{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "7f9508e6",
   "metadata": {},
   "source": [
    "# Exercise: Derivatives\n",
    "\n",
    "- by Börge Göbel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "04fafd2c",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1d7d137f",
   "metadata": {},
   "source": [
    "### Load data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "eeb0d15b",
   "metadata": {},
   "outputs": [],
   "source": [
    "data = np.loadtxt(\"04b_Exercise_velocity_acceleration_data_file.dat\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c635be8b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0.        ,   1.        ],\n",
       "       [  0.1       ,   1.01980001],\n",
       "       [  0.2       ,   1.03920011],\n",
       "       ...,\n",
       "       [ 99.8       ,  15.25565932],\n",
       "       [ 99.9       ,  15.33160759],\n",
       "       [100.        ,  15.40808206]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "dc649d6e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x221e63d15e0>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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za0/eeg5+SVOAW5r74xbCwT84eu3JdmsQvwH0ErpTBBe+s/890m7/nQYx/NP8lFPl3j5MHPyJQ9kR8QywTVJ9Jrla10a2jDK0+upcQh+eXYpg7Ujn2RplsfzC67sO/WUL5vCTj67IJJg2rVne1SDwZRvvG5i/65ak2v6yBXMqHfqdpK3xfxd4DfB94HfTDyLiqOya9iz3+MstrweOJlL2WnS3ves876fbn0IGpeef5mtyx/krcmpNcSbq8aed1XNun9tjFVGG+edlfQK1l9JOv2r5aX3kmFcy/KI5qQd+L9t4H8MvKndPeWTLaGLoV32jlSTdDO7Oo9HrB/h+ROzKrFXjuMdfTr2G/kT1+X4NCucdnu308ndTdG+6m59MylwbX7T2m4lr8pS5/f00qVk9kt4JrKPxtK6A1wJnRsSX+tzOthz85dNtsHU7SNmPbwJFBOmgT5vsJvyL/kbVTpq//0GcGNCryQb/NmB5q5ff3Hz9XyLi4B4bsw/waeAgGj+pvzsibprofAd/uXRb059sQEymnLT3HlNzexCpl7GOebNnsGnN8oxa1JtBDv+XfOgannpm4kybAvykBrX9lp5n9bTOG1fa+XkXn9vOJ4BrI+JlwMHA7ZN4L8vR2pHtqcNt3uwZ7Dh/xaSDYcPpR3DRcUt6+oJ75PGnGVp9NYvWfjNxg+1etWY0dRv6yxbMKV3oQ2PGz8L9ZqU697KN92X299qtEy++qWPoA1x43JJ8GlNyaXv864DFNJ7cBTiOxtz+rjdgl7Q3sA14caQcYHCPvxy6Kb9kVWef7AqW/e5h9/LcQplKO510c29Fz5BJ83UxfQrc9Xf16e1Df57cfRvwJzS+bnt+clfSEmA98CMavf2bgVMjYsKf5R38xesm9POooRY5BjCyZZQzv7iVJzuPH7ZVxtJOJ2mfMi76m1lSiQfqM6A7Vj+Cfx7wxzRq8j3P6pE0DGwElkXEJkmfAB6JiL8ad94qYBXA/PnzX33vvff2cjnrg2562XnXfPvxDEGnBc/6ea1BHVQs+2wfbzo/sdLM6pH0R8DGiBhqfvxaYHVETPgzmHv8xUrTm4Li/nONbBnl9Cu30kMHPBdF94b7oZsB9jzLPmk6JXUb0B1rsoO7a4DXRMRJEfEXNHr+f5XwOW1FxM+A+yUtah56I42yj5VQmgEzaNT0i+pRHXPI/vzk/BVcVMKBu4X7zeKe87NZdiFPG04/gnmzZ6Q698DVV+c24Hv6F7YmnuMB3T9U1Kye9wKXS7oFWAL83STeyzKS5glIKMcDU9D4BjDZjUj6ZYoaZY8y/L30y6Y1y9l7j6mJ57WepM56bZ/F51xLUp+kzuvxdDKZWT3bI+KDGbbtd1zqKcaLz7468T9WWUK/nSyXh+6kbHPb+62bZaWz+rtIO+5Q9GyjopVmVk8vHPz5S1PTHZTaaV6LyNVpALGb8O/330va8YaqfwNOo6fgl/QSYF5E3Dju+OuA0Yi4u+8tbcPBn6+0s3gGbXpcVj8B1Cnwx+om/Ps1jTVtT7/MP4nmqdfg/wbwoYi4ZdzxYeCciPizvre0DQd/vtKUeAa9NzXZbwJ1Dfvxugn/yc5uetmaa/jt08kVikF7ViJLvQb/rRPtvCVpe0Tk8j/fwZ+fND0qh56N1e3Ty932xrt9Wrvudf2xep3OuWeH12ZOrklWNidefFOq/8AOfRtr05rlqad6Aty169epd0879LwNXYV+Gaf0llFS8P9A0injD0o6mcZSC1YRaadu+j+WtdNt+ENjgbeh1Vfzkg9d83vz/teObGdo9dVdb+W5cun8gRpzKlJSqWce8BXgCZ4N+mFgBvDnzYexMudST/bSbF7hEo8lKWobzkEfc8pKT6WeiHgwIg6nsfXijuavcyPisLxC37I3smU0MfTnzZ7h0LdEl59yGCuXzs/1mhcdt8Sh36VUe+5GxHXAdRm3xQpyxhe3JZ7jWRKWVrf7+PZqz6nijvPekuk1qmoyyy5YBaRZi8d1fetWa/mMrHr/82bPcOhPgoO/xtIM6E6fggfMrGcfOeaV7Dh/RdcDv52sXDrfP4FOUqpSj1XTWV++JfGcdccuyb4hVnmb1iyf1AY24MkF/eTgr6k0A7pe2dD66ZhD9v/d11PaJ6enTREXHHuwvw77LPUibUXydM7+S9pcZVAWYDOziU12IxarkDQDut68wqy6HPw14wFdM3Pw10yaOfse0DWrNgd/jawd2Z5Y4vGArln1OfhrJGkWxRS88qZZHRQW/JKmStrS3OzFMnbixTclnuMBXbN6KLLHfypwe4HXr400A7ou8ZjVRyHBL+kAYAXw6SKuXzdpBnRd4jGrj6J6/BcBHwR6fHjb0kozZz/vZXTNrFi5B7+ktwK7IqLjDl6SVknaLGnz7t27c2pdtaQp8UwBr2VuVjNF9PiXAUdJ2gF8HniDpMvGnxQR6yNiOCKG586dm3cbKyFNiccDumb1k3vwR8TZEXFARAwBxwPfjYiVebej6tKUeDyga1ZPnsdfQWk3TveArlk9Fbosc0RcD1xfZBuqKE2JxwO6ZvXlHn/FpFmWYeF+szyga1ZjDv6KSbMsw4bTj8ilLWZWTg7+CvGyDGaWhoO/ItaObPeyDGaWioO/Aka2jKbav9SzeMwMHPyVcNaXb0k8x7N4zKzFwT/g1o5s5/GnOi955GUZzGwsB/8AS1vi8YCumY3l4B9gp39ha+I5HtA1s/Ec/APq0PM2kPCclrdSNLO2HPwD6MSLb+LBR59IPM8lHjNrx8E/YNLM1weXeMxsYg7+AZJ2MHfhfrNc4jGzCTn4B0iawdx5s2d4LR4z68jBPyDSDOYCbFqzPPvGmNlAc/APgEPP25BqMNdP55pZGoVuxGLJFp9zLY88/nTieV5j38zScvCX2MvWXMNvn06u7yzcb5br+maWWu6lHkkvlHSdpNsl3Sbp1LzbUHYjW0YZWn11qtAHb6xiZt0posf/FPCBiPihpNnAzZI2RMSPCmhL6Zx48U2p5um3XOSHtMysS7kHf0Q8ADzQ/POjkm4H9gdqH/xpB3FbVi6d74e0zKxrhdb4JQ0BhwCbimxH0daObE/1YFaLgI8ft8Shb2Y9KSz4JT0H+DJwWkQ80ub1VcAqgPnzqzlNcWTLKKddubWrz9lzqrjjvLdk0yAzq4VCgl/SdBqhf3lEXNXunIhYD6wHGB4eTjfKOSC6reO3OPTNrB9yD35JAj4D3B4RF+Z9/SL0GvRjzZs9w0/lmllfFNHjXwb8V2C7pK3NYx+KiGsKaMuE+hHW/eJ5+mbWT0XM6vm/NMYnS6PbwdU8rVw630/kmllf1fbJ3ZEto5z5xa082Xmf8sK4l29mWall8Hc7Xz5PnqppZlmrVfCXqW7fjss6ZpaHWgR/L/Pl87LHtCl87O2L3cM3s9xUPvjL1st30JtZ0Sod/P2q5S9bMMd72JpZZVQ2+CcT+q61m1mVVTL4145s7zr0HfZmVheVDP4rNt2f+lyXccysbioZ/E9H8ppue+8xlVvOPTKH1piZlUvuWy/mYao6rwixbMEch76Z1VYlg/+EQ1/Y9vgUNbYqdGnHzOqskqWe1iDtFZvu5+kIpkqccOgLPXhrZgYoUtTDizY8PBybN28uuhlmZgNF0s0RMTz+eCVLPWZmNjEHv5lZzTj4zcxqxsFvZlYzDn4zs5oZiFk9knYD9/b46fsCD/WxOYPA91wPvud6mMw9vygi5o4/OBDBPxmSNrebzlRlvud68D3XQxb37FKPmVnNOPjNzGqmDsG/vugGFMD3XA++53ro+z1XvsZvZma/rw49fjMzG6PSwS/pSEl3SvqxpNVFt6ffJL1Q0nWSbpd0m6RTm8fnSNog6a7m788tuq39JmmqpC2SvtH8uNL3LGkfSV+SdEfz3/uwGtzz+5tf17dKukLSnlW7Z0mXSNol6dYxxya8R0lnN/PsTklv6vW6lQ1+SVOBfwTeDLwcOEHSy4ttVd89BXwgIv4jsBT4y+Y9rga+ExELge80P66aU4Hbx3xc9Xv+BHBtRLwMOJjGvVf2niXtD7wPGI6Ig4CpwPFU754vBcbvCtX2Hpv/t48HXtH8nE81c65rlQ1+4I+BH0fETyLiCeDzwNEFt6mvIuKBiPhh88+P0giD/Wnc52ebp30WOKaQBmZE0gHACuDTYw5X9p4l7Q28DvgMQEQ8ERG/oML33DQNmClpGrAX8FMqds8RcQPw8LjDE93j0cDnI+LxiLgH+DGNnOtalYN/f2Dsrus7m8cqSdIQcAiwCZgXEQ9A45sDsF+BTcvCRcAHgWfGHKvyPb8Y2A3872Z569OSZlHhe46IUeAC4D7gAeCXEfFtKnzPY0x0j33LtCoHf7uNdys5hUnSc4AvA6dFxCNFtydLkt4K7IqIm4tuS46mAa8C/ikiDgF+zeCXODpq1rWPBg4EXgDMkrSy2FYVrm+ZVuXg3wmM3Xz3ABo/KlaKpOk0Qv/yiLiqefhBSc9vvv58YFdR7cvAMuAoSTtolO/eIOkyqn3PO4GdEbGp+fGXaHwjqPI9/ylwT0TsjogngauAw6n2PbdMdI99y7QqB/8PgIWSDpQ0g8agyNcKblNfSRKNuu/tEXHhmJe+BpzU/PNJwFfzbltWIuLsiDggIoZo/Jt+NyJWUu17/hlwv6RFzUNvBH5Ehe+ZRolnqaS9ml/nb6QxhlXle26Z6B6/BhwvaQ9JBwILge/3dIWIqOwv4C3AvwF3A2uKbk8G9/cnNH7UuwXY2vz1FuB5NGYD3NX8fU7Rbc3o/o8AvtH8c6XvGVgCbG7+W48Az63BPZ8L3AHcCnwO2KNq9wxcQWMM40kaPfqTO90jsKaZZ3cCb+71un5y18ysZqpc6jEzszYc/GZmNePgNzOrGQe/mVnNOPjNzGrGwW9mVjMOfqs0Sc+TtLX562eSRpt//pWkT2VwvWMmWgVW0qWS7pH0ng6f/1pJPxq7TK9Zv3kev9WGpA8Dv4qICzK8xqU0Hir7UjevjTtvqHneQVm00cw9fqslSUeM2cTlw5I+K+nbknZIepukv5e0XdK1zfWQkPRqSd+TdLOkb7XWUxnznocDRwHrmj9VLEhow7HNTUa2Sbohq3s1G8/Bb9awgMYa/0cDlwHXRcQrgceAFc3w/wfgHRHxauAS4LyxbxAR/4/GeipnRsSSiLg74Zp/DbwpIg6m8Q3DLBfTim6AWUl8MyKelLSdxm5P1zaPbweGgEXAQcCGxpphTKWxxspk3AhcKukLNFafNMuFg9+s4XGAiHhG0pPx7ODXMzT+nwi4LSIO69cFI+I9kg6l8ZPGVklLIuLn/Xp/s4m41GOWzp3AXEmHQWMfBEmvaHPeo8DsNG8oaUFEbIqIvwYe4vfXWjfLjIPfLIVo7Nv8DuBjkrbRWAL78Danfh44s7lFYsfBXRqDwNubUzdvALb1s81mE/F0TrOceDqnlYV7/Gb5+SXwt0kPcAFfp1H6McuEe/xmZjXjHr+ZWc04+M3MasbBb2ZWMw5+M7OacfCbmdXM/wdRlttBtlKXCgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.xlabel('Time t [s]')\n",
    "plt.ylabel('Coordinate x [m]')\n",
    "plt.scatter(data[:,0],data[:,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb7db420",
   "metadata": {},
   "source": [
    "### Task"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96ae330b",
   "metadata": {},
   "source": [
    "Calculate the velocity \\\\(v(t)=\\dot{x}(t)\\\\) and the acceleration \\\\(a(t)=\\ddot{x}(t)\\\\) for the loaded data set that describes a set of points \\\\((t_i,x_i)\\\\) for \\\\(i=0,\\dots,1000\\\\).\n",
    "\n",
    "1. Calculate \\\\(v_i\\\\) and \\\\(a_i\\\\) using the forward-differences, central-differences and Richardson methods\n",
    "2. Determine the maximum value of the acceleration and the corresponding time.\n",
    "\n",
    "Be careful, now we do not know the function \\\\(x(t)\\\\) but only its values for specific points. You have to define the functions for the derivativesa bit differently."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c9c52bff",
   "metadata": {},
   "source": [
    "### Solution"
   ]
  },
  {
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   "execution_count": null,
   "id": "830aa249",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "84015415",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "65ded37e",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3d42bac5",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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