{"id":296,"date":"2012-02-08T00:30:57","date_gmt":"2012-02-07T15:30:57","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=296"},"modified":"2021-09-14T15:54:12","modified_gmt":"2021-09-14T06:54:12","slug":"iks200803","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks200803\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272008\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\u5fae\u5206\u53ef\u80fd\u306a\u95a2\u6570 \\(f(x) , g(x)\\) \u304c\u6b21\u306e \\(4\\) \u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u308b.<\/p>\r\n

    \r\n

  1. (a)\u3000\u4efb\u610f\u306e\u6b63\u306e\u5b9f\u6570 \\(x\\) \u306b\u3064\u3044\u3066, \\(f(x) \\gt 0 , \\ g(x) \\gt 0\\)<\/p><\/li>\r\n

  2. (b)\u3000\u4efb\u610f\u306e\u5b9f\u6570 \\(x\\) \u306b\u3064\u3044\u3066, \\(f(-x) =f(x) , \\ g(-x) = -g(x)\\)<\/p><\/li>\r\n

  3. (c)\u3000\u4efb\u610f\u306e\u5b9f\u6570 \\(x\\) , \\(y\\) \u306b\u3064\u3044\u3066, \\(f(x+y) = f(x)f(y) +g(x)g(y)\\)<\/p><\/li>\r\n

  4. (d)\u3000\\(\\displaystyle\\lim _ {x \\rightarrow 0} \\dfrac{g(x)}{x} = 2\\)<\/p><\/li>\r\n<\/ol>\r\n

    \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\\(f(0)\\) \u304a\u3088\u3073 \\(g(0)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\\(\\left\\{ f(x) \\right\\}^2 -\\left\\{ g(x) \\right\\}^2\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    3. (3)<\/strong>\u3000\\(\\displaystyle\\lim _ {x \\rightarrow 0} \\dfrac{1-f(x)}{x^2}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    4. (4)<\/strong>\u3000\\(f(x)\\) \u306e\u5c0e\u95a2\u6570\u3092 \\(g(x)\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

    5. (5)<\/strong>\u3000\u66f2\u7dda \\(y = f(x)g(x)\\) , \u76f4\u7dda \\(x = a \\ ( a \\gt 0 )\\) \u304a\u3088\u3073 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u56f3\u5f62\u306e\u9762\u7a4d\u304c \\(1\\) \u306e\u3068\u304d, \\(f(a)\\) \u306e\u5024\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

      \u6761\u4ef6 (b) \u306b \\(x=0\\) \u3092\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\ng(0) & = -g(0) \\\\\r\n\\text{\u2234} \\quad g(0) & = \\underline{0}\n\\end{align}\\]\r\n\u6761\u4ef6 (c) \u306b \\(x=y=0\\) \u3092\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\nf(0) = \\left\\{ f(0) \\right\\}^2 & -\\left\\{ g(0) \\right\\}^2 = \\left\\{ f(0) \\right\\}^2 \\\\\r\n\\text{\u2234} \\quad f(0) & = 0, 1\n\\end{align}\\]\r\n\\(f(0)=0\\) \u3068\u4eee\u5b9a\u3059\u308b\u3068, \u6761\u4ef6 (c) \u306b \\(y=0\\) \u3092\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\r\nf(x) = f(0) f(x) +g(0) g(x) =0\n\\]\r\n\u3053\u308c\u306f\u6761\u4ef6 (a) \u306b\u77db\u76fe\u3059\u308b\u306e\u3067, \u4e0d\u9069.
      \r\n\u3088\u3063\u3066\r\n\\[\r\nf(0) = \\underline{1}\n\\]\r\n

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

      \u6761\u4ef6 (c) \u306b \\(y =-x \\ ( x \\gt 0 )\\) \u3092\u4ee3\u5165\u3059\u308c\u3070, \u6761\u4ef6 (b) \u3092\u7528\u3044\u3066\r\n\\[\r\nf(0) = \\left\\{ f(x) \\right\\}^2 -\\left\\{ g(x) \\right\\}^2 = \\underline{1}\n\\]\r\n

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

      (2)<\/strong> \u306e\u7d50\u679c\u3068, \u6761\u4ef6 (d) \u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{1 -f(x)}{x^2} & = \\dfrac{1 -\\left\\{ f(x) \\right\\}^2}{x^2} \\cdot \\dfrac{1}{1 +f(x)} \\\\\r\n& = -\\left\\{ \\dfrac{g(x)}{x} \\right\\}^2 \\cdot \\dfrac{1}{1 +f(x)} \\\\\r\n& \\rightarrow -2^2 \\cdot \\dfrac{1}{1+1} = -2 \\quad ( \\ x \\rightarrow 0 \\text{\u306e\u3068\u304d} )\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {x \\rightarrow 0} \\dfrac{1-f(x)}{x^2} = \\underline{-2}\n\\]\r\n

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

      \u6761\u4ef6 (c) , (d) \u3068 (3)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{f(x+h) -f(x)}{h} & = \\dfrac{f(x)f(h) +g(x)g(h) -f(x)}{h} \\\\\r\n& = -\\dfrac{1 -f(h)}{h^2} \\cdot h f(x) +\\dfrac{g(h)}{h} g(x) \\\\\r\n& \\rightarrow -(-2) \\cdot 0 \\cdot f(x) +2 g(x) \\quad ( \\ h \\rightarrow 0 \\text{\u306e\u3068\u304d} ) \\\\\r\n& = 2g(x)\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nf'(x) = \\underline{2g(x)}\n\\]\r\n

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

      \u6761\u4ef6 (a) \u3088\u308a, \\(f(x)g(x) \\gt 0\\) \u306a\u306e\u3067, \u4e0e\u3048\u3089\u308c\u305f\u9818\u57df\u306e\u9762\u7a4d\u3092 \\(S\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nS =\\displaystyle\\int _ 0^a f(x) g(x) \\, dx\n\\]\r\n(4)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{4} \\displaystyle\\int _ 0^a 2 f(x) f'(x) \\, dx \\\\\r\n& = \\dfrac{1}{4} \\left[ \\left\\{ f(x) \\right\\}^2 \\right] _ 0^a =\\dfrac{\\left\\{ f(a) \\right\\}^2 -1}{4}\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\dfrac{\\left\\{ f(a) \\right\\}^2 -1}{4} & = 1 \\\\\r\n\\text{\u2234} \\quad f(a) & = \\underline{\\sqrt{5}} \\quad ( \\ \\text{\u2235} \\ f(a) \\gt 0\\ )\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5fae\u5206\u53ef\u80fd\u306a\u95a2\u6570 \\(f(x) , g(x)\\) \u304c\u6b21\u306e \\(4\\) \u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u308b. (a)\u3000\u4efb\u610f\u306e\u6b63\u306e\u5b9f\u6570 \\(x\\) \u306b\u3064\u3044\u3066, \\(f(x) \\gt 0 , \\ g(x) \\gt 0\\) (b)\u3000\u4efb\u610f\u306e\u5b9f\u6570 \\ … \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":[52],"tags":[145,16],"class_list":["post-296","post","type-post","status-publish","format-standard","hentry","category-ikashika_2008","tag-ikashika","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/296","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=296"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/296\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=296"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=296"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=296"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}