{"id":850,"date":"2013-08-03T13:37:41","date_gmt":"2013-08-03T04:37:41","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=850"},"modified":"2021-09-15T09:33:15","modified_gmt":"2021-09-15T00:33:15","slug":"ngr201302","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201302\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2013\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(x \\gt 0\\) \u3068\u3057, \\(f(x) = \\log x^{100}\\) \u3068\u304a\u304f.<\/p>\r\n

(1)<\/strong>\u3000\u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u8a3c\u660e\u305b\u3088.\r\n\\[\r\n\\dfrac{100}{x+1} \\lt f(x+1) -f(x) \\lt \\dfrac{100}{x}\r\n\\]\r\n

(2)<\/strong>\u3000\u5b9f\u6570 \\(a\\) \u306e\u6574\u6570\u90e8\u5206\uff08 \\(k \\leqq a \\lt k+1\\) \u3068\u306a\u308b\u6574\u6570 \\(k\\) \uff09\u3092 \\([a]\\) \u3067\u8868\u3059.\r\n\u6574\u6570 \\(\\left[ f(1) \\right] , \\left[ f(2) \\right] , \\cdots , \\left[ f(1000) \\right]\\) \u306e\u3046\u3061\u3067\u7570\u306a\u308b\u3082\u306e\u306e\u500b\u6570\u3092\u6c42\u3081\u3088.\r\n\u5fc5\u8981\u306a\u3089\u3070 \\(\\log 10 = 2.3026\\) \u3068\u3057\u3066\u8a08\u7b97\u305b\u3088.<\/p>\r\n

\r\n\r\n

\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

\\(g(x) = \\dfrac{100}{x}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(x) = g(x) \\quad ... [1] \\ .\r\n\\]\r\n\u5b9f\u6570 \\(t \\ ( t \\gt 0 )\\) \u306b\u3064\u3044\u3066, \\(t \\lt x \\lt t+1\\) \u306b\u304a\u3044\u3066 \\(g(x)\\) \u306f\u5358\u8abf\u6e1b\u5c11\u306a\u306e\u3067\r\n\\[\r\ng(t+1) \\lt g(x) \\lt g(t) \\ .\r\n\\]\r\n\u3053\u306e\u533a\u9593\u3067, \u8fba\u3005\u3092\u7a4d\u5206\u3059\u308b\u3068, [1] \u3082\u7528\u3044\u3066\r\n\\[ \\begin{align}\r\ng(t+1) & \\lt \\displaystyle\\int_{t}^{t+1} g(x) \\, dx \\lt g(t) \\\\\r\n\\dfrac{100}{t+1} & \\lt \\left[ f(x) \\right]_{t}^{t+1} \\lt \\dfrac{100}{t} \\\\\r\n\\text{\u2234} \\quad \\dfrac{100}{t+1} & \\lt f(t+1) -f(t) \\lt \\dfrac{100}{t} \\ .\r\n\\end{align} \\]\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

    \r\n

  1. 1*<\/strong>\u3000\\(1 \\leqq n \\leqq 99\\) \u306e\u3068\u304d
    \r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nf(n+1) -f(n) \\gt \\dfrac{100}{99+1} = 1 \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(\\left[ f(n) \\right]\\) \u3068 \\(\\left[ f(n+1) \\right]\\) \u306f\u5fc5\u305a\u7570\u306a\u308b\u6574\u6570\u3068\u306a\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066, \\([f(1)] , \\cdots [f(99)]\\) \u306e\u3046\u3061, \u7570\u306a\u308b\u3082\u306e\u306e\u500b\u6570\u306f\r\n\\[\r\n99 \\ \\text{\u500b} \\ .\r\n\\]<\/li>\r\n

  2. 2*<\/strong>\u3000\\(100 \\leqq n \\leqq 1000\\) \u306e\u3068\u304d
    \r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nf(n+1) -f(n) \\lt \\dfrac{100}{100} = 1 \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(\\left[ f(n) \\right]\\) \u3068 \\(\\left[ f(n+1) \\right]\\) \u306f, \u7b49\u3057\u3044\u304b, \u5dee\u304c \\(1\\) \u3067\u3042\u308b\u304b, \u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b.\r\n\\[ \\begin{align}\r\n[ f(100) ] & = [ 200 \\log 10 ] = 460 \\ , \\\\\r\n[ f(1000) ] & = [ 300 \\log 10 ] = 690 \\ .\r\n\\end{align} \\]\r\n\u306a\u306e\u3067, \\([f(100)] , \\cdots , [f(1000)]\\) \u306f, \\(460\\) \u304b\u3089 \\(690\\) \u307e\u3067\u306e\u6574\u6570\u3092\u6f0f\u308c\u306a\u304f\u53d6\u308a\u306a\u304c\u3089\u5927\u304d\u304f\u306a\u308b.\r\n\u3057\u305f\u304c\u3063\u3066, \u7570\u306a\u308b\u3082\u306e\u306e\u500b\u6570\u306f\r\n\\[\r\n690 -460 +1 = 231 \\ \\text{\u500b} \\ .\r\n\\]<\/li>\r\n<\/ol>\r\n

    \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u500b\u6570\u306f\r\n\\[\r\n99 +231 = \\underline{330} \\ \\text{\u500b} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(x \\gt 0\\) \u3068\u3057, \\(f(x) = \\log x^{100}\\) \u3068\u304a\u304f. (1)\u3000\u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u8a3c\u660e\u305b\u3088. \\[ \\dfrac{100}{x+1} \\lt f(x+1) -f(x) \\lt \\dfrac{1 … \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":[98],"tags":[143,111],"class_list":["post-850","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2013","tag-nagoya_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/850","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=850"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/850\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=850"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=850"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=850"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}