{"id":275,"date":"2012-01-28T17:15:42","date_gmt":"2012-01-28T08:15:42","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=275"},"modified":"2021-09-10T21:14:13","modified_gmt":"2021-09-10T12:14:13","slug":"tok200801","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok200801\/","title":{"rendered":"\u6771\u5de5\u59272008\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\u6b63\u306e\u5b9f\u6570 \\(a , b\\) \u306b\u5bfe\u3057, \\(x \\gt 0\\) \u3067\u5b9a\u7fa9\u3055\u308c\u305f \\(2\\) \u3064\u306e\u95a2\u6570 \\(x^a , \\ \\log bx\\) \u306e\u30b0\u30e9\u30d5\u304c \\(1\\) \u70b9\u3067\u63a5\u3059\u308b\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u63a5\u70b9\u306e\u5ea7\u6a19 \\(( s , t )\\) \u3092 \\(a\\) \u3092\u7528\u3044\u3066\u8868\u305b. \u307e\u305f, \\(b\\) \u3092 \\(a\\) \u306e\u95a2\u6570\u3068\u3057\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(0 \\lt h \\lt s\\) \u3092\u307f\u305f\u3059 \\(h\\) \u306b\u5bfe\u3057, \u76f4\u7dda \\(x = h\\) \u304a\u3088\u3073 \\(2\\) \u3064\u306e\u66f2\u7dda \\(y = x^a , \\ y = \\log bx\\) \u3067\u56f2\u307e\u308c\u308b\u9818\u57df\u306e\u9762\u7a4d\u3092 \\(A(h)\\) \u3068\u3059\u308b. \\(\\displaystyle\\lim _ {h \\rightarrow 0} A(h)\\) \u3092 \\(a\\) \u3067\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\(2\\) \u3064\u306e\u95a2\u6570\u3092\u305d\u308c\u305e\u308c\u5fae\u5206\u3059\u308b\u3068\r\n\\[\r\ny' = a x^{a-1} , \\quad y' = \\dfrac{1}{x}\r\n\\]\r\n\u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\ns^a = \\log bs & = t \\quad ... [1] , \\\\\r\na s^{a-1} & = \\dfrac{1}{s} \\quad ... [2]\r\n\\end{align}\\]\r\n[2] \u3088\u308a\r\n\\[\\begin{align}\r\ns^a & =\\dfrac{1}{a} \\\\\r\n\\text{\u2234} \\quad s & = \\underline{\\dfrac{1}{a^{\\frac{1}{a}}}}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 [1] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\r\nt = \\left( \\dfrac{1}{a^{\\frac{1}{a}}} \\right)^a = \\underline{\\dfrac{1}{a}}\r\n\\]\r\n\u307e\u305f\r\n\\[\r\nb =\\dfrac{e^t}{s} = \\underline{(ea)^{\\frac{1}{a}}}\r\n\\]\r\n

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

    \u4e0e\u3048\u3089\u308c\u305f\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"\"\r\n

    \u4e0d\u5b9a\u7a4d\u5206 \\(F(x) = \\displaystyle\\int \\left( x^a -\\log bx \\right) \\, dx\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nA(h) = F(s) -F(h)\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nF(x) & = \\dfrac{x^{a+1}}{a+1} -x \\log bx +\\displaystyle\\int x \\cdot \\dfrac{1}{x} \\, dx \\\\\r\n& = \\dfrac{x^{a+1}}{a+1} -x \\log bx +x +C \\quad ( C \\text{\u306f\u7a4d\u5206\u5b9a\u6570} )\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nF(s) & = \\dfrac{1}{(a+1) a^{\\frac{a+1}{a}}} -\\dfrac{1}{a^{\\frac{1}{a}}} \\cdot \\log e^{\\frac{1}{a}} +a^{\\frac{1}{a}} \\\\\r\n& = \\dfrac{1}{a^{\\frac{1}{a}}} \\left\\{ \\dfrac{1}{a(a+1)} -\\dfrac{1}{a} +1 \\right\\} \\\\\r\n& = \\dfrac{a}{(a+1) a^{\\frac{1}{a}}}\r\n\\end{align}\\]\r\n\\(F(h)\\) \u306b\u3064\u3044\u3066\u8003\u3048\u308b\u3068\r\n\\[\r\n\\displaystyle\\lim _ {h \\rightarrow 0} \\dfrac{h^{a+1}}{a+1} +h = 0\r\n\\]\r\n\u307e\u305f, \\(h \\log bh\\) \u306b\u3064\u3044\u3066, \\(h = \\dfrac{1}{t}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\displaystyle\\lim _ {h \\rightarrow 0} h \\log bh = \\displaystyle\\lim _ {t \\rightarrow \\infty} -\\dfrac{\\log bt}{t} = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\r\n\\displaystyle\\lim _ {h \\rightarrow 0} A(h) = \\underline{\\dfrac{a}{(a+1) a^{\\frac{1}{a}}}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b63\u306e\u5b9f\u6570 \\(a , b\\) \u306b\u5bfe\u3057, \\(x \\gt 0\\) \u3067\u5b9a\u7fa9\u3055\u308c\u305f \\(2\\) \u3064\u306e\u95a2\u6570 \\(x^a , \\ \\log bx\\) \u306e\u30b0\u30e9\u30d5\u304c \\(1\\) \u70b9\u3067\u63a5\u3059\u308b\u3068\u3059\u308b. (1)\u3000\u63a5\u70b9\u306e\u5ea7\u6a19 \\(( s , … \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":[50],"tags":[141,16],"class_list":["post-275","post","type-post","status-publish","format-standard","hentry","category-toko_2008","tag-toko","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/275","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=275"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/275\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=275"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=275"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=275"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}