{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 17のpre試験\n",
    "\n",
    "1234 西谷滋人\n",
    "\n",
    "# 1(a)\n",
    "(Einstein 結晶のエネルギー) 次の関数 E(x) を求めて x=0..2 でプロットせよ.\n",
    "\\begin{align*}\n",
    "Z(x) = \\frac{\\exp(1/x)}{1-\\exp(-1/x)} \\\\\n",
    "E(x) = x^2 \\frac{\\rm d}{{\\rm d}x} \\log \\left(Z(x)\\right)\n",
    "\\end{align*}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "from sympy import *\n",
    "from sympy.plotting import plot\n",
    "\n",
    "x=symbols('x')\n",
    "\n",
    "ZZ = exp(1/x)/(1-exp(-1/x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "EE = x**2*diff(log(ZZ),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fbc4917f0a0>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(EE,(x,0,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1(b)\n",
    "資料を参考にして，次の2重積分を求めよ．(15点)\n",
    "\\begin{equation*}\n",
    "\\int \\int_D \\sqrt{2x^2-y^2}dxdy,\\hspace{5mm} D:0\\leqq y \\leqq x \\leqq 1 \n",
    "\\end{equation*}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "x, y = symbols('x y', positive = True)\n",
    "\n",
    "f = sqrt(2*x**2-y**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{6} + \\frac{\\pi}{12}$"
      ],
      "text/plain": [
       "1/6 + pi/12"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(integrate(f,(y,0,x)),(x,0,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2(a)\n",
    "\n",
    "行列$\\displaystyle A= \n",
    "\\left( \\begin {array}{ccc} \n",
    "1&1&3\\\\ \n",
    "-1&0&1\\\\ \n",
    "1&2&1\n",
    "\\end {array} \\right) $\n",
    "の対角化行列を求めて，対角化せよ．(15点)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 1 & 3\\\\-1 & 0 & 1\\\\1 & 2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 1, 1, 3],\n",
       "[-1, 0, 1],\n",
       "[ 1, 2, 1]])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "A=Matrix([[2,0,1],[0,3,0],[1,0,2]])\n",
    "A=Matrix([[1,1,3],[-1,0,1],[1,2,1]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}7 & 1 - \\sqrt{3} & 1 + \\sqrt{3}\\\\-2 & -1 & -1\\\\3 & 1 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 7, 1 - sqrt(3), 1 + sqrt(3)],\n",
       "[-2,          -1,          -1],\n",
       "[ 3,           1,           1]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P,D=A.diagonalize()\n",
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & 0 & 0\\\\- \\frac{14 \\left(2 \\sqrt{3} + 5\\right)}{12 + 10 \\sqrt{3}} + \\frac{4 \\left(-2 + 7 \\sqrt{3}\\right)}{12 + 10 \\sqrt{3}} - \\frac{546}{- 70 \\sqrt{3} - 84} & - \\frac{2 \\left(2 \\sqrt{3} + 5\\right)}{12 + 10 \\sqrt{3}} + \\frac{91}{- 70 \\sqrt{3} - 84} + \\left(1 - \\sqrt{3}\\right) \\left(- \\frac{2 \\sqrt{3} + 5}{12 + 10 \\sqrt{3}} + \\frac{-2 + 7 \\sqrt{3}}{12 + 10 \\sqrt{3}} - \\frac{91}{- 70 \\sqrt{3} - 84}\\right) - \\frac{-2 + 7 \\sqrt{3}}{12 + 10 \\sqrt{3}} & - \\frac{2 \\left(2 \\sqrt{3} + 5\\right)}{12 + 10 \\sqrt{3}} + \\frac{91}{- 70 \\sqrt{3} - 84} - \\frac{-2 + 7 \\sqrt{3}}{12 + 10 \\sqrt{3}} + \\left(1 + \\sqrt{3}\\right) \\left(- \\frac{2 \\sqrt{3} + 5}{12 + 10 \\sqrt{3}} + \\frac{-2 + 7 \\sqrt{3}}{12 + 10 \\sqrt{3}} - \\frac{91}{- 70 \\sqrt{3} - 84}\\right)\\\\0 & 0 & \\sqrt{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                                                                                                    2,                                                                                                                                                                                                                   0,                                                                                                                                                                                                                   0],\n",
       "[-14*(2*sqrt(3) + 5)/(12 + 10*sqrt(3)) + 4*(-2 + 7*sqrt(3))/(12 + 10*sqrt(3)) - 546/(-70*sqrt(3) - 84), -2*(2*sqrt(3) + 5)/(12 + 10*sqrt(3)) + 91/(-70*sqrt(3) - 84) + (1 - sqrt(3))*(-(2*sqrt(3) + 5)/(12 + 10*sqrt(3)) + (-2 + 7*sqrt(3))/(12 + 10*sqrt(3)) - 91/(-70*sqrt(3) - 84)) - (-2 + 7*sqrt(3))/(12 + 10*sqrt(3)), -2*(2*sqrt(3) + 5)/(12 + 10*sqrt(3)) + 91/(-70*sqrt(3) - 84) - (-2 + 7*sqrt(3))/(12 + 10*sqrt(3)) + (1 + sqrt(3))*(-(2*sqrt(3) + 5)/(12 + 10*sqrt(3)) + (-2 + 7*sqrt(3))/(12 + 10*sqrt(3)) - 91/(-70*sqrt(3) - 84))],\n",
       "[                                                                                                    0,                                                                                                                                                                                                                   0,                                                                                                                                                                                                             sqrt(3)]])"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DD=(P.inv()*A*P)\n",
    "DD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "-sqrt(3)\n",
      "sqrt(3)\n"
     ]
    }
   ],
   "source": [
    "print(DD[1,0].simplify())\n",
    "print(DD[1,1].simplify())\n",
    "print(DD[2,2].simplify())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2(b)\n",
    "\n",
    "資料を参考にして，行列$\\displaystyle  \n",
    "\\left[ \\begin {array}{cc} \n",
    "1/\\sqrt{2}&a\\\\ \n",
    "b&-1/\\sqrt{2}\n",
    "\\end {array} \\right] $が直交行列であるとき，$a,b$を求めよ．(15点)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & a\\\\b & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[sqrt(2)/2,          a],\n",
       "[        b, -sqrt(2)/2]])"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "a,b=symbols('a b')\n",
    "A=Matrix([[1/sqrt(2),a],[b,-1/sqrt(2)]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\b\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[sqrt(2)/2],\n",
       "[        b]])"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "col_0 = A[:,0]\n",
    "col_0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a\\\\- \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[         a],\n",
       "[-sqrt(2)/2]])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "col_1 = A[:,1]\n",
    "col_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[{a: -sqrt(2)/2, b: -sqrt(2)/2},\n",
       " {a: -sqrt(2)/2, b: sqrt(2)/2},\n",
       " {a: sqrt(2)/2, b: -sqrt(2)/2},\n",
       " {a: sqrt(2)/2, b: sqrt(2)/2}]"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ss=solve({col_0.T.dot(col_0)-1,col_1.T.dot(col_1)-1},{a,b})\n",
    "ss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 0 0\n",
      "1 -1 -1\n",
      "2 1 1\n",
      "3 0 0\n"
     ]
    }
   ],
   "source": [
    "for i in range(4):\n",
    "    ss10=col_0.T.dot(col_1).subs(ss[i])\n",
    "    ss01=col_1.T.dot(col_0).subs(ss[i])\n",
    "    print(i,ss10,ss01)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{a: -sqrt(2)/2, b: -sqrt(2)/2}\n",
      "{a: -sqrt(2)/2, b: sqrt(2)/2}\n"
     ]
    }
   ],
   "source": [
    "print(ss[0])\n",
    "print(ss[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3(a)\n",
    "\n",
    "$p$を実数とし，$f(x)=x^3-p\\,\\, x$とする．\n",
    "\n",
    "関数$f(x)$が極値をもつための$p$の条件を求めよう．$f(x)$の導関数は，\n",
    "\\begin{equation*}\n",
    "f'(x) = \\fbox{ ア }\\,\\,  x^{\\,\\,\\fbox{ イ }}-p\n",
    "\\end{equation*}\n",
    "である．したがって，$f(x)$が$x=a$で極値をとるならば，\n",
    "\\begin{equation*}\n",
    "\\fbox{ ア }\\,\\,  a^{\\,\\,\\fbox{ イ }}-p=\\fbox{ ウ }\n",
    "\\end{equation*}\n",
    "が成り立つ．さらに$x=a$の前後での$f'(x)$の符号の変化を考えることにより，\n",
    "$p$が条件$\\fbox{ エ $(p>0)$}$を満たす場合は$f(x)$は必ず極値を持つことがわかる．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - p x + x^{3}$"
      ],
      "text/plain": [
       "-p*x + x**3"
      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "x, p, k=symbols('x p k')\n",
    "\n",
    "f = x**3-p*x\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - p + 3 x^{2}$"
      ],
      "text/plain": [
       "-p + 3*x**2"
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = diff(f,x)\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 a^{2} - p$"
      ],
      "text/plain": [
       "3*a**2 - p"
      ]
     },
     "execution_count": 178,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.subs({x:a})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3(b)\n",
    "関数$f(x)$が$\\displaystyle x=\\frac{p}{3}$で極値をとるとする．また，曲線$y=f(x)$を$C$とし，$C$上の点$\\displaystyle \\left(\\frac{p}{3}, f\\left(\\frac{p}{3}\\right) \\right)$をAとする．\n",
    "\n",
    "$f(x)$が$\\displaystyle x=\\frac{p}{3}$で極値をとることから，$p=\\fbox{ オ }$であり，$f(x)$は$x=\\fbox{ カキ }$で極大値をとり，$x=\\fbox{ ク }$で極小値をとる．\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{p^{2}}{3} - p$"
      ],
      "text/plain": [
       "p**2/3 - p"
      ]
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = p/3\n",
    "df.subs({x:x0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 3]"
      ]
     },
     "execution_count": 180,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pp = solve(df.subs({x:x0}),p)\n",
    "pp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {},
   "outputs": [],
   "source": [
    "p0 = pp[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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gvig0tnfjpZ8v0dIchOhIW5cKeTU916CEggrUEFQ2deDblBLcEy2FjyOdPWlLiIctnp4XgD0XK/HL+QrWcQgxCpcrmqHhgUgvKlB6aVtKCcI8bLGWzp607k8zx2GCzxj87edLqGhsZx2HEIN3oawRABDuac84yf9QgRqklk4l/nOiCD6OVvBysGQdx+CZiDi8vzQS4VI7/OOXTBrqI0TLLpQ1wcPOHM42Zqyj9KICNUjbU8vQ2qXCqmljWUcxGj6OVpgT5IqDmdX4OaOcdRxCDNqFskaES4UzvAdQgRoUtYbHf08VYqLvGMH9AA3dQ1N9Md7bHq/uykRtaxfrOIQYpKZ2JYrq2hEhFc7wHkAFalAOZVWjtL6Dzp4YMBFxeHtJBNq61HhlVybrOIQYpIvlTQCASCpQ+ufLE4XwtLfAvBBX1lGMkr+rDdbMkmHX+QocyqxmHYcQg3O+d4KEsEaIqEAN4FJ5E84U1mPlVF+ITehwsfLYbeMQ6GqDl36+hOZOJes4hBiUi2VN8HW0hJ0l+yU2rkXvuAPYdLIIlhITLJ3oxTqKUZOIRdhwTwRqWjqxYV826ziEGJSeCRLCGt4DqEDdVE1LJ3adr8C9E6SwsxDWJwtjFOVlj1XTxuJcSQPOFtazjkOIQVC0dKGiqRORApwARgXqJr45XYJutQYraXKEYPx5bgCaO1T4688XoVRrWMchRO9dLBfm9SeAClS/OpVqfHOmGHOCXKgprIBYm4vxSlworlS34j8nClnHIUTvnS9tgoiDYNaAuhYVqH7sOl+B2tZurJpOZ09CMzfEFfNDXPHhoSsorac2SISMxMXyJshcrGFlJrwF1qlA9YHneXx5sgiBrjaYOs6RdRzSh3/EhYIDh1d2XWYdhRC9xfN8zwQJAfXfuxYVqD6cK2mAWMTh0ZljBbMuCrmep70Fnp7nj0NZNThwuYp1HEL0UkVTJ2pbuwXVwfxaVKD68H1qGfJqWhET5s46CrmJh6eNRaCrDV7+5TLaulSD/nerVq2Ci4sLwsLC+nye53k8+eSTkMlkiIiIwLlz50YrMiGCclGgN+heRQXqd9q7Vdh9oRKx4e6wFuCYLPkfUxMR3lgchoqmTnx0OHfQ/27lypVITEzs9/l9+/YhNzcXubm5SEhIwGOPPTYacQkRnPNlTRCLOAS727KO0icqUL+TeKkKrV0qLJVLWUchgyD3dcDqW/2w63wF8mpaBvVvZs6cCQcHh36f37lzJ1asWAGO4zBlyhQ0NjaisrJytCITIhgXy5oQ6GYDc1MT1lH6RAXqd7anlsLH0RKTxvb/BkaE5dEZfmjrUuHlUVo3qry8HF5e/+scIpVKUV7e93IfCQkJkMvlkMvlUCgUI943IbpydYKE0DqYX4sK1DVK6tpxuqAe90RLaXKEHnG0NsOz8wNxIq8WiZdGPmGiryLX3+9DfHw8UlNTkZqaCmdn5xHvmxBdKaprR3OnChEC7CBxFRWoa+w4VwaOA5ZMoOE9fbN8sjeC3Gzw+p4sdHSrR/RaUqkUpaWlvY/Lysrg4eEx0oiECMrVJd6pQOkBjYbHD2llmC5zgoe9Bes4ZIjEJiK8EheK8sYOfHo0b0SvFRcXhy1btoDneZw+fRp2dnZwd6cZncSwXChrgplYhABXG9ZR+kXT1H5zKr8O5Y0dWL8wiHUUMkyT/RwRF+mBz44V4J4JXvB2tOxzu/vvvx9Hjx5FbW0tpFIpXnnlFSiVPUt4rF69GrGxsdi7dy9kMhksLS2xadMmXX4bhOjExbImhHjYwlTAywhRgfrN9tRS2JqLaVFCPfdibDAOZVXj1d2Z+OIheZ/bbN269aavwXEcPvnkE23EI0QQ1BoelyqasFQu7GWEhFs6daipQ4n9l6twZ5SnYKdbksFxszPHE7P9kVnZhBO5NKuOkL7kK1rR3q0W7A26V1GBQk9j2C6VRvCfJsjgrJrmCyuJGC/vyqQlOQjpw/nSngkSQm1xdBUVKADfp5YiyM0GYZ7CvJuaDI2ZqQmejwlCXk0rvj1TwjoOIYJzoawJVhITjHWyZh3lpoy+QF2pbsH5sibcK/eie58MyNxgF0yTOeKDQ1fQ2N7NOg4hgnKhvAlhnnYwEQn7Pc/oC9ThrGpIx1jgrii6z8WQcByHlxaFoLlDOaQ+fYQYum6VBlkVzYj0Em4HiauMukDxPI+vT5fA38UajtZmrOOQURbsbotlk7zxVXIx8mpaWcchRBCuVLegW60R/AQJwMgLVEZpI8obO7Aogs6eDNWz8wJgYWqCN/Zkso5CiCCc/62DRKSAe/BdZdQFau/FSpiacHTvkwFztDbDk3P8kZSjwNGcGtZxCGGuQNEKJ2sJvByE3zHHaAsUz/PYe7EKM/ydYWdhyjoO0aKHpvrC19ESr+/JgoqmnRMjdzy3FhFSO72YFGa0Berq8F5sOPVYM3QSsQgvxgajvrUbP6WXsY5DCDNN7UpcqW7FeK8xrKMMitEWKBreMy7zQlwRLrXDm/ty0NypZB2HECbOlTYAACb4UIESLBreMz4cx2HdgkDUt3Xj06P5rOMQwkRaUQNMRJxeTDEHjLRA0fCecQrztMPd4z3xnxOFKG/sYB2HEJ1LK25AsLsNrMz0o0+4URYoGt4zXs8uCAQH4N39OayjEKJTKrUGGaWNkPs4sI4yaEZXoGh4z7h52lvgkelj8VN6OS6WNbGOQ4jOZFW2oEOpRrSeXH8CjLBA0fAeeey2cXC0kuCNvZngeZ51HEJ0Iq24HgAgpwIlXDS8R2zMTfHnuf44XVCPw1l08y4xDqnFDXC3M4eHvfBv0L3KqArU1eG96TInGt4zcssmecPP2Qr/3JdFa0YRo3CuuEFvppdfZVQFinrvkatMTUR4YWEwChRt+Cm9nHUcQrSqorEDFU2dVKCEjIb3yLXmBrvgjkh3vJ2YjbYuFes4hGhNWnHPDbr6NIMPMKICRcN75FqJiYkICgrC3g/Woba1G58fL7ju+aNHj8LOzg5RUVGIiorCq6++yigpISOXVtwAC1MTBLnbsI4yJPpxt9YouDq89/S8ANZRCGNqtRpr1qzBwYMHIZVKERL/Af59lMPyyT5wtvnfumAzZszA7t27GSYlZHSkFTcgyssepib6dU6iX2lH4HRBHfycrGh4jyAlJQUymQx+fn6QSCRYIhOjU6XBx0do5V1ieNq7VcisbNa760+AERWoHWll8LC3oOE9gvLycnh5efU+DvNxgbeyFN+eKUFhbVvv15OTkxEZGYmFCxfi8uXLLKISMmIZpY1Qa3hM8KUCJUglde3IV7RhVpAL6yhEAPq6OTdAmQeJWIR3D/S0QIqOjkZxcTHOnz+PJ554AnfddVefr5WQkAC5XA65XA6FQqHV3IQMx7nfJkhE68kSG9cyigJ1JLsaADCbChQBIJVKUVpa2vu4rKwMfu5O+OMMP+y5UInzpY2wtbWFtbU1ACA2NhZKpRK1tbU3vFZ8fDxSU1ORmpoKZ2dnnX0PhAxWanED/F2sYWepf6NHxlGgchQY62SFsU5WrKMQAZg4cSJyc3NRWFiI7u5ubNu2DXFxcYif6QdHKwne3JeFysrK3jOtlJQUaDQaODo6Mk5OyNBoNDzOFTdArofDe4ARzOJr71bhdEEdHpzswzoKEQixWIyNGzdiwYIFUKvVWLVqFUJDQ/HZZ58h2swaBwu68c7Xmdj/3/chFothYWGBbdu26cUS2YRcK0/RiuZOFaK9qUAJ0qm8OnSrNDS8R64TGxuL2NjY6762evVqdKs0mPfBr8g09ceFi5dgIqKiRPRX7w26vvp1g+5VBj/EdySnBlYSE0waq58/IKJbErEIz80PRHZVC36mFkhEz6UVN8DRSgJfR0vWUYbFoAsUz/NIyq7BdH8nSMQG/a2SUbQo3B1Txzli36VKdCrVrOMQMmxpxQ2I9hmjt8PTBv2unV3VgsqmThreI0MiEnFYM0uGQ1k1+OZMCes4hAxLbUsX6tu69fIG3asMukAdye5Z62dWIBUoMjTTZE6YLnPCJ0l5aOlUso5DyJClFNWjqUOJyXp8ecOgC1RSdg3CPG3hYmvOOgrRQ+sWBKK+rRtfHC9kHYWQIUvOr4OlxARhnnasowybwRaoxvZunCtpwGw6eyLDFOllj9hwN3xxvAC1rV2s4xAyJMkFdZjo66B3DWKvpb/JB/DrFQU0PKi9ERmRZ+cHolOlwcYjeayjEDJoNS2dyKtpxS3j9PvmcoMtUEnZNXC0kiBSas86CtFj45ytsVQuxTdnilFa3846DiGDcrqgHgBwix8VKMFRa3j8ekWBWwOcIaIbLckIPTnHHyKOwweHrrCOQsigJOfXwcZMjFAPW9ZRRsQgC1RGaQMa2pU0vEdGhbudBVZO9cVP6eXIqWphHYeQAZ0uqMOksQ4Q6/H1J8BAC9SR7BqYiDjMDKDu0mR0PHbbOFibifHO/hzWUQi5qaqmThTWtun99SfAYAuUAhN8xtDihGTU2FtKsPrWcTiUVY204nrWcQjpV3JBz7IwU/T8+hNggAWqsqkDWZXN1D2CjLqHp/nC2cYMHxzM7XPRQ0KEIDm/DnYWpghx1+/rT4ABFqijOT2rmlKBIqPNUiLGX2ICcamiCUev0Oq5RJiSC+oweayDQUwQM7gCdSS7Bp72FvB3sWYdhRiguEhP2JiL8U5iDjQaOosiwlLW0I7S+g6DuP4EGFiB6lKp0dKpwoJQN73t3kuETSIW4Zl5AcisbMaei5Ws4xByneT8OgCgAiVE6SWNOF1Qhyl++tsckQhfXKQnAl1t8P7BK1CqNazjENIruaAOYyxNEeBiwzrKqDCoAnUqrxYiDphsALNXiHCZiDg8tyAQhbVt2JFWxjoOIQB61r87nV+HKX6OBnH9CTC0ApVfh3BPO5peTrRubrALor3t8dGhXFrUkAhCSX07Kpo6DWZ4DzCgAtXWpUJGaSOmypxYRyFGgOM4rFsQhKrmTmxJLmIdh5D/XX8yoBEkgylQKYX1UGl4TBtHBYroxi3jHDHDv2dRw+YOWtSQsJVcUAcnazPIDGgGs8EUqJN5tZCIRZD76u/yxkR3EhMTERgYCJlMhrfeeuuG53mex5NPPgmZTIaIiAicO3euz9d5fkEQ3O0s8MXxAm1HJqRfPM8jOb9ngpghzWA2nAKVX4cJ3mNgbmrCOgoROLVajTVr1mDfvn3IzMzE1q1bkZmZed02+/btQ25uLnJzc5GQkIDHHnusz9cKl9rBz9kKX5wohIruiyKMFNS2oaaly6CuPwEGUqDq27qRVdmMaTLD+uEQ7UhJSYFMJoOfnx8kEgmWLVuGnTt3XrfNzp07sWLFCnAchylTpqCxsRGVlX3f9/TMvEB0KtWoaenURXxCbmCI158AgLtZT7GYmBi+trZ2VHakUCjg7Kyd7uJNHUqU1LdjnLM1LCWjewalzdzapK+5Ae1nb2hoQHNzM3x8fAAAdXV1aGtrg7e3d+82eXl5cHNzg7V1z3j+lStXIJVKYWlpeUPW2tpaqM3t0d1ch7CwEEj0bIkD+l3RvdHOXVLfjrYuFYK13H9vNHOnpaXt53k+5qYb8Tx/sz+jZsKECaP5ctd54ccLfOjfE3mlSj3qr63N3Nqkr7l5XvvZt2/fzj/yyCO9j7ds2cKvXbv2um1iY2P548eP9z6ePXs2n5qa2u9rljW08xLXcfxz2zNGP7CW0e+K7o1mbo1Gw0947QD/1NZzo/aa/Rnl4z1Q/TGMIb5TebWYbACLcxHdkEqlKC0t7X1cVlYGDw+PIW9zLU97C3DKNvxwrgx5NbSoIdGd3JpW1LZ2G9z1J8AArkGVN3agqK6d7n8igzZx4kTk5uaisLAQ3d3d2LZtG+Li4q7bJi4uDlu2bOm5O//0adjZ2cHd3f2mryvqaoWFqQneO0BLwxPd+d/1J8N7DxTrakfx8fFaed1TeT3XyLQ1QUJbubVNX3MD2s8uFouxceNGLFiwAGq1GqtWrUJoaCg+++wzAMDq1asRGxuLvXv3QiaTwdLSEps2bRrwdZ0dHfDIDD/83+FcXChrRITUXqvfx2ih3xXdG83cBbWtmDzWAV4OFqP2mv3R9fG+6SQJAIKfN/v0dxk4nqvA2b/ONeLoLsQAAB/0SURBVKj5/0T/yOVyJJ1Ixsy3kxDmaYevHpnMOhIxcN0qDaJfO4g7Ij3w5t3hrOMM1YBv2Ho9xMfzPE7m1eKWcU5UnIgg2Jib4vHbZDhTUIczBXWs4xADl1pcj9YuFWYF6t9MxsHQ6wKVr2hFTUsXphrgxUGivx6c4g1/Vxu8uS+bloYnWpWUXQNTEw7TDPQavNYK1Lp16xAUFISIiAgsXrwYjY2NfW43UMuZmzmZ1/MJdTT7733//fcIDQ2FSCRCampqv9v5+voiPDwcUVFRkMvlo7b/4Rps7pEcb22pr6/HvHnz4O/vj3nz5qGhoaHP7YRyzAc6huamJhAXnkJGaSNC5t/fb5skXRso99GjR2FnZ4eoqChERUXh1VdfZZDyRqtWrYKLiwvCwsL6fJ4fZFsqXRso92gc76QcBSaPdYSV2ehNJygtLcWsWbMQHByM0NBQfPTRRzdso7NjPsA89GHbv38/r1QqeZ7n+eeff55//vnnb9hGpVLxfn5+fH5+Pt/V1cVHRETwly9fHvQ+4rec5ae9dXgkMW+QmZnJZ2dn87feeit/9uzZfrfz8fHhFQrFqO57JAaTe6THW1vWrVvHv/nmmzzP8/ybb77Z5+8KzwvjmN/sGF69R2TPnj38gpiF/G3vJPHTXt/HT5w0mWVknucH97NPSkriFy1axChh/3799Vc+LS2NDw0N7fP5PXv28DExMbxGo+GTk5P5SZMm6Thh3wbKPdLjXVLXxvv8ZTf/xfGCYb9GXyoqKvi0tDSe53m+ubmZ9/f3v+F3ZZSOObv7oObPnw+xuKeqT5kyBWVlNy7sNpiWM/1Ra3qaI4529/Lg4GAEBgaO6mvqwmByj+R4a9POnTvx0EMPAQAeeugh/Pzzz4wT9W+wbZIeWvEHPDc/EGUtatTbjuu3TZKuCPVnPxgzZ86Eg0P/q2QPpS2VLg2Ue6SO5tQAwKhff3J3d0d0dDQAwMbGBsHBwSgvL79uG10dc51cg/ryyy+xcOHCG75eXl4OLy+v3sdSqfSGA9GfyxVNaO5UYSqj/nscx2H+/PmYMGECEhISmGQYqpEcb22qrq7uvcfI3d0dNTU1fW4nhGM+mGN4dZuFYW4I97QDwm5HUQnblXcH+7NPTk5GZGQkFi5ciMuXL+sy4rAJ9fd6MEZyvI9k18DH0RJjnay0lA4oKipCeno6Jk++fkaqro75iAYu586di6qqqhu+/sYbb+DOO+/s/btYLMby5ctv2I7v4wLyYGfjpRU3YLyX/bDunh5M7oGcPHkSHh4eqKmpwbx58xAUFISZM2cOOctQjDT3SI73SN0s+2CxOOa/N5hjeHUbkYjDugWBWPFlEw4WduAWhrPOB5M7OjoaxcXFsLa2xt69e3HXXXchNzdXVxGHjeXv9UiM5Hh3KtU4lV+H+yd5a+17bW1txZIlS/Dhhx/C1vb6Hn+6OuYjKlCHDh266fObN2/G7t27cfjw4T7DD7WdzLVO5tWisUMJFxvzoYXGwLkH42pOFxcXLF68GCkpKVp/sxxp7pEc75G6WXZXV1dUVlbC3d0dlZWVcHFx6XM7Fsf894baJmmGvxM4RS5+ygnGM10qWI/ixeyhGEzua9+EYmNj8fjjj6O2thZOTsKeIcby93okRnK8kwvq0KXSYFZQ3/9XRkqpVGLJkiVYvnw57r777hue19Ux19oQX2JiIjZs2IBffvnlhg7QVw2m5UxfNBoeqcUNmMhoccK2tja0tLT0/v3AgQP9ztQRkuEeb22Li4vD5s2bAfR8qOnrbFAox3yobZLOnDkD++Jf0dChwpcnCnWe96rB5K6qqur9ZJySkgKNRgNHR+HfwjGctlRCMJLjnZRdA3NTESaPHf1rXDzP45FHHkFwcDCeeeaZPrfR2TEfYBbFsI0bN46XSqV8ZGQkHxkZyf/pT3/ieZ7ny8vL+YULF/Zut2fPHt7f35/38/PjX3/99UG9dk5VM+/zl9389rMlI4nYpx9//JH39PTkJRIJ7+Liws+fP/+G3Pn5+XxERAQfERHBh4SEDDq3Ng0mN88P73hrW21tLT979mxeJpPxs2fP5uvq6nieF+4x7+sYfvrpp7y3tzfP8z3dpR9//HHez8+PDwsL48+ePcs/uvksH/r3RL6utUtwuT/99FOe53n+448/5kNCQviIiAh+8uTJ/MmTJ5llvdayZct4Nzc3XiwW856envwXX3xxXe6+jrcQDJR7uMdbo9Hw0946zK/alKKV3MePH+cB8OHh4b3v33v27NHGMR9wFp9etjr6+nQxXvr5Eo4+dxt8tXiBkJChkMvl/d6DdqW6BTEfHsMj08fir4tCdJyMGJK8mlbMff9XvH5XGB6c4sM6zkgYZqujs0X1cLYxg49j30OHhAhNgKsNHpzsg4vlTaho7GAdh+ixq9PLbzPQ9kbX0ssClVrUgEm+DnoxU4eQq+Jv9cO54kZ8eIiW4yDDdyS7BgGu1pCOMfwP6HpXoMobO1De2AE5owkShAyXdIwl/nCLD3aklSG3mhY1JEPX0qnE2aJ6rc3eExq9K1BnC+sBABN9tXeHNiHasmaWDFYSMd7en8M6CtFDJ/PqoFTzmBVIBUqQzhbVw9pMjGB324E3JkRgHKwkiJ/ph4OZ1Ugrrmcdh+iZpOwa2JiJMcHHOEaQ9LJARfuMgYmIrj8R/fTIjLFwsjbDhn05tBwHGTSe55GUU4MZAU4wNdG7t+5h0avvsqGtG1eqWzGJrj8RPWYpEeOpuf5IKapHUk7ffQcJ+b3MymbUtHQZzfAeoGcFKq24Z50guv5E9N2yiV7wdbTE24k5UGvoLIoM7GiOAgBwqx5PLz979iwiIiLQ2dkJjuOsOI67zHFcvy1h9KpAnS2qh6kJh0gve9ZRCBkRUxMRnp0fiOyqFuzM0I/O24StI9k1CPe0G1b/UaGYOHEi4uLi8NJLLwHA2wC+5nn+Un/b61WBSimqR4TUHuamJqyjED0lpNV7F4W7Y7y3PfZdqkKnUq2VfRDD0NDWjfSSBoOYXv73v/8dBw8eBAA5eopUv/SmQHV0q3GpvImG98iIvPXWW5gzZw5yc3MxZ86cmy57n5SUhIyMjH7bF42USMTh2XmBOJhZja9PF2tlH8QwHMtVQMOP/uKELNTX16O1tRUAbADc9HRQbwpURmkjlGqeWQdzYhiEtnrvdH8nzAxwxsdH8tDUrmSahQhXUnYNHKwkiJDq/+WN+Ph4vPbaawDwDYANN9tWbwrU2aJ6cBwg96EzKDJ8o716b0JCAuRyOeRyORQKxbAyrY8JQnOnEv86mjesf08Mm1rD49crCtwa4Kz3t9ds2bIFYrEYDzzwAAC8BWAix3Gz+9uezeppw3C2qB6BrjawszRlHYUInC5X742Pj0d8fDwADPtaVYiHLe4eL8WmU0VYMdUXnvYWw3odYpjSihsgc7HGglA31lFGbMWKFVixYgUAgOd5NYCbrjOtFwVKpdbgXHEDFkd7so5C9IA+rt777PwA7LpQgfcO5OD9pVFa2w/RP3suVOBCWROm+wt7ZWNt0IshvqzKFrR1q2mCBBkxoa7e62FvgYen+eKn9HJcrmjS6r6I/lBreOy9VIXZQS6wNtOL84lRpRcF6mxRT8+ySVpY3pgYl/Xr1+PgwYPw9/fHwYMHsX79egBARUUFYmNjAfRcp5o+fToiIyMxadIkLFq0CDExMVrP9vhtMthZmOKtfdla3xfRD2cK66Bo6cLtER6sozChFyW5sLYNMhdruNvR2DwZGUdHRxw+fPiGr3t4eGDv3r0AAD8/P5w/f17X0WBnYYq1s2R4fU8WjucqMMNf/6cUk5HZfaESlhITzDaA+5+GQy/OoA5mViPYzYZ1DEK07g+3+EA6xgJv7s2GhlogGTWlWoPES1WYE+wKC4lxNicQfIGqbOpAVXMnoo2kvTwxbmZiE6xbEIjMymbsPE8tkIzZqfw61Ld14/YId9ZRmBF8gcooaQQAjPemAkWMwx0RHpguc8Su85XUAsmI7T5fARszMW4NMN6hXuEXqNJGSExECHanIT5iHEQiDo/PkuFIdg2+PFnIOg5hoFulwf7LVZgX6mrUvUcFX6DSSxoR6mkLM7Hx/pCI8Zk6zglzg13xr6R8KFq6WMchOnY8V4HmThXuMNLZe1cJukAp1RpcKG9EFC2vQYzQC7FB6FSq8cGhK6yjEB3bfaESdhammCYzvptzryXoApVT1YJOpYauPxGjNM7ZGg9O8cG2lBLkVLWwjkN0pFOpxsHMasSEukEiFvRbtNYJ+rtPL/1tggSdQREj9dQcf1ibifHG3izWUYiOHM2pQWuXCndEGvfwHiDwApVR0ggnawmkY+gGXWKcxlhJ8OQcfxy7osDRnL47rxPDsutCJRytJJjiR51zBF2g0ksbEOU1Bhyn3y3mCRmJP9ziAx9HS/xzbxZUag3rOESL2rtVOJJVg4XhbhCbCPrtWScEewSa2pUoULRhvDcN7xHjZiY2wQsLg3CluhXfpZayjkO06HBWDTqUaqPtvfd7gi1QGWV0/YmQqxaEumGSrwM2Hs5Fc0c36zhES3ZfqICLjRmt3PAbwRao9JIGcBwQLrVjHYUQ5jiOw99uDwE4Dp8czWcdh2hBS6cSSTkKxIa76/3KuaNFsAUqo7QRAS42sDGnFXQJAXo+rE2TOeHLE4UorG1jHYeMsoOZ1ehWaWj23jUEWaB4nkdGaSNdfyLkd56PCYSZ2ASv785kHYWMst0XKuFpb4Foet/rJcgCVVTXjsZ2JXWQIOR3XGzM8cRsGQ5n19C0cwPS1K7E8VwFFkW406zlawiyQKWXNACgDuaE9OXhaWMx1skKr+7OhJKmnRuE/ZeroFTzRr20Rl8EWaAyShthJTGBzMWadRRCBEciFuGlRcEoULRh86ki1nHIKNh1oQLeDpYI96RJYdcSZIFKL2lEpJc9zWQhpB+zg1xwa4AzPjqUi9pW6nauzyoaO5BT1YKHbvGl4b3fEVyB6lSqkVXZTNefCLmJq9POO5RqvLs/h3UcMgLbzpZC0dqF+aGurKMIjuAK1KXyJqg0PF1/Ilrx/fffIzQ0FCKRCKmpqf1ul5iYiMDAQMhkMrz11ls6TDh4MhdrrJzqi+9SS3HhtxvbiX5RqTX47mwJbg1whpeDJes4giO4ApX+2xLvdAZFtCEsLAw//vgjZs6c2e82arUaa9aswb59+5CZmYmtW7ciM1OY07qfnOuPWwOc8cquTGg0POs4ZIgOZ9egurkLD0zyZh1FkARXoDJKGyEdYwFnGzPWUYgBCg4ORmBg4E23SUlJgUwmg5+fHyQSCZYtW4adO3fqKOHQ2Jqb4vYID6QVN+D7NOrTp2++PVMCN1tzzA5yYR1FkARXoCqbOjA3mH5YhJ3y8nJ4eXn1PpZKpSgvL+9z24SEBMjlcsjlcigUCl1FvM7d4z0h9xmDt/Zlo7Gd+vTpi9L6dhzLVeC+iV7UubwfgjoqDW3dOFfSCHc7Wv+JDN/cuXMRFhZ2w5/BngXx/I1DZf3NroqPj0dqaipSU1Ph7Ow8otzDJRJxeO2uMDR3qvA2TZjQG1tTSsABWDbJa8BtjZWYdYBrXSxvAgCE0b0AZAQOHTo0on8vlUpRWvq/4bKysjJ4eAi7P1qwuy0eusUXm04V4j65FyLpGq6gdas02J5ahtlBrvSB/CYEdQbVW6A8qEARdiZOnIjc3FwUFhaiu7sb27ZtQ1xcHOtYA3p6nj+crc3w0s+XoKYJE4J2MLMata1dWD6ZJkfcjKAK1KXyJng7WMLOkjqYE+346aefIJVKkZycjEWLFmHBggUAgIqKCsTGxgIAxGIxNm7ciAULFiA4OBhLly5FaGgoy9iDYmNuir8uCsbF8iZsTSlhHYfcxLcpxfC0t8DMADbDwvqC62u8/Ro6/Rg2fcMRRErt8cnyaF3ulpBRIZfLb3pvlS7wPI8HPj+D8sZ27Fg9FS625kzzkBsV1rZh1rtH8dz8AKyd7c86DksDts0QzBlUQ1s3yho66PoTISPAcRxeXxyGti41Xt+TxToO6cPWlBKIRRyWymlyxEAEU6AuVfRcf6JmiYSMzDhnazw4xQe/nK9AEi3JIShdKjV2pJVhbrArnd0OgnAKVHkzACDM05ZxEkL03+OzxkHmYo2XfrqEti4V6zjkN4mXqlDf1o3lU2hyxGAIqEA1wcvBAvaWEtZRCNF7ZmITvHl3OMobO/DegSus45DffHOmBN4Olpg2zol1FL0gmAJ1sbyJhvcIGUUTfR3w4BRv/PdUIc6XUjNZ1vJqWpBSWI/7J3lDREsJDYogClRTuxIl9e00QYKQUfZ8TBCcbczwlx8u0Oq7jP2YXo4oL3vcK5eyjqI3BFGgaIIEIdpha26KV+8MQ3ZVCxKOFbCOY7QULV34z/FCBLrawMmaGmEPliAKFHWQIER7FoS6ISbUDR8dzkVhbRvrOEbpPycKoVRrsPq2cayj6BXBFChPewuMsaIJEoRowyt3hsJMLMILP1yARkNDfbrU1K7E16eLsSjCA2OdrFjH0SuCKFCXaIIEIVrlamuOv90eguZOFb5JoXWjdGlzchFau1R4nM6ehox5gWrqUKK4rh3hUipQhGjTvROkcLIxwz/3ZNFQn460danw5clCzA12QbA73eM5VMwL1OUKWmKDEF3gOA5vL4mARCzCs9szoKJZfVr37ZkSNLYr8fgsGesoeol5gbpUTjP4CNEVNztzvHZXGM6VNOLfNKtPqzqVanx+vABTxzki2nsM6zh6iXmBuljeDE97CzjQBAlCdCIu0gO3R7jjg4NXej8gktG3I60MNS1dWENnT8PGvEBdKm+i/nuE6Nhrd4bBwUqCZ7efR6dSzTqOwVGqNfjs13xEedlj6jhH1nH0FtMC1dypRGFtGw3vEaJjY6wk2HBPBHKqW/D+QerVN9p2na9AWUMH1sySgeOordFwMS1Ql3s7mFOBIkTXZgW6YPlkb3x+vABnCupYxzEYGg2Pfx3NR5CbDeYEubCOo9eYFqir499UoAhh48XYYHg7WOLZ78+jlZblGBUHMquQV9OKx24bR01hR4htgapogrudOfWmIoQRKzMx3rs3EhWNHXh3fw7rOHpPpdbgm9MlmBvsgtsjPFjH0XtMC9TF8iY6eyKEMbmvA16IDcY3Z4rxU3oZ6zh6bevZUhzPq8WSaClM6OxpxJgVqNYuFU2QIEQgHp7qi/FeY/Dij5eQW93COo5eampX4v0DOZji54CYMDfWcQwCswJ1ubwJPE836BLd+v777xEaGgqRSITU1NR+t/P19UV4eDiioqIgl8t1mJANsYkIHz8wHpYSEzz2zTm0d9P1qKH68PAVNHUo8ffbQ2nm3ihhVqAu0gQJwkBYWBh+/PFHzJw5c8Btk5KSkJGRcdNCZkhcbc3x0bLxyFe04q8/XQLP86wj6Y28mhZ8lVyMZZO8EeJB93WOFmYFqrq5ExFSOzjb0AQJojvBwcEIDAxkHUOwpvs74c9zAvBTejm2naWu54P12u4sWEhM8Oy8ANZRDAqzAnXsSi3N3iOCxXEc5s+fjwkTJiAhIaHf7RISEiCXyyGXy6FQKHSYUHvWzpZhhr8T/vHLZWqFNAhJ2TX49YoCT83xhyO9p40qJgWqS6VGvqIVwe42LHZPDNzcuXMRFhZ2w5+dO3cO+jVOnjyJc+fOYd++ffjkk09w7NixPreLj49HamoqUlNT4ezsPFrfAlMmIg4f3hcFB0sJ1nx7Ds2dStaRBKtbpcFrezLh52SFFbf4so5jcMQsdppb3QqVhqf1UYhWHDp0aMSv4eHRcw+Li4sLFi9ejJSUlEFdtzIUjtZm+PiB8ViWcBov77yMd++NpJtO+7AluQgFijZ8uVIOiZh5a1ODw+SIZlX2tDgKcqMCRYSnra0NLS0tvX8/cOAAwsLCGKfSvYm+DnjtzlAczKrGB4eoX9/v1bV24aPDubg1wBmzAqmlkTYwKVDZVS0wNxVhrJMVi90TI/bTTz9BKpUiOTkZixYtwoIFCwAAFRUViI2NBQBUV1dj+vTpiIyMxKRJk7Bo0SLExMSwjM3M/ZO8sSjcHR8fycP3qTRp4lqf/ZoPc1MT/O32YJpWriXcAFNJtTLP9IHPT6OtS4Wda6dr4+UJYUIulxvklHSlWoOHN53F6YI6bFk1CVNlTqwjMXfgchXiv0rDU3P88TTN3BuuAau6zs+geJ5HVmUzXX8iRE+YmojwrwejMdbJCqu/TkNejXF3mlC0dOGFHy8i1MOWFiPUMp0XqOrmLjS0KxHkRjP4CNEXtuam+HLlREjEJnj4v2dR29rFOhITPM/jLz9cQGuXCh/eF0UTI7RM50c3q6pnggSdQRGiX7wcLPHFQ3IoWrrw6JZUo1yJ99uUEhzJrsH6hUHwd6UP2dqm+wJ1dQYfFShC9E6Ulz0+vG88Mkob8ez289BojKcdUmFtG17fnYUZ/k54iO550gkGBaoFnvYWsLMw1fWuCSGjICbMDS8uDEZKYT3e3p9tFD37VGoNnv4uAxKxCO/cQ/eE6YrOb9TNrmymDhKE6Lk/zhiLlk4l/u9IHpRqHi8tMuyp1p8fL0BGaSM2PjAebnbmrOMYDZ0WqE6lGgW1bbRWCiF6juM4PD0vAM2dKvznRCFEXM/y8YZYpBIvVWJDYg7W3DaOVsnVMZ0WqNzqVqipxREhBoHjOPzjjhDwPI/PjxeC4zi8sDDIoIrU5YomPP3deUR52eOJOf6s4xgdnRaoqxMkqEARYhg4jsPLcaHQ8EDCsQJwHLA+xjCKlKKlC49uToWdhSkS/jAB5qYmrCMZHd0WqKpmWJiawMfBUpe7JYRoEcdxePXOUPDg8e9fC2BpaoInZvvr9USC9m4VXtudifr2bnz/p6lwsaXrTizo/Awq0M1Gr39xCSE34jgOr8aFwcXGHO8fvIKiunZsWBKhlzeydirViN+ShlP5tfjPQxMRLqVVv1nRWYHqaXHUgthwd13tkhCiQyIRhydmy2Ai4vDO/hzUtnbhkwfGw9ZCwjraoCnVGqz99hxO5NXi3XsjMSuIupSzpLOPN5VNnWjqUCKEppgTYrA4jsOaWTK8c08E2rpUuOezZBTWtrGONShKtQYv/ngRh7Jq8NqdobhngpR1JKOnswKVTS2OCDEa98q98Nz8QNS2diNu4wkcya5mHemmWrtUeGRzKk7kKfBKXCj+QJ0iBEFnBSqrsqcDciA1iSXEKEyVOeGXtdPg7WCJRzan4rOj+VCpNaxj3aCmpRPLEpJxMq8Wf54bgIem+rKORH6jswKVWdkMLwcL2JhTiyNCjIV0jCV+eGwqVtzig/87kot7PktGgaKVdaxeWZXNeH7HBeTXtOHzFRNw30Rv1pHINXQ3xFfZjGBa4p0Qo2NuaoJX4sLw1pIIFNa2Ifb/jmPzqUJoNOzOpniex8/p5Vj8r5MorG3Dd3+agtlBrszykL7ppEB1KtUorG2j60+EGLG4SA8ceHomJo91xA9p5Vj679O4XNGk8xy1rV1Y9/0F/Pm7DMRFemLH6qmIkNrrPAcZmE4KVE5VCzQ8qEksIUbO1dYc/314Ih6e7ouC2jas/+Ei1n1/HuUN7Vrft1rD4+vTxbj7X6eQr2jF+oVBePPucDjbmGl932R4dHIfFLU4IoRcxXEcFo+XYnagCzYm5WHX+UokXq7C7RHu+OP0sRjnMrofZDUaHoeyqvHd2VIczq7BUrkU8TP9IBvl/ZDRp5MClV3VAiuJCbzGUIsjwta6deuwa9cuSCQSjBs3Dps2bYK9/Y3DO4mJiXjqqaegVqvxxz/+EevXr2eQ1rDZWUrw10UhWDnNF58ezUdudSvmvH8MS+VemOnvhNnBLrCUDP8tqralCzvPV+BUfi0KFW3gOOCLFRMwJ9jVIHoFGgNugMXGRmUlsqX/ToZaw+OHx6aOxssRMmwHDhzA7NmzIRaL8Ze//AUAsGHDhuu2UavVCAgIwMGDByGVSjFx4kRs3boVISEhN31tuVyO1NRUrWU3dDXNndieWoodaWVwsTXHhbJGzA9xQ7SPPcZ7jUGgqw3MJf03bO3oViO7qhlnC+uRWdmMw1nV6FbzmBvsgpgwdywMdYNYD1svGbABPyVo/QyK53nkVLXgjkhqcUTYmz9/fu/fp0yZgh07dtywTUpKCmQyGfz8/AAAy5Ytw86dOwcsUGRkXGzNsXa2Px6/TYbU4noczq5B4sUqlNS34eVfMuHvag1bM1N4jrGAjbkYTR1K2FtK0NzRjQ6lBjlVLSipb4e3gwUipPZYMdUXd0Z6IoDuvdRbWi9QHMfh5PrZ6FSqtb0rQobkyy+/xH333XfD18vLy+Hl5dX7WCqV4syZM32+RkJCAhISEgAACoVCO0GNjEjEYdJYR0wa64gXFgajorEDGaWNKKlrQ3VLF9q6VGjrUqGisQMcAAuJCVxtLRDqYYtAVxtEe9vD1c6C9bdBRoFOrkFZm4lhbabz1eWJkZo7dy6qqqpu+Pobb7yBO++8s/fvYrEYy5cvv2G7voa9+7tmER8fj/j4eAA9Q3xk9HnYW8DDngqOMaKqQQzOoUOHbvr85s2bsXv3bhw+fLjPwiOVSlFaWtr7uKysDB4etNQ3IbpGVwyJUUlMTMSGDRvwyy+/wNKy71mlEydORG5uLgoLC9Hd3Y1t27YhLi5Ox0kJIVSgiFFZu3YtWlpaMG/ePERFRWH16tUAgIqKCsTGxgIAxGIxNm7ciAULFiA4OBhLly5FaGgoy9iEGCWdTDMnxBjQNHNChmTAaeZ0BkUIIUSQqEARQggRJCpQhBBCBIkKFCGEEEGiAkUIIUSQqEARQggRpIGmmRNCBonjuESe52NY5yDEUFCBIoQQIkg0xEcIIUSQqEARQggRJCpQhBBCBIkKFCGEEEGiAkUIIUSQ/h8tkVuLELwJkAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fbc49eaa4f0>"
      ]
     },
     "execution_count": 182,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "from sympy import *\n",
    "from sympy.plotting import plot\n",
    "\n",
    "yC = f.subs({p:p0})\n",
    "plot(yC,(x,-2,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xx = solve(df.subs({p:p0}),x)\n",
    "xx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yC.subs({x:xx[0]})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -2$"
      ],
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yC.subs({x:xx[1]})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "曲線$C$の接線で，点Aを通り傾きが0でないものを$l$とする．$l$の方程式を求めよう．$l$と$C$の接点の$x$座標を$b$とすると，$l$は点$(b, f(b))$における$C$の接線であるから，$l$の方程式は$b$を用いて\n",
    "\\begin{equation*}\n",
    "y= \\left(\\fbox{ ケ }\\,\\,  b^2 - \\fbox{ コ }\\right)(x-b)+f(b)\n",
    "\\end{equation*}\n",
    "と表すことができる．また，$l$は点Aを通るから，方程式\n",
    "\\begin{equation*}\n",
    "\\fbox{ サ }\\,\\, b^3-\\fbox{ シ }\\,\\, b^2+1=0\n",
    "\\end{equation*}\n",
    "を得る．この方程式を解くと，\n",
    "\\begin{equation*}\n",
    "b = \\fbox{ ス }\\,\\,, \\frac{\\fbox{ セソ }}{\\fbox{ タ }}\n",
    "\\end{equation*}\n",
    "であるが，$l$の傾きが0でないことから，$l$の方程式は\n",
    "\\begin{equation*}\n",
    "y = \\frac{\\fbox{ チツ }}{\\fbox{ テ }}\\,\\, x+\\frac{\\fbox{ ト }}{\\fbox{ ナ }}\n",
    "\\end{equation*}\n",
    "である．\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle b^{3} - 3 b + \\left(- b + x\\right) \\left(3 b^{2} - 3\\right)$"
      ],
      "text/plain": [
       "b**3 - 3*b + (-b + x)*(3*b**2 - 3)"
      ]
     },
     "execution_count": 186,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yl=df.subs({p:p0}).subs({x:b})*(x-b)+yC.subs({x:b})\n",
    "yl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 b^{3} - 3 b^{2} + 1$"
      ],
      "text/plain": [
       "2*b**3 - 3*b**2 + 1"
      ]
     },
     "execution_count": 187,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq = -expand(yl.subs({x:xx[1]})-yC.subs({x:xx[1]}))\n",
    "eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1/2, 1]"
      ]
     },
     "execution_count": 188,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bb = solve(eq,b)\n",
    "bb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{4} - \\frac{9 x}{4}$"
      ],
      "text/plain": [
       "1/4 - 9*x/4"
      ]
     },
     "execution_count": 189,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b0=bb[0]\n",
    "yl.subs({b:b0})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "点Aを頂点とし，原点を通る放物線を$D$とする．$l$と$D$で囲まれた図形のうち，不等式$x \\geqq 0$の表す領域に含まれる部分の面積$S$を求めよう．$D$の方程式は，\n",
    "\\begin{equation*}\n",
    "y = \\fbox{ ニ }\\,\\, x^2 -\\fbox{ ヌ }\\,\\, x\n",
    "\\end{equation*}\n",
    "であるから，定積分を計算することにより，$\\displaystyle S=\\frac{\\fbox{ ネノ }}{24}$となる．(10点)\n",
    "\n",
    "(2014年度大学入試センター試験　本試験　数学II・B第2問)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k*(x - 1)**2 - 2\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 211,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yD = k*(x-xx[1])**2+yC.subs({x:xx[1]})\n",
    "print(yD)\n",
    "kk = solve(yD.subs({x:0}),k)\n",
    "k0 = kk[0]\n",
    "k0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 2 x^{2} + \\frac{7 x}{4} + \\frac{1}{4}$"
      ],
      "text/plain": [
       "-2*x**2 + 7*x/4 + 1/4"
      ]
     },
     "execution_count": 215,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(yl.subs({b:b0}) - yD.subs({k:k0}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{11}{24}$"
      ],
      "text/plain": [
       "11/24"
      ]
     },
     "execution_count": 216,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(yl.subs({b:b0}) - yD.subs({k:k0}),(x,0,xx[1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4\n",
    "\n",
    "前問3(b)の$C$上の頂点Aの座標を$\\displaystyle \\left(\\frac{p}{4}, f\\left(\\frac{p}{4}\\right) \\right)$と変えて問題を解け．ただし数値を変えたので，それほど複雑な数字にはならないが，\\fbox{ オ }，\\fbox{ カキ }等には箱にこだわらず数字がはいる．最後は$\\displaystyle S=\\frac{11}{27}$ではなく，$\\displaystyle S=\\frac{352}{243}$になる．(30点)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - p x + x^{3}$"
      ],
      "text/plain": [
       "-p*x + x**3"
      ]
     },
     "execution_count": 234,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "x, p, k=symbols('x p k')\n",
    "\n",
    "f = x**3-p*x\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - p + 3 x^{2}$"
      ],
      "text/plain": [
       "-p + 3*x**2"
      ]
     },
     "execution_count": 235,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = diff(f,x)\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 236,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 a^{2} - p$"
      ],
      "text/plain": [
       "3*a**2 - p"
      ]
     },
     "execution_count": 236,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.subs({x:a})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4(b)\n",
    "関数$f(x)$が$\\displaystyle x=\\frac{p}{4}$で極値をとるとする．また，曲線$y=f(x)$を$C$とし，$C$上の点$\\displaystyle \\left(\\frac{p}{4}, f\\left(\\frac{p}{4}\\right) \\right)$をAとする．\n",
    "\n",
    "$f(x)$が$\\displaystyle x=\\frac{p}{4}$で極値をとることから，$p=\\fbox{ オ }$であり，$f(x)$は$x=\\fbox{ カキ }$で極大値をとり，$x=\\fbox{ ク }$で極小値をとる．\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{3 p^{2}}{16} - p$"
      ],
      "text/plain": [
       "3*p**2/16 - p"
      ]
     },
     "execution_count": 237,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = p/4\n",
    "df.subs({x:x0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 16/3]"
      ]
     },
     "execution_count": 238,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pp = solve(df.subs({x:x0}),p)\n",
    "pp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "metadata": {},
   "outputs": [],
   "source": [
    "p0 = pp[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fbc4b389bb0>"
      ]
     },
     "execution_count": 240,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "from sympy import *\n",
    "from sympy.plotting import plot\n",
    "\n",
    "yC = f.subs({p:p0})\n",
    "plot(yC,(x,-2,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-4/3, 4/3]"
      ]
     },
     "execution_count": 241,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xx = solve(df.subs({p:p0}),x)\n",
    "xx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{128}{27}$"
      ],
      "text/plain": [
       "128/27"
      ]
     },
     "execution_count": 242,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yC.subs({x:xx[0]})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{128}{27}$"
      ],
      "text/plain": [
       "-128/27"
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yC.subs({x:xx[1]})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "曲線$C$の接線で，点Aを通り傾きが0でないものを$l$とする．$l$の方程式を求めよう．$l$と$C$の接点の$x$座標を$b$とすると，$l$は点$(b, f(b))$における$C$の接線であるから，$l$の方程式は$b$を用いて\n",
    "\\begin{equation*}\n",
    "y= \\left(\\fbox{ ケ }\\,\\,  b^2 - \\fbox{ コ }\\right)(x-b)+f(b)\n",
    "\\end{equation*}\n",
    "と表すことができる．また，$l$は点Aを通るから，方程式\n",
    "\\begin{equation*}\n",
    "\\fbox{ サ }\\,\\, b^3-\\fbox{ シ }\\,\\, b^2+1=0\n",
    "\\end{equation*}\n",
    "を得る．この方程式を解くと，\n",
    "\\begin{equation*}\n",
    "b = \\fbox{ ス }\\,\\,, \\frac{\\fbox{ セソ }}{\\fbox{ タ }}\n",
    "\\end{equation*}\n",
    "であるが，$l$の傾きが0でないことから，$l$の方程式は\n",
    "\\begin{equation*}\n",
    "y = \\frac{\\fbox{ チツ }}{\\fbox{ テ }}\\,\\, x+\\frac{\\fbox{ ト }}{\\fbox{ ナ }}\n",
    "\\end{equation*}\n",
    "である．\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 244,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle b^{3} - \\frac{16 b}{3} + \\left(- b + x\\right) \\left(3 b^{2} - \\frac{16}{3}\\right)$"
      ],
      "text/plain": [
       "b**3 - 16*b/3 + (-b + x)*(3*b**2 - 16/3)"
      ]
     },
     "execution_count": 244,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yl=df.subs({p:p0}).subs({x:b})*(x-b)+yC.subs({x:b})\n",
    "yl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 245,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 b^{3} - 4 b^{2} + \\frac{64}{27}$"
      ],
      "text/plain": [
       "2*b**3 - 4*b**2 + 64/27"
      ]
     },
     "execution_count": 245,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq = -expand(yl.subs({x:xx[1]})-yC.subs({x:xx[1]}))\n",
    "eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-2/3, 4/3]"
      ]
     },
     "execution_count": 246,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bb = solve(eq,b)\n",
    "bb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 247,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{16}{27} - 4 x$"
      ],
      "text/plain": [
       "16/27 - 4*x"
      ]
     },
     "execution_count": 247,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b0=bb[0]\n",
    "yl.subs({b:b0})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "点Aを頂点とし，原点を通る放物線を$D$とする．$l$と$D$で囲まれた図形のうち，不等式$x \\geqq 0$の表す領域に含まれる部分の面積$S$を求めよう．$D$の方程式は，\n",
    "\\begin{equation*}\n",
    "y = \\fbox{ ニ }\\,\\, x^2 -\\fbox{ ヌ }\\,\\, x\n",
    "\\end{equation*}\n",
    "であるから，定積分を計算することにより，$\\displaystyle S=\\frac{\\fbox{ ネノ }}{24}$となる．(10点)\n",
    "\n",
    "(2014年度大学入試センター試験　本試験　数学II・B第2問)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k*(x - 4/3)**2 - 128/27\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{8}{3}$"
      ],
      "text/plain": [
       "8/3"
      ]
     },
     "execution_count": 248,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yD = k*(x-xx[1])**2+yC.subs({x:xx[1]})\n",
    "print(yD)\n",
    "kk = solve(yD.subs({x:0}),k)\n",
    "k0 = kk[0]\n",
    "k0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{8 x^{2}}{3} + \\frac{28 x}{9} + \\frac{16}{27}$"
      ],
      "text/plain": [
       "-8*x**2/3 + 28*x/9 + 16/27"
      ]
     },
     "execution_count": 249,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(yl.subs({b:b0}) - yD.subs({k:k0}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 250,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{352}{243}$"
      ],
      "text/plain": [
       "352/243"
      ]
     },
     "execution_count": 250,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(yl.subs({b:b0}) - yD.subs({k:k0}),(x,0,xx[1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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