{"id":1197,"date":"2015-07-29T10:04:20","date_gmt":"2015-07-29T01:04:20","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1197"},"modified":"2021-10-23T02:58:39","modified_gmt":"2021-10-22T17:58:39","slug":"wsr201404","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201404\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52014\uff1a\u7b2c4\u554f"},"content":{"rendered":"
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\u95a2\u6570 \\(f(x)\\) \u3092\u6b21\u306e\u7a4d\u5206\u3067\u5b9a\u7fa9\u3059\u308b.\r\n\\[\r\nf(x) = \\displaystyle\\int _ x^{x +\\log 2} \\left| e^{2t} -e^t -2 \\right| \\, dt\r\n\\]\r\n\u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(g(t) = e^{2t} -e^t -2\\) \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u3051.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(f(x)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(f(x)\\) \u304c\u6975\u5024\u3092\u3068\u308b \\(x\\) \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

    \\[\r\ng'(t) = 2 e^{2t} -e^t = e^t ( 2 e^t -1 )\r\n\\]\r\n\\(g'(t) = 0\\) \u3092\u3068\u304f\u3068 \\(e^t \\gt 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\ne^t & = \\dfrac{1}{2} \\\\\r\n\\text{\u2234} \\quad t & = -\\log 2\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {t \\rightarrow -\\infty} g(t) & = 0 -0 -2 = -2 \\\\\r\n\\displaystyle\\lim _ {t \\rightarrow \\infty} g(t) & = \\displaystyle\\lim _ {t \\rightarrow \\infty} e^t \\left( e^t -1 \\right) -2 = \\infty \\\\\r\ng( -\\log 2 ) & = \\dfrac{1}{4} -\\dfrac{1}{2} -2 = -\\dfrac{9}{4}\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \\(g(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & ( -\\infty ) & \\cdots & -\\log 2 & \\cdots & ( \\infty ) \\\\ \\hline g'(t) & & - & 0 & + & \\\\ \\hline g(t) & ( -2 ) & \\searrow & -\\dfrac{9}{4} & \\nearrow & ( \\infty ) \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \\(g(t)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0b\u56f3.<\/p>\r\n\"wsr20140401\"\r\n

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

    \\(g(t)\\) \u306e\u4e0d\u5b9a\u7a4d\u5206\u3092 \\(G(t)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\ng(t) & = \\displaystyle\\int \\left( e^{2t} -e^t -2 \\right) \\, dt \\\\\r\n& = \\dfrac{e^{2t}}{2} -e^t -2t +C \\quad ( C : \\text{\u7a4d\u5206\u5b9a\u6570} )\r\n\\end{align}\\]\r\n

    (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u3066, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(x \\lt 0\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(x) & = -\\displaystyle\\int _ x^{x +\\log 2} g(t) \\, dt \\\\\r\n& = G(x) -G( x +\\log 2 ) \\\\\r\n& = \\left( \\dfrac{e^{2x}}{2} -e^x -2x \\right) -\\left( 2 e^{2x} -2 e^x -2x +2 \\log 2 \\right) \\\\\r\n& = -\\dfrac{3 e^{2x}}{2} +e^x -2 \\log 2\r\n\\end{align}\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(0 \\leqq x \\lt \\log 2\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(x) & = -\\displaystyle\\int _ x^{\\log 2} g(t) \\, dt +-\\displaystyle\\int _ {\\log 2}^{x +\\log 2} g(t) \\, dt \\\\\r\n& = G(x) +G( x +\\log 2 ) -2 G( \\log 2 ) \\\\\r\n& = \\left( \\dfrac{e^{2x}}{2} -e^x -2x \\right) +\\left( 2 e^{2x} -2 e^x -2x +2 \\log 2 \\right) \\\\\r\n& \\qquad -2 \\left( 2 -2 -2 \\log 2 \\right) \\\\\r\n& = \\dfrac{5 e^{2x}}{2} -3 e^x -4x +2 \\log 2\r\n\\end{align}\\]<\/li>\r\n

    3. 3*<\/strong>\u3000\\(x \\geqq \\log 2\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(x) & = \\displaystyle\\int _ x^{x +\\log 2} g(t) \\, dt \\\\\r\n& = \\dfrac{3 e^{2x}}{2} -e^x +2 \\log 2\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a\r\n\\[\r\nf(x) = \\underline{\\left\\{ \\begin{array}{ll} -\\dfrac{3 e^{2x}}{2} +e^x -2 \\log 2 & ( \\ x \\lt 0 \\ \\text{\u306e\u3068\u304d} ) \\\\ \\dfrac{5 e^{2x}}{2} -3 e^x -4x +2 \\log 2 & ( \\ 0 \\leqq x \\lt \\log 2 \\ \\text{\u306e\u3068\u304d} ) \\\\ \\dfrac{3 e^{2x}}{2} -e^x +2 \\log 2 & ( \\ x \\geqq \\log 2 \\ \\text{\u306e\u3068\u304d} ) \\end{array} \\right.}\r\n\\]\r\n

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

        \r\n

      1. 1*<\/strong>\u3000\\(x \\lt 0\\) \u306e\u3068\u304d\r\n\\[\r\nf'(x) = -3 e^{2x} +e^x = -e^x \\left( 3 e^x -1 \\right)\r\n\\]\r\n\\(f'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\\begin{align}\r\ne^x & = \\dfrac{1}{3} \\\\\r\n\\text{\u2234} \\quad x & = -\\log 3\r\n\\end{align}\\]<\/li>\r\n

      2. 2*<\/strong>\u3000\\(0 \\leqq x \\lt \\log 2\\) \u306e\u3068\u304d\r\n\\[\r\nf'(x) = 5 e^{2x} -3 e^x -4\r\n\\]\r\n\\(f'(x) = 0\\) \u3092\u3068\u304f\u3068 \\(e^x \\gt 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\ne^x & = \\dfrac{3 +\\sqrt{89}}{10} \\\\\r\n\\text{\u2234} \\quad x & = \\log \\dfrac{3 +\\sqrt{89}}{10}\r\n\\end{align}\\]\r\n\uff08 \\(9 \\lt \\sqrt{89} \\lt 10\\) \u306a\u306e\u3067, \u3053\u308c\u306f2*<\/strong>\u306e\u5834\u5408\u3092\u307f\u305f\u3057\u3066\u3044\u308b. \uff09<\/p><\/li>\r\n

      3. 3*<\/strong>\u3000\\(x \\geqq \\log 2\\) \u306e\u3068\u304d
        \r\n\\[\r\nf'(x) = 3 e^{2x} -e^x = e^x \\left( 3 e^x -1 \\right) \\gt 0\r\n\\]\r\n\u306a\u306e\u3067, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0<\/p><\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a, \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccc} x & \\cdots & -\\log 3 & \\cdots & ( 0 ) & \\cdots & \\log \\dfrac{3 +\\sqrt{89}}{10} & \\cdots & ( \\log 2 ) & \\cdots \\\\ \\hline f'(x) & + & 0 & - & & - & 0 & + & & + \\\\ \\hline f(x) & \\nearrow & \\text{\u6975\u5927} & \\searrow & & \\searrow & \\text{\u6975\u5c0f} & \\nearrow & & \\nearrow \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6975\u5024\u3092\u3068\u308b\u5024\u306f\r\n\\[\r\nx = \\underline{-\\log 3 , \\ \\log \\dfrac{3 +\\sqrt{89}}{10}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\(f(x)\\) \u3092\u6b21\u306e\u7a4d\u5206\u3067\u5b9a\u7fa9\u3059\u308b. \\[ f(x) = \\displaystyle\\int _ x^{x +\\log 2} \\left| e^{2t} -e^t -2 \\right| \\, dt \\] \u6b21\u306e\u554f … \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":[120],"tags":[147,112],"class_list":["post-1197","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2014","tag-waseda_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1197","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=1197"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1197\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1197"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1197"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}