{"id":555,"date":"2012-12-29T23:05:45","date_gmt":"2012-12-29T14:05:45","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=555"},"modified":"2021-10-31T07:44:13","modified_gmt":"2021-10-30T22:44:13","slug":"tkb200904","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkb200904\/","title":{"rendered":"\u6771\u5927\u6587\u7cfb2009\uff1a\u7b2c4\u554f"},"content":{"rendered":"
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\\(2\\) \u6b21\u4ee5\u4e0b\u306e\u6574\u5f0f \\(f(x) = ax^2+bx+c\\) \u306b\u5bfe\u3057\r\n\\[\r\nS = \\displaystyle\\int _ 0^2 \\left| f'(x) \\right| \\, dx\r\n\\]\r\n\u3092\u8003\u3048\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(f(0) = 0\\) , \\(f(2) = 2\\) \u306e\u3068\u304d \\(S\\) \u3092 \\(a\\) \u306e\u95a2\u6570\u3068\u3057\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(f(0) = 0\\) , \\(f(2) = 2\\) \u3092\u307f\u305f\u3057\u306a\u304c\u3089 \\(f\\) \u304c\u5909\u5316\u3059\u308b\u3068\u304d, \\(S\\) \u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\nf(0) & = c = 0 , \\\\\r\nf(2) & = 4a+2b+c = 2 \\\\\r\n\\text{\u2234} \\quad b & = 1-2a , \\ c = 0\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x) = ax^2 +(1-2a) x\\) \u306a\u306e\u3067\r\n\\[\r\nf'(x) = 2ax -2a+1\r\n\\]\r\n\\(f'(1) = 1\\) \u3067\u50be\u304d \\(2a\\) \u306e\u76f4\u7dda\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(a \\leqq -\\dfrac{1}{2}\\) \u306e\u3068\u304d\r\n\"tokyo_b_2009_04_01\"\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{2} ( 1-2a ) \\left( 1 -\\dfrac{1}{2a} \\right) +\\dfrac{1}{2} ( -1-2a ) \\left( 1 +\\dfrac{1}{2a} \\right) \\\\\r\n& = -\\dfrac{(1-2a)^2}{4a} -\\dfrac{(1+2a)^2}{4a} \\\\\r\n& = -2a -\\dfrac{1}{2a}\r\n\\end{align}\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(\\dfrac{1}{2} \\lt a \\lt -\\dfrac{1}{2}\\) \u306e\u3068\u304d\r\n\"tokyo_b_2009_04_02\"\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{2} \\left\\{ (1+2a) +(1-2a) \\right\\} \\cdot 2 \\\\\r\n& = 2\r\n\\end{align}\\]<\/li>\r\n

    3. 3*<\/strong>\u3000\\(a \\geqq \\dfrac{1}{2}\\) \u306e\u3068\u304d\r\n\"tokyo_b_2009_04_03\"\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{2} ( -1+2a ) \\left( 1 -\\dfrac{1}{2a} \\right) +\\dfrac{1}{2} ( 1+2a ) \\left( 1 +\\dfrac{1}{2a} \\right) \\\\\r\n& = \\dfrac{(1-2a)^2}{4a} +\\dfrac{(1+2a)^2}{4a} \\\\\r\n& = 2a +\\dfrac{1}{2a}\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      \u3088\u3063\u3066, 1*<\/strong>\uff5e3*<\/strong>\u3088\u308a\r\n\\[\r\nS = \\underline{\\left\\{ \\begin{array}{ll} -2a -\\dfrac{1}{2a} & \\left( \\ a \\leqq -\\dfrac{1}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ 2 & \\left( \\ -\\dfrac{1}{2} \\lt a \\lt \\dfrac{1}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ 2a +\\dfrac{1}{2a} & \\left( \\ a \\geqq \\dfrac{1}{2} \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\r\n\\]\r\n

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

      (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(|a| \\geqq \\dfrac{1}{2}\\) \u306e\u3068\u304d, \u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nS & = |2a| +\\dfrac{1}{|2a|} \\\\\r\n& \\geqq 2 \\sqrt{|2a| \\cdot \\dfrac{1}{|2a|}} = 2\r\n\\end{align}\\]\r\n\u7b49\u53f7\u304c\u6210\u7acb\u3059\u308b\u306e\u306f\r\n\\[\\begin{align}\r\n|2a| & = \\dfrac{1}{|2a|} \\\\\r\n\\text{\u2234} \\quad a & = \\pm \\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(S\\) \u306f \\(|a| \\leqq \\dfrac{1}{2}\\) \u306e\u3068\u304d, \u6700\u5c0f\u5024 \\(\\underline{2}\\) \u3092\u3068\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(2\\) \u6b21\u4ee5\u4e0b\u306e\u6574\u5f0f \\(f(x) = ax^2+bx+c\\) \u306b\u5bfe\u3057 \\[ S = \\displaystyle\\int _ 0^2 \\left| f'(x) \\right| \\, dx \\] \u3092\u8003\u3048\u308b. (1)\u3000\\ … \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":[81],"tags":[149,15],"class_list":["post-555","post","type-post","status-publish","format-standard","hentry","category-tokyo_b_2009","tag-tokyo_b","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/555","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=555"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/555\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=555"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=555"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=555"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}