{"id":86,"date":"2011-11-26T23:16:40","date_gmt":"2011-11-26T14:16:40","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=86"},"modified":"2021-10-20T15:20:21","modified_gmt":"2021-10-20T06:20:21","slug":"ykr201104","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201104\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2011\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(xy\\) \u5e73\u9762\u4e0a\u306e \\(2\\) \u66f2\u7dda \\(C _ 1 : \\ y= \\dfrac{\\log x}{x}\\) \u3068 \\(C _ 2 : \\ y = ax^2\\) \u306f\u70b9 P \u3092\u5171\u6709\u3057, P \u306b\u304a\u3044\u3066\u5171\u901a\u306e\u63a5\u7dda\u3092\u3082\u3063\u3066\u3044\u308b.\r\n\u305f\u3060\u3057, \\(a\\) \u306f\u5b9a\u6570\u3068\u3059\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\u95a2\u6570 \\(y = \\dfrac{\\log x}{x}\\) \u306e\u5897\u6e1b, \u6975\u5024, \u30b0\u30e9\u30d5\u306e\u51f9\u51f8, \u5909\u66f2\u70b9\u3092\u8abf\u3079, \\(C _ 1\\) \u306e\u6982\u5f62\u3092\u63cf\u3051. \u305f\u3060\u3057, \\(\\displaystyle\\lim _ {x \\rightarrow \\infty} \\dfrac{\\log x}{x}=0\\) \u306f\u8a3c\u660e\u306a\u3057\u306b\u7528\u3044\u3066\u3088\u3044.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000P \u306e\u5ea7\u6a19\u304a\u3088\u3073 \\(a\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\u4e0d\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int \\left( \\dfrac{\\log x}{x} \\right)^2 \\, dx\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(4)<\/strong>\u3000\\(C _ 1 , C _ 2\\) \u304a\u3088\u3073 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u3092, \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d \\(V\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\u95a2\u6570 \\(y =\\dfrac{\\log x}{x}\\) \u306e\u5b9a\u7fa9\u57df\u306f, \\(x \\gt 0\\) .\r\n\\[\\begin{align}\r\ny' & = \\dfrac{\\dfrac{1}{x} \\cdot x -1 \\cdot \\log x}{x^2} \\\\\r\n& = \\dfrac{1 -\\log x}{x^2} , \\\\\r\ny'' & = \\dfrac{-\\dfrac{1}{x} \\cdot x^2 -2x( 1-\\log x )}{x^4} \\\\\r\n& = \\dfrac{2 \\log x -3}{x^3}\r\n\\end{align}\\]\r\n\\(y'=0\\) \u3092\u89e3\u304f\u3068\r\n\\[\r\nx = e\r\n\\]\r\n\\(y''=0\\) \u3092\u89e3\u304f\u3068\r\n\\[\r\nx = e^{\\frac{3}{2}}\r\n\\]\r\n\\(t = \\dfrac{1}{x}\\) \u3068\u304a\u304f\u3068, \\(x \\rightarrow +0\\) \u306e\u3068\u304d, \\(t \\rightarrow +\\infty\\) \u306a\u306e\u3067\r\n\\[\r\nf(x) = t \\log \\dfrac{1}{t} = -t \\log t \\rightarrow -\\infty\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\displaystyle\\lim _ {x \\rightarrow +0} f(x) = -\\infty\r\n\\]\r\n\u307e\u305f\r\n\\[\r\n\\displaystyle\\lim _ {x \\rightarrow \\infty} \\dfrac{\\log x}{x} = 0\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \\(f(x)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccccc} x & (0) & \\cdots & e & \\cdots & e^{\\frac{3}{2}} & \\cdots & ( +\\infty ) \\\\ \\hline f'(x) & & + & 0 & - & & - & \\\\ \\hline f''(x) & & - & & - & 0 & + & \\\\ \\hline f(a) & (-\\infty) & \\nearrow (\\cap) & \\dfrac{1}{e} & \\searrow (\\cap) & \\dfrac{3}{2 e^{\\frac{3}{2}}} & \\searrow (\\cup) & ( 0 ) \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066<\/p>\r\n<ul>\r\n<li><p>\u6975\u5024\u306f, \\(\\underline{x=e}\\) \u306e\u3068\u304d, \u6975\u5927\u5024 \\(\\underline{\\dfrac{1}{e}}\\)<\/p><\/li>\r\n<li><p>\u5909\u66f2\u70b9\u306f, \\(\\underline{\\left( e^{\\frac{3}{2}} , \\dfrac{3}{2 e^{\\frac{3}{2}}} \\right)}\\)<\/p><\/li>\r\n<\/ul>\r\n<p>\u3067\u3042\u308a, \u30b0\u30e9\u30d5\u306e\u6982\u5f62\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/yokokoku_2011_04_01.png\" alt=\"yokokoku_2011_04_01\" class=\"aligncenter size-full\" \/>\r\n<p><strong>(2)<\/strong><\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/yokokoku_2011_04_02.png\" alt=\"yokokoku_2011_04_02\" class=\"aligncenter size-full\" \/>\r\n<p>\\(C _ 2 : \\ y = ax^2\\) \u306b\u3064\u3044\u3066\r\n\\[\r\ny' = 2ax\r\n\\]\r\nP \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(p\\) \u3068\u304a\u3051\u3070, \\(C _ 1\\) \u3068 \\(C _ 2\\) \u306e \\(y\\) \u5ea7\u6a19, \u63a5\u7dda\u306e\u50be\u304d\u304c\u3068\u3082\u306b\u7b49\u3057\u3044\u306e\u3067\r\n\\[\r\n\\left\\{ \\begin{array}{l} ap^2 = \\dfrac{\\log p}{p} \\\\ 2ap = \\dfrac{1 -\\log p}{p^2} \\end{array} \\right.\r\n\\]\r\n\u3053\u308c\u3088\u308a\r\n\\[\\begin{align}\r\nap^3 = \\log p & = \\dfrac{1 -\\log p}{2} \\\\\r\n\\text{\u2234} \\quad \\log p & = \\dfrac{1}{3} \\\\\r\n\\text{\u2234} \\quad p & = e^{\\frac{1}{3}}\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, P \u306e \\(y\\) \u5ea7\u6a19\u306f\r\n\\[\\begin{gather}\r\n\\dfrac{\\log p}{p} = \\dfrac{1}{3 e^{\\frac{1}{3}}} \\\\\r\n\\text{\u2234} \\quad \\text{P} \\ \\underline{\\left( e^{\\frac{1}{3}} , \\dfrac{1}{3 e^{\\frac{1}{3}}} \\right)}\r\n\\end{gather}\\]\r\n\u307e\u305f\r\n\\[\r\na = \\dfrac{\\log p}{p^3} = \\underline{\\dfrac{1}{3e}}\r\n\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p>\\[\\begin{align}\r\n\\displaystyle\\int \\left( \\dfrac{\\log x}{x} \\right)^2 \\, dx & = -\\dfrac{( \\log x )^2}{x} +\\displaystyle\\int \\dfrac{2 \\log x}{x^2} \\, dx \\\\\r\n& = -\\dfrac{( \\log x )^2}{x} -\\dfrac{2 \\log x}{x} +2 \\displaystyle\\int \\dfrac{1}{x^2} \\, dx \\\\\r\n& = \\underline{-\\dfrac{( \\log x )^2}{x} -\\dfrac{2 \\log x}{x} -\\dfrac{2}{x^2} +C}\r\n\\end{align}\\]\r\n\u305f\u3060\u3057, \\(C\\) \u306f\u7a4d\u5206\u5b9a\u6570.<\/p>\r\n<p><strong>(4)<\/strong><\/p>\r\n<p>\u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u56de\u8ee2\u4f53 \\(V _ 1 , V _ 2\\) \u3092\u7528\u3044\u3066\r\n\\[\r\nV = V _ 1 -V _ 2\r\n\\]\r\n\u3068\u8868\u305b\u308b.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/yokokoku_2011_04_03.png\" alt=\"yokokoku_2011_04_03\" class=\"aligncenter size-full\" \/>\r\n<p>\\[\\begin{align}\r\nV _ 1 & = \\pi \\displaystyle\\int _ 0^{e^{\\frac{1}{3}}} \\left( \\dfrac{x^2}{3e} \\right)^2 \\, dx \\\\\r\n& = \\dfrac{1}{9e^2} \\left[ \\dfrac{x^5}{5} \\right] _ 0^{e^{\\frac{1}{3}}} = \\dfrac{1}{9e^2} \\cdot \\dfrac{e^{\\frac{5}{3}}}{5} \\\\\r\n& = \\dfrac{\\pi}{45 e^{\\frac{1}{3}}} , \\\\\r\nV _ 2 & = \\pi \\displaystyle\\int _ 1^{e^{\\frac{1}{3}}} \\left( \\dfrac{\\log x}{x} \\right)^2 \\, dx \\\\\r\n& = \\pi \\left[ -\\dfrac{( \\log x )^2}{x} -\\dfrac{2 \\log x}{x} -\\dfrac{2}{x^2} \\right] _ 1^{e^{\\frac{1}{3}}} \\\\\r\n& = \\left( 2 -\\dfrac{\\dfrac{1}{9} +2 \\cdot \\dfrac{1}{3} +2}{e^{\\frac{1}{3}}} \\right) \\pi \\\\\r\n& = \\left( 2 -\\dfrac{25}{9 e^{\\frac{1}{3}}} \\right) \\pi\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\r\nV = V _ 1 -V _ 2 = \\underline{\\left( \\dfrac{14}{5 e^{\\frac{1}{3}}} -2 \\right) \\pi}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306e \\(2\\) \u66f2\u7dda \\(C _ 1 : \\ y= \\dfrac{\\log x}{x}\\) \u3068 \\(C _ 2 : \\ y = ax^2\\) \u306f\u70b9 P \u3092\u5171\u6709\u3057, P \u306b\u304a\u3044\u3066\u5171\u901a\u306e\u63a5\u7dda\u3092\u3082\u3063\u3066\u3044\u308b. &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/ykr201104\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[44],"tags":[13,9],"class_list":["post-86","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2011","tag-13","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/86","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=86"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/86\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=86"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=86"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=86"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}