xquy82W2c+d9qbu7q9IDzN8fA8yf/GRvhEvKc335CaPKzTWWnzCq3KzB8hNGVVDV/9IKo8otwqjt27enyy67LD9TPY26BiOAimN4eDir/InqrcP5+G3/kf7l1k/m9x5WhFHFTKwilGprbkuvfvql6WmP3ZLdLzNvhMvNC9TyswbLzRqshqpfozCq3KzB8qvnGjQzCqDBxGcE0TY4NDSU+vv70549e1JPT8+MgqjwE4//sXTq2sek5qaH29WKiqiJs6C627rS0097ciWCKAAAoDyEUQANojaEirbB3bt3p76+vqwyajaaljWlV1300nTRYy5Ix606NrU2tz4qhIpqqA1rTkzPP+PH0q897ZXZOQAAgIUijAJYZNFCF1VP0YYXpb1RCRW359JFHW14LS0taePRJ6Xffvar0wvP/x/pKac+KZ229pR0yrqN6awTz8iqoV5x8YvTJRf+Qv6rAAAAFo4wCmCRFLOg9u/f/0h/eVRFzTaEioqn2EGura0tdXZ2prVr12ZDvGNnuec+4ZnpN5/1q+my8y9Jl170y+mPnv/arBrqgpM25b8aAABgYQmjABbQxHlQEUDFPKgIpWYrZkBF4BQhVIRPa9asSR0dHdn5iTpbO9NJq9en9ub2/AwAAMDiEEYBLICJ86CiFS/mQcW52YqwKVrx2tvbU3d3dxZCxe1iJhQAAEAjE0YBzKMIoYp5UFEBVcyDijlRsxFBUzEPKlrxii1/W1tb858AAAAoB2EUwDwohpLXzoMaHByc8zyoCJ2iFS/mQUUYFecAAADKSBgFUEcRQhVDySOAiiDqSOZB1bbirVy5ctJ5UAAAAGXiXQ1AHUwMoaIlLyqjZqtoxYvgKdrwoh0vBpSbBwUAAFSFMArgCIyNjWUhVAwjL0Ko2Q4lr23Fi93wogoqqqEilAIAAKgaYRTAHBQhVH9//5x3xptqHlRURwEAAFSVMApgFiKEGhoaysKn3bt3Z1/j3GwU86CKVryohIrZUFrxAACApUAYBXAYsQNeVD1FCNXb25uFUFERFXOiZqOYBxWteDELKoKoqIoSQgEAAEuJMApgCpOFUDGgPM7PRm0IFVVQ0ZIXlVEAAABLkTAKYIIIm2InvMHBwWwgecyEGhgYmFUIFdVOEUJF5VPMgTIPCgAA4GHCKIDcxBAqdseL27MNoSYOJY+KqJgTBQAAgDAK4FEh1L59+7JKqGjNO5IQKtrxYkC5eVAAAACPJowClqwYQB4hVMyBikqoOIaHh/PvzkyETTH/qa2tLXV3d9sZDwAA4DCEUcCSEyFUhE4RQkUrXlRDRSg1G9F2FyFUVD/FrngRQkUgJYQCAACYnjAKWDLGxsayEKq/vz8LoaISai4hVOyMV4RQccR9AAAAZkYYBVRehFAxAypCqJgH1dvbm0ZHR/PvzkzsgleEUKtXr85a8oRQAAAAsyeMAiqrCKH6+vqyECq+xrnZiBAqhpLHjnhFCBWDygEAAJgbYRRQORNDqKiIijlRsxEhVMyA6uzszOZBxVchFAAAwJETRgGVcaQhVAwfj8ApQqiurq4shIqKqJgTBQAAQH14hwWUXr1CqGjHK0KomA1lZzwAAID6E0YBpVWPEKq5uTm1t7dnbXgxDypuC6EAAADmjzAKKJ16hlARQMVg8mjNE0IBAADMP2EUUBr1CqGiBW/VqlVZCBWteQAAACwcYRTQ8OodQsXR0tKSfxcAAICFJIwCGlaEUIODg0IoAACAChFGAQ2nCKF6e3uFUAAAABUjjAIaxsQQav/+/enAgQP5dw+vCKFiMLkQCgAAoDEtGxkZmfk7vUXyd3/3d9lx8cUXpw9+8IP52WrZt29f2rt3b9q4cWN+plp27NiRDYuOcKCqqnyN27ZtSzfeeGO69NJL8zP1E2FThFCjo6NZEBXHbKqgQoRQK1aseCSImstQcmuw/JbCNb7zne9M5557btq8eXN+pjqqvgaDx5nyswbLzTWWX5XXYKHKj6XWYPnVcw0ue/DBBxs+jHrrW9+aHXHBV155ZX62Wnp6erK/uBs2bMjPVMs999yTPaDGNvpVVeVrvOGGG9K3vvWt9NKXvjQ/Ux8TQ6jZihCqqakpC6IihDqSKihrsPyWwjVeffXV6ayzzkrnnXdefqY6qr4Gg8eZ8rMGy801ll+V12Chyo+l1mD51XMNLvvhD3/Y8GHU2972tuy44IIL0hVXXJGfrZb4SxvH+vXr8zPVcu+992YPqFV+gVrla9y+fXu6+eab0yWXXJKfmbuohIrKp5GRkWyHvDjmWgkV4VNUQdWjFc8aLL+lcI3vec970plnnpk2bdqUn6mOqq/B4HGm/KzBcnON5VflNVio8mOpNVh+9VyDyw6+MWz4MOryyy/Pji1btqRrr702P1stkZ5WuZxPi1C5RZteBFKXXXZZfmZuogqqqIQaGBiY1TyoUNuOF8PJ6zkPyhosv6VwjVu3bs2e/LUIlZPHmfKzBsvNNZZflddgocqPpdZg+dVzDRpgDsy7YjB5X1/fEQ8mj0+J4gnaYHIAAIByEkYB8yZCqGjDO9IQqq2t7ZEQai7DyQEAAGgcwiig7mIG1PDwcOrv789CqPg62xAq2vGEUAAAANUjjALqZmIIFRVRsxlOPlUIFecBAACoBmEUcMSi6ilCqGjDi4F9vb29WYveTBUhVO1MqAikhFAAAADVI4wC5ixCqJGRkSyEip0jYhvT2C1vNmpDqNg1RAgFAABQbcIoYNaKEGpgYCALoeKI+7MxsR1PCAUAALA0CKOAGYsQKiqfIoSKKqhoyYv2vNloamrKgqfOzs60Zs2a7DYAAABLhzAKmJEYRD44OJiFUFEJNTQ0lH9nZiKEimHkEUJFJdTKlStVQgEAACxBwihgWjGIPKqfIoiKSqj4GhVSM7V8+fLU0tKSOjo6skqo+BrnAAAAWJq8IwQmFSFUBE+xM15UQ0UgNdcQKiqhoiJKCAUAAIB3hsCjRDtetOD19fVllVCxU95sQ6jm5uZsh7wIobq6urJh5QAAABCEUUAmQqiofurv789CqPga52Yq5j8VIdSqVauyQwgFAADARMIoWOKKECoqoPbs2ZO15UWL3kxNFkJFex4AAABMRhgFS1S03o2MjKSBgYGsEirmQs02hIrKp7a2tqwVTwgFAADATAijYAkaHR3NQqh9+/ZlR9yfjSKE6u7uzuZCxe0IpwAAAOBwhFGwhNTukBchVLTnzUZTU1MWPMXOeEIoAAAA5kIYBUtAsUNehFAxFyqqoma7Q15UQ0U7XoRQK1euFEIBAAAwJ8IoqLAInGp3yIsh5bMNoVpbW7PwqaOjI/sa5wAAAGCuvKuECiqGk0f4FCHUbHfIi8AphpFHABWVULFTnkooAAAA6kEYBRUTw8hjLlTsjjfb4eRFJVQRQkVbXsyJAgAAgHoRRkFF1A4nj2qomBE1U1H1VFsJFSFUzIgCAACAehNGQckVw8n7+vqyEGo2w8kjhGpubs5mQa1atUolFAAAAPNOGAUlNXEuVAwpj2BqpqLyqa2tLQugIoiKUAoAAADmmzAKSqY2hIqZUDEbarbDySOE6uzszFry4jYAAAAsFGEUlEgxnDzmQkUQNTw8nH/n8Iod8ooQKlrz7JAHAADAQhNGQQkUw8mLuVBxe6aKuVDt7e1ZCBVhVARTAAAAsBi8I4UGFy15MQ8qQqhozZvpcPJQzIXq7u7O5kLZIQ8AAIDFJoyCBlVUQ0VLXlREzWY4eeyI19ra+khLXtwGAACARiCMggYToVPMgooAas+ePbNqySvmQnV0dKQ1a9aYCwUAAEDDEUZBgyh2yRsYGMhCqGjNm2lLnrlQAAAAlIV3q9AAalvyYpe8uD9T5kIBAABQJsIoWETRkjc0NJSFULPdJS/mQkUI1dXVZS4UAAAApSGMgkVQtOTF7niz3SWvdi5UhFDRmmcuFAAAAGUhjIIFNjo6mlVA9fT0ZMdMW/IihIrqpyKEMhcKAACAMvJOFhZI0ZIXu+RFNVTcnoliOHnsjBchVLTlmQsFAABAWQmjYJ4dSUte7VyoGFAe9wEAAKDMhFEwj6Ilb2BgINshb7YteTEXKlrxohoqAikAAACoAmEUzINoyYu5UNGSF0HU8PBw/p3pTWzJi/lQhpMDAABQJcIoqKNov4vgqb+//4hb8syFAgAAoIqEUVAntS15vb29WXXUTETlU7TkFbvkackDAACgyoRRcIRiDlS05EU1VMyFimHlMxXVT0VLXsyH0pIHAABA1QmjYI4mtuRFIDVTRUtetOOtWrVKSx4AAABLhjAK5iCqoYqWvBhSPpeWvDVr1mjJAwAAYMkRRsEsFNVQMRMqgigteQAAADA7wiiYoRhQHrvjzXWXPC15AAAAIIyCw4oWvJgHFdVQMaA8QqmZsEseAAAAHEoYBVOIyqdow4sqqGjJixlRM6mGihCqubk5a8mLSqhoyVu+3FIDAACA4B0yTCIGlEc1VFRCxRH3Z6JoyYsQKo4IpQAAAIAfEUZBjWJAeX9/fzYbamhoKP/O9Ca25MVtAAAA4FDCKMjFLKhoxYuWvL6+vhkPKC92yYsB5fHVLnkAAAAwNWEUS14MKI8KqAigIoiKOVEzEXOgoiWvq6srC6KiRQ8AAACYnjCKJa0YUB4tefF1LgPK29vbVUMBAADADAmjWJKiGioGlPf29s5pQHlUQqmGAgAAgNkTRrGkFAPKi2qoCKRmohhQ3tnZmQ0ob21tzb8DAAAAzIYwiiWjqIaKSqg44v5MxIDyaMWLlrzYLU9LHgAAAMydMIolIWZD9ff3Z9VQURk1EzGgPCqgohoqgqiYEwUAAAAcGWEUlVZbDRXzoWYyoDwUA8qjJS++qoYCAACA+hBGUVm11VBDQ0P52elFNVQMKO/q6jKgHAAAAOaBMIrKqd0pL46ZzIaKyqeohoqZUFENFYEUAAAAUH/CKCpldHQ02ylv3759M94pr6iGikqoqIiK+wAAAMD88K6bSohZUNGKF5VQMR9qbGws/870aquhYlg5AAAAML+EUZReUQ0Vs6EGBgbys9ObWA1lQDkAAAAsDGEUpVVUQ/X19c2qGmrFihXZDnmrVq1SDQUAAAALTBhFKUXwFFVQMRsqqqIimDqcqH6yUx4AAAAsLmEUpRKh0/DwcFYNFUFUtOjNRG01VHt7e34WAAAAWGjCKEojqqFih7wIofr7+2dcDRWteKqhAAAAoDEIo2h4RTVUBFAxpHxkZCT/zvSiGiqqoIpqKEPKAQAAYPEJo2ho4+Pjj1RDRWveTKuhWlpaUmdnZxZERSgFAAAANAZhFA0r5kFFNVQEUTOthoo2vBhSHiFUzIhSDQUAAACNRRhFw4nqp6GhodTb25sdUR11OLXVUKtXr07Nzc35dwAAAIBGIoyiocSQ8oGBgawaKr7OxPLly7NqqBhQ3tHRoRoKAAAAGpgwioYRrXhFW1606M1EVEBFABXVUFEZBQAAADQ2YRSLLtryYkh5T0/PjIeUF7Ohohqqq6tLNRQAAACUhDCKRRVtefv378+qoWJO1OFMnA3V2tqafwcAAAAoA2EUiyKqn4aHh7NKqKiIilDqcIrZULFTXrTmxX0AAACgXBr63XwEFjFHKFqyjjnmmCyImEloQeMYGhlKu3v3pAd7Hkr37/lhdntgeDBry4tqqJgRNZO2vBUrVmQBVARRdsoDAACA8mrYMCrCih/84Adpx44d2UygZz3rWenUU09Nd999dxZizCTAYHHtuP/udPv3v5Nu+PaN6dt33nbw6/bs9m133JZuu/P2LGg8nKItL/4OxKEaCgAAAMqtId/Z79mzJ915553pW9/6Vrr11luzipgzzjgjC6NuvvnmdNttt6X77rsva/OiMd15z53pC9uuS9tvvDHtvOvedGBgPI30Dme3v3nwz3X7bTel+x/8Qf7Tk6tty2tvb8/PAgAAAGXWcGFUb29vuuWWW9Idd9wx5UDrBx54IN1www1Z1ZS2vcZz58670ue+8oUsgJpK08jydNOt30z37bo/P/No0YoXIWQMKdeWBwAAANWx7EAD9bvFb+XGG29M9957b37mYf19fdmOaxFKRJVMYfmyZem8c85JR3V05GfKK0K4fT370voT1+dnyumB/n3pM9/4clo+vCw/c/DPNf86mdGm0XT+Y5+Y1rY8/GcYbXkrmlekle0rs6qoMrl3571pVfeqrJ2wam666aasKvElL3tpal61Oj9bLdH+u3fv3rRx48b8TLVEeB/hbu1jaNUshWvcunVr2rRpU9q8eXN+pjqqvgaDx5nyswbLzTWWX5XXYKHKj6XWYPnVcw02VBgVlVCf+9znHjVL6MDoaBrq739kkPnErfy7li9Lj92zO79XXkPDw2locDB1d3fnZ8rpxjScBjseHVYc7m9Yc++udPrAwxVuRWteGWdDxa6ArQd/760tLfmZ6oj5bVGRuOmii1L3uZtS91ln59+pjqo/cQijqsEb4XLzOFN+1mC5ucbyE0aVmzVYfnUNo/r6+homjIo/uOuvvz6/d9CBA2mstzcdmKYVb9nwUDruIx9Ky0ZH8zMspod+7Hlp+Ljj83sz0/KD+9NRn/6P/B6NruOxj02P+b9/kZZ3duZnqiHCxHgM2rBhQ36mWu65557sRU3ZA+/pLIVrvPLKK9PZZ5+dzj///PxMdVR9DQaPM+VnDZabayy/Kq/BQpUfS63B8qvnGly2c+fOhgmjHnzwwXTXXXfl91I6MDKSxvv783tTO+bj/5xWHPwDZ/H94Od+MY3Pctj48oGBdNyHP5jfowyO+Z3fT21nPDG/Vw1Zq+zBx5H168vdKjuVaH+OFzZVbCMtLIVrvOaaa9KZZ56Zzj333PxMdVR9DQaPM+VnDZabayy/Kq/BQpUfS63B8qvnGlzW39/fMGFULLzYKa9wYHQkjfX25femdvS/fSw1V6BVrwp+8IIXpfEJrZSHs3xoKB33offn9yiDU/7iDamzYq168aQRx0knnZSfqZa77747e2ETR1UthWt897vfnc4555x0wQUX5Geqo+prMHicKT9rsNxcY/lVeQ0WqvxYag2WXz3XYEPNjNq9e3f68pe/nN9L6cD4WBZGTdemt3xkJB370WvT8gq06cUfxI/GfpdL8Zdo1zOemYaOn10K3Hr/znTU5z+b3yuvMv/5HU48TMTR1Nqa2tevT4//izeklnVH5d+thnjSqHJ/t5lR1WBeTbl5nCk/a7DcXGP5mRlVbtZg+dV1ZlQjhVExwPy6665LAwMD+ZmDb4JHRtLw/v3ZAPMYat3a+ujh0GuamtL6u+7M75XX8MhwGh4eTp0d5ZrDMz4+nv2+xw+MZ/fvPWpN6ulem92eqXWDvenkPYevgGt0ff19qaWlJbU0V2+A+UO7Hkq7du1KZ5xzblp94UXZUTVVf+IQRlWDN8Ll5nGm/KzBcnON5SeMKjdrsPwqG0aFaNP77ne/m997WH9/f+rre/iN/po1a/KzKTU3N2e9iscdd1x+przK+Jd2bGwsCw7jzyZCqXDXg3en279ze1o+PsPd8Fak9KSLnpQev/60/ER5VfmJY9u2bWn79u3psssuy89UT9WfOIRR1eCNcLl5nCk/a7DcXGP5CaPKzRosv3quwYbbPz8GfcXCW7Zs+oan2P4//oCrEESVUVSqRQgVA9qKICo85uiT0jHHHZMONOUnprGsZXk6+ZSTKxFEAQAAADPTcGFUZ2dnesITnpCOPfbYLHCK1rxaUQ0VSfEpp5yS/RwLKwrpop0ytqyMirXJCus2nXpOOvb4Y1JbR1saSz8Kqgrjy8ZT1+qudMqpp6SnbXpKfhYAAABYChoujApHH310Ouuss7LAacOGDWlwcDB9//vfT/fff39WDRUh1Kmnnpr/NAslKqDizyJKDyOQms45J5+VTn7MyWnDSevTSQf/DFd2rkzdq7qz2/FneNaZZ6WnnH1x/tMAAADAUtGQYVSIqqgInM4+++y0Z8+e9G//9m/phhtuyIKoo46q1i5eZRDzofbv358FUaMz2LmwqakpnX7S49KzL3pmeupFT0kbTzopnf74x2e3n3XRM9JpJwoTAQAAYClq2DCqVlTkPPjgg1kgwsKbaj7UVKKVMtotu7q6sjbLFU0r0rHdx6QT1h6f3QYAAACWrlKEUSyOmAcVbXnTzYeqFUPnW1tbU3d3d+ro6DjsEHoAAABg6RFGMamogBoYGJjRfKgQFVDt7e3ZcPkIpAAAAAAmI4ziELXzoWbSGlm05a1atSqbFQUAAAAwFWEUjxLDyaMlL1rzDteWF1paWrLZUBFGacsDAAAADkcYxSOGh4ezIeUxrPxwIniKHQ+jGiq+AgAAAMyEMIqsAirmQkUQFXOiDifmQ61cuTILoqJFDwAAAGCmhFFLXO2OeTMZVB4zoWKnvNgxz3woAAAAYLaEUUtYsWNeBFEjIyP52alFFZT5UAAAAMCREEYtUbFLXgwqn+mOecWg8mjPE0QBAAAAcyWMWoKiCiqGlMdxuB3zDCoHAAAA6kkYtcQUO+ZFVdThgqgYVF4EUQaVAwAAAPUgjFoiikHl0ZYXXw+ndsc8g8oBAACAehFGLQHFoPIIomYyqHzFihXZkPKYERWhFAAAAEC9SBoqLoKo/fv3z2pQeQRRdswDAAAA5oMwqsIiiIrZUDEj6nDzoUJra2vq7u7O2vMAAAAA5oMwqqKiCmq2O+ZFEBWVUQAAAADzRRhVQaOjo1lF1EyCqJgJ1d7ebsc8AAAAYEEIoyomBpQXFVGHE4PKOzo67JgHAAAALBhhVIVEEBXzoWJg+eFEFVSxY55B5QAAAMBCEUZVxPDwcOrp6UmDg4P5manFXCiDygEAAIDFIIyqgCKIGhoays9MrQiiYuc8AAAAgIUmjCq5CKD27duXBVKHEwFUzIeyYx4AAACwWIRRJRW75EVLXgRRMStqOjETqq2tzY55AAAAwKITRpVQbRA1Ojqan51cbRAVu+cBAAAALCZhVMlEEBW75cWMqLGxsfzs5CKIam9vz2ZENTU15WcBAAAAFo8wqkRmE0QtX7482y1PEAUAAAA0EmFUSdQGUXF7OrVBVNwGAAAAaBSSihIogqje3t7DBlFRBdXR0ZG6urqyNj0AAACARiKManARPg0MDGRB1Pj4eH52ckUQ1dnZKYgCAAAAGpIwqoEVQVS05h0uiIqd8iKEijBKEAUAAAA0KmFUg4oganBwcEZBVHNzc9aWJ4gCAAAAGp0wqgHNJoiKiqgIotrb2/MzAAAAAI1LGNVgaoOosbGx/Ozkita8tra2/AwAAABAYxNGNZC5BFErV67MzwAAAAA0PmFUAxkaGppxEBWteYIoAAAAoGyEUQ2gqIjq6+ubcUWUGVEAAABAGQmjFlkRRPX29h42iGpqatKaBwAAAJSaMGqRzXRGlCAKAAAAqAJh1CKabUVUR0dHfgYAAACgnIRRi2R4eDiriBodHc3PTE4QBQAAAFSJMGoRjIyMCKIAAACAJUkYtcAigIrWvKiMmo4gCgAAAKgiYdQCitlQEUTFrKjpCKIAAACAqhJGLZAIovr6+tLAwEB+ZnLLly/PQihBFAAAAFBFwqgFMD4+nvr7+7NjOsuWLUsrV64URAEAAACVJYyaZwcOHEj79++fURDV1taWurq6stsAAAAAVSSMmkdFEBVzouL2dCKIijlRgigAAACgyoRR8ygGlc8miIp5UQAAAABVJv2YJ0NDQ6mnpyebFzWd1tbW1N3dne2gBwAAAFB1wqh5MDo6mlVExQ5602lpacmCqBUrVuRnAAAAAKpNGFVnEUBFEDU8PJyfmVxzc3MWRMVXAAAAgKVCGFVHxcDygYGB/MzkohIqds2LyigAAACApUQYVUcRQvX19eX3JhezoWJYeQwtBwAAAFhqhFF1EgPLD7dz3rJly9LKlSuzAwAAAGApEkbVwcjISLZz3uEGlkc1VFRFAQAAACxVwqgjFAFUtOZFIDWd1tbWbE5UVEcBAAAALFXCqCMw24Hl8RUAAABgKRNGHYGZDCxfvny5nfMAAAAAcsKoORocHMzmRB1uYHlHR0dqb2/PzwAAAAAsbcKoOSgGlo+Pj+dnJhchlIHlAAAAAD8ijJqlGFgeQdTo6Gh+ZnIGlgMAAAAcShg1C9GSFzOihoaG8jOTa25uzoKopqam/AwAAAAAQRg1CzGwPHbPm04EUNGaZ2A5AAAAwKGEUTMUc6KiKsrAcgAAAIC5E0bNQAwq7+3tPeycqAihIowCAAAAYHLCqBno7+9Pg4OD+b3JtbW1pe7ubgPLAQAAAKYhjDqMGFYeYdR0VqxYkQ0sX77cf04AAACA6UhPpjE2Npa150Wb3lQigIqB5bGDHgAAAADTE0ZNIQaVx8Dy4eHh/MzkYk6UgeUAAAAAMyOMmsLAwEDav39/fm9yLS0tWXueOVEAAAAAMyOMmsTIyEhWFRXVUVNpamoyJwoAAABgliQpE8R8qAiiRkdH8zOHikqojo6O1Nramp8BAAAAYCaEURPEznnRojedtra2LIwCAAAAYHaEUTViWHmEUdOJXfPMiQIAAACYm2VDQ0NTD0ZqEG9605uy40lPelL6wAc+kJ+tr2jP27t3bzr43yM/c6iYD7Vq1aqsMqre9u3blx0nnXRSfqZa7r777uy/XRxVVeVrvP7669ONN96YXvnKV+ZnqscaLL+lcI3vete70rnnnpsuuOCC/Ex1VH0NBo8z5WcNlptrLL8qr8FClR9LrcHyq+caXPbAAw80fBj11re+Nb3tbW9LmzdvTldccUV+tr5i57yenp783qGiEqq9vT11d3fnZ+or/t3xl3bDhg35mWq55557sgfU+frv1wiqfI033HBD+ta3vpVe+tKX5meqxxosv6VwjVdffXU666yz0nnnnZefqY6qr8Hgcab8rMFyc43lV+U1WKjyY6k1WH71XIPLHnrooVKEUW95y1uyMOqqq67Kz9ZP7J4XVVFjY2P5mUO1tLSk1atXz9vuefGXNo7169fnZ6rl3nvvzR5Qq/wCtcrXuH379vTNb36z8mGUNVhuS+Ea4wXA2WefnTZt2pSfqY6qr8Hgcab8rMFyc43lV+U1WKjyY6k1WH71XIPLDhyU325Yl19+eXZs2bIlXXvttfnZ+ojLjyBquqHlEUCtWbNmXnfPi/Q0fh8bN27Mz1TLjh07sjAvUv6qqvI1btu2LQukLrvssvxM9ViD5bcUrnHr1q3Zk398OFM1VV+DweNM+VmD5eYay6/Ka7BQ5cdSa7D86rkGl/wA88HBweyYzsqVK+c1iAIAAABYKpZ0GBVteb29vVl11FRi97zOzs78HgAAAABHYkmHUf39/Wl0dDS/d6gYWh5B1HzNiQIAAABYapZsyjI8PJyFUdOJ3fPa2tryewAAAAAcqSUZRo2Pjx+2PW/FihVZVVRURwEAAABQH0syjIqd84aGhvJ7k+vo6MgCKQAAAADqZ8mFUTG0/HDtebFzXuygBwAAAEB9Lbkw6nBDy2NYeVdXl/Y8AAAAgHmwpMKoCKH279+f35tcVES1tLTk9wAAAACopyUVRvX19WXDy6fS3NycDS0HAAAAYH4smTBqeHg4G1w+nQiiok0PAAAAgPmxJJKXAwcOZFVR8XUq0ZrX1taW3wMAAABgPiyJMGpoaCg7phLDyqMqytByAAAAgPlV+TAqZkQdriqqtbU1OwAAAACYX5UPowYHB7N5UVOJGVGqogAAAAAWRqXDqLGxsawqajoxJyrmRQEAAAAw/yodRu3fvz+Njo7m9w4VVVEdHR35PQAAAADmW2XDqKiK6u/vz+9Nrr29PTU3N+f3AAAAAJhvlQ2joioqhpdPpampSVUUAAAAwAKrZBgVIdTAwEB+b3IrV65MK1asyO8BAAAAsBAqGUbFDnrTzYqKECrCKAAAAAAWVuXCqAMHDhx2VlQEUdGmBwAAAMDCqlwYNTw8nEZGRvJ7h4oQSlUUAAAAwOKoXBg1kx30li+vZHciAAAAQMOrVCoTFVFDQ0P5vUNFCKUqCgAAAGDxVCqMiqqomBk1ldbWVjvoAQAAACyiyoRRY2Nj2S56U1m2bFnq6OjI7wEAAACwGCoTRg0MDKTx8fH83qFaWlqyAwAAAIDFU4kwKkKo/fv35/cmpyoKAAAAYPFVIoyKoeWjo6P5vUM1Nzdn86IAAAAAWFyVCKNicPl0oioqZkYBAAAAsLhKH0ZFRdTIyEh+71Cxe15bW1t+DwAAAIDFVPowKgaXHzhwIL93qJUrV6blyytRAAYAAABQeqVOaSKEijBqKtGapyoKAAAAoHGUOoyK9rzpBpe3tLRkbXoAAAAANIZSh1H79+/Pb00uWvQAAAAAaBylDaPGx8fT0NBQfu9QMSeqtbU1vwcAAABAIyhtGDU8PJzGxsbye4eKIMrgcgAAAIDGUtq0RoseAAAAQPmUMoyKiqjpWvSampqy4eUAAAAANJZShlGDg4PpwIED+b1Dtbe3p2XLluX3AAAAAGgUpQyjBgYG8luTa2try28BAAAA0EhKF0bFLnojIyP5vUM1NzdnBwAAAACNp3RhVMyK0qIHAAAAUE6lDKOm09ramt8CAAAAoNFUKoxasWJFdgAAAADQmEoVRkV73tjYWH7vUC0tLVr0AAAAABpYqcKo6YKooEUPAAAAoLGVKoyKnfSmEhVRURkFAAAAQOMqXZveVGJWVFNTU34PAAAAgEZUmTBKix4AAABA4ytNGDVdEBWEUQAAAACNrxJh1PLly82LAgAAACiBUoRRh6uKam5uzgaYAwAAANDYShFGxS565kUBAAAAlF8pwqixsbH81uSEUQAAAADlUJrKqKlEe96KFSvyewAAAAA0stJXRkUQZV4UAAAAQDk0fBgVs6Kmq4yK4eUAAAAAlEPDh1FRFTXd8HJhFAAAAEB5NHwYNTIykt+anDAKAAAAoDxKH0YZXg4AAABQHqUOoyKIWr684S8BAAAAgFypwygtegAAAADl0tBh1OF20tOiBwAAAFAuDR1GxU5602lqaspvAQAAAFAGwigAAAAAFowwCgAAAIAFI4wCAAAAYMGUNoxatmxZdgAAAABQHqUNo+ykBwAAAFA+pQ2jtOgBAAAAlE9pw6jlyxv6tw4AAADAJBo20Tlw4EB2TEVlFAAAAED5NGwYNT4+nt+anDAKAAAAoHwaujJqOnbSAwAAACgfYRQAAAAAC0YYBQAAAMCCEUYBAAAAsGCEUQAAAAAsGGEUAAAAAAtGGAUAAADAglm2devW6VOfRTI8PJz279+f3b7uuuvS5z//+fSYxzwmXXLJJdm5VatWVSqQGhgYyI61a9fmZ6pl9+7dqb29PTuqqsrXeNddd6V77rknPfWpT83PVI81WH5L4Rrj+XDDhg3Z82HVVH0NBo8z5WcNlptrLL8qr8FClR9LrcHyq+caXHb88cc3ZBgVlVHj4+PZ7fjD7O/vT83NzVkIFZqamrKvVVFcb9WuqzA2NpaWL19e6Yq2Kl/jyMhIdqxcuTI/Uz3WYPkthWuMD2niuTCOqqn6GgweZ8rPGiw311h+VV6DhSo/llqD5VfPNbjs0ksvbcgwKhbh4OBgdvu2225Lt956azrqqKOyyoxYnFVLiqMSbGhoKHV1deVnqqW3tze1tramlpaW/Ez1VPkaH3zwwbRr1650+umn52eqxxosv6VwjfF8uG7dunT00UfnZ6qj6msweJwpP2uw3Fxj+VV5DRaq/FhqDZZfPdfgsjvvvLNhw6goUYxU8X3ve19673vfm84+++z0hje8IavOqNofbjzo7Nu3L61fvz4/Uy333ntvVtVW5QeeKl/jTTfdlL797W+nF73oRfmZ6rEGy28pXOP73//+dMYZZ6RzzjknP1MdVV+DweNM+VmD5eYay6/Ka7BQ5cdSa7D86rkGlx2IOrIGFX+QkSq+5S1vSW9961vThRdemK666qrU3d1dudLM+Au7d+/etHHjxvxMtezYsSOtXr06e2Ctqipf47Zt29L27dvTZZddlp+pHmuw/JbCNW7dujVt2rQpbd68OT9THVVfg8HjTPlZg+XmGsuvymuwUOXHUmuw/Oq5Bht2N73Q2dmZJcJRphiteStWrMjuV7lHGAAAAKDKGjqMiqFtEURF+BS3YwhYlecQAAAAAFRdQ4dRAAAAAFSLMAo4rNg04LjjjsvvAYslWtZjPQKL49hjj630gHZodPF61BqEahBGAYd1wgknpAsuuCC/ByyWU089NVuPwOKI58ITTzwxvwcstPPPP98ahEVUz8H6wijgsNasWeMNMDSAeBEe6xFYHPEmWFUGLB5rEBbXYx7zmGyedz0sWBg1NjaWbrvttvT+978/vfGNb0xXXXVVtl384OBg/hNAI+rp6Ulf+MIX0rve9a5s7V5zzTXZ2h0YGMh/Algo8QJ89+7d6T3veU/6r//6r/wsMN9GRkbSTTfddMjrWM+FsDAmW4Nf+cpXrEGYZ/fdd1/64Ac/mPbs2ZPdjw9F29rastvhwIED6e67707XXntttjbjPWO8d+zt7c1/YmpN//eg/Pa8GR0dTZ///OfTm9/85vS5z30u3XLLLenGG29MN9xwQ7ZD3imnnJLtmDeVr371q9mxYcOG9Au/8Av52WoZGhrKgrnVq1fnZ6pl37592V/a2r+4VVPFa7z33nvTlVdemb3xjfUaaze+xpvgWLPRMjTd2i0Ta7D8lsI1xovxj3zkI+n3fu/30tq1a9PTn/70/DvlV/U1GDzOlFO82f2P//iP7HXsl770pUe9jo05blV5LrQGq6GK1xjX8+///u+PWoNxxOvR4eHhdPLJJ6f29vb8p8uvyq9nrMFy6evrywoRtm7dmp773Oemo446Ku3duzd1dnam7u7uLIj6xje+kd7+9renf/3Xf00333zzo9ZmPD9OtzYXpDLqrrvuSn/913+dduzYkX7pl34pRf516aWXZt/7m7/5m3T99ddnt4HGEQ+iH/7wh9PVV1+dPcn/2q/9WrZ2f/mXfzmNj4+nyy+/PF133XXZgxCwMOIJP16Mx6dUwMKI16l/9Vd/lfbv359e/OIXP+p1bDwXRoUUMH++/e1vpz/7sz9LDzzwwCPvJV/1qldls2ve9KY3ZVUYXo9CfUUl/oc+9KGs0ikCtsKKFSseaZV98MEH01ve8pYsfPrpn/7pbG3Ge8bYaOCtb31r+sxnPpN1yE1l3sOo+JfHb+KOO+5Iv/Ebv5H+5//8n+nHfuzH0kte8pL0O7/zO9nP/PM//7MSS2gw8Wb3ve99bzaj5o//+I/Tz/3cz2VrN9bwH/7hH2YVGhFWWbuwMH7wgx9kn0xFn/7RRx+dnwXmU7Sqf+ADH8je6MaL7NrXsb/1W7+V+vv7s9e58ZwIzI94oxttQL/+67/+yBp8wQtekL2XjC6bqFyMD1GBI1eMV4oQKrpj4nmuVmtr6yNVe1//+tfTF7/4xaxY4X//7/+dnvOc56Sf//mfz94rRldbhFkRak1l3sOoKO2K3+ATn/jEtGXLlqycOcSL6U2bNqVnPetZWXXF/fffn50HGkNUNMab33jCj3S7eJKPtXvWWWels88+O3thIIyC+RfrLNrzdu7cmX71V3/VhgKwQKKq/xOf+ER64QtfmM4777zsOTDE13PPPTf95m/+ZvZ8qCoD5k/MqomWp8c97nGPei8Zr0+PPfbY9NBDD2UtQcCRi9ear3/969O//Mu/pGc/+9lZXlMr2tKjOirW3Je//OW0cuXK7P1iS0tL/hMpG8P0kz/5k9n3v/e97+VnDzXvYVQ8eHz/+9/PHjwm7gAUv+HHPvax6bvf/a4wChrMMccck/7P//k/6UlPelL2hB/tCYUImWNtR69wPBgB8yfaYuNT4Y9+9KPZG+LYWh5YGBFGxXPehRdemH1A87GPfSz94z/+YzY8OVqHLrnkkuxT4NoX4UB9FSFUzCCOeTUhPiSNFtl44xxFD1WcrwSLIXKZXbt2pZe//OXZB6AbN27Mv/OweO9XvDeMHOf000/PQuFasV5PO+20rNPmzjvvzM8eat7DqBjAFk/eEURN9kR9/PHHZyXQkWgDjePMM8/MPvGNFwDRflAMZ43bsRHBt771rXTRRRdlaTgwf2IjgSuuuCJ7sv+pn/qpymwaAGUQVcLxWjXe8Ma8tmhfj/ESMaz1T//0T7PbtR/WAPX3tKc9LZvX9qlPfSr9v//3/7Idu2JeW6zH6LT5mZ/5max1CDhyJ554YvqLv/iL9LKXveyQYqIQQVQc0b4XM6PWrVs36fqL8zHwPLKgqcx7GBVP0NNVPRUXGJ86AY0pKjM6OjqynTFjG90YVPf4xz8+K7/0xhjmTzyHxla58YFNzMkwKwoWVlQBx/qLuRkx9+Jnf/Zn0+///u9nA8zjufFv//ZvbeYB8yx27ooqiwiFIwCOSuF/+qd/yiqj4kNTretQP+vXr8/a0qfqfilaZeM1am9vb3Z7MvHeMdbmdDnPvIdR8eQ83RN0pGpAY4uqxmIzgr/8y7/M1vRv//ZvZy8AgPkRb3Qj/I3BrDEsOdoQgIUXH6rGi+/f/d3fTS960YuyGRqveMUr0h/90R9l3483xT5UhfkRH4TG82BUCEfV/h/8wR9kmwnE+othyfG997///dYgLKDIcA73IcxMcp55D6OihSdmz0wl2viCVh9oXJF6xxC7KI2OFwWxu94zn/nMR5JxoP6iPehtb3tbNhz5x3/8x81ng0VQPM9Fi2wMZI3nwNjiOl5kxyfHT33qU7PZUTFfA6i/Bx54INvVK3bNiwAq2vVioHLMaovXo7Hrc2wh/53vfCf/FcB8KoqNIr+J6qepROVUtPFNl/PM+zvJaMOLUq9Iq+MJfKIYahUXEf2EQOP54Q9/mLUn/P3f/31au3ZtNiPjuc99rvY8mGfR+hPzMWKu4pVXXpnNx4gjbseTe1RNxf0Ybg7Mj2KcROzkFQFUBFFFW0LMyIiZGLHbZZwH6i/eK0Y7XmwiEBtfxfvJYg3GbnoxMuKOO+7INswC5l9U7oeurq5sDcbr1MlynviQJiqLJw43r7UgYdRjHvOYbNJ6sftBIbYDjAeOU089NbsQoLHEfIyrr746e/N7zjnnpD/8wz9MT37yk1VEwQKInYJiMOutt96aVSYWx2c/+9msqvj222/P7t900035rwDqLebUxGvZ2FUv2tXj+S8qNEKs0fjAJj5UbW9vz84B9RUVGLH24s1u3I43wrXhb7EeY4MdYP7Feoy1GM97kePE69GoYKwV3//e976XzTqNLGgq8/6OMp6gN2/enO28Fal2/OYLUdb8hS98IW3ZssXgOWgwxeDkj3zkI1kl1Ote97oskDLnDRZG7B4UczEmHq9+9auzD3CiTSHux88B8yN2sYx2vJiZGB+gRjVUd3d39kL7lltuySoTo5U2KqSA+ouqinj9+c1vfjMLhaMyPyoVQ1RkfPWrX812/4oDmH8R/EamEzOF4/kxgqgvfvGLWaFR4Z577snmuV188cVZYDWVpoMvZP9vfntexCdIkZrFJ7k333xzamtry0or45Pcq666Kntif81rXpM92U/1JjceZOLYsGFD+oVf+IX8bLVEwh+fsBUPrlUTn+LHn30cVVW1a7z++uuz3vx4wIn5ULFuo0Kj9ogt5+PJvwqzbKzB8qvaNcanSSeffPIhR1zfxz72sewJPrbdrcoOe1Vfg8HjTPnErIt40R2vY2Mnr3itGpX+X//617OByrFldQTE072OLQtrsBqqdo3xPjLe5H7605/O3vTG69J4TRrVGB//+Mez58MYZB7vEatSoVjl12zWYPlE0BTv+17wghdko5WicyaCqDiiVe/GG29MX/rSl7K/r9G2ftttt6Vrrrkmfe1rX8t2go7Xq1N11cx7GBVizkyUOEdlVBzxm40jHkjihXT0+sYnTVMRRpWfN8LlEqXQ8SDygQ98IHvSj8rGeCEenwzXHvEiPKqmqvDkbw2W31K4xhA9+FG1eMYZZ6SnP/3p+dny8wK1/Kq4BqMF6KSTTspeSMdr0XgdGy/M43ZUEF966aXZBgPxorzsrMFqqNo1xhqMNp+oRoxKxDjifeSXv/zlbFZUzJKK3S03btxYmer9Kr+esQbLZ2IYFa9DYz12dnZmYVRUL8Z7xQifirUZVYz//b//9/TCF74w+5mpLEgYFQ8isQX84x//+Ky0OYKnKGn+xV/8xSyIiguZjjCq/LwRLpf4FDgeaOIFeJRfxie+sX4nHtGCG9+vwjBza7D8lsI1hngBEC+4L7jggmx3r6rwArX8qroGozrqrLPOyioT4zVrfABz0UUXZbt6Pe95z6tMNYY1WA1VvMYY+3Luuedma3DVqlXZmovXobHLZbzZjfNVCaJClV/PWIPlE4UJMR7iSU96UrYWo0IxKqCiMj/WXYTF8RwZazPeE8bajCAqwqvDtbAvO/ii9kB+u2EVOwjFbKn4NLiK4kEnyr4j1a+iSEdjQcZf0qpaCtdYZdZg+S2Fa4xWhSpUYEym6msweJyhkVmD1bAUrrHqqvxYag2WX1RAxYiW888/Pz8zd7bEAg6rv78/20oeWFwxI6Ovry+/Byy0GJhcO6QVWFgx5sUahMVz9913Z7ta1oMwCjis++67L91www35PWCxxOYfMUQZWBzbt29P999/f34PWGjWICyuPXv25LeOnDAKOKwoNY2SYWBxxcDkqMwAFkdUJ8a8DGBxxE5d1iBUgzAKAAAAgAUjjAIAAABgwQijAAAAAFgwwigAAAAAFowwCgAAAIAFI4wCAAAAYMEIowAAAABYICn9/9Vozc3sPCpeAAAAAElFTkSuQmCC)"
      ],
      "metadata": {
        "id": "vs8aWEAs0cQq"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Approximation with lines:"
      ],
      "metadata": {
        "id": "RA4GjO3r0ffW"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here >>"
      ],
      "metadata": {
        "id": "Tc7BrYuB0kg8"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Approximation with quadratics (using second derivatives)"
      ],
      "metadata": {
        "id": "X1qvO7-90ojQ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here >>"
      ],
      "metadata": {
        "id": "2b3exFcX0vDS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "[ Optional ]  Approximating with the cubic (using a third derivative )"
      ],
      "metadata": {
        "id": "G02QgE1q0w-y"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here >>"
      ],
      "metadata": {
        "id": "UoUJEqua03eC"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        ""
      ],
      "metadata": {
        "id": "eCVItxDn066R"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here  >>"
      ],
      "metadata": {
        "id": "aDhDEnO70iWD"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Approximating with just one interval\n",
        "\n",
        "In this part of the activity, we're going to approximate a function\n",
        "\n",
        "$$ y = sin(x)$$\n",
        "\n",
        "Our prior approach, breaking up the interval into several small segments, would work fine.  We'll use a different approach, one which uses more derivatives but fewer intervals.\n",
        "\n",
        "For this process, we'll need to find several derivatives.  We'll successively add them into the expression, generating:\n",
        "\n",
        "  1.  A linear approximation  (using a derivative)\n",
        "  2.  A quadratic approximation (using a 1st and 2nd derivative)\n",
        "  3.  A cubic approximation (using a 1st, 2nd, and 3rd derivative)\n",
        "  4.  A quartic, or 4th order, approximation\n",
        "  5.  A quintic (5th order) approximation\n",
        "\n",
        "Again, you'll provide screenshots of the graphs below.  For each graph, you'll also provide an interval over which the approximation works well for the function."
      ],
      "metadata": {
        "id": "keec69IL1ehX"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        ""
      ],
      "metadata": {
        "id": "XPa_fXkm2Sje"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "DesmosURL = \"https://www.desmos.com/calculator/nvbgdvmbk1\"\n",
        "#  Uncomment these lines to display here:\n",
        "#from IPython.display import IFrame\n",
        "#IFrame(src=DesmosURL, width=1200, height=575)"
      ],
      "metadata": {
        "id": "apK897IB2Yk1"
      },
      "execution_count": 6,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Linear (1st order) approximation:\n",
        "\n",
        "Q:  When $a = 0$, how well does this approximate the $y = sin(x)$ function?  Give the interval.\n",
        "\n",
        "A:  "
      ],
      "metadata": {
        "id": "k-aQMO703HQs"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here >>"
      ],
      "metadata": {
        "id": "GMQ8LpuC37ua"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Quadratic (2nd order) approximation\n",
        "\n",
        "Q:  When $a = 0$, how well does this approximate the $y = sin(x)$ function?  Give the interval.\n",
        "\n",
        "A:  "
      ],
      "metadata": {
        "id": "7Axte1dW3KnB"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here >>"
      ],
      "metadata": {
        "id": "OTM-_lon39Ij"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Cubic (3rd order) approximation\n",
        "\n",
        "Q:  When $a = 0$, how well does this approximate the $y = sin(x)$ function?  Give the interval.\n",
        "\n",
        "A:  "
      ],
      "metadata": {
        "id": "dCO4tGLQ3N9o"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Quartic (4th order) approximation\n",
        "\n",
        "Q:  When $a = 0$, how well does this approximate the $y = sin(x)$ function?  Give the interval.\n",
        "\n",
        "A:  "
      ],
      "metadata": {
        "id": "7-qv5YYF3Qh0"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here >>"
      ],
      "metadata": {
        "id": "4gM64Ny43_qR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Quintic (5th order) approximation\n",
        "\n",
        "Q:  When $a = 0$, how well does this approximate the $y = sin(x)$ function?  Give the interval.\n",
        "\n",
        "A:  "
      ],
      "metadata": {
        "id": "cDTTWCv93VID"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here >>"
      ],
      "metadata": {
        "id": "ZGIM-XZ24Cax"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Challenge:  Keep expanding the approximation.  How far out can you get it?\n",
        "\n",
        "Q:  When $a = 0$, how well does this approximate the $y = sin(x)$ function?  Give the interval.\n",
        "\n",
        "A:  "
      ],
      "metadata": {
        "id": "r8otyIRZ3Yqg"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<< Paste the picture here >>"
      ],
      "metadata": {
        "id": "h3EREGVo4D-j"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Summary\n",
        "\n",
        "Answer the following five questions on the attached packet.  They are included here for reference:\n",
        "\n",
        "1.  What is an advantage of the first method, using several segments, compared to the second method, which uses several derivatives?  \n",
        "\n",
        "\n",
        "2.  What is an advantage of the second method, using several derivatives, compared to the first method, using several segments?\n",
        "\n",
        "\n",
        "3.  Provide a scenario where you'd prefer to use the first method to approximate a function.\n",
        "\n",
        "\n",
        "4.  Provide a scenario where you'd prefer to use the second method to approximate a function.\n",
        "\n",
        "\n",
        "5.  What happened to the intervals on the second method as you included more derivatives?  How wide do you think the intervals could become?"
      ],
      "metadata": {
        "id": "LKplBxqU4Ogc"
      }
    }
  ]
}