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Table of Contents

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\n", " 2020年度　数式処理演習　pair試験問題 \n", "
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\n", " cc by Shigeto R. Nishitani, 2020/11/26 実施 \n", "
\n", "\n", "* file: ~/symboic_math/exams/20_pre_ans.ipynb\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．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 微積分\n", "## ソフトマックス関数の概形(15点)\n", "ソフトマックス関数\n", " \\begin{equation*}\n", " f(x) = \\frac{1}{1+e^{-x}}\n", " \\end{equation*}\n", " の増減，極値，凹凸を調べ，曲線$y=f(x)$の概形を描け．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "| x | $-\\infty$ | $\\cdots$ | 0 | $\\cdots$ | $\\infty$ |\n", "| ---- | ---- | ---- | ---- | ---- | ---- |\n", "| f(x) | 0 | $\\nearrow$ | 0.5 | $\\nearrow$ | 0 |\n", "| f'(x) | 0 | + | + | + | 0 |\n", "| f''(x) | 0 | + | 0 | - | 0 " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3D関数のプロット(15点)\n", "\n", "３変数のシグモイド関数で，１変数を固定すると次のような関数となる．\n", "``` python\n", "import numpy as np\n", "\n", "def softmax(x,y):\n", " return np.exp(-x)/(np.exp(-x)+np.exp(-y)+np.exp(-1))\n", "```\n", "\n", "この関数を\n", "``` python\n", "x = np.arange(-4, 4, 0.5)\n", "y = np.arange(-4, 4, 0.5)\n", "```\n", "で３次元プロットせよ．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 線形代数\n", "## 線形結合の確認(p.173, 5-39)(15点)\n", "\n", "sympyを使って，$w^T x$で線形結合が得られることを確認せよ．\n", "1. $ w=\\left(\\begin{array}c w_0\\\\w_1\\\\w_2\\end{array}\\right)$, $ x=\\left(\\begin{array}c x_0\\\\x_1\\\\x_2\\end{array}\\right)$を作る．\n", "1. wを転置する\n", "1. `ww.T*xx`で線形結合となることを確認する．\n", "1. `ww*xx.T`では3x3の行列が得られることも確認せよ．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 解析解の確認(p.177, 5-60)(15点)\n", "\n", "``` python\n", "xdata=np.array([1,2,3,4])\n", "ydata=np.array([0,5,15,24])\n", "```\n", "を対象データとして，(5-53)にしたがって，N=4, n=3で\n", "$$\n", "y=a_0 + a_1\\,x +a_2\\, x^2\n", "$$\n", "に対するfittingを行う．得られたデザイン行列$X$は\n", "$$X=\\left(\n", "\\begin{array}{@{\\,}ccc@{\\,}}\n", "1 & 1 & 1\\\\\n", "1 & 2 & 4\\\\\n", "1 & 3 & 9\\\\\n", "1 & 4 & 16\\\\\\end{array}\n", "\\right)$$\n", "となる．(5-59)式の左辺の$X^TX$が3x3行列になることを確認せよ．\n", "\n", "ヒント：https://nbviewer.jupyter.org/github/daddygongon/jupyter_num_calc/blob/master/numerical_calc/least_square_fit.ipynb\n", "の「正規方程式(Normal Equations)による解」の「python codeによる具体例」を参照せよ．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# センター試験原題(10点)\n", "(2018大学入試センター試験　追試験 数学II・B 第2問)\n", "\n", "$a$ を正の実数とし，\n", "放物線$y=3x^2$ を$C_1$,\n", "放物線$y=2x^2+a^2$ を$C_2$ とする．\n", "$C_1$ と$C_2$ の二つの共有点を $x$ 座標の小さい順にA,Bとする．\n", "また，$C_1$ と$C_2$ の両方に第1象限で接する直線を$l$ とする．\n", "\n", "(1) Bの座標を$a$ を用いて表すと\n", "$(\\fbox{ ア }\\,, \\fbox{ イ }\\,a^{ \\fbox{ ウ }})$\n", "である\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "直線$l$ と二つの放物線$C_1, C_2$ の接点の$x$ 座標をそれぞれ$s,t$ とおく．\n", "$l$ は$x=s$ で$C_1$ と接するので，$l$ の方程式は\n", "\\begin{equation*}\n", " y = \\fbox{ エ }\\,sx - \\fbox{ オ }\\,s^{ \\fbox{ カ }}\n", "\\end{equation*}\n", "と表せる．\n", "同様に，$l$ は$x=t$ で$C_2$ と接するので，$l$ の方程式は\n", "\\begin{equation*}\n", " y = \\fbox{ キ }\\,tx - \\fbox{ ク }\\,t^{ \\fbox{ カ }} + a^2\n", "\\end{equation*}\n", "とも表せる．これらにより，$s,t$ は\n", "\\begin{equation*}\n", " s = \\frac{\\sqrt{\\fbox{ ケ }}}{\\fbox{ コ }}a , \\,\\,\\,\\, \n", "t= \\frac{\\sqrt{\\fbox{ ケ }}}{\\fbox{ サ }}a \n", "\\end{equation*}\n", "である．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "放物線$C_1$ の$s \\leqq x \\leqq {\\fbox{ ア }}$ の部分\n", "放物線$C_2$ の${\\fbox{ ア }} \\leqq x \\leqq t$ の部分,\n", "$x$ 軸，\n", "および2直線$x=s, x=t$で囲まれた図形の面積は\n", "\\begin{equation*}\n", " \\frac{\\fbox{ シ }\\sqrt{\\fbox{ ス }}-\\fbox{ セ }}{\\fbox{ ソ }}a^{\\fbox{ タ }}\n", "\\end{equation*}\n", "である．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 数値改変(30点)\n", "\n", "問3.において，放物線$C_1$ が\n", "$$\n", "y = 2.9 x^2\n", "$$\n", "である場合について解きなさい．\n", "ただし，係数が浮動小数点数に変わったので，$\\fbox{ ア }\\,, \\fbox{ イ }$などには浮動小数点数が入る．最後の図形の面積は，$1.284186\\ldots a^3$ となる．(30点)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "toc": { "base_numbering": 1, "nav_menu": { "height": "12.666666984558105px", "width": "252.6666717529297px" }, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": true, "toc_position": { "height": "538.3287963867188px", "left": "0px", "right": "1189.3333740234375px", "top": "60.801631927490234px", "width": "295.28533935546875px" }, "toc_section_display": "block", "toc_window_display": true } }, "nbformat": 4, "nbformat_minor": 2 }