{"id":201,"date":"2011-12-02T23:16:07","date_gmt":"2011-12-02T14:16:07","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=201"},"modified":"2021-09-16T06:26:52","modified_gmt":"2021-09-15T21:26:52","slug":"ngr200902","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr200902\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2009\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u95a2\u6570 \\(f(x)\\) \u3068 \\(g( \\theta )\\) \u3092\r\n\\[\\begin{align}\r\nf(x) & = \\displaystyle\\int _ {-1}^x \\sqrt{1-t^2} \\, dt \\quad ( -1 \\leqq x \\leqq 1 ) , \\\\\r\ng( \\theta ) & = f( \\cos \\theta ) -f( \\sin \\theta ) \\quad ( 0 \\leqq \\theta \\leqq 2\\pi )\r\n\\end{align}\\]\r\n\u3067\u5b9a\u3081\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u5c0e\u95a2\u6570 \\(g'( \\theta )\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

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

  3. (3)<\/strong>\u3000\\(y = g( \\theta )\\) \u306e\u30b0\u30e9\u30d5\u3092\u304b\u3051.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\(f'(x) = \\sqrt{1-x^2}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\ng'( \\theta ) & = -\\sin \\theta f' \\left( \\cos \\theta \\right) -\\cos \\theta f' \\left( \\sin \\theta \\right) \\\\\r\n& = -\\sin \\theta \\big| \\sin \\theta \\big| -\\cos \\theta \\big| \\cos \\theta \\big|\n\\end{align}\\]\r\n\\(\\sin \\theta , \\cos \\theta\\) \u306e\u6b63\u8ca0\u306b\u6ce8\u610f\u3057\u3066, \u5834\u5408\u5206\u3051\u3059\u308c\u3070\r\n\\[\r\ng'( \\theta )= \\underline{\\left\\{ \\begin{array}{ll} -1 & \\ \\left( 0 \\leqq \\theta \\lt \\dfrac{\\pi}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ \\cos 2\\theta & \\ \\left( \\dfrac{\\pi}{2} \\leqq \\theta \\lt \\pi \\text{\u306e\u3068\u304d} \\right) \\\\ 1 & \\ \\left( \\pi \\leqq \\theta \\lt \\dfrac{3 \\pi}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ -\\cos 2\\theta & \\ \\left( \\dfrac{3 \\pi}{2} \\leqq \\theta \\leqq 2 \\pi \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\n\\]\r\n

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

    (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\ng( \\theta ) = \\left\\{ \\begin{array}{ll} -\\theta +C _ 1 & \\ \\left( 0 \\leqq \\theta \\lt \\dfrac{\\pi}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ \\dfrac{1}{2}\\sin 2\\theta +C _ 2 & \\ \\left( \\dfrac{\\pi}{2} \\leqq \\theta \\lt \\pi \\text{\u306e\u3068\u304d} \\right) \\\\ \\theta +C _ 3 & \\ \\left( \\pi \\leqq \\theta \\lt \\dfrac{3 \\pi}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ -\\dfrac{1}{2} \\sin 2\\theta +C _ 4 & \\ \\left( \\dfrac{3 \\pi}{2} \\leqq \\theta \\leqq 2 \\pi \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.\n\\]\r\n\u305f\u3060\u3057, \\(C _ 1 , C _ 2 , C _ 3 , C _ 4\\) \u306f\u7a4d\u5206\u5b9a\u6570.
    \r\n\u3053\u3053\u3067, \\(f(x)\\) \u306e\u5024\u306f\u534a\u5f84 \\(1\\) \u306e\u534a\u5186\u306e\u9762\u7a4d\u306b\u7740\u76ee\u3059\u308c\u3070<\/p>\r\n\"\"\r\n

    \\[\r\nf(-1) = 0 , \\ f(0) = \\dfrac{\\pi}{4} , \\ f(1) = \\dfrac{\\pi}{2}\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\ng(0) & = f(1) -f(0) = \\dfrac{\\pi}{4} = 0+C _ 1 \\\\\r\n& \\text{\u2234} \\quad C _ 1 = \\dfrac{\\pi}{4} \\\\\r\ng \\left( \\dfrac{\\pi}{2} \\right) & = f(0) -f(1) = -\\dfrac{\\pi}{4} = 0+C _ 2 \\\\\r\n& \\text{\u2234} \\quad C _ 2 = -\\dfrac{\\pi}{4} \\\\\r\ng \\left( \\pi \\right) & = f(-1) -f(0) = -\\dfrac{\\pi}{4} = \\pi +C _ 3 \\\\\r\n& \\text{\u2234} \\quad C _ 3 = -\\dfrac{5 \\pi}{4} \\\\\r\ng \\left( \\dfrac{3 \\pi}{2} \\right) & = f(0) -f(-1) = \\dfrac{\\pi}{4} = 0+C _ 4 \\\\\r\n& \\text{\u2234} \\quad C _ 4 = -\\dfrac{\\pi}{4}\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\ng( \\theta ) = \\underline{\\left\\{ \\begin{array}{ll} -\\theta +\\dfrac{\\pi}{4} & \\ \\left( 0 \\leqq \\theta \\lt \\dfrac{\\pi}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ \\dfrac{1}{2}\\sin 2\\theta -\\dfrac{\\pi}{4} & \\ \\left( \\dfrac{\\pi}{2} \\leqq \\theta \\lt \\pi \\text{\u306e\u3068\u304d} \\right) \\\\ \\theta -\\dfrac{5 \\pi}{4} & \\ \\left( \\pi \\leqq \\theta \\lt \\dfrac{3 \\pi}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ -\\dfrac{1}{2} \\sin 2\\theta -\\dfrac{\\pi}{4} & \\ \\left( \\dfrac{3 \\pi}{2} \\leqq \\theta \\leqq 2 \\pi \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\n\\]\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \u30b0\u30e9\u30d5\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"\"\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\(f(x)\\) \u3068 \\(g( \\theta )\\) \u3092 \\[\\begin{align} f(x) & = \\displaystyle\\int _ {-1}^x \\sqrt{1-t^2} \\, dt \\quad ( … \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":[33],"tags":[143,15],"class_list":["post-201","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2009","tag-nagoya_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/201","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=201"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/201\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=201"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=201"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=201"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}