{"id":1175,"date":"2015-07-21T23:48:22","date_gmt":"2015-07-21T14:48:22","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1175"},"modified":"2021-09-29T23:02:23","modified_gmt":"2021-09-29T14:02:23","slug":"tbr201402","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201402\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2014\uff1a\u7b2c2\u554f"},"content":{"rendered":"
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\\(xy\\) \u5e73\u9762\u4e0a\u306e\u66f2\u7dda \\(C : \\ y = x \\sin x +\\cos x -1 \\ ( 0 \\lt x \\lt \\pi )\\) \u306b\u5bfe\u3057\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.\r\n\u305f\u3060\u3057, \\(3 \\lt \\pi \\lt \\dfrac{16}{5}\\) \u3067\u3042\u308b\u3053\u3068\u306f\u8a3c\u660e\u306a\u3057\u3067\u7528\u3044\u3066\u3088\u3044.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u66f2\u7dda \\(C\\) \u3068 \\(x\\) \u8ef8\u306e\u4ea4\u70b9\u306f\u305f\u3060 \\(1\\) \u3064\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u66f2\u7dda \\(C\\) \u3068 \\(x\\) \u8ef8\u306e\u4ea4\u70b9\u3092 \\(A \\ ( \\alpha , 0 )\\) \u3068\u3059\u308b. \\(\\alpha \\gt \\dfrac{2}{3} \\pi\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u66f2\u7dda \\(C\\) , \\(y\\) \u8ef8\u304a\u3088\u3073\u76f4\u7dda \\(y = \\dfrac{\\pi}{2} -1\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S\\) \u3068\u3059\u308b.\r\n\u307e\u305f, \\(xy\\) \u5e73\u9762\u306e\u539f\u70b9 \\(O\\) , \u70b9 \\(A\\) \u304a\u3088\u3073\u66f2\u7dda \\(C\\) \u4e0a\u306e\u70b9 \\(B \\ \\left( \\dfrac{\\pi}{2} , \\dfrac{\\pi}{2} -1 \\right)\\) \u3092\u9802\u70b9\u3068\u3059\u308b\u4e09\u89d2\u5f62 \\(OAB\\) \u306e\u9762\u7a4d\u3092 \\(T\\) \u3068\u3059\u308b. \\(S \\lt T\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\(f(x) = y\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf'(x) & = \\sin x +x \\cos x -\\sin x \\\\\r\n& = x \\cos x\r\n\\end{align}\\]\r\n\\(f'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nx = \\dfrac{\\pi}{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} x & (0) & \\cdots & \\dfrac{\\pi}{2} & \\cdots & ( \\pi ) \\\\ \\hline f'(x) & & + & 0 & - & \\\\ \\hline f(x) & (0) & \\nearrow & \\dfrac{\\pi}{2} -1 & \\searrow & ( -2 ) \\end{array}\r\n\\]\r\n\u3086\u3048\u306b, \\(C\\) \u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308a, \\(x\\) \u8ef8\u3068\u305f\u3060 \\(1\\) \u70b9\u3067\u4ea4\u308f\u308b.<\/p>\r\n\"tbr20140201\"\r\n

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

    \\[\\begin{align}\r\nf \\left( \\dfrac{2 \\pi}{3} \\right) & = \\dfrac{2 \\pi}{3} \\cdot \\dfrac{\\sqrt{3}}{2} -\\dfrac{1}{2} -1 \\\\\r\n& = \\dfrac{2 \\sqrt{3} \\pi -9}{6}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(3 \\gt \\left( \\dfrac{3}{2} \\right)^2 = \\dfrac{9}{4}\\) \u306a\u306e\u3067\r\n\\[\r\n\\sqrt{3} \\gt \\dfrac{3}{2}\r\n\\]\r\n\u307e\u305f, \\(\\pi \\gt 3\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf \\left( \\dfrac{2 \\pi}{3} \\right) & \\gt \\dfrac{2 \\cdot \\frac{3}{2} \\cdot 3 -9}{6} \\\\\r\n& = 0 = f( \\alpha ) \\quad ... [1]\r\n\\end{align}\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(\\dfrac{2 \\pi}{3} \\lt x \\lt \\pi\\) \u306b\u304a\u3044\u3066, \\(f(x)\\) \u306f\u5358\u8abf\u6e1b\u5c11\u306a\u306e\u3067, [1] \u3088\u308a\r\n\\[\r\n\\alpha \\gt \\dfrac{2 \\pi}{3}\r\n\\]\r\n

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

    \\[\\begin{align}\r\nS & = \\dfrac{\\pi}{2} \\left( \\dfrac{\\pi}{2} -1 \\right) -\\displaystyle\\int _ 0^{\\frac{\\pi}{2}} f(x) \\, dx \\\\\r\n& = \\dfrac{{\\pi}^2}{4} -\\dfrac{\\pi}{2} -\\left[ -x \\cos x +2 \\sin x -x \\right] _ 0^{\\frac{\\pi}{2}} \\\\\r\n& = \\dfrac{{\\pi}^2}{4} -\\dfrac{\\pi}{2} -\\left( 2 -\\dfrac{\\pi}{2} \\right) \\\\\r\n& = \\dfrac{{\\pi}^2}{4} -2\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\r\nT = \\dfrac{\\alpha}{2} \\left( \\dfrac{\\pi}{2} -1 \\right)\r\n\\]\r\n(2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nT -S & = \\dfrac{\\pi}{3} \\left( \\dfrac{\\pi}{2} -1 \\right) -\\left( \\dfrac{{\\pi}^2}{4} -2 \\right) \\\\\r\n& = 2 -\\dfrac{{\\pi}^2}{12} -\\dfrac{\\pi}{3} \\\\\r\n& \\gt 2 -\\dfrac{1}{12} \\left( \\dfrac{16}{5} \\right)^2 -\\dfrac{1}{3} \\cdot \\dfrac{16}{5} \\quad ( \\ \\text{\u2235} \\ \\pi \\lt \\dfrac{16}{5} \\ ) \\\\\r\n& = 2 -\\dfrac{64}{75} -\\dfrac{16}{15} \\\\\r\n& = \\dfrac{2}{25} \\gt 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nS \\lt T\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306e\u66f2\u7dda \\(C : \\ y = x \\sin x +\\cos x -1 \\ ( 0 \\lt x \\lt \\pi )\\) \u306b\u5bfe\u3057\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057, \\(3 \\lt \\pi \\lt \\d … \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":[121],"tags":[144,112],"class_list":["post-1175","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2014","tag-tsukuba_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1175","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=1175"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1175\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1175"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1175"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}