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ロルの定理と平均値の定理
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{
"cells": [
{
"metadata": {
"toc": "true"
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"cell_type": "markdown",
"source": " # Table of Contents\n<div class=\"toc\" style=\"margin-top: 1em;\"><ul class=\"toc-item\" id=\"toc-level0\"><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#2章-微分法の応用\" data-toc-modified-id=\"2章-微分法の応用-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>2章 微分法の応用</a></span><ul class=\"toc-item\"><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#ロルの定理\" data-toc-modified-id=\"ロルの定理-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>ロルの定理</a></span><ul class=\"toc-item\"><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#例:sin-x\" data-toc-modified-id=\"例:sin-x-1.1.1\"><span class=\"toc-item-num\">1.1.1&nbsp;&nbsp;</span>例:sin x</a></span></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#x^3-x\" data-toc-modified-id=\"x^3-x-1.1.2\"><span class=\"toc-item-num\">1.1.2&nbsp;&nbsp;</span>x^3-x</a></span></li></ul></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#平均値の定理\" data-toc-modified-id=\"平均値の定理-1.2\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>平均値の定理</a></span><ul class=\"toc-item\"><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#x^3\" data-toc-modified-id=\"x^3-1.2.1\"><span class=\"toc-item-num\">1.2.1&nbsp;&nbsp;</span>x^3</a></span></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#関数-f(x)=x^2\" data-toc-modified-id=\"関数-f(x)=x^2-1.2.2\"><span class=\"toc-item-num\">1.2.2&nbsp;&nbsp;</span>関数 f(x)=x^2</a></span></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#応用1(不等式の証明)\" data-toc-modified-id=\"応用1(不等式の証明)-1.2.3\"><span class=\"toc-item-num\">1.2.3&nbsp;&nbsp;</span>応用1(不等式の証明)</a></span></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#応用2(漸化式と極限)\" data-toc-modified-id=\"応用2(漸化式と極限)-1.2.4\"><span class=\"toc-item-num\">1.2.4&nbsp;&nbsp;</span>応用2(漸化式と極限)</a></span></li></ul></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Math/Derivative/%E3%83%AD%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86%E3%81%A8%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86.ipynb#参考資料\" data-toc-modified-id=\"参考資料-1.3\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>参考資料</a></span></li></ul></li></ul></div>"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "# 2章 微分法の応用"
},
{
"metadata": {
"trusted": true,
"collapsed": true,
"ExecuteTime": {
"start_time": "2017-12-09T12:32:47.514296Z",
"end_time": "2017-12-09T12:32:48.755816Z"
}
},
"cell_type": "code",
"source": "import scipy as sp\nimport matplotlib.pyplot as plt",
"execution_count": 1,
"outputs": []
},
{
"metadata": {},
"cell_type": "markdown",
"source": "[gist](https://gist.github.com/Cartman0/9481554ddc7905eace4b81c5794b4e2a)"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## ロルの定理"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "新訂微分積分I p.38-39\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$f(x)$が開区間$(a,b)$で微分可能とし,$f(a) = f(b)$が成り立つとする.\n$f(x)$が定数関数でない場合は,$a,b$間のいずれかの点$c$で最大値または最小値をとり,\n\n$$\nf'(c) = 0\n$$\n\nが成り立つ.\n\nまた,$f(x)$が定数関数の場合は,\nすべての点で$f'(x)=0$である."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "以上からロルの定理が成り立つ.\n\n\\begin{theorem}\\label{theo:}\nロルの定理\n\n関数$f(x)$が閉区間$[a,b]$で連続で,開区間$(a,b)$で微分可能であり,さらに,\n$f(a)=f(b)$であるとき,\n\n\\begin{eqnarray}\nf'(c) = 0, (a < c < b)\n\\end{eqnarray}\nを満たす点が少なくとも1つ存在する.\n\\end{theorem}\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 例:sin x"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "関数$f(x) = \\sin{x}$ は,$[0, 2\\pi]$でロルの定理の条件を満たす.\n実際,$f'(x) = \\cos{x}$ より,$f'(c) = 0$を満たす点は$\\frac{\\pi}{2}, \\frac{3}{2}\\pi$ である."
},
{
"metadata": {
"trusted": true,
"ExecuteTime": {
"start_time": "2017-12-09T12:49:29.263235Z",
"end_time": "2017-12-09T12:49:29.570942Z"
}
},
"cell_type": "code",
"source": "x = sp.linspace(0, 2 * sp.pi)\ny = sp.sin(x)\n\nplt.plot(x, y)\n\nplt.plot(x, sp.cos(x), \"--\")\n\nplt.hlines(\n [sp.sin(sp.pi / 2), sp.sin(3 * sp.pi / 2)],\n [sp.pi / 2 - 1, 3 * sp.pi / 2 - 1], [sp.pi / 2 + 1, 3 * sp.pi / 2 + 1],\n colors=\"red\")\n\nplt.grid()\nplt.show()",
"execution_count": 17,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
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ecequ0hLIOaP9+/VuqHeaOsHqwiClLAYmA0uB/cBcKWWcEOJ1IcSFq4ymAd7A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\n8mhHAAAAAElFTkSuQmCC\n",
"text/plain": "<matplotlib.figure.Figure at 0x2d7c1755080>"
}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### x^3-x"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "関数$f(x)=x^3-x$ について$f(-1) = f(1)$であることを確認せよ.\n\n$$\nf(1) = 1^3 - 1 = 0\\\\\nf(-1) = (-1)^3 - (-1) = 0\\\\\n$$\n"
},
{
"metadata": {
"trusted": true,
"ExecuteTime": {
"start_time": "2017-12-09T12:56:52.088999Z",
"end_time": "2017-12-09T12:56:52.099506Z"
}
},
"cell_type": "code",
"source": "f = lambda x: x**3 - x\nf(1), f(-1)",
"execution_count": 21,
"outputs": [
{
"output_type": "execute_result",
"metadata": {},
"data": {
"text/plain": "(0, 0)"
},
"execution_count": 21
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$f'(c) = 0, (-1<c<c)$を満たす点$c$を求める.\n\n$$\nf'(x) = 3x^2 - 1 = 0\\\\\n3x^2=1\\\\\nx=\\pm \\frac{1}{\\sqrt{3}}\n$$"
},
{
"metadata": {
"trusted": true,
"ExecuteTime": {
"start_time": "2017-12-09T13:11:45.108297Z",
"end_time": "2017-12-09T13:11:45.488053Z"
}
},
"cell_type": "code",
"source": "x = sp.linspace(-1.5, +1.5)\ny = x**3 - x\n\nplt.plot(x, y)\n\nplt.plot(x, 3 * x**2 - 1, \"--\")\n\nplt.hlines(\n [f(-1. / sp.sqrt(3)), f(1. / sp.sqrt(3))], \n [-1. / sp.sqrt(3) - .3, 1. / sp.sqrt(3) - .3], \n [-1. / sp.sqrt(3) + .3, 1. / sp.sqrt(3) + .3],\n colors=\"red\")\n\nplt.grid()\nplt.show()",
"execution_count": 28,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"image/png": 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bdTSiFJz5f8bY5K//APVVMOVflg2FVErRMzaMnrFhnD2sO/BDsV+7u5w1u8tY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\n",
"text/plain": "<matplotlib.figure.Figure at 0x2d7bd1bc438>"
}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## 平均値の定理"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "関数$f(x)$ が閉区間$[a, b]$ で連続で,開区間$(a,b)$で微分可能であるとする.\n2点$A(a, f(a)), B(b, f(b))$ を通る直線の傾きを$m$とおく.\n\n$$\nm = \\frac{f(b) - f(a)}{b - a}\n$$\n\nこのとき,\n$$\nf'(c)=m, (a<c<b)\n$$\n\nを満たす点$c$が存在することを,ロルの定理を用いて示す.\n\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "2つの関数$y = f(x)$と$y=f(a) + m(x-a)$の差を$F(x)$とおくと,\n\n$$\nF(x) = f(x) - (f(a) + m(x-a))\n= f(x) - f(a) - m(x-a)\\\\\n= f(x) - f(a) - \\frac{f(b)-f(a)}{b-a}(x-a)\n$$\n\n$F(x)$は区間$[a, b]$ で連続で,区間$(a,b)$ で微分可能であり,\n\n$$\nF'(x) = f'(x) - m = f'(x) - \\frac{f(b)-f(a)}{b-a} \n$$\n\nまた,\n$$\nF(a) = f(a) - f(a) - m(a-a)=0\n$$\n\n$$\nF(b) = f(b) - f(a) - m(b-a)\n= f(b) - f(a) - \\frac{f(b)-f(a)}{b-a}(b-a)\n= f(b) - f(a) - (f(b)-f(a))\n= 0\n$$\n\nであるから,ロルの定理より,\n\n$$\nF'(c) = f'(c) - m = f'(c) - \\frac{f(b)-f(a)}{b-a} = 0, (a < c < b)\n$$\nを満たす点$c$が少なくとも1つ存在する.\n\n平均値の定理より,\n\\begin{theorem}\\label{theo:}\n平均値の定理\n\n関数$f(x)$が閉区間$[a,b]$で連続で,開区間$(a,b)$で微分可能であるとき,\n\n$$\n\\frac{f(b)-f(a)}{b-a} = f'(c), (a < c <b)\n$$\nを満たす点$c$が少なくとも1つ存在する.\n\n平均値の定理は,接線の傾きが直線$AB$の傾きと一致するような点が少なくとも1つ存在する.\n\\end{theorem}\n\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "資料:[平均値の定理とその応用例題2パターン](https://mathtrain.jp/meanvalue)\n\n平均値の定理は1次微分だが,\nn次微分まで拡張すると,**テイラーの定理**になる."
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### x^3"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "関数$f(x) = x^3$ を区間$[1,4]$で考え,平均値の定理を満たす点$c$を求める.\n\n$$\n\\frac{4^3 - 1^3}{4-1} = \\frac{63}{3} = 21\\\\\nf'(x)=3x^2 \\\\\nf'(c) = 3c^2 = 21\\\\\nc^2 = 7 \\\\\nc = \\sqrt{7}\\\\\n$$"
},
{
"metadata": {
"trusted": true,
"scrolled": false,
"ExecuteTime": {
"start_time": "2017-12-09T15:12:07.576895Z",
"end_time": "2017-12-09T15:12:07.974663Z"
}
},
"cell_type": "code",
"source": "f = lambda x: x**3\n\nx = sp.linspace(1, 4)\ny = f(x)\nplt.plot(x, y)\n\nplt.plot(x, 3 * x**2, \"--\")\n\n# 直線\na, b = 1, 4\nm = (f(b)-f(a))/(b-a)\ny = f(a) + m*(x-a)\nplt.plot(x, y, \"--\")\n\nplt.scatter(sp.sqrt(7), f(sp.sqrt(7)))\n\n# 直線\ny = f(sp.sqrt(7)) + m*(x-sp.sqrt(7))\nplt.plot(x, y, \"--\")\n\nplt.grid()\nplt.show()",
"execution_count": 33,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"image/png": 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DCfRaNcvvDCPh2N9g+c8IjRcF4jYs/p3xD2qFe6Ag6vO1uLVr5+iypZtki1E6\ngYCpOuzdgEHAm8A24C6sI3UmAhtq25Yk3YiC8gLOFZ2jU1AnWvm3Ynyr8YyNHUuE55/M79KIXcgt\n4enPUzh8/gr3xph57J6RBLoKxMFiCvUTMGw+geniAdy7qvB7xIKiUsmwr6dscYYfCqyovo6vAtYK\nIb5RFOUYkKgoyj+AQ8BHNmhLkv5QUWURK4+uZNWxVXhoPPj+ru/RqDQ81eUpR5fmlIQQfLInnYXf\n7ma8egurvH5EV6xHcR9NaXIKmRv0VJ7biq51HBELXsAjIaFRTvnckNhilM5hoNM1Pj8LdKvt8SXp\neoyVRj5J+4SVR1dSbCpmSNMhPNLxETSqG1xKrxG6VFDGu4nfEX9xJdtcdqDFhGh6K5YOU1Cr1Ki9\nvVG0WsLnfIDnoEEoKpWjS5ZsQD5pK9V7hw2HmZc8j/6R/Xm046PE+sU6uiSnZTGb+el8CY9u3U4/\n0hir3YWq4wSKLV3IWbkO7d7viPjgVnQxMTT7ar08o29gZOBL9U55VTlrT6yl3FzO9PbTuSX0Ftbf\nvp4Y3xhHl+a8Kksw7FpJ+c/z8K2Ip1PUIzw3+nHKD3UiZ8kKyg+/jqZJEzwH/meRdRn2DY8MfKne\nqDRX8sXJL1iaupScshz6RfRDCIGiKDLs/0hBBpa9SzDt/ZiAqmKOEk1wRAyrpnQjf9UnZPzzn2jC\nwgj9x//hPWoUikZeBmvIZOBL9cIvl3/hpV0vkVWSReegzrzZ9026hnR1dFnO6aqJzIq+fh79mW/5\nt7kLqZH3MaFVLGFnjqMoCl4jhqNoXPAZMwZF27hnAW0sZOBLTqvKUkWJqQRvnTdB7kGE6kN5redr\n9AjtIS83XEtlKaR+DnsXUz5qCR8cVvFD2gBcXW9jVrsQxm9aQ/G7b+LRrh1MnoyLnx++997r6Kql\nOiQDX3I6ZouZ785/x8KUhcT6xvKvhH/R3Kc5K4etdHRpzin/POz/GA6uhLJ8jD6t+OuKJDYVNuWh\nsADGpn5L5ZqfKPfxIeiZp0mNjHR0xZKDyMCXnIZFWNhyYQsLkhdwpvAMLX1bMjx6uKPLcm6VpbCg\nF5jKqGhxG4vKB/HuyQCiAzz4dFo7YnduIiclmcAnZuE74X7UHnpISnJ01ZKDyMCXnMbyo8t578B7\nRHtH806/dxjcdDAqRY7//i9l+XBoNVzcB3cvB607ltEL2ZAdzMtJ+fjnZvJx7gbadR5CYHN/LJH3\n4H3HaNSKDaw9AAAgAElEQVSef7KwuNRoyMCXHEYIwY5LO/DUetIxqCOjY0YT5B7EsKhhqFVqR5fn\nXC4dgH0fw5EvoaoMIntARREpOYKXtvqSfTyNv13aTocTe1BcXVGX9QZApdWCvCErVZOBL9U5IQS7\nM3czL3keKTkpDG46mI5BHfFz9WNE9AhHl+d8jm2EtfeDxh3aj4WuU8n3asVbm06QuC+dSee2c/fh\nb1FpNPhOnIj/tKm4+Pk5umrJCcnAl+pUcnYy7x98nwNXDhDsHszfe/ydO2LucHRZziX7OBxYBkFx\nED8JYgbBbe9A+7FYtF58tj+DJeu+Ises5sF+cTzUX4dlrzf+06ehCQpydPWSE5OBL9WJXx+QOmI4\nwoWiCzzX7TnuankXOrXO0aU5B1MZHNsAB5ZD+i+g0kDPx6zf07pDt2kcuJDPu4mbaLv9K96/sBvN\nfQ8QN2IM0BqGDHJk9VI9IQNfsqv0inQe/vfDDGoyiDEtxzA2dixjWo7BzcXN0aU5ly8ehBObwC8a\nBr8GHe8DfQAAmYVlvP/FXjzWf8oz53ahFWZ8Ro8m4IHx1zmoJP03GfiSXZzIO8G85Hlsy9qGt86b\ngU0GAqBVyxuIVJbCsa/g0Cdw18fgGQK9noDuMyCqD1TPTFlWaWbxT2dZsP00z+xaxi2Xj6IfPpzQ\nmY+ibdrUwZ2Q6iMZ+JLNzTk4hyWpS/DUeDLcezgv3vYiHtqGse5orVxOtj4clfo5VBSBX3MoyLAG\nfpPuv20mhODb3ac48v4ivgrqwIAebRh05yuEeOnQNW/uwA5I9Z0tVryKBFYCIYAFWCyE+EBRFD/g\nMyAKOA+MFULk17Y9yTldKLqAn6vfb0Msp7WbxsQ2Ezn0y6HGHfa/zmtTnAWLE8BFB61HQ+cHoGnP\n3+a8+dW+tIvsfHM+PQ98zyhTGUMejqXzffGOqV1qcGxxhl8F/EUIcVBRFE/ggKIoW4BJwI9CiNmK\nojwHPAc8a4P2JCdysfgiiw4v4uszXzO9/XQe6fgIfSP60jeir6NLcxyLBc7/ZL1kYyqDcautZ/Hj\n1lhD3s3nd7ucyTGy+bUP6Lx9PbdWllDUsTtNXnwGfds2DuiA1FDZYsWrTCCz+utiRVHSgHBgFJBQ\nvdkKIAkZ+A1GVkkWiw4v4qtTX6FSVNzb6l7Gxo51dFmOVZBufQo2eQ0UpoOrN3S49z9n+a1u+90u\nOQVGPkg6x6d7M3j8zBmqmrck9IVniOvyu0XkJKnWbHoNX1GUKKzLHe4Bgqt/GSCEyFQURQ4QbkBe\n3/M6Oy7tYEzLMUxrN41gfbCjS3KMCiOoNdZLNUfWwfY3IToBBr0MrUaAxvWauxUby9j89mJCNn7K\noW73M35YP8b/dQ6BPo348pdkd4oQwjYHUhQPYDvwuhBinaIoBUIIn6u+ny+E8L3GftOB6QDBwcHx\niYmJNWrfaDTi4dEw/mdxxr4Um4v5d9G/6ePRhwBNANmmbFwUF/xc/vyJTmfsS0391hdhwbvwGCFZ\nPxKUvYsTsY+QHdwPF1MxanMZFa5/fG5TWVlF+qZdtPjpe4JK87kQHEX5uLH4xDWr+340ALIvVv37\n9z8ghOhyve1scoavKIoG+BJYLYRYV/3xFUVRQqvP7kOB7GvtK4RYDCwG6NKli0hISKhRDUlJSdR0\nX2fjTH0pKC9g+dHlrDm+hgpzBf3a9iOhRcIN7+9Mfamt7Vv/TT/LTjj8GRRcAK0ndLib1t3vpHVI\nu99t/9WhS7y9+QSXC8oI9XalT0wAfT94ll6GC1wOaUbliy9x6x231vnc/g3pZyL7cnNsMUpHAT4C\n0oQQ7171rY3ARGB29Z8batuWVLcWpSxi2dFllJpKGdZsGDM6zKCZd92diTqF0jzISoXofghFDcc3\ngV8zGPCi9ZKN1v2au3116BLPr0ulvNJEt6w09oo4PjtwERHXj6CE1gwYf7tcxEWqc7Y4w+8F3A+k\nKoqSXP3Z37AG/VpFUaYA6cDdNmhLsrPyqnJcXazXna+UXqFnWE8e6fBI41oz1lQOpzbD4bVwcrP1\n+vwzp603Xqdttb6/jre+S6PThWQmHN9Ms6IsXu0+id2hbdkR3Y237htYB52QpN+zxSidHcAfnarI\nv9n1RKmplMQTiSw7sowPB3xIx6COvNjjxcY3H/2RdfD1E1BRCPog6DYdOoyD6l+C1wt7s9nC9pXr\neX7DAmIKL5HhEcjsLvexJ6Q1AJmF5fbugST9IfmkbSNXXlXO5yc/Z2nqUvLK8+gV3gu9Rg/Q8MNe\nCOvlmtTPrZdnmnQH/+bW4ZPt7oZm/UB9Y/+LmC2C745kMnfLcZ5e8z56BO90Hse2iE5YrprbP8xH\nziEkOY4M/EbMIizc++29nC44TffQ7jza8VE6BTWC8d+G03DkC+tiIoaToHIBr3Br4Id2gDsW3vCh\nTGYLW1ZvonD1J7zSbiwRoX5Uvv4vyn2D2PXNcSwm82/bumnUPHNrrD16JEk3RAZ+I2OymNhyfgtD\nmw1FpaiY2m4qQe5BdA3p6ujS7KuyBLR6sJhh2TAoyYGmvaDHw9apDtxvbsGQcpOZTZ9+D8sWE5d5\nkkK9D3P7BDBgZB/UKusVTpVW89sonTAfN565NZbRncLt0TtJuiEy8BuJKksV3579loUpC7lovIin\n1pM+EX0a9iLhRZnWWSmPrrd+PSsFVGoYsxQCWoBX2M0fstxE4vYTeM9+ifaZaRjdvCh68DG6znwQ\ntdt/X64Z3SlcBrzkVGTgN3Bmi5nvz3/PwpSFnC86T5xfHPMGzqN3eG9Hl2Y/Z7bB9resC4kgILgd\nxE8EcyWoXCG6300fMrOwjI0HDDy2bSvGchPvebhTNvlhOj82BbVeb/s+SJIdyMBv4ASC+cnz0bno\neL//+wyIHNDwxn8XZULa19B8AATEQFUFlOVBwvPQ5g4IbFnjQ6dlFvHF59sJ+2oV9185jnbW+0wY\n3oO24XLtXan+kYHfwAgh2JaxjcTjibzf/33cNe4sHbKUYH1wwxp1U3gJ0jbC0a8gYw8gYOhsa+C3\nvBVih9b40EIIfjplYP2GXcR+n8idl1Iw61zJHziI18d3Q91AHuWXGh8Z+A2EEIIdl3YwL3keR3OP\n0sSzCZeNl4nxjSHUI9TR5dlGhRF0HtYHoz6Mh6oyCG4L/f8GrUdBYPUImBr+C6a0sop1By+xbOc5\n8jIyWfnD6yguLnhOnkzY9KnsSEmRYS/VazLwG4DiymIe/vfDpOSkEKYP47WerzGy+UhcVPX8xysE\nXDlqvVyT9jW4aGF6knUGyjsWWK/NB9T+CeCL+aWs+uUCW7YlE5t+FPfeQ3lsYh/C+v8T7969cPH3\nr3UbkuQM6nkiNG6XjZcJ8wjDQ+NBmEcYtze/nTti7kCj1ji6tNo7uBJ+esc6SRmKdeGQuJH/mVu+\nzR21OrzFIth1JpdVu89z8MBJxp78kTnn96BSq2kx53E0wUHQKcI2fZEkJyEDvx5KyUlh7qG5HMo+\nxKY7NxHkHsRbfd9ydFk1V1kKZ7fB8W9h4EvW1aEUNQS2gj5PQext4GGb5RQKS018fiCD1XvSMVzO\nZvLZbcw8vRO1EPjcfRcBMx6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UUW6y/Nc+QZ46Iv3cuaW5P8389TQN0NPMX08Tf3e83erP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\nQHjDd865AvFfOIqad0gmP5YAAAAASUVORK5CYII=\n",
"text/plain": "<matplotlib.figure.Figure at 0x2d7c1b98588>"
}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 関数 f(x)=x^2"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "関数$f(x) = x^2$ を区間$[a,b]$で考え,平均値の定理を満たす点$c$を求める.\n\n$$\n\\frac{b^2 - a^2}{b-a} = \\frac{(b-a)(b+a)}{b-a}=b+a\\\\\nf'(x)=2x \\\\\nf'(c) = 2c = a+b\\\\\nc = \\frac{a+b}{2} \\\\\n$$"
},
{
"metadata": {
"trusted": true,
"ExecuteTime": {
"start_time": "2017-12-09T15:21:02.463029Z",
"end_time": "2017-12-09T15:21:02.868299Z"
}
},
"cell_type": "code",
"source": "f = lambda x: x**2\n\na, b = -1, +2\nx = sp.linspace(a, b)\ny = f(x)\nplt.plot(x, y)\n\nplt.plot(x, 2 * x, \"--\")\n\n# 直線\nm = (f(b)-f(a))/(b-a)\ny = f(a) + m*(x-a)\nplt.plot(x, y, \"--\")\n\nplt.scatter((a+b)/2, f((a+b)/2))\n\n# 直線\ny = f((a+b)/2) + m*(x-(a+b)/2)\nplt.plot(x, y, \"--\")\n\nplt.grid()\nplt.show()",
"execution_count": 36,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"image/png": 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l5BTkYLVbuU5/HTq1jocGP0RCYAIpoSmoFHlrQYdlbYLlr8DyV+GKedDrAki+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\nj4lnRlo03l7yjMht7Pwefn0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vKxNt165QXQTzroP8HIga4mjaFZro6vIlSXIDMtjdgLDb\nadiyBdOibEw5OViKiwl/7FGCr7sO/6wsDKNGoQkJ+e8XVRdC4WqY+AIMueW/mnZJktS5yWB3MXtT\nEwcumIi1pARFq8V3xAhC7rgDw/h0AFQ+Pqh8TuxaVJEPh5dB2o2OPUfv2wHeQS6sXpIkdySDvR2J\n5mbq1q7FlJ2Nvb4BJl/s2C7uskvx6h6PIX0caoPhf19os8LqNyD3OfDyhb6XgXegDHVJkk5LBns7\nqN+wgeqvv8GUm4u9thaVjw9+mRmOHYuA0LvvPvOLS7Y7mnYd2wpJF8NFrzhCXZIk6QxksLcBe309\n5mXL8B01CrXBQMPWrZjy8vAbPx6/rCx8R45ApdOxLy/vzwdqqIYPJ4LWB2Z8Cn2mtEv9kiR5Nhns\nTmIzmTDn5WHKzsa8bDmiqYnIV1/B/8ILCbrySoKvvRZFqz23wcr3QWgvx5n5tA8dm0qf2ABDkiTp\nbGSwt4IQAkVRsBQXc+CCiQiLBU1YGIHTpuGXlYVPWirAb29+nk2TGZY8A2vfhSvmQdJF0OuCNpyB\nJEkdUauCXVGU6cD/Ab2BIUKIDc4oyp1Zy8sxGY3UZmej7RpBt+f+iSYigpA77sBn6BC8+/dHUanO\nf+D8xfDTvVBTCENmQ/cxzi9ekqROobVn7DuAy4B3nVCLW6v54Qeqvvqahk2bQAi84uLwHTYccGxK\nEXLbrS0ffNFfYfWb0KUn3LAQYoc7qWpJkjqjVgW7EGI30CE7BzYXFmJasoTgWbNQ1Goad+/BbjIR\ncscd+GVlouvZs/XzPnFVDJGDYPQDMOZh0OpbX7wkSZ2aXGP/naYDBzBlZ1ObnUPT7t0A+AwahHdK\nCmEP3I/y6CPOOZCpFH59kKjmUCAdki93PCRJkpxAESfPGs/0BEUxAl1P86W/CiF+OPGcPODBP1tj\nVxRlNjAbIDw8PHX+/PktKthsNmM43U08LSEEWK2g1aLNzyf45VcAaI6Pp2nQQBoHDMQe0sU5xzpx\nvPDSXBLyP0Bta2J35EzKE6Y7b3wXcur3xYU6yjxAzsVdtWYu6enpG4UQaWd9ohCi1Q8gD0g71+en\npqaKlsrNzW3xa4UQwm63i/otW0TJiy+K/RmZouSFFx2ft1hE5ZdfiuaS0laNf0ZVBUJ8eqkQT/kL\nMTdLiLK9rZ6LO+koc+ko8xBCzsVdtWYuwAZxDhnbqZZiyl5/nZrvvsdaUgIaDb7DhuGdkgKAotEQ\ndMUVbXfw6iIoWgcXvgxpN4FKBRS33fEkSeq0Wnu546XAG0Ao8IuiKFuEEG5x4bWwWqlft476jZsI\nvetOAKxlZeh798bv3nvwS09HHRDQtkWU73M07Rp88++adsl2AJIkta3WXhXzHfCdk2ppNXtzM3Wr\nVmHKzsG8eDG2mhoUb28CZ85AGxZGxD/+0T5X8NgssPJ1WPoC6PwgedqJpl0y1CVJansevxRjb2xE\nWG2oDb6YFmVT/NBDqAwGDOnp+GVlYhg1CpW3Y9/Pdgn14i3w452O5l19psCkl2SgS5LUrjwy2G3m\nOsxL8xxn5suWEXrXXXS58QYM6eOIfvcdfIYPR+Xl1f6FNVTDxxc5WuvO+Az6XNL+NUiS1Ol5VLAL\nu52Ad95l/657EM3NqENCCJhyyameLGqDAcPYse1fWNkeCEtynJlP/xii0mSvdEmSXMajgl1RqRA6\nLwKvmIl/VhbeAweiqF24JVyTCYxPw/r34YovHE27ema6rh5JkiQ8LNgBam+4gUHjxrm6DNhvhJ/v\nhZojMPQ26O6CvxQkSZJOw+OC3S0sfBzWzIGQRLgpG6KHuLoiSZKkU2Swn6uTrRcUxbGGPuYhx0Oj\nc21dkiRJfyCD/VyYSuCXByB2BAy/A5Ivw9GtWJIkyf20YEeITkQI2PQZvDkE8o2guPCNWkmSpHMk\nz9jPpKoAfrobDuZBzAi45A0ISXB1VZIkSWclg/1Maovh6Ca46BVI/f/t3WuMHXUZx/HvD5BupEVo\nC6UWEAuEIl67TW2rXKq8qE2geIHgG9uAkoZAFH1hTaMvfIOVxKgBuXiJJTHdRkStSCNVtoGorZam\n9M7SIsbSdYugLRuBUnx8MVM92e7ZM7tzzpk5s79PcrKzZ/5n9nnmv/vsnP/M/M9N6aRdZmbl58Je\n69BeeP5JmPu55OPp7tgJXS2eKMzMrMlc2AGOHYXffxueuAsmnA7vuT65i9RF3cw6kAv7C1th3e0w\nsDP5eLpFqzxpl5l1tPFd2F/9F6y+Jpla98Y1MGtx0RGZmeU2Pgv7wG44+9LkyPyG1TBjjo/Szawy\nxtelHq8dgUe+CPfOh72/Tp676GoXdTOrlPFzxN73WDJp1yv9MP82uHBh0RGZmbXE+Cjs61fA5nvh\nrFlww4PJXC9mZhVV3cIekTxOOimZfbHrdLj8S560y8wqr5qF/cjB/0/ateD2dNIuM7PxoVonTyPg\nqa9RJdUAAAXbSURBVB/DPR+E/b1wso/OzWz8qc4R+8t/SW40ev5JuOByuOY7MOXCoqMyM2u76hT2\nwQH4+/akoM9emnwghpnZOJRrKEbSXZL2Stou6eeS2ntB+MBu2Hx/snz+PLhjF3Qvc1E3s3Et7xj7\nBuDdEfFeoA/4Sv6QMjh2FDZ+A+6/Ipm467XDyfMTJrXlx5uZlVmuwh4Rj0XEsfTbTcC5+UMa2aQj\nffDAlbDxTrjsOrh1k2dhNDOr0cwx9puAtU3c3ole/Sfv3/ZVOG0yfLoHLvlYS3+cmVknUkSM3ED6\nLXDOMKtWRsQv0zYrgTnAJ6LOBiXdAtwCMG3atO6enp4xBdz1wh94Y9r7ePOU08b0+jIZHBxk4sSJ\nRYfRFFXJpSp5gHMpqzy5LFy48KmIaHzrfETkegBLgT8Cb836mu7u7hir3t7eMb+2bJxL+VQljwjn\nUlZ5cgG2RIYam2soRtIi4MvAlRHx7zzbMjOz5sh7VczdwCRgg6Rtku5rQkxmZpZDriP2iLioWYGY\nmVlzVGuuGDMzc2E3M6saF3Yzs4pxYTczqxgXdjOziml452lLfqj0IvDXMb58KvCPJoZTJOdSPlXJ\nA5xLWeXJ5R0RcVajRoUU9jwkbYkst9R2AOdSPlXJA5xLWbUjFw/FmJlVjAu7mVnFdGJhf6DoAJrI\nuZRPVfIA51JWLc+l48bYzcxsZJ14xG5mZiMofWGXdL2kXZL+I6numWRJiyQ9I2mfpBXtjDErSZMl\nbZD0bPr1zDrt3kxny9wmaV2746yn0T6WNEHS2nT9ZkkXtD/KbDLkskzSizX98Nki4mxE0o8kHZK0\ns856Sfpumud2SbPbHWNWGXK5StLhmj75WrtjzELSeZJ6Je1Ja9fnh2nT2n7JMml7kQ/gUuASYCMw\np06bk4H9wEzgVOBp4F1Fxz5MnN8EVqTLK4BVddoNFh3rWPYxcCtwX7p8I7C26Lhz5LIMuLvoWDPk\ncgUwG9hZZ/1iYD0gYB6wueiYc+RyFfBI0XFmyGM6MDtdngT0DfP71dJ+Kf0Re0TsiYhnGjSbC+yL\niOci4ijQAyxpfXSjtgRYnS6vBq4rMJbRyrKPa/N7CPioJLUxxqw65feloYh4Anh5hCZLgAcjsQk4\nQ9L09kQ3Ohly6QgR0R8RW9PlV4A9wIwhzVraL6Uv7BnNAP5W8/0BTtyRZTAtIvoh6Xzg7DrtuiRt\nkbRJUlmKf5Z9/L82EXEMOAxMaUt0o5P19+WT6dvkhySd157Qmq5T/jaymi/paUnrJV1WdDCNpMOR\nHwA2D1nV0n7J9UEbzZLlA7MbbWKY5wq53GekXEaxmfMj4qCkmcDjknZExP7mRDhmWfZxafqhgSxx\n/gpYExGvS1pO8k7kIy2PrPk6pU+y2EpyS/2gpMXAL4CLC46pLkkTgZ8BX4iII0NXD/OSpvVLKQp7\nRFydcxMHgNojqnOBgzm3OSYj5SJpQNL0iOhP33YdqrONg+nX5yRtJPmPX3Rhz7KPj7c5IOkU4G2U\n8611w1wi4qWab78PrGpDXK1Qmr+NvGqLY0Q8Kul7kqZGROnmkJH0FpKi/pOIeHiYJi3tl6oMxfwZ\nuFjSOyWdSnLirjRXk9RYByxNl5cCJ7wbkXSmpAnp8lTgQ8DutkVYX5Z9XJvfp4DHIz1TVDINcxky\n3nktyThpJ1oHfCa9CmMecPj4cGCnkXTO8XM2kuaS1K+XRn5V+6Ux/hDYExHfqtOstf1S9BnkDGeY\nP07y3+11YAD4Tfr824FHh5xl7iM5sl1ZdNx1cpkC/A54Nv06OX1+DvCDdHkBsIPkSo0dwM1Fxz3S\nPga+DlybLncBPwX2AX8CZhYdc45c7gR2pf3QC8wqOuY6eawB+oE30r+Tm4HlwPJ0vYB70jx3UOfK\nsjI8MuRyW02fbAIWFB1znTw+TDKssh3Ylj4Wt7NffOepmVnFVGUoxszMUi7sZmYV48JuZlYxLuxm\nZhXjwm5mVjEu7GZmFePCbmZWMS7sZmYV818CuQ1b82nVFAAAAABJRU5ErkJggg==\n",
"text/plain": "<matplotlib.figure.Figure at 0x2d7c17d1a90>"
}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 応用1(不等式の証明)"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "引用:https://mathtrain.jp/meanvalue"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "平均値の定理を使って不等式の証明を行う問題は超頻出です。\n\n2001年名古屋大学理系第1問\n\n\\begin{problem}\\label{prob:}\n例題\n$e$ を自然対数の底とする。 $e< p < q$ のとき以下の不等式を証明せよ:\n\n$$\n\\log{(\\log{q})} − \\log{(\\log{p})} < \\frac{q−p}{e}\n$$\n\\end{problem}\n\n"
},
{
"metadata": {
"trusted": true,
"ExecuteTime": {
"start_time": "2017-12-09T17:09:37.611479Z",
"end_time": "2017-12-09T17:09:38.084293Z"
}
},
"cell_type": "code",
"source": "q = sp.linspace(3.1, 6)\np = 3\ny = sp.log(sp.log(q)) - sp.log(sp.log(p)) \n\nplt.plot(q, y, label=\"$\\log{(\\log{q})} − \\log{(\\log{p})}$\")\nplt.plot(q, (q-p)/sp.e, label=\"$\\\\frac{q-p}{e}$\")\n\nplt.xlabel(\"q\")\nplt.legend()\nplt.grid()\nplt.show()",
"execution_count": 43,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"image/png": 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5vvFf0eaq62wcyCxhd3ohu88Usf1oBfkbNgHQwUsR3cefG2P7cmW/HlzZvwcR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\nhRDCgv4/kE7iRpaos0oAAAAASUVORK5CYII=\n",
"text/plain": "<matplotlib.figure.Figure at 0x2d7bd231a58>"
}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "解答\n\n$$\nf(x)=\\log{(\\log{x})}\n$$\n\nとおくと,証明すべき不等式は,\n\n$$\nf(q)−f(p)<\\frac{q−p}{e}\\\\\n\\frac{f(q)−f(p)}{q-p} < \\frac{1}{e}\n$$\n\nとなる。\nここで平均値の定理より,\n$$\n\\frac{f(q)−f(p)}{q-p} = f'(c)\n$$\nとなる $c$ が $p$ と $q$ の間に存在する。\n\n合成関数の微分公式を用いて $f(x)$ の導関数を計算すると \n$$\nf'(x)=\\frac{d\\log{y}}{dy}\\frac{d\\log{x}}{dx}=\\frac{1}{\\log{x}}\\frac{1}{x}\n= \\frac{1}{x \\log{x}}\n$$ \n\nであるので,証明すべき不等式は,\n$$\nf'(c) = \\frac{1}{c \\log{c}} < \\frac{1}{e}\\\\\nc \\log{c} > e\n$$\n\nここで $g(x)=x\\log{x}$ について考える.\nいま,$e < p < x < q$ より,\n\n$$\nx > e \\\\\n\\log{x} > 1\n$$\n\nよって,\n\n$$\nc \\log{c} > e\n$$\n\nが成り立つ."
},
{
"metadata": {
"trusted": true,
"ExecuteTime": {
"start_time": "2017-12-09T17:37:41.060736Z",
"end_time": "2017-12-09T17:37:41.441991Z"
}
},
"cell_type": "code",
"source": "x = sp.linspace(sp.e+0.1, 10)\ny = x * sp.log(x)\n\nplt.plot(x, y, label=\"$x \\log{x}$\")\n\nplt.hlines(sp.e, 0, 10)\n\nplt.ylim(0, 30)\nplt.legend()\nplt.grid()\nplt.show()",
"execution_count": 46,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
"image/png": 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"text/plain": "<matplotlib.figure.Figure at 0x2d7c1d58550>"
}
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 応用2(漸化式と極限)"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "$a_{n+1}=f(a_n)$ 型の漸化式で表される数列の極限を求める問題にも平均値の定理は活躍します。このタイプの問題の基本的な考え方は漸化式で表される数列の極限で詳しく解説しています。\n\n平均値の定理を使うことで所望の不等式:\n$|a_n+1−\\alpha|< k |a_n−\\alpha|$ を示しにいきます。\n\n2005年東京大学理系第3問です。\n\n問題\n\n$$\nf(x)=\\frac{x}{2}(1+e^{−2(x−1)})\n$$\n\nとする。\n"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "- (1)$x > \\frac{1}{2}$ ならば $0 \\leq f'(x)<\\frac{1}{2}$ を示せ。\n\n$$\nf(x) = \\frac{1}{2}(x+xe^{−2(x−1)})\\\\\nf'(x) = \\frac{1}{2}(1 + e^{−2(x−1)} + xe^{−2(x−1)}(-2))\\\\\n= \\frac{1}{2}(1 + e^{−2(x−1)} -2 xe^{−2(x−1)})\\\\\n= \\frac{1}{2}(1 + e^{−2(x−1)}(1 - 2x))\\\\\n= \\frac{1}{2} + (\\frac{1}{2} - x)e^{−2(x−1)}\\\\\n= \\frac{1}{2} - (x-\\frac{1}{2})e^{−2(x−1)}\\\\\n$$\n\n$x > \\frac{1}{2}$のとき,\n$$\nx-\\frac{1}{2} > 0\\\\\ne^{−2(x−1)} > 0\n$$\n\nよって,$x > \\frac{1}{2}$のとき,\n\n$$\n0 <= f'(x) < \\frac{1}{2}\n$$"
},
{
"metadata": {
"trusted": true,
"scrolled": true,
"ExecuteTime": {
"start_time": "2017-12-09T19:18:47.888371Z",
"end_time": "2017-12-09T19:18:48.468257Z"
}
},
"cell_type": "code",
"source": "x = sp.linspace(0, 5)\ny = x / 2. * (1 + sp.exp(-2. * (x - 1)))\nplt.plot(x, y, label=\"$f(x)$\")\n\ny_=1./2 - (x - 1./2)*sp.exp(-2*(x-1))\nplt.plot(x, y_, label=\"$f'(x)$\")\n\nplt.hlines(1./2, 0, 5)\n# plt.ylim(0, 30)\nplt.legend()\nplt.grid()\nplt.show()",
"execution_count": 58,
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"data": {
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dwP1a63KlVDxQppR6W2u9L8C1XVj/4B5zmamlCCHOTWvN9opTrC2tYOPOatpcbsamxvCD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\nhLCZ/w8Zod7LvkiyIgAAAABJRU5ErkJggg==\n",
"text/plain": "<matplotlib.figure.Figure at 0x2d7c1d58d30>"
}
}
]
},
{
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"source": "- (2)$x_0 > \\frac{1}{2}$ 正の数とし,\n数列 $\\{x_n\\}$ を $x_{n+1}=f(x_n)$ で定める。\nこのとき $\\lim_{n\\to \\infty}x_n=1$を証明せよ。\n\n解答\n\n$f(1)$を計算する.\n\n$$\nf(1) = \\frac{1}{2}(1 + 1\\cdot e^{−2(1−1)})\n= \\frac{1}{2}2 = 1\n$$\n\n平均値の定理より,($x_n\\neq 1$ のもとで)\n$[x_n, 1]$の範囲で,\n\n$$\n\\frac{f(x_n)−f(1)}{x_n−1}=f'(c)\n$$\n\nとなる $c$ が $1$と$x_n$ の間に存在する。\nよって,(1)と合わせると「 $x_n>\\frac{1}{2},x_n\\neq 1$ 」のもとで\n\n$$\n0 \\leq f'(c) < \\frac{1}{2} \\\\\n0 \\leq \\frac{f(x_n)−f(1)}{x_n−1} < \\frac{1}{2}\\\\\n0 \\leq \\frac{x_{n+1}−1}{x_n − 1} < \\frac{1}{2}\\\\\n|\\frac{x_{n+1}−1}{x_n − 1}| < \\frac{1}{2}\\\\\n\\frac{|x_{n+1}−1|}{|x_n − 1|} < \\frac{1}{2}\\\\\n|x_{n+1}−1| < \\frac{1}{2} |x_n − 1|\\\\\n$$\n\n- $x_0=1$ のとき,数列 $\\{x_n\\}$ の各項は全て $1$ なのでOK。\n\n- $x_0 \\neq 1$ のとき,数列 $\\{x_n\\}$ の各項は$x_n > 1/2$\n\n\n「 $x_n>1/2,x_n\\neq 1$」を満たす(数学的帰納法で簡単に証明できる)ので,\n上記の不等式を繰り返し適用できる:\n\n$$\n|x_{1}−1| < \\frac{1}{2} |x_0 − 1|\\\\\n|x_{2}−1| < \\frac{1}{2} |x_1 − 1| < \\frac{1}{2} \\frac{1}{2} |x_0 − 1|\\\\\n\\vdots \\\\\n|x_{n}−1| < \\frac{1}{2^n} |x_0 − 1| \\\\\n$$\n\nここで $n\\to \\infty$ とすると,右辺 →0 なので \n\n$$\n\\lim_{n\\to \\infty}|x_{n}−1| < \\frac{1}{2^n} |x_0 − 1| \\\\\n=|x_n|=1\\\\\n$$"
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"source": "x = 9/10\nfor i in sp.arange(10):\n x = x / 2. * (1 + sp.exp(-2. * (x - 1)))\n print(x)",
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"text": "0.999631241172\n0.999999999983\n1.0\n1.0\n1.0\n1.0\n1.0\n1.0\n1.0\n1.0\n"
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"source": "## 参考資料"
},
{
"metadata": {},
"cell_type": "markdown",
"source": "- [新訂微分積分I p.38-39](https://www.amazon.co.jp/%E6%96%B0%E8%A8%82-%E5%BE%AE%E5%88%86%E7%A9%8D%E5%88%86-1/dp/4477016506)\n\n- [平均値の定理とその応用例題2パターン](https://mathtrain.jp/meanvalue)"
}
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