{"id":433,"date":"2012-06-20T00:35:16","date_gmt":"2012-06-19T15:35:16","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=433"},"modified":"2021-09-30T08:41:16","modified_gmt":"2021-09-29T23:41:16","slug":"kbr201203","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201203\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2012\uff1a\u7b2c3\u554f"},"content":{"rendered":"
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\u66f2\u7dda \\(C : \\ y =\\log x \\ ( x \\gt 0 )\\) \u3092\u8003\u3048\u308b.\r\n\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \u66f2\u7dda \\(C\\) \u4e0a\u306b\u70b9 \\(P ( e^n , n ) , \\ Q ( e^{2n} , 2n )\\) \u3092\u3068\u308a, \\(x\\) \u8ef8\u4e0a\u306b\u70b9 \\(A ( e^n , 0 ) , \\ B ( e^{2n} , 0 )\\) \u3092\u3068\u308b. \u56db\u89d2\u5f62 \\(APQB\\) \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3055\u305b\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092 \\(V(n)\\) \u3068\u3059\u308b. \u307e\u305f, \u7dda\u5206 \\(PQ\\) \u3068\u66f2\u7dda \\(C\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3055\u305b\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092 \\(S(n)\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(V(n)\\) \u3092 \\(n\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{S _ n}{V _ n}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \u56de\u8ee2\u4f53\u306f, \u5e95\u9762\u3068\u4e0a\u9762\u306e\u5186\u306e\u534a\u5f84\u306e\u6bd4\u304c \\(2:1\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nV(n) & = \\dfrac{2^3 -1^3}{2^3} \\cdot \\dfrac{1}{3} \\cdot \\pi (2n)^2 \\cdot 2 \\left( e^{2n} -e^n \\right) \\\\\r\n& =\\underline{\\dfrac{7 \\pi}{3} n^2 e^n \\left( e^n-1 \\right)}\r\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\n\\displaystyle\\int \\left( \\log x \\right)^2 \\, dx & = x \\left( \\log x \\right)^2 -\\displaystyle\\int x \\cdot 2 \\log x \\cdot \\dfrac{1}{x} \\, dx \\\\\r\n& = x \\left( \\log x \\right)^2 -2x \\log x +2 \\displaystyle\\int x \\cdot \\dfrac{1}{x} \\, dx \\\\\r\n& = x \\left( \\log x \\right)^2 -2x \\log x +2x +C \\quad ( \\ C \\text{\u306f\u7a4d\u5206\u5b9a\u6570} )\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nS(n)+V(n) & = \\pi \\displaystyle\\int _ {e^n}^{e^{2n}} \\left( \\log x \\right)^2 \\, dx \\\\\r\n& = \\pi \\left[ x \\left\\{ \\left( \\log x \\right)^2 -2 \\log x +2 \\right\\} \\right] _ {e^n}^{e^{2n}} \\\\\r\n& = \\pi \\left\\{ e^{2n} \\left( 4n^2-4n+2 \\right) -e^n \\left( n^2-2n+2 \\right) \\right\\}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\dfrac{S(n)+V(n)}{V(n)} & = \\dfrac{e^{2n} \\left( 4n^2-4n+2 \\right) -e^n \\left( n^2-2n+2 \\right)}{\\dfrac{7}{3} n^2 e^n \\left( e^n-1 \\right)} \\\\\r\n& =\\dfrac{4 -\\dfrac{4}{n} +\\dfrac{2}{n^2} -e^{-n} \\left( 1 -\\dfrac{2}{n} +\\dfrac{2}{n^2} \\right)}{\\dfrac{7}{3} \\left( 1 -e^{-n} \\right)} \\\\\r\n& \\rightarrow \\dfrac{12}{7} \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{S _ n}{V _ n} = \\dfrac{12}{7} -1 = \\underline{\\dfrac{5}{7}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u66f2\u7dda \\(C : \\ y =\\log x \\ ( x \\gt 0 )\\) \u3092\u8003\u3048\u308b. \u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \u66f2\u7dda \\(C\\) \u4e0a\u306b\u70b9 \\(P ( e^n , n ) , \\ Q ( e^{2n} , 2n )\\ … \u7d9a\u304d\u3092\u8aad\u3080 →<\/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":[65],"tags":[144,68],"class_list":["post-433","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2012","tag-tsukuba_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/433","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=433"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/433\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=433"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=433"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}