{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"text-align: center;\">\n",
    "   <font size=\"5\"> 2023年度　数式処理演習　pair試験問題 </font>\n",
    "</div>\n",
    "   <div style=\"text-align: right;\">\n",
    "   <font size=\"3\"> cc by Shigeto R. Nishitani, 2023/11/30実施 </font>\n",
    "</div>\n",
    "\n",
    "- file: ~/symbolic_math/23_pair_ans.ipynb\n",
    "- make problem: pick_works_from_ans 23_pair_ans.ipynb -1 '' '4, 11'\n",
    "\n",
    "以下の問題を python で解き，LUNA へ提出せよ．LUNA へは ipynb と pdf 形式の２種類を提出すること．\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 問 1 微積分\n",
    "\n",
    "## 1(a) データ点のプロット(15 点)\n",
    "\n",
    "以下の3点のデータ点をxy平面上にプロットせよ．\n",
    "```\n",
    "import numpy as np\n",
    "xdata = np.array( [1,2,3])\n",
    "ydata = np.array([0,5,15])\n",
    "```\n"
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1(b) 関数の微分(15 点)\n",
    "上記のデータ点に対して，一次関数\n",
    "$$\n",
    "F(x) = a0 + a1*x\n",
    "$$\n",
    "でのフィッティングを考える．\n",
    "\n",
    "データと関数との誤差の2乗で得られたsum関数のa0, a1に対する最安定(最小値)点を求めよ．\n",
    "微分して連立方程式で解けばもとまります．\n",
    "\n",
    "また，得られた関数と1(a)のデータ点を同時にプロットせよ．\n",
    "うまくいけば，次のplotのようになります．\n",
    "\n",
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 a_{0}^{2} + 12 a_{0} a_{1} - 40 a_{0} + 14 a_{1}^{2} - 110 a_{1} + 250$"
      ],
      "text/plain": [
       "3*a0**2 + 12*a0*a1 - 40*a0 + 14*a1**2 - 110*a1 + 250"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "xdata = np.array( [1,2,3])\n",
    "ydata = np.array([0,5,15])\n",
    "\n",
    "from sympy import *\n",
    "a0, a1, x,y = symbols('a0, a1, x, y')\n",
    "\n",
    "def func(x):\n",
    "    return a0+a1*x\n",
    "\n",
    "sum = 0\n",
    "for i in range(0,3):\n",
    "  sum += (ydata[i]-func(xdata[i]))**2\n",
    "\n",
    "expand(sum)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 問 2 線形代数\n",
    "\n",
    "## 2(a) ノルム(15 点)\n",
    "\n",
    "次のデータ\n",
    "\n",
    "```\n",
    "edata = [0.1, -0.2, 0.4]\n",
    "\n",
    "```\n",
    "の平均と，　ユークリッドノルム \n",
    "$$\n",
    "||x|| = \\sqrt{x_1^2+x_2^2+\\cdots + x_n^2}\n",
    "$$　をベクトルの内積を使って求めよ．\n",
    "\n",
    "平均は，  $n$をベクトルのサイズとしたとき，全ての要素を$1/n$で満たす(full)ベクトルとの内積で，\n",
    "ユークリッドノルムはそのベクトル自身との内積の平方根で求められる．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2(b) ヴァンデルモンド行列(15 点)\n",
    "\n",
    "ヴァンデルモンド(Vandermonde)行列　$V$　を\n",
    "\n",
    "$$\n",
    "V = \\begin{bmatrix}\n",
    "1 & x_1 & x_1^2\\\\\n",
    "1 & x_2 & x_2^2\\\\\n",
    "1 & x_3 & x_3^2\\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "として，\n",
    "$$\n",
    "V \\begin{bmatrix}\n",
    "a_0\\\\\n",
    "a_1\\\\\n",
    "a_2\\\\\n",
    "\\end{bmatrix}\n",
    "=\\begin{bmatrix}\n",
    "y_0\\\\\n",
    "y_1\\\\\n",
    "y_2\\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "の逆行列から, 2次のフィッティング関数\n",
    "$$\n",
    "f(x)　= a_0 + a_1 x+a_2x^2\n",
    "$$\n",
    "を求めよ．\n",
    "\n",
    "データは\n",
    "``` python\n",
    "xdata = np.array([1,2,3])\n",
    "ydata = np.array([0,5,15])\n",
    "```\n",
    "を使え．\n",
    "\n",
    "ただし，以下に添付したBingの解答例(2023/11/28)は，題意から正解ではありません．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a0: 2.499999999999999\n",
      "a1: -2.499999999999998\n",
      "a2: -1.3322676295501882e-15\n"
     ]
    }
   ],
   "source": [
    "# Bingの解答例(2023/11/28)\n",
    "import numpy as np\n",
    "\n",
    "# データを定義します\n",
    "xdata = np.array([1,2,3])\n",
    "ydata = np.array([0,5,15])\n",
    "\n",
    "# ヴァンデルモンド行列を生成します\n",
    "V = np.vander(xdata, N=3)\n",
    "\n",
    "# 逆行列を計算し、フィッティングパラメータを求めます\n",
    "a = np.linalg.solve(V, ydata)\n",
    "\n",
    "print(\"a0:\", a[0])\n",
    "print(\"a1:\", a[1])\n",
    "print(\"a2:\", a[2])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 問 3 センター試験原題(20 点)\n",
    "\n",
    "(2021 大学入試センター試験　数学 II・B/本試験 第 2 問)\n",
    "\n",
    "$a$ を実数とし，\n",
    "$f(x)=(x-a)(x-2)$とおく．\n",
    "また，\n",
    "$F(x)= \\int_0^x f(t)dt$とする．\n",
    "\n",
    "(1) $a=1$のとき，　$F(x)$は\n",
    "$x = \\fbox{ ア }$で極小になる．\n",
    "\n",
    "(2) $a = \\fbox{ イ }$のとき，$F(x)$はつねに増加する．\n",
    "また，$F(0) = \\fbox{ ウ }$であるから，\n",
    "$a = \\fbox{ イ }$のとき，$F(2)$ の値は$\\fbox{ エ }$である．\n",
    "\n",
    "$\\fbox{ エ }$の解答群:[0，　 正，　負]\n",
    "\n",
    "(3) $a > \\fbox{ イ }$とする．\n",
    "\n",
    "$b$を実数とし，$G(x)= \\int_b^x f(t)dt$とおく．\n",
    "\n",
    "関数$y=G(x)$のグラフは，\n",
    "$y=F(x)$のグラフを\n",
    "$\\fbox{ オ }$方向に\n",
    "$\\fbox{ カ }$だけ並行移動したものと一致する．\n",
    "また，$G(x)$は$x = \\fbox{ キ }$で極大になり，\n",
    "$x = \\fbox{ ク }$で極小になる．\n",
    "\n",
    "$G(b) = \\fbox{ ケ }$であるから，\n",
    "$b = \\fbox{ キ }$のとき，\n",
    "曲線$y=G(x)$と\n",
    "$x$軸との共有点の個数は$\\fbox{ コ }$個である．\n",
    "\n",
    "$\\fbox{ オ }$の解答群:[$x$軸，　$y$軸]\n",
    "\n",
    "$\\fbox{ カ }$の解答群:[$b$, $-b$, $F(b)$, $-F(b)$, $F(-b)$, $-F(-b)$]"
   ]
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   "source": [
    "# 問 4 センター試験改変(20 点)\n",
    "\n",
    "問３で$f(x)=(x-a)(x-\\sqrt{5})$として同様に求めよ．\n",
    "さらに， 得られたG(x)関数をsubs({a: 3}), (x,2,3.5)でプロットせよ．"
   ]
  }
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