{"id":702,"date":"2013-03-26T21:24:05","date_gmt":"2013-03-26T12:24:05","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=702"},"modified":"2021-10-04T19:15:16","modified_gmt":"2021-10-04T10:15:16","slug":"kbr200703","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr200703\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2007\uff1a\u7b2c3\u554f"},"content":{"rendered":"<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(\\displaystyle\\int _ 0^\\pi x^2 \\cos^2 x \\, dx\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u5b9a\u6570 \\(a\\) \u306b\u5bfe\u3057\u3066,\r\n\\[\r\nf(x) = ax \\sin x +x +\\dfrac{\\pi}{2}\r\n\\]\r\n\u3068\u304a\u304f. \u3053\u306e\u3068\u304d, \u4e0d\u7b49\u5f0f\r\n\\[\r\n\\displaystyle\\int _ 0^\\pi \\left\\{ f'(x) \\right\\}^2 \\, dx \\geqq f \\left( \\dfrac{\\pi}{2} \\right)\r\n\\]\r\n\u3092\u6e80\u305f\u3059 \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(f'(x)\\) \u306f \\(f(x)\\) \u306e\u5c0e\u95a2\u6570\u3068\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\[\\begin{align}\r\nx^2 \\cos^2 x & = \\dfrac{x^2}{2} +\\dfrac{x^2 \\cos 2x}{2} \\\\\r\n& = \\dfrac{\\left( x^3 \\right)'}{6} +\\dfrac{1}{4} \\left( x^2 \\sin 2x \\right)' -\\dfrac{1}{2} x \\sin 2x \\\\\r\n& = \\dfrac{\\left( x^3 \\right)'}{6} +\\dfrac{1}{4} \\left( x^2 \\sin 2x \\right)' +\\dfrac{1}{4} \\left( x \\cos 2x \\right)' +\\dfrac{1}{2} \\cos 2x \\\\\r\n& = \\left( \\dfrac{x^3}{6} +\\dfrac{x^2 \\sin 2x}{4} +\\dfrac{x \\cos 2x}{4} +\\dfrac{\\sin 2x}{4}\\right)'\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 0^\\pi x^2 \\cos^2 x \\, dx & = \\left[ \\dfrac{x^3}{6} +\\dfrac{x^2 \\sin 2x}{4} +\\dfrac{x \\cos 2x}{4} +\\dfrac{\\sin 2x}{4}\\right] _ 0^\\pi \\\\\r\n& = \\underline{\\dfrac{{\\pi}^3}{6} +\\dfrac{\\pi}{4}}\r\n\\end{align}\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\nf \\left( \\dfrac{\\pi}{2} \\right) & = \\pi \\left( \\dfrac{a}{2} +1 \\right) , \\\\\r\nf'(x) & = a \\left( \\sin x +x \\cos x \\right) +1\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\left\\{ f'(x) \\right\\}^2 & = a^2 \\left( \\sin^2 x +2x \\sin x \\cos x +x^2 \\cos^2 x \\right) \\\\\r\n& \\qquad +2a \\left( \\sin x +x \\cos x \\right) +1 \\\\\r\n& = a^2 \\left( x \\sin^2 x \\right)' +2a \\left( x \\sin x \\right)' +a^2 x^2 \\cos^2 x +1\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 0^\\pi \\left\\{ f'(x) \\right\\}^2 \\, dx & = \\left[ a^2x \\sin^2 x +2ax \\sin x +x \\right] _ 0^\\pi \\\\\r\n& \\qquad +a^2 \\left( \\dfrac{{\\pi}^3}{6} +\\dfrac{\\pi}{4} \\right) \\\\\r\n& = a^2 \\left( \\dfrac{{\\pi}^3}{6} +\\dfrac{\\pi}{4} \\right) +\\pi\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \\(\\displaystyle\\int _ 0^\\pi \\left\\{ f'(x) \\right\\}^2 \\, dx \\geqq f \\left( \\dfrac{\\pi}{2} \\right)\\) \u3092\u3068\u304f\u3068\r\n\\[\\begin{gather}\r\na^2 \\left( \\dfrac{{\\pi}^3}{6} +\\dfrac{\\pi}{4} \\right) +\\pi \\geqq \\pi \\left( \\dfrac{a}{2} +1 \\right) \\\\\r\na \\left\\{ ( 2{\\pi}^2 +3 ) a -6 \\right\\} \\geqq 0 \\\\\r\n\\text{\u2234} \\quad \\underline{a \\leqq 0 , \\ \\dfrac{6}{2{\\pi}^2 +3} \\leqq a}\r\n\\end{gather}\\]\r\n","protected":false},"excerpt":{"rendered":"(1)\u3000\\(\\displaystyle\\int _ 0^\\pi x^2 \\cos^2 x \\, dx\\) \u3092\u6c42\u3081\u3088. (2)\u3000\u5b9a\u6570 \\(a\\) \u306b\u5bfe\u3057\u3066, \\[ f(x) = ax \\sin x +x +\\dfrac{\\ &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/kbr200703\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[101],"tags":[144,109],"class_list":["post-702","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2007","tag-tsukuba_r","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/702","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=702"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/702\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=702"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=702"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}