{"id":124,"date":"2011-11-30T00:51:46","date_gmt":"2011-11-29T15:51:46","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=124"},"modified":"2021-09-24T18:02:51","modified_gmt":"2021-09-24T09:02:51","slug":"tok201001","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201001\/","title":{"rendered":"\u6771\u5de5\u59272010\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(f(x) = 1 -\\cos x -x \\sin x\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(0 \\lt x \\lt \\pi\\) \u306b\u304a\u3044\u3066, \\(f(x) = 0\\) \u306f\u552f\u4e00\u306e\u89e3\u3092\u6301\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(J = \\displaystyle\\int _ 0^{\\pi} \\left| f(x) \\right| \\, dx\\) \u3068\u3059\u308b. (1)<\/strong> \u306e\u552f\u4e00\u306e\u89e3\u3092 \\(\\alpha\\) \u3068\u3059\u308b\u3068\u304d, \\(J\\) \u3092 \\(\\sin \\alpha\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000(2)<\/strong> \u3067\u5b9a\u7fa9\u3055\u308c\u305f \\(J\\) \u3068 \\(\\sqrt{2}\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u305b\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\[\\begin{align}\r\nf'(x) & = \\sin x -\\sin x -x \\cos x \\\\\r\n& = -x \\cos x\r\n\\end{align}\\]\r\n\\(f'(x) = 0\\) \u3092\u89e3\u304f\u3068\r\n\\[\r\nx = 0 , \\dfrac{\\pi}{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} x & 0 & \\cdots & \\dfrac{\\pi}{2} & \\cdots & \\pi \\\\ \\hline f'(x) & 0 & - & 0 & + & \\\\ \\hline f(x) & 0 & \\searrow & 1 -\\dfrac{\\pi}{2} & \\nearrow & 2 \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f'(x) = 0\\) \u306f \\(\\dfrac{\\pi}{2} \\lt x \\lt \\pi\\) \u306b \\(1\\) \u3064\u306e\u89e3\u3092\u3082\u3064.
    \r\n\u3086\u3048\u306b\u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n1 -\\cos \\alpha & -\\alpha \\sin \\alpha = 0 \\\\\r\n\\text{\u2234} \\quad \\alpha & = \\dfrac{1 -\\cos \\alpha}{\\sin \\alpha} \\quad ... [1]\r\n\\end{align}\\]\r\n\u307e\u305f, \\(F(x) = \\displaystyle\\int f(x) \\, dx\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nF(x) & = x -\\sin x +x \\cos x -\\displaystyle\\int \\cos x \\, dx \\\\\r\n& = x -2\\sin x +x \\cos x +C \\quad ( C \\text{\u306f\u7a4d\u5206\u5b9a\u6570} )\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a\r\n\\[\\begin{align}\r\nJ & = -\\displaystyle\\int _ 0^{\\alpha} f(x) \\, dx +\\displaystyle\\int _ {\\alpha}^{\\pi} f(x) \\, dx \\\\\r\n& = 2 F( \\alpha ) -F(0) -F( \\pi ) \\\\\r\n& = 2 \\left( \\alpha -2\\sin \\alpha +\\alpha \\cos \\alpha \\right) -0 -( \\pi -\\pi ) \\\\\r\n& = 2 \\left( \\dfrac{1 -\\cos \\alpha}{\\sin \\alpha} -2\\sin \\alpha + \\dfrac{1 -\\cos \\alpha}{\\sin \\alpha} \\cdot \\cos \\alpha \\right) \\quad ( \\ \\text{\u2235} \\ [1] \\ ) \\\\\r\n& = 2 \\cdot \\dfrac{1 -\\cos \\alpha -2\\sin^2 \\alpha +\\cos \\alpha -\\cos^2 \\alpha}{\\sin \\alpha} \\\\\r\n& = 2 \\cdot \\dfrac{\\sin^2 \\alpha}{\\sin \\alpha} \\\\\r\n& = \\underline{2\\sin \\alpha}\r\n\\end{align}\\]\r\n

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

    \\[\\begin{align}\r\nf \\left( \\dfrac{3\\pi}{4} \\right) & = 1 +\\dfrac{\\sqrt{2}}{2} -\\dfrac{3\\pi}{4} \\cdot \\dfrac{\\sqrt{2}}{2} \\\\\r\n& = \\dfrac{8 +4\\sqrt{2} -3\\pi \\sqrt{2}}{8} \\\\\r\n& \\gt \\dfrac{8 +4 \\cdot 1.4 -3 \\cdot 3 \\cdot 1.5}{8} \\quad \\left( \\ \\text{\u2235} \\ 1.4 \\lt \\sqrt{2} \\lt 1.5 , \\ 3 \\lt \\pi \\right) \\\\\r\n& = \\dfrac{8 +5.6 -13.5}{8} = \\dfrac{0.1}{8} \\gt 0\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306f \\(\\dfrac{\\pi}{2} \\lt x \\lt \\pi\\) \u306b\u304a\u3044\u3066\u5358\u8abf\u5897\u52a0\u3067\u3042\u308a, \\(f \\left( \\dfrac{3\\pi}{4} \\right) \\gt f( \\alpha )\\) \u306a\u306e\u3067\r\n\\[\r\n\\dfrac{3\\pi}{4} \\gt \\alpha\r\n\\]\r\n\u3055\u3089\u306b, \\(\\sin x\\) \u306f \\(\\dfrac{\\pi}{2} \\lt x \\lt \\pi\\) \u306b\u304a\u3044\u3066\u5358\u8abf\u6e1b\u5c11\u3067\u3042\u308a, \\(\\dfrac{3\\pi}{4} \\gt \\alpha\\) \u306a\u306e\u3067\r\n\\[\r\n\\sin \\alpha \\gt \\dfrac{\\sqrt{2}}{2}\r\n\\]\r\n\u3088\u3063\u3066, (2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n\\underline{J \\gt \\sqrt{2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(f(x) = 1 -\\cos x -x \\sin x\\) \u3068\u3059\u308b. (1)\u3000\\(0 \\lt x \\lt \\pi\\) \u306b\u304a\u3044\u3066, \\(f(x) = 0\\) \u306f\u552f\u4e00\u306e\u89e3\u3092\u6301\u3064\u3053\u3068\u3092\u793a\u305b. (2)\u3000\\(J = \\displ … \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":[25],"tags":[141,14],"class_list":["post-124","post","type-post","status-publish","format-standard","hentry","category-toko_2010","tag-toko","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/124","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=124"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/124\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=124"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=124"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=124"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}