Higher Order Derivatives.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# MSDS 400 Higher Order Derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from numpy import poly1d, linspace"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first example shows how to generate and print a second degree\n",
    "polynomial with coefficients, 1, -2 and 4.  The software is not limited\n",
    "to second degree polynomials.  Higher order can be generated.  The critical\n",
    "thing is to have all the coefficients in the right sequence.\n",
    "\n",
    "Second Degree Polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "poly1d([ 5, -3,  2])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = poly1d([5, -3, 2])\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fourth Degree Polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "poly1d([ 2,  1,  4, -2,  3])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = poly1d([2, 1, 4, -2, 3])\n",
    "q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is possible to combine p and q algebraically."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "poly1d([ 10,  -1,  21, -20,  34, -16,   8])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = p + p*q\n",
    "g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Derivatives of different orders may be calculated.  This next section will\n",
    "show determination of the first and second derivatives of p."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " \n",
      "10 x - 3\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([0.3])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = p.deriv(m=1)  # First derivative with m=1.\n",
    "print(h)\n",
    "h.roots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Second Derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " \n",
      "10\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([], dtype=float64)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = p.deriv(m=2)  # Second derivative with m=2.\n",
    "print(t)\n",
    "t.roots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using t, the original function p can be restored if the missing\n",
    "coefficients -2 and 4 are supplied.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   2\n",
      "5 x - 3 x + 2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 5., -3.,  2.])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = t.integ(m=2, k=[-3, 2])\n",
    "print(w)\n",
    "w.coeffs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Different coefficients would result in a different function instead of the original p."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   2\n",
      "5 x - 2 x + 4\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "poly1d([10., -2.])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new = t.integ(m=2, k=[-2, 4])\n",
    "print(new)\n",
    "new.deriv(m=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Roots may also be found.  This is useful when locating the maxima, minima and inflection points of a function from the first and second derivatives."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.3+0.55677644j, 0.3-0.55677644j])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w.roots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting requires defining a domain for the polynomial.  The linspace function is used to set boundaries and define the number of points used for calculation. A new polynomial p will be defined."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = poly1d([.3333, 0, -1, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a final example, we will find the first and second derivatives of the polynomial p, find the roots of the derivatives and plot the functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Roots of First Derivative\n",
      "[ 1.00005 -1.00005]\n",
      "\n",
      "Roots of Second Derivative\n",
      "[0.]\n"
     ]
    }
   ],
   "source": [
    "g = p.deriv(m=1)\n",
    "\n",
    "print('\\nRoots of First Derivative')\n",
    "print(g.roots)\n",
    "\n",
    "print('\\nRoots of Second Derivative')\n",
    "q = p.deriv(m=2)\n",
    "print(q.roots)\n",
    "\n",
    "x = linspace(-4, 4, 101)\n",
    "y = p(x)\n",
    "yg = g(x)  # These statements define points for plotting.\n",
    "yq = q(x)\n",
    "y0 = 0*x   # This statement defines the y axis for plotting."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is shown below is a different way to legends using a label.  Python will pick the colors to assign to the labels and the plotted points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(x, y, label='y=p(x)')\n",
    "plt.plot(x, yg, label='First Derivative')\n",
    "plt.plot(x, yq, label='Second Derivative')\n",
    "plt.legend(loc='best')\n",
    "\n",
    "plt.plot(x, y0)\n",
    "plt.xlabel('x-axis')\n",
    "plt.ylabel('y-axis')\n",
    "plt.title('Plot Showing Function, First and Second Derivatives')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This next example calculates and plots the higher order derivatives for f(x) = 3x<sup>4</sup> - 4x<sup>3</sup> - 12x<sup>2</sup> + 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Fourth Degree Polynomial\n",
      "   4     3      2\n",
      "3 x - 4 x - 12 x + 2\n",
      "\n",
      "First Derivative\n",
      "    3      2\n",
      "12 x - 12 x - 24 x\n",
      "\n",
      "Second Derivative\n",
      "    2\n",
      "36 x - 24 x - 24\n"
     ]
    },
    {
     "data": {
      "image/png": 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acMstt3DllVeybt06PD09C+bVrl2bmJgYnTS0MnG2lNHAvwHXX3K90/ajk4bmEi5UInCmO++8k3nz5nHy5Eluv/12UlNT6dOnT5HLFi5pFL5s9OxzHx+ff91/sHPnToKCgjh16tQ507OysvDx8SnLQ9GqsN+O/sbepL1M6zUND5vz7rTXSUOr8kaMGMGUKVPIzc1l4cKF2Gw2h0oaixYtYuLEiSxatIgePXoA0KpVq4L2DYBvvvmGxMREVq1axfDhw9m4cSNBQUEAHDhwgDZtrEuWWuWRZ8/j7W1v0zSwKcObDnfqvnTS0Ko8T09P+vfvT1BQEDabzeH1srOz6datG3a7nc8//xyAYcOGccsttwCQkJDAxIkTWb58OY0aNeL+++/noYceYv78+Zw6dQofHx/q1avnlGPSqpZvD31LVEoUb/Z/E5ub49/hi6GThlbl2e121q9fz5dfflmq9e677z6eeeaZc6aFhIRQo0YNDh48SIsWLThw4EDBvAcffLDg+cKFC7n77rv/W+CaBmTmZfL+9vfpUKsDAxoNcPr+9CW3WpW2Z88emjdvzsCBA2nRokWZbHP69OnExsaWuExQUBBjx44tk/1pVdvn+z4nLjOOhzs/XC7ds+iShlaltW7dmsjIyFKvFxUVVey8sLAwwsLCSlx/3Lhxpd6npp3vTPYZPtr5EX0a9CG8bni57FOXNDRN0yqoubvmkpaTxkOdHyq3feqkoWmaVgHFpsXy6Z5PuaLpFYQFl1yyLUs6aWiaplVA72x/B4XigU4PlOt+LUsaItJIRP4Qkb0isltEHjKnB4vIbyJy0PxbvdA6k0TkkIjsF5EhVsWuaZpmpf1J+/nh8A+MaTWG+v71y3XfVpY08oDHlFKtgO7AfSLSGpgILFdKtQCWm68x540G2gBDgfdExLkXJGuVns1mo2PHjgWPqKgoevbsWaptvPnmm2RkFD1uQb9+/QgLC6N9+/a0bNmS+++/n+Tk5FLHeeedd7Jnz55Srwfw4osvnvO6tMenuZ43tr5BNc9q3NnuzvLfuVLKJR7Ad8BlwH6gnjmtHrDffD4JmFRo+V+AHhfabpcuXZTmmvbs2WN1CMrPz8+h5fLy8oqdFxISouLj44ucd+mll6pNmzYppZTKzs5Wjz76qOrbt2+pYixp345w9BjLmyt8/hXRuph1qu28turjnR87dT/AZlXEOdUl2jREJBToBGwA6iilYgHMv7XNxRoAxwutFm1OK2p740Vks4hsjo+Pd1rcWuXk7+8PwMqVK+nfvz833XQT7dq1Iz09nSuuuIIOHTrQtm1bFi1axMyZM4mJiaF///7079+/xO16enryyiuvcOzYMf7++28APv30U7p27UrHjh25++67yc/PL4hhypQpdOvWjXXr1tGvXz82b97M+++/z5NPPlmwzXnz5vHAA0ad9jXXXEOXLl1o06YNs2fPBmDixIlkZmbSsWNHxowZc87xjRo1ip9++qlgW7fddhtff/01+fn5PPHEE0RERNC+fXs++OCDsnhbtTJgV3ZmbJ5BPb963NjqRktisPw+DRHxB74GHlZKpZRwc0pRM1RRCyqlZgOzAcLDw4tcRnMxP0+EkzvLdpt128Gw6SUucvaECtCkSROWLFlyzvyNGzeya9cumjRpwtdff039+vX58ccfAThz5gyBgYG8/vrr/PHHH9SsWfOCIdlsNjp06MC+ffvw9PRk0aJFrF27Fg8PD+69914+++wzbr31VtLT02nbti3PPffcOetfd9119OjRo2AAp0WLFhWM0TF37lyCg4PJzMwkIiKCa6+9lunTp/POO+8U2ZfW6NGjWbRoEZdffjk5OTksX76c999/nzlz5hAYGMimTZvIzs6mV69eDB48mCZNmlzw+DTnWhq5lH1J+5jeZzpeNi9LYrA0aYiIB0bC+Ewp9Y05+ZSI1FNKxYpIPSDOnB4NNCq0ekMgpvyi1SojHx+fEjsn7Nq1a8HJsl27djz++ONMmDCB4cOHF9sT7oUYJX9Yvnw5W7ZsISIiAjASWO3aRsHaZrNx7bXX/mvdWrVq0bRpU9avX0+LFi3Yv38/vXr1AmDmzJkFSe/48eMcPHiQGjVqFBvHsGHDePDBB8nOzmbZsmX07dsXHx8ffv31V3bs2MFXX30FGMnx4MGDOmlYLDMvk5lbZ9KmRhuGNRlmWRyWJQ0xihRzgL1KqdcLzfoeGAtMN/9+V2j6QhF5HagPtAA2ll/EmlNdoERgFT8/v4Lnl1xyCVu2bOGnn35i0qRJDB48mClTppRqe/n5+ezcuZNWrVoRFxfH2LFjeemll/61nLe3d7GdJ44aNYrFixfTsmVLRowYgYiwcuVKfv/9d9atW4evry/9+vX7VxftRe2jX79+/PLLLyxatKhgfBClFG+//TZDhugLFF3JJ3s+4VTGKab3mY6bWNeyYGWbRi/gFmCAiGw3H5djJIvLROQgRsP4dACl1G5gMbAHWAbcp5TKtyZ0rSqKiYnB19eXm2++mccff5ytW7cCUK1aNVJTUy+4fm5uLpMmTaJRo0a0b9+egQMH8tVXXxEXZxSmk5KSOHr06AW3M3LkSL799ls+//xzRo0aBRilgerVq+Pr68u+fftYv359wfIeHh7k5uYWua3Ro0fz8ccfs3r16oIkMWTIEN5///2CdQ4cOEB6evoF49KcJyEzgTk75zCg0YBy6y6kOJaVNJRSayi6nQJgYDHrvAC84LSgNK0EO3fu5IknnsDNzQ0PDw/ef/99AMaPH8+wYcOoV68ef/zxx7/WGzNmDF5eXmRnZzNo0CC++84oPLdu3Zpp06YxePBg7HY7Hh4evPvuu4SEhJQYR/Xq1WndujV79uyha9euAAwdOpRZs2bRvn17wsLC6N69e8Hy48ePp3379nTu3JnPPvvsnG0NHjyYW2+9lauuuqpgVME777yTqKgoOnfujFKKWrVq8e233170+6b9d+9tf4+c/Bwe6fKI1aEgZ+tXK6vw8HC1efNmq8PQirB3795zhkbVqhb9+TvmwOkDXP/D9YwOG82kbpPKbb8iskUp9a9ijUtccqtpmqb9m1KK1za9hr+HP/d2vNfqcACdNDRN01zW6hOrWRe7jv91+B+BXoFWhwPopKFpmuaScu25vLrpVUIDQhnVcpTV4RSw/OY+TdM07d8W719MVEoU7wx4Bw83D6vDKaBLGpqmaS7mdNZp3t3+Lt3rdadvw75Wh3MOnTQ0TdNczLvb3yUjN4MJERPKZdzv0tBJQ6vSXnjhBdq0aUP79u3p2LEjGzZsKPcYVq5cyfDhw4ucHhgYSKdOnQgLC6Nv374sXbq01NvfvHkzDz744EXH9tdffxW8njVrFgsWLLiobWmO2Z+0ny8PfMmosFE0r97c6nD+RbdpaFXWunXrWLp0KVu3bsXLy4uEhARycnKsDuscffr0KUgU27dv55prrsHHx4eBA4u8//Vf8vLyCA8PJzz84u4iXrlyJf7+/gVjcNxzzz0XtR3NMUopXtn0CtU8q7nMJbbn0yUNrcqKjY2lZs2aeHkZvYXWrFmT+vWNUdC2bNnCpZdeSpcuXRgyZAixsbEAHDp0iEGDBtGhQwc6d+7M4cOHUUrxxBNP0LZtW9q1a8eiRYsA44Tbr18/rrvuOlq2bMmYMWMKOitctmwZLVu2pHfv3nzzzTdFRPdvHTt2ZMqUKbzzzjsAxMfHc+211xIREUFERARr164FYOrUqYwfP77gbu+zJRm73U5oaOg5g0A1b96cU6dO8cMPP9CtWzc6derEoEGDOHXqFFFRUcyaNYs33niDjh07snr1aqZOncprr73G3r17C+5GB4iKiqJ9+/Ylvnfahf1+7Hc2ntzI/R3vd5lLbM+nSxqaS3h548vsS9pXpttsGdySCV0nFDt/8ODBPPfcc1xyySUMGjSIUaNGcemll5Kbm8sDDzzAd999R61atQq6H587dy5jxoxh4sSJjBgxgqysLOx2O9988w3bt2/n77//JiEhgYiICPr2NRovt23bxu7du6lfvz69evVi7dq1hIeHc9ddd7FixQqaN29e0H+UIzp37syrr74KwEMPPcQjjzxC7969OXbsGEOGDGHv3r2AceJes2YNPj4+rFy5EgA3NzeuvvpqlixZwrhx49iwYQOhoaHUqVOH3r17s379ekSEjz76iFdeeYUZM2Zwzz334O/vz+OPPw4YPfMCtGrVipycHCIjI2natCmLFi3ihhtuKPG900qWlZfFa5teo0X1Flx3yXVWh1MsnTS0Ksvf358tW7awevVq/vjjD0aNGsX06dMJDw9n165dXHbZZYDRM229evVITU3lxIkTjBgxAjB6iQVYs2YNN954IzabjTp16nDppZeyadMmAgIC6Nq1Kw0bNgQoGE7W39+fJk2a0KJFCwBuvvnmgkGTLqRwtz+///77OUPApqSkFHSceNVVV+Hj4/Ov9UeNGsVzzz3HuHHj+OKLLwoSVnR0NKNGjSI2NpacnByHukG/4YYbWLx4MRMnTmTRokUsWrSI/fv3F/neaRc2d9dcYtJjmNt7Lu5urntqdt3ILGZXdnLyc/B297Y6lCqhpBKBM9lsNvr160e/fv1o164d8+fPLxj9bt26decsm5KSUuQ2Suq/7WzV19l95eXlAVz0FTHbtm0r6K/Jbrezbt26IpND4S7dC+vRoweHDh0iPj6eb7/9lsmTJwPwwAMP8Oijj3LVVVexcuVKpk6desFYRo0axfXXX8/IkSMREVq0aMHOnTuLfO+0kkWnRjNn5xyGhg4lom6E1eGUSLdpFCHPnse4ZeN4bfNrVoeiOdH+/fs5ePBgwevt27cTEhJCWFgY8fHxBSe+3Nxcdu/eTUBAAA0bNizo8TU7O5uMjAz69u3LokWLyM/PJz4+nlWrVp1T33++li1bcuTIEQ4fPgzA559/7lC8O3bs4Pnnn+e+++4DjOq1s+0bZ+O/EBFhxIgRPProo7Rq1apgkKYzZ87QoIExevL8+fMLli+p2/dmzZphs9l4/vnnC0osxb13Wsle3fQqNjcbj4U/ZnUoF6STRhHc3dxpVaMVi/cvZk/inguvoFVIaWlpjB07ltatW9O+fXv27NnD1KlT8fT05KuvvmLChAl06NCBjh07Flx2+sknnzBz5kzat29Pz549OXnyJCNGjKB9+/Z06NCBAQMG8Morr1C3bt1i9+vt7c3s2bO54oor6N27d4ldoa9evbrgktv77ruPmTNnFlw5NXPmTDZv3kz79u1p3bo1s2bNcui4R40axaeffnpOW8rUqVO5/vrr6dOnzznD1l555ZUsWbKkoCG8uG3dcMMNACW+d1rR1p5Yy4rjKxjffjx1/Yr/3rgK3TV6MVJyUrhyyZU0rNaQT4Z9YulIWZWV7hq7atOfP+Tk53Dt99diV3aWXL0ET5un1SEV0F2jl1KAZwCPhT/GjvgdfHfouwuvoGmaVkrzd88nKiWKSd0muVTCKIlOGiW4sumVdKrdiTe2vMGZ7DNWh6NpWiVyIu0Es3fMZlDjQfRu0NvqcBymk0YJRISnuz3NmZwzvL3tbavDqZQqe/WoVjT9uRv3JomIZVcOXiydNC4gLDiMG1veyOL9i9mVsMvqcCoVb29vEhMT9QmkilFKkZiYWHCfS1W0KnoVfxz/g7vb310hGr8L0w3hDkjLSeOqb6+ipk9NPr/ic2xutjKKrmrLzc0lOjqarKwsq0PRypm3tzcNGzbEw8N1xokoL1l5WYz4bgQeNg++vvJrPGyu+R4U1xCub+5zgL+nP09GPMkTq57gi/1fMKbVGKtDqhQ8PDwcuvNY0yqT2TtmE50WzZzBc1w2YZTE0uopEZkrInEisqvQtKkickJEtpuPywvNmyQih0Rkv4gMKc9Yh4QOoUe9Hryz7R3iM+LLc9eaplUSkcmRfLz7Y65qdhVd6xV/A6grs7pNYx4wtIjpbyilOpqPnwBEpDUwGmhjrvOeiJRbPZGI8HT3p8nJz+GVTa+U1241TasklFI8t/45fN19ebTLo1aHc9EsTRpKqVVAkoOLXw18oZTKVkodAQ4B5ZqqQwJCuLP9nSyLWsaaE2vKc9eaplVw3x/+ni2ntvBIl0eo4VPD6nAumtUljeLcLyI7zOqr6ua0BsDxQstEm9P+RUTGi8hmEdkcH1+2VUl3tL2DJoFNmLZ+Ghm5GWW6bU3TKqfTWad5bfNrdKzVkZEtRlodzn/iiknjfaAZ0BGIBWaY04vqFrTIS7+UUrOVUuFKqfBatWqVaXCeNk+mdJ/CibQTzNrhWF8/mqZVba9tfo20nDSm9JhS4bskcrnolVKnlFL5Sik78CH/VEFFA40KLdoQiCnv+ADC64YzovkIFuxewP6k/VaEoGlaBbE+dj3fH/6ecW3H0aJ6i/LZaUoMHF0HTrilwuWShogUHrFlBHD2yqrvgdEi4iUiTYAWwMbyju+sx8IfI9ArkGfXPUu+Pd+qMDRNc2FZeVk8t+45QgJCuLvD3eW343XvwrwrIPVkmW/a6ktuPwfWAWEiEi0idwCviMhOEdkB9AceAVBK7QYWA3uAZcB9SinLztaBXoE8GfEkOxN28vk+x8ZD0DStavlgxwccTz3OlO5T8LJ5XXiFspCVAlsXQJtrIKDsR0209OY+pdSNRUyeU8LyLwAvOC+i0rm8yeUsjVzKzG0z6d+4Pw38i2yX1zStCtqXtI95u+ZxdbOry/eejG2fQHYK9LjPKZt3ueqpikREmNJ9CgDPr3te96GkaRpgjP45Ze0UAr0CeSLiifLbcX4erJ8FjXtCgy5O2YVOGv9RPf96PNT5IdbGrGVp5FKrw9E0zQV8sucT9ibt5aluTxHoFVh+O977PZw55rRSBuikUSZGh42mQ60OvLzpZRIyE6wOR9M0Cx1NOcq7299lQKMBXBZyWfntWClY9w4EN4WwYU7bjU4aZcDmZuO5ns+RkZvBSxtesjocTdMsYld2nl33LJ5unjzd/WlEirq9zEmOb4ATW6D7veDEnrh10igjTYOacm/He/n16K/8dvQ3q8PRNM0CX+7/kk0nN/F4xOPU9q1dvjtfOxO8g6DjTU7djU4aZWhsm7G0Cm7FtPXTSM5KtjocTdPK0Ym0E8zYMoOe9XsyovmI8t15/H7Y/yN0HQ+efk7dlU4aZcjDzYPnez1PSnYK0zdNtzocTdPKiVKKZ/56BjdxY2qPqeVbLQWw9i1w94Fuzr+BUCeNMhYWHMZd7e/ix8gfWXFshdXhaJpWDr46+BUbYjfwaJdHqedf9jfUlejMCdixGDrfAn41nb47nTSc4K52d9EyuCXPrnuW01mnrQ5H0zQnOpF2ghmbZ9Ctbjeuv+T68g9g/Xug7NDj/nLZnU4aTuBh82Bar2mk5KTw4oYXrQ5H0zQnsSs7U9YaN/g+2+vZ8q+WykiCzR9D22uheki57FInDScJCw7jfx3+x7KoZfwS9YvV4Wia5gSf7/ucjSc38kT4E9Z0I7RpDuSmQ++Hy22XOmk40e1tb6d1jda8sP4FfdOfplUyUWeieHPLm/Ru0NuagZWyU2H9u3DJUKjTptx2q5OGE7m7ufNi7xdJz01n6l9Tdd9UmlZJ5NnzmLx2Mp42T57taUG1FBiljMzT0PfJct2tThpO1iyoGQ93eZg/o/9kyaElVoejaVoZ+HjXx/wd/zdPdXuq/G/iA8jJMLoMaTYAGjqnY8Li6KRRDsa0GkPXul15eePLHE89fuEVNE1zWXsS9/De9vcYGjqUy5tcbk0QW+dDeny5lzJAJ41y4SZuTOs1DTdxY/KayXqkP02roLLyspi0ehLB3sFM7j7Zmmqp3CzjZr7QPhDSo9x3r5NGOannX4+nuj3F1ritfLz7Y6vD0TTtIry19S0iz0TyfO/ny7fL88K2fwqpsdC3HMfpKEQnjXI0vOlwhoQO4d1t77I7YbfV4WiaVgp/nfiLT/d+yk0tb6Jn/Z7WBJGbBatmQKPu0KSvJSHopFGORIT/6/5/1PCpwcTVE8nIzbA6JE3THJCUlcTTa5+meVBzHunyiHWBbJ0PqTHQ/ymwomoMnTTKXaBXIC/2fpGjKUd5bfNrVoejadoFKKV4Zu0znMk+w/Q+0/F297YmkJwMWD3DaMtoeqk1MaCThiW61uvKbW1u48sDX7L82HKrw9E0rQSL9y9mZfRKHunyCGHBYdYFsnkOpJ0yShkWsjRpiMhcEYkTkV2FpgWLyG8ictD8W73QvEkickhE9ovIEGuiLhsPdHqA1jVaM2XtFE6mn7Q6HE3TinDo9CFe3fwqver3YkyrMdYFkp0Ga96Apv0hxKL2FJPVJY15wNDzpk0EliulWgDLzdeISGtgNNDGXOc9EXHemIZO5mHz4JW+r5Bnz2Pi6on6MlxNczFZeVk8seoJ/Dz8mNbbuGTeMhs/gIxE6P+0dTGYLE0aSqlVQNJ5k68G5pvP5wPXFJr+hVIqWyl1BDgEdC2POJ0lJCCEyd0ns+XUFj7c+aHV4WiaVsirm17lUPIhXur9EjV9nD9ORbEyTxv3ZbQYDI0irIvDZHVJoyh1lFKxAObfs/foNwAK304dbU77FxEZLyKbRWRzfHy8U4P9r65sdiXDmw7n/b/fZ/PJzVaHo2ka8GvUryw+sJhxbcfRs4G11UGseQOyUmDgM9bGYXLFpFGcoq4vK7IHQKXUbKVUuFIqvFatWk4O67+b3H0yjao1YsKqCSRlnV/w0jStPJ1IO8HUv6bSrmY7Huj0gLXBpMTAhg+g/Q1Qt621sZhcMWmcEpF6AObfOHN6NNCo0HINgZhyjs0p/Dz8eO3S10jOTuapNU9hV3arQ9K0Kik3P5fHVz4OwMt9X8bDzcPagFZOB3u+5VdMFeaKSeN7YKz5fCzwXaHpo0XES0SaAC2AjRbE5xQtg1vyZMSTrD2xlo936W5GNM0Kr295nV2Ju3i+1/M0qtbowis4U8JB2PYpRNwB1UOtjaUQqy+5/RxYB4SJSLSI3AFMBy4TkYPAZeZrlFK7gcXAHmAZcJ9SqlJdcnRD2A0MDhnM29veZsupLVaHo2lVyvKjy/l076fc3OpmBoYMtDocWP4sePhAn8etjuQcUtkHBgoPD1ebN1ecBua0nDRG/ziajNwMFl+52NqrNjStijieepxRP4wiJCCEBcMW4GGzuFrq6F/w8TDjEttLy7/7cwAR2aKUCj9/uitWT1Vp/p7+zLh0Bik5KUxYNUHfv6FpTpaVl8WjKx9FRHj10letTxh2O/zyNFSrDz3utzaWIuik4YLCgsOY3H0yG09u5N3t71odjqZVai9ueJF9Sft4qc9LNKzW0OpwYNdXELMVBk4BT1+ro/kXnTRc1DXNr2Fki5F8uPNDVh5faXU4mlYpfXPwG5YcWsLd7e+mb0Nruho/R24m/P4s1OsA7UdZHU2RdNJwYZO6TqJVcCueWv0UR1OOWh2OplUquxN388L6F+hRrwf/6/A/q8MxrHsXUqJhyIvg5pqn5wtGJSKviEiAiHiIyHIRSRCRm8sjuKrO292bN/q/gZubGw//8bAef0PTykhSVhIP//EwNXxqML3vdGxuLtCNXUoMrH4dWg6H0N5WR1MsR1LZYKVUCjAc4wa7SwBrxhmsghr4N+CVvq8QeSaSqX9NpbJf7aZpzpZnz+PJP5/kdNZp3uz/JsHewVaHZPj1/8CeB4OnWR1JiRxJGmcvJbgc+Fwppfu5KGc96/fkgU4P8HPUz8zfPf/CK2iaVqw3t7zJhpMbmNJjCq1rtLY6HMPRv4wG8F4PQXATq6MpkSNJ4wcR2QeEA8tFpBaQ5dywtPPd0fYOBocM5o2tb7DmxBqrw9G0CunHyB+Zv2c+N7W8iauaXWV1OAZ7Pvz0JAQ2gt4WDiXroAsmDaXURKAHEK6UygXSMbop18qRiPB8r+dpHtScJ/98kqgzUVaHpGkVyq6EXTzz1zOE1wnn8QgXust681w4tdOolnLBS2zPV2zSEJEB5t+RQH/gavP5UMDivoKrJl8PX2YOmInNzcaDfzxIak6q1SFpWoUQnxHPQyseoqZPTWb0m2F9R4RnpcXBiuehSV9oXTF+i5dU0jg7cvmVRTyGOzkurRgN/Bvwer/XOZ5ynCdXPanvGNe0C8jOz+bhPx4mNTeVt/q/5ToN32Dc+Z2bCVe8DlLU6A+ux724GUqpZ8y/48ovHM0REXUjmNRtEs+vf54ZW2bwZIQ1fdNomqtTSvHMX8+wI2EHb/Z7k7DgMKtD+sfhP2DnYrh0AtRsYXU0DnPkPo1PRCSw0OsQEVnu3LC0C7kh7AbGtBrDJ3s+4esDX1sdjqa5pA92fMCPkT/yYKcHXaPn2rNys+DHxyC4KfR+1OpoSsWRq6fWABtE5HIRuQv4DXjTqVFpDnk8/HF61e/FtPXT2BhbaYYW0bQysezIMt7d/i5XNbuKO9vdaXU451rzBiQdhitmgIe31dGUiiNXT30A3IkxGNJzQF+l1A/ODky7MHc3d1699FVCAkJ4eOXDRCZHWh2SprmEHfE7mLx2Mp1rd+aZHs8grtReELcXVs+AttdBswFWR1NqjlRP3QLMBW4F5gE/iUgHJ8elOaiaZzXeHfQunm6e3Lv8XhIzE60OSdMsdTzlOA+seIBaPrV4s/+beNo8rQ7pH/Z8+O4+8A6AYS9bHc1FcaR66lqgt1Lqc6XUJOAeQN+W7EIa+Dfg7QFvk5iZyIMrHiQrT997qVVNyVnJ3Lv8XvJVPu8Pep/q3tWtDulc69+DE1tg2CvgVzEHWHOkeuoapVRcodcbga5OjUortXa12jG9z3R2JuzUgzdpVVJ2fjYP/fEQMWkxzOw/k9DAUKtDOlfiYVgxDS4ZBm2vtTqai+ZI9ZS3iNwnIu+JyFwRmQvMKofYtFIaGDKQCV0nsOL4Cl7a+JLu3FCrMuzKzlOrn2Jr3FZe6P0Cnet0tjqkc9nt8P2DYPOE4RXnnoyiOFI99QlQFxgC/Ak0BPStyC5qTKsxjGszjkX7FzFn1xyrw9E0p1NK8fLGl/n16K881uUxhjYZanVI/7bhfTi6xhgnI6C+1dH8J44kjeZKqf8D0pVS84ErgHbODUv7Lx7u8jCXN7mct7a+xXeHvrM6HE1zqjm75rBw30JubX0rt7W9zepw/i1unzEaX9jl0KniD0VU7B3hheSaf5NFpC1wEgh1WkTaf+YmbkzrNY2krCSe+esZAr0C6deon9VhaVqZW3JwCW9tfYthTYbxWPhjVofzb/m5sGQ8ePnDlW9V6GqpsxwpacwWkerAZOB7YA/g9GvFRCRKRHaKyHYR2WxOCxaR30TkoPnXxS6NcB0eNg/e6v8WrYJb8fifj7P55GarQ9K0MrX86HKmrptKj3o9eKHXC7iJCw6PuupViP3bSBj+ta2Opkw4cvXUR0qp00qpVUqppkqp2uYNf+Whv1Kqo1Iq3Hw9EViulGoBLDdfa8Xw9fDlvUHvUd+/Pg+seIC9iXutDknTysT62PU8seoJ2tZsy5v938TD5iK91hZ2bL2RNDrcCK2utDqaMlOq1CwiS50ViIOu5p97ROYD11gXSsVQ3bs6sy+bjb+nP/f8fg+RZ/Rd41rFtjN+Jw+ueJCQgBDeG/gevh4uOAZF5mn4+k4ICjHuyahESluea+CUKIqmgF9FZIuIjDen1VFKxQKYf4ss74nIeBHZLCKb4+Pjyylc11XXry4fDf4IQbjrl7s4nnrc6pA07aLsS9rH3b/fTbB3MB9c9gGBXoEXXqm8KWVcXpsaC9fNMe7+rkQcuU/jfhEJMl9uc2445+illOoMDAPuE5G+jq6olJqtlApXSoXXqlXLeRFWICEBIcwePJtsezZ3/XoXJ9NPWh2SppXK4eTDjP91PL7uvswZMofavi7aRrBlHuz9HgY+Aw26WB1NmXOkpFEX2Cwii4HFUk49fymlYsy/ccASjLvQT4lIPQDzb1zxW9DOd0n1S5g1aBbJ2cnc+eudxGfoUphWMRxLOcZdv96Fzc3GnCFzaOBfnpUepXByJyybaHRE2ON+q6NxCkcawicDLYA5wG3AQRF5UUSaOSsoEfETkWpnnwODgV0YV2+NNRcbi9HzrlYKbWu2ZdagWcRnxHPHr3eQkJlgdUiaVqLjKce5/ZfbybXn8uFlHxISEGJ1SEXLTIZFt4BPMIyYDW4ueDVXGXDoqJTRH8VJ85EHVAe+EhFntfDUAdaIyN/ARuBHpdQyYDpwmYgcBC4zX2ul1LF2R94b9B4n009y5y936p5xNZd1PPU4t/96O9n52Xw0+COaV29udUhFU8rovfbMcbh+HvhX3mpxuVD/RCLyIMav+gTgI+BbpVSuiLgBB5VSTitxlIXw8HC1ebO+R6Eom05u4t7f76WBfwM+GvIRNX0qZq+bWuV0PNUoYWTmZTJn8BzXGqr1fGvfgt+mwJCXoMe9VkdTJkRkS6HbHQo4UtKoCYxUSg1RSn2plMoFUErZgeFlHKdWjiLqRvDeoPeISY9h3LJxxGXoJiLNNUSdiWLcsnFk5Gbw0eCPXDthHF4Bv0+F1ldD9/9ZHY3TOdKmMUUpdbSYefpusQouom4EswbNIi4jjnHLxumrqjTLHU4+zLhfxpGTn8PcIXNpGdzS6pCKl3gYvhwHtVrC1e9Vim5CLqRyttSUgcj4NM5k5F54wUqgc53OfHDZByRlJTH257EcT9H3cWjW2J+0n3HLxgHw8dCPXbuEkZUCn98I4gY3fm70L1UF6KRRhJw8O7d9vIlRs9cRl1I1RsHrWLsjHw35iIy8DMYuG8uh04esDkmrYrbHbWfcL+PwtHkyb+g8mgW5cHOpPR++uQsSD8EN86F6qNURlRudNIrg6e7GCyPaciwpg+tmreNoYrrVIZWLNjXa8PGQjwG47Zfb2JWwy+KItKpi7Ym1jP9tPNW9qrNg2ALXvaz2rF+eggPLjHG+mzh833GloJNGMfq0qMXCu7qTkpXLte+vY09MitUhlYvm1Zszf+h8/D38ueOXO/gr5i+rQ9IquV+ifuH+FffTuFpj5g+bT31/Fx+kaP37sGEWdL8Put5ldTRFysmzcyTBOT92ddIoQcdGQXx1Tw88bML1s/5ixb5TVodULhoFNGLBsAU0qNaA+5bfx89HfrY6JK2SWrh3IU/8+QTtarZj7tC5rn/Z996lsGwStBwOg5+3OppizfhtP5e/tZqY5Mwy37ZOGhfQvHY1ltzbiya1/Lhz/mY+Wh1ZJcberu1bm3lD59G+ZnueXPUkn+751OqQtEpEKcVbW9/ipY0v0b9Rf2ZfNpsATxfv2O/YeqPn2gZdYOSH4GazOqIirTmYwAd/RjKicwPqB/mU+fZ10nBA3UBvFt/dg8Gt6zLtx71M+HoHWbn5VofldAGeAXxw2QcMbDyQlze9zMsbXybfXvmPW3Ou3PxcJq+dzEc7P+L6S67n9X6v4+3ubXVYJTu5Ez67AQIbwI1fgKcLdscOJKZl8+ji7TSv7c//XdHaKfvQScNBvp7uvDemM/f3b87izdGMfO+vKtFA7u3uzYxLZ3Bzq5v5dO+nPPbnY2TmlX2RV6sazmSf4e7f7+b7w99zX8f7+L/u/4fNRX+xF0g8DJ+MNC6pveVbl+0iRCnFhK93kJyRy8zRnfDxdM77qpNGKbi5CY8PCWPO2HBOJGcy/O01LNsVa3VYTmdzszGh6wQmRExgxbEV3L7sdt1DrlZqx1OOc/NPN7M9bjsv9XmJezrcQzl1mn3xzpyAT0aAyjcSRlAjqyMq1ry/ovh9bxyTLm9J6/rOq+rTSeMiDGxVh6UP9KZJTT/u+XQrT3z5N6lZlf9GwJtb38yb/d/k8JnDjP5xtB4+VnPYppObuOmnmzidfZoPB3/I8KYVoAeilBiYd4UxCt+Yr6DWJVZHVKzNUUm88ONeBrWqw209Q526L500LlKjYF++uqcn9/dvztdboxn21mrWR1b+3mIHNB7AgmELcBM3xi4by29Hf7M6JM3FLd6/mPG/jqe6d3U+u/wzutSpAAMTpcTAvOGQngC3LIEGna2OqFhxqVnc+9lWGlb3YcYNHZxeetNJ4z/wdHfj8SFhfHlPD2xuwujZ65n49Q6SM3KsDs2pWga35PMrPqdF9RY8uvJRZm6dqRvItX/Jzc9l2vppPL/+eXrU78Fnl3/m+jftgZEw5l8Jaafg5q+h4b86enUZufl27v9sGylZucy6pQuBPh5O36dOGmWgS0gwPz/Uh7v7NuXLLdEMnPEnX2+Jxm6vvJfm1vSpycdDPmZki5F8uPND7l9xPyk5VeMGSO3C4jLiGPfLOBbtX8S4NuN4e8DbVPOsZnVYF5YUCXOHQOopo0qqcTerIyrRtKV72BiVxPSR7WlZt3wuWb7geBoVXXmPp7EnJoVJS3by9/Fk2jcMZPIVrenaJLjc9l/elFJ8eeBLXtr4EvX86vF6v9ddu1dSzem2nNrCYysfIyMvg+d6PcfQ0KFWh+SYuL2w4BrIz4abv3HpKimAeWuPMPWHPdzZuwmTh5f95bX/ZTwNrRRa1w9gyf96MuP6DsSlZHPDB+sYv2Aze2Mr569wEeGGsBv4eMjHZOdnM+bHMXx94OsqcQOkdi67sjN311zu+OUO/D39WXj5woqTMI5vgo8vN56P+9nlE8aKfad4bukeBrWqw6TLW5XrvnVJw4kyc/L5cHUkH66KJDU7j2Ft6/LgwBa0qufid75epKSsJCaumsi62HVc2fRKJnefjK+Ha94EpZWt5Kxknl77NKuiV3FZyGU82/PZilEdBUbXIF/fAdXqwS3fQHBTqyMq0Z6YFK6f9RehNf1YfHcP/LzcnbKf4koaOmmUg+SMHOauOcLHa6NIzc6jT4ua3N67CZe2qIWbm4tfp15K+fZ8Zu+Yzawds2hUrRGv9H2F1jXOKzorZXQtrezg5g5uusBbkW06uYlJqyeRlJXEExFPMDpstOvff3HWhg/g5wlG1yA3LQI/1+776mhiOtfNWodNhG/v60XdQOfdSa+Thgs4k5HLpxuOsmBdFKdSsmlay48bIxozsnMDavh7WR3exVPKuOIkKRLOHIfkY2xK2sOk1F0kqlwezrZxS1oWbtlpkJcJ9rzzNiBg8wBPP/D0Nx4+1cE32Hj414VqdaBafePmqqDG4B1oyaFq/8i15/LutneZu2sujQMa83Lfl2lTo43VYTkmP9foeHDThxB2BVz7kct2DXLWqZQsrpv1F6lZeXx5dw9a1HFuSU4nDReSk2fnx50xfLLuKFuPJeNhEwa2rMNVHevTP6y2027/LxN5ORC3G2K2GY9TuyH+AOSknrucXy2S/WrxjJ+dFWTQ1T2IaYGdqecdbCQIN3djaEy73Ugi+dmQkwG5GZB1xnhkJEFGAqTHG6WSwrwDIbgZ1GgGNZobw23WbmVULdicf9lhVReZHMlTa55id+Jurm1xLU9GPFlxqiLTE+HLsRC1Gno+AIOeddnOB886nZ7DDR+sIyY5k4V3dadDoyCn77PSJA0RGQq8BdiAj5RS00ta3hWTRmEHT6WyaNNxvt1+goS0HHw9bQxsVYdBrWpz6SW1CPL1tDbA/DyI3ghHVhv/ZNGbIM8czdCnOtRpa5ysa4UZJ/GgxhDYENyNkpNSim8Pfcv0jdNxEzcmdp3IVc2uKl31RX6ekThSYuDMMUg+DqejIOkwJJqlG8zvsc3TSCB120PddlC/k/HXxX9FVhT59nw+3fspM7fOxNfDlyk9pnBZyGVWh+W4mO2w+Bbjktqr3oYOo6yO6IJOp+dwy9wNHDiVxrxxEfRsVj5VaJUiaYiIDTgAXAZEA5uAG5VSe4pbx9WTxll5+XY2HEli6Y5Yft19ksT0HNwEOjWuTs9mNejetAadG1cvn1JIVgrs/9kYmezwcuNXPwJ120JIb2jU1bi6JCjEKC04IDo1mqfXPM3WuK30adCHKT2mUNev7kWFZ7cr0nPySMvOIy0rj7T0VIg/gHviPrySDlAteR9BKXvxyUkylseNeJ8mHPdtwzGf1kT5tOaUZ2PycUMp4xAEcBPBZhM83AR3mxue7m54u9vw8nDDx8OGj6cNP093/LxsVPP2INDHnQAfD4J8PPF0r/ztMlFnonjmr2fYGreVfo368UyPZ1x//IuzlIJNHxkj7vnVglGfGO0YLi4xLZsxH20gMiGdD27uQv+Wtctt35UlafQApiqlhpivJwEopV4qbp2LTRonX3yR7L37LjbU/0QB6dl5nM7I5UxmDunZ+SilEBF8PW34e7nj5+WOr6dxIrOVRaOjskNGotFtQuZp47XN02xbqA7eQUaV0n86LkVcRhzRaScQoKF/Q4K8apJnV+TlK/Ls9vP+KvLy7eTZFfl2VeivvaBgURIP8vCXLPwlE3+y8CMTd4w71/OwkS6+pOFLmviRgTcKQSlj00op7AqHLx22mYnGwyZ42NzwsLnhaXPDw13wtLnh5W7D090Nm5tQQZqIC9hRnEo/SUxaDCJuNK7WiBo+NSvOcdjzjLG80xOM73LNsP/8XS4POfl29samkJ1rJ6xutVLf7e3VqiV1n3rqovdfXNJw/XfuXA2A44VeRwP/umVTRMYD4wEaN25cPpGVIQH8vdzx93KnUXUf8uyKtOw8UjJzSc/OIyEtm1MpWQULe7nb8PEwfhF7mycnT3fzpGUT3EpKKjnpkHYS0uKNfy53L6hW1/g15uVYQ5tdqX+f2M2TfeETv5EIvHGzNyLP7RRHU48SlXwKe05NsJ9bDSciuNsEdzfB3c04Fh9P47XNfLi7GSdhmwhuboLNzSwtuBnH7CbGds45+txMyE7BPTuFwKwUAnNPGVnCzWYcr3egkSC9/AFBmcdnV5BvV9jtinxV6FjNY8vNN44z124nO9dOWlYeuUUkN5ub4GWWXozPzQ0vDxveHja83N1c7kSclpvG0ZSjZORlUN0rmJCAxni4VaA2o6xkSDgI+TlQPdSoOq0AsvLy2RebSk6+nbB61Qj0dp33vKIljaL+p/71U1ApNRuYDUZJ42J29F8ytLPZ7YqoxHQOnEpl/8k0DsSlcjQxnaMJGaRmn39lEvh52qju50k1bw+qeblTzRO65axn4JmvaZa5k1zxZE9gX7bVvJJIv87kI9iVUWWWm28nJ984EWbm5huPnHzSsvPIyMknLSuPnHx7EVH+w9/LnSBfD6r7ehb8re7rQQJr2ZQyn2x7PAPqX8dNl9xBvYBAqvt64utpK5/LNtMT4OhaiFoLUWsgbqsx3dMfQnpB037Q9FKo3drhqrizcvPtJKRlc/JMFjHJWcQkZ3IiOZPjSRkcP53BsaQMsnL/ee+83N1oUtOP5rX9aVG7GpfU8eeSutUIreGHrZwvzT6ddZo3t77JNwe/obZvbZ7q+goDQwaWawz/SW4mLH8e1n9nXCgx4lNo6PrVUQDbjydzx7xN2FspPhobQfuQ6laHdA5dPVWJKKVIzsgl9kwWp1KyOJmSRVJ6TsEjMyODiOQfGZ66mDr2OGKlNovdLud76Uey8idfqYJf6CB42ARPdzfc3QRvD6M04+1hK6gi8/Wy4eflToC3BwHe7mY9vwcBPu4E+ngQ5OtJoI8HHrbi6/uTs5J5fcvrLDm0hJo+NXmkyyMMbzocN7GojSA9wUgeR1bBkT+Nag0A/zrQtD8062/8rVbnP+9KKUVcajZRCelEJqQTGZ/G4fh0Dsalcjzpn4GuvD3cuKRONVrVDaBNgwDa1A+gVb0AfD3L/jdfnj2PxfsX897f75Gek87NrW/mng734OfhV+b7cprIlbD0EeMS8Ii74LLnKsyFEL/vOcUDn2+jZjVP5o/rStNa/pbFUlnaNNwxGsIHAicwGsJvUkrtLm6dqpQ0ipWbBVs+hrVvQWosNOwKvR6CsGEuc6nhjvgdTN84nZ0JO2lfsz2PhT9G5zou0JVD8nHjJBS5EiL/MNp9AOq0g+YDoNlAaNy94GqxspKRk8fhuHT2nUxh/8lU9p1MZU9sCknpRg/KItC8lj/tGgbSvkEgHRoF0bp+AF7uF/95rj2xllc3vcrhM4fpVrcbE7pOoEX1FmV1SM6Xngi/Toa/FxqXXg9/0yglVgB2u+KdPw7xxu8HaNcgkDljI6hVzdp7typF0gAQkcuBNzEuuZ2rlHqhpOWrdNKw58OOxfDHC8ZlqSG94dInoMmlpa5qKQ92ZeeHwz8wc+tM4jLjGNBoAA91eYimgS7SrYPdDif/hsMr4PAfcGw92HPBww9Ce0PzQdB8oHHviBMopTiZksXuEynsPHGGXSfO8Hf0GRLSsgHwsAmt6wXQqXF1OjUOonPj6jSs7nPBar7dCbt5Y+sbbIjdQKNqjXg8/HH6N+pfce7qzs+FjR/CyumQm278IOr7BHj4WB2ZQ1Kzcnls8d/8uucUIzo14KWR7fD2sP7HXKVJGqVVZZNG1Br4eSKc2gn1Ohg3MDXrb3VUDsnIzeDTvZ8yd9dcMvMyubLpldzd4W4aVXOxoTazU433+dDvcGg5nD5iTK/e5J8EEtrHbFR3jrOJ5O/jyWw/foZtx06zI/oMmbnGVWJ1ArzoElKd8JBgIkKDaVWvGu5mdeHh5MO8t/09fj36K0FeQYxvP55RYaPwtFl8b5CjlDIuDf/9GUg4AM0GwJCXoHbF6WV5R3QyD32xnWNJGUy+ohW39Qx1mWStk0ZVkRJrFNF3fQWBjWHQM9BmZIXs3ykxM5E5u+awaN8i7MrO1c2v5o52d7he8jgr8bBRCjn4m3EjZG4GuHkY1Vdnk0idtk4v5eXl29l3MpWtx06zOeo0W46e5kSy0Ubi52mjdUgmuQG/EpmxFm93b25tfSu3tbkNf0/r6s9LLXKl0dB9YrPR0D34BbhkiEuWoItitytmr47ktV/2U6uaF2+O6ki3pjWsDuscOmlUdnY7bJ4Dv081iuu9H4ZeD1eYBsCSxGXE8eGOD/n64Nfkq3yGNRnGHW3vcO369rxsOLbOLIWsMLpeAfCrbfwibjbAuDKrDBrUHRF7JpOvd//F90c+JTZ3C8ruQc7pnnDmUjo1aED3pjXo1jSYzo2ru0TVSJGUgoO/wpo34dhfENAQ+k2ADjeBreJcCHokIZ1J3+xgfWQSl7ery4sj2lnf80MRdNKozBIOwfcPGP9IzQbAFTNcvnvnixGXEceC3QtYfGAxmXmZ9Krfi5tb30yv+r1cpkhfrJRYsy1kxbkN6rXbmJf19oOQnmVelZVnz2Pl8ZV8sucTtsZtJcAzgNEtR3Nl6A0cjIX1kYlsOJLInpgU7MoYwrhToyC6N61Bj2Y16NQ46D81rpeJ3EzY9Q2se9dIvoGNjD6jOo8FD+f18lrWcvPtzF4VyVvLD+Ll7sb/XdGa68Mbuux3VyeNykgp2DgbfptiXL0z5EXoOKbCFNEvVnJWMosPLOaLfV8QnxlPk8AmjAobxfCmwwn0qgC9355tUI9c+U+Den62cZdygy7QpK/RFtIw4qJLiomZiXx76FsW7V9EbHos9fzqcWvrWxnZYmSRHQueycxlc1QS6yMTWR+ZxK6YMyhl3DvSJaQ63ZvWoGezGrRvGFR+XaYkHoat82HrAqOXglotjdJzu+sqXKeUqw7EM+3HPRw4lcawtnV59qo21A5w7YSnk0ZlkxYH394Lh36D5pcZna8F1LM6qnKVm5/LsqhlLNy7kF2Ju/CyeTEkdAjXNL+GLnW6WHevR2nlZsLxDRD5p3F/SMxWc6wRDyOJhPSAxj2N8apL6BI+357P+tj1fH3wa/44/gd59jy61u3KTS1v4tJGl+Jeiq4zzmTmsvGIkUTWHU5k78kUlAIfDxvhodULSiLtGgSWeB9OqWWeht3fwt+fG++J2KDlFdD1LiORVrAfRAdOpfLiT3tZuT+exsG+TL6iFYPbXFyfa+VNJ43K5PAf8M1dRseCg6cZ/1AV7J+prO1N3MuXB77kx8gfycjLoJ5fPYY3Hc7QJkNpEdTCZasAipSVYpwwo1Ybd6rHbjfHIBGjR+FGXY17bRp0QdVowZ7T+1gauZRlUctIyEwgyCuIq5pdxcgWI2kWVDaX/55Oz2FDoSSy/5TRFb6fp43w0OCCNpGLSiIpsbD/J9j7g3HM9jyjf6iON0L7URBQv0yOoTztO5nC2ysO8dPOWPy93HloYAtu6RFifVVfKeikURnY7bBmBqx4weiK/Pp5xklEK5CZl8mKYyv44fAPrItdh13ZCQ0IZXDoYPo36k/rGq0rTgnkrJx0iN5sVGNFbyT/+CZ2kMnvfr4s9/PjhLsND9zoG9iCy5tcQb9Wo/B08gUQCWnZbIhMYsORRNZHJnLgVBoAvp42uoRUp2toMBFNgunYKOjfDeuZyUYX+5ErjTaeOLOT6uCm0OoqaH210aV9RUr0GJc/r49M4uO1R/h1zyn8PG2M7RnKnX2aEuzneg3dF6KTRkWXmQzfjIeDv0C76427XZ14/X9lkJCZwPKjy/nt6G9sOrUJu7JTw7sGfRr2oUe9HnSt17XCdO19Mv0kG2I3sPbEWv6K+YszOWfwEBvd3aszKCOLgScjCTw7zonN0/ilXqc11GwBNVoYf6uHGqMjOkFCWjYbjySxITKRDUeS2HfSKIkE2bIYWvs0/QJiaet2lLqpu3BP2AcoI87GPYxLkZtfZvwAqmCJAowROb/fEcOn646y/1QqQb4e3NI9hDt6N3HJq6IcpZNGRZZ4GBaOMgYeGvoSRNxZIf+5rJSclcyamDWsOr6KNTFrSDVHGmwe1JxOtTvRvlZ7OtTqQEhAiOUlkXx7PofPHGZXwi62xW1j88nNRKdFA1DDuwa9GvSid4Pe9GnQ5597K/JzjRvcTu40RlOM2wNxeyHlxLkb96ttJI+A+hDQwGgH86sFvjXBr4bRZuIVYHTY6O5V9PdMKeOS4twMoxfZjNPG1WBpJ82BsqLJS4gkP/4gXllxBaslKz922JtywLstmXW7EtC8O21D69GmfoDrXuZbjPTsPFYdiOe77TGs2BdHTr6d1vUCuK1nKFd1rF/hjqcoOmlUVJF/wuJbQdxg9GfGZZnaf5Jvz2df0j42nNzAxtiN7IjfQWqukUR83X0JCw7jkuqX0CyoGaEBoTQJbEJt39plnkzy7HnEpsVyPPU4R1KOcPD0QQ4lH+LA6QNk5hk34wV6BdK5dmfC64QTUTeCsOCw0sWRk250uphwEJKPGj88TkcZJ/eUGOPEXxKbp/EAo3Fe2Y2EUdKAJmcTU80WRpcqtVqSVbMtu1L92Xo8mW3Hktl67DSnUozuT9zdhBZ1qtGmfgBt6wfQun4gYXWqEejrOldIKaU4HJ/GusgkVuw9xdrDieTk2anp78VVHeozolMD2jYIqFhtZxegk0ZFtO1T+OEh447XG7+A4CZWR1Qp2ZWdyORIdiTsYG/iXg6cPsD+0/tJz00vWMbdzZ06vnWo41uHmj41qe5dnSCvIPw9/PFx98HHwwcPNw8EQUTIt+eTY88hJz+H9Nx0UnJSSMlO4XT2aeIy4ojPiCcuI4489U9X9oFegbQIakFYcBhta7alXc12NK7W2HknIqUgO8Xo2Tc9wRiPPSvF6B4lO8UYgyI/xxgX/mwMIuDuY/Tr5OFjjD3iUx18g42egKvVA3fHqmROnsni7+hkdkQns/NECntizpCQllMwv26ANy3q+NO8tj/NavnTtKYfjYJ9qR/k49Su4pVSJKbnsCcmhV0xRh9fG48kFcQWUsOXgS2NIZm7Ngku6JalstFJoyJRCla/BiumGd1w37AAvAOsjqpKUUoRnxlP1JkoolKiiE2PJTY9lpPpJ0nKSuJ01mnOZJ9BOTKEIGATGwGeAQR5B1Hbtza1fWpT168ujao1omG1hoQGhFLTp2al+qVaWme7it8Ta/Tse+BkKgfiUomMTycjJ79gOXc3oX6QD3UDvakX6E2dAG+C/TyNh68nfuYAZj6eNjxs/wzadXY0yNx8O6lZeaRm5ZKSlUdcShbxqdnEnsniaGI6RxLSScn6J5k3CvYhPCSYbk2C6da0BqE1fKvE56STRkVhz4efJ8CmD6HdDXD1uw7/ctPKV749n8y8zIJHnj0PhcKu7NjcbHjZvPB088TXwxdf96pxonGGs50yRsanczzJGLwq+nQmJ1OyOGmOHZOdV/JAYBfiaXOjdoAXoTX8CK3pS2gNP1rXD6BNvUCXqiYrT5VluNfKLT8Pvr0Hdn5pdJMw6LkK2dFgVWFzs+Hv6V+xOvqrgESEeoE+1AssuqtzpRQZOfkkpedwOiOHtOw80rPzycjJIy/fGJo3XylsbsbAYu5ubvh7uxPgbQwgVquaF4E+HjqpO0gnDVeRlwNf327c4DTwGejzqNURaVqFICL4ebnj5+VOo+CK30Gnq9NJwxXkZhlXSB38BYZOh+7/szoiTdO0IumkYbW8bFh8i9Hl8/A3IPx2qyPSNE0rlk4aVsrPha9uNxLGlW9Bl9usjkjTNK1EupXVKvl5RqeD+5bCsFd0wtA0rULQScMKSsHSh2H3Erjseeh2t9URaZqmOUQnDSuseB62fQJ9n4BeD1odjaZpmsN00ihv62fB6hnGUJX9n7Y6Gk3TtFJxuaQhIlNF5ISIbDcflxeaN0lEDonIfhEZYmWcF2XXN7BsIrQcDle8rnuq1TStwnHVq6feUEq9VniCiLQGRgNtgPrA7yJyiVIqv6gNuJzjG2HJPdCoG1z7Edhc9a3XNE0rnsuVNEpwNfCFUipbKXUEOAR0tTgmx5yOgs9vNMYwGL3Q6B1U0zStAnLVpHG/iOwQkbkiUt2c1gA4XmiZaHPav4jIeBHZLCKb4+PjnR1rybLOGAMo2XNhzJfGQDeapmkVlCVJQ0R+F5FdRTyuBt4HmgEdgVhgxtnVithUkV30KqVmK6XClVLhtWrVcsYhOMaeD1/dYQyCM+pTY1AaTdO0CsySinWl1CBHlhORD4Gl5stooFGh2Q2BmDIOrWz98QIc+s1o9G7S1+poNE3T/jOXq54SkXqFXo4AdpnPvwdGi4iXiDQBWgAbyzs+h+357p9LayPusDoaTdO0MuGKl/C8IiIdMaqeooC7AZRSu0VkMbAHyAPuc9krp07tgSX/g4Zd4fJXrY5G0zStzLhc0lBK3VLCvBeAF8oxnNLLTjV6rfXyN4ZpdfeyOiJN07Qy43JJo0JTCn54CJIiYexSCKh34XU0TdMqEJdr06jQNs+FXV/DgMkQ2svqaDRN08qcThplJWa70UVI88ug1yNWR6NpmuYUOmmUhew0YzAlv1ow4gNw02+rpmmVk27TKAvLJhjtGLf9qO/41jStUtM/if+r3d/Ctk+hz6O6HUPTtEpPJ43/4ky0cbVUgy7Qb5LV0WiapjmdThoXy26Hb/8H9jwY+SHYPKyOSNM0zel0m8bF2vQRHFkFV86EGs2sjkbTNK1c6JLGxUg8DL8/A80HQedbrY5G0zSt3OikUVr2fPj2XnDzMEoZeshWTdOqEF09VVrr34Pj6+GaWRBY5BhQmqZplZYuaZRG4mFYMQ0uGQYdRlsdjaZpWrnTScNRSsHSh8HmCcNf19VSmqZVSbp6ylHbPjWulhr+BgTUtzoaTdM0S+iShiNST8GvT0PjntD5Nquj0TRNs4xOGo5YNgFys+CqmbozQk3TqjR9BryQQ7/D7iXQ93Go2cLqaDRN0yylk0ZJcrPgpycguBn0esjqaDRN0yynG8JLsvYto8vzW5bosb41TdPQJY3iJUXC6hnQZiQ0G2B1NJqmaS7BkqQhIteLyG4RsYtI+HnzJonIIRHZLyJDCk3vIiI7zXkzRZx4o4RS8NOTxj0ZQ1502m40TdMqGqtKGruAkcCqwhNFpDUwGmgDDAXeExGbOft9YDzQwnwMdVp09nyo3QoGToGAek7bjaZpWkVjSZuGUmovQBGFhauBL5RS2cARETkEdBWRKCBAKbXOXG8BcA3ws1MCtLnD4OedsmlN07SKzNXaNBoAxwu9jjanNTCfnz+9SCIyXkQ2i8jm+Ph4pwSqaZpWFTmtpCEivwN1i5j1tFLqu+JWK2KaKmF6kZRSs4HZAOHh4cUup2mappWO05KGUmrQRawWDTQq9LohEGNOb1jEdE3TNK0cuVr11PfAaBHxEpEmGA3eG5VSsUCqiHQ3r5q6FSiutKJpmqY5iVWX3I4QkWigB/CjiPwCoJTaDSwG9gDLgPuUUvnmav8DPgIOAYdxViO4pmmaVixRqnJX+YeHh6vNmzdbHYamaVqFIiJblFLh5093teopTdM0zYXppKFpmqY5rNJXT4lIPHD0IlevCSSUYTgVgT7mqqGqHXNVO17478ccopSqdf7ESp80/gsR2VxUnV5lpo+5aqhqx1zVjhecd8y6ekrTNE1zmE4amqZpmsN00ijZbKsDsIA+5qqhqh1zVTtecNIx6zYNTdM0zWG6pKFpmqY5TCcNTdM0zWE6aVyAiLwqIvtEZIeILBGRIKtjcraShuOtTERkqDms8CERmWh1PM4mInNFJE5EdlkdS3kRkUYi8oeI7DW/0w9ZHZOziYi3iGwUkb/NY362LLevk8aF/Qa0VUq1Bw4AkyyOpzwUORxvZWIOI/wuMAxoDdxoDjdcmc3DmcMku6Y84DGlVCugO3BfFfics4EBSqkOQEdgqIh0L6uN66RxAUqpX5VSeebL9Zw7rkelpJTaq5Tab3UcTtYVOKSUilRK5QBfYAw3XGkppVYBSVbHUZ6UUrFKqa3m81RgLyWM+lkZKEOa+dLDfJTZFU86aZTO7egu2SuL4oYW1iopEQkFOgEbLA7F6UTEJiLbgTjgN6VUmR2z00buq0gcGZpWRJ7GKOp+Vp6xOctFDsdbmZRqCGGtYhMRf+Br4GGlVIrV8TibOQ5RR7MNdomItFVKlUlblk4aXHhoWhEZCwwHBqpKcmPLRQ7HW5kUN7SwVsmIiAdGwvhMKfWN1fGUJ6VUsoisxGjLKpOkoaunLkBEhgITgKuUUhlWx6OVmU1ACxFpIiKewGiM4Ya1SsQcHnoOsFcp9brV8ZQHEal19ipPEfEBBgH7ymr7Omlc2DtANeA3EdkuIrOsDsjZihuOtzIxL264H/gFo3F0sTnccKUlIp8D64AwEYkWkTusjqkc9AJuAQaY/7/bReRyq4NysnrAHyKyA+PH0W9KqaVltXHdjYimaZrmMF3S0DRN0xymk4amaZrmMJ00NE3TNIfppKFpmqY5TCcNTdM0zWE6aWiaCxKRe0TkVqvj0LTz6UtuNU3TNIfpkoam/UciEmGOt+ItIn7mGAZtz1vmShHZICLbROR3EaljTp8pIlPM50NEZJWIuInIVBF53Jz+oIjsMffxRfkfoab9Q5c0NK0MiMg0wBvwAaKVUi+dN786kKyUUiJyJ9BKKfWYiPhi3LV7PzALuFwpdVhEpgJpSqnXRCQGaKKUyhaRIKVUcjkemqadQ3dYqGll4zmMk38W8GAR8xsCi0SkHuAJHAFQSmWIyF0YA149opQ6XMS6O4DPRORb4NuyD13THKerpzStbAQD/hj9lHmLyAtn+zoy578NvKOUagfcjVEqOasdkAjUL2bbV2CMMtgF2CIi+seeZhmdNDStbMwG/g9jvJWXlVJPK6U6KqU6mvMDgRPm87FnVxKREOAxjMGBholIt8IbFRE3oJFS6g/gSSAIIzlpmiX0LxZN+4/MS2PzlFILzbHH/xKRAUqpFYUWmwp8KSInMIYNblKo2+7HlVIxZq+z80QkotB6NuBTEQnEGDjqDd2moVlJN4RrmqZpDtPVU5qmaZrDdNLQNE3THKaThqZpmuYwnTQ0TdM0h+mkoWmapjlMJw1N0zTNYTppaJqmaQ77f+PRusl7KrdlAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "p = poly1d([3, -4, -12, 0, 2])\n",
    "print('\\nFourth Degree Polynomial')\n",
    "print(p)\n",
    "print('\\nFirst Derivative')\n",
    "g = p.deriv(m=1)  # First derivative with m=1.\n",
    "print(g)\n",
    "print('\\nSecond Derivative')\n",
    "q = p.deriv(m=2)  # Second derivative with m=2.\n",
    "print(q)\n",
    "x = linspace(-2, 3, 101)\n",
    "y = p(x)\n",
    "yg = g(x)  # These statements define points for plotting.\n",
    "yq = q(x)\n",
    "y0 = 0*x  # This statement defines the y axis for plotting.\n",
    "plt.plot(x, y, label='y=p(x)')\n",
    "plt.plot(x, yg, label='First Derivative')\n",
    "plt.plot(x, yq, label='Second Derivative')\n",
    "plt.legend(loc='best')\n",
    "\n",
    "plt.plot(x, y0)\n",
    "plt.xlabel('x-axis')\n",
    "plt.ylabel('y-axis')\n",
    "plt.title('Plot Showing Function, First and Second Derivatives')\n",
    "plt.show()"
   ]
  }
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