{"id":1310,"date":"2015-10-19T00:46:03","date_gmt":"2015-10-18T15:46:03","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1310"},"modified":"2021-10-20T13:30:15","modified_gmt":"2021-10-20T04:30:15","slug":"ykr201501","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201501\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2015\uff1a\u7b2c1\u554f"},"content":{"rendered":"
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\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

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  1. (1)<\/strong>\u3000\u5b9a\u7a4d\u5206\r\n\\[\r\n\\displaystyle\\int _ 0^{\\log 3} \\dfrac{dx}{e^x +5 e^{-x} -2}\r\n\\]\r\n\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(x \\gt 0\\) \u306e\u3068\u304d, \u4e0d\u7b49\u5f0f\r\n\\[\r\n\\log x \\geqq \\dfrac{5x^2 -4x -1}{2x (x+2)}\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \u6c42\u3081\u308b\u7a4d\u5206\u5024\u3092 \\(I\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nI & = \\displaystyle\\int _ 0^{\\log 3} \\dfrac{e^x}{e^{2x} -2 e^x +5} \\, dx \\\\\r\n& = \\displaystyle\\int _ 0^{\\log 3} \\dfrac{( e^x -1 )'}{( e^x -1 )^2 +4} \\, dx\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(2 \\tan \\theta = e^x -1 \\ \\left( -\\dfrac{\\pi}{2} \\lt \\theta \\lt \\dfrac{\\pi}{2} \\right)\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{gather}\r\n\\dfrac{2}{\\cos^2 \\theta} \\, d \\theta = e^x \\, dx , \\\\\r\n\\begin{array}{c|ccc} x & 0 & \\rightarrow & \\log 3 \\\\ \\hline \\theta & 0 & \\rightarrow & \\dfrac{\\pi}{4} \\end{array}\r\n\\end{gather}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nI & = \\displaystyle\\int _ 0^{\\frac{\\pi}{4}} \\dfrac{1}{4 ( \\tan^2 \\theta +1 )} \\cdot \\dfrac{2}{\\cos^2 \\theta} \\, d \\theta \\\\\r\n& = \\dfrac{1}{2} \\big[ \\theta \\big] _ 0^{\\frac{\\pi}{4}} = \\underline{\\dfrac{\\pi}{8}}\r\n\\end{align}\\]\r\n

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

    \\(f(x) = \\log x -\\dfrac{5x^2 -4x -1}{2x (x+2)}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf'(x) & = \\dfrac{1}{x} -\\dfrac{(10x-4) x (x+2) -( 5x^2 -4x -1 ) (2x+2)}{2 x^2 (x+2)^2} \\\\\r\n& = \\dfrac{x (x+2)^2 -( 5x^3 +8x^2 -4x ) +( 5x^3 +x^2 -5x -1 )}{x^2 (x+2)^2} \\\\\r\n& = \\dfrac{x^3 -3x^2 +3x -1}{x^2 (x+2)^2} \\\\\r\n& = \\dfrac{(x-1)^3}{x^2 (x+2)^2}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(x \\gt 0\\) \u306b\u304a\u3051\u308b \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|cccc} x & (0) & \\cdots & 1 & \\cdots \\\\ \\hline f'(x) & & - & 0 & + \\\\ \\hline f(x) & & \\searrow & \\text{\u6700\u5c0f} & \\nearrow \\end{array}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nf(x) \\geqq f(1) = 0\r\n\\]\r\n\u306a\u306e\u3067, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u5b9a\u7a4d\u5206 \\[ \\displaystyle\\int _ 0^{\\log 3} \\dfrac{dx}{e^x +5 e^{-x} -2} \\] \u3092\u6c42\u3081\u3088. (2)\u3000\\(x \\gt 0\\) \u306e\u3068\u304d, … \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":[133],"tags":[137,9],"class_list":["post-1310","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2015","tag-137","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1310","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=1310"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1310\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1310"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1310"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1310"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}