{"id":523,"date":"2012-12-13T21:15:07","date_gmt":"2012-12-13T12:15:07","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=523"},"modified":"2021-09-29T20:16:13","modified_gmt":"2021-09-29T11:16:13","slug":"thr200806","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr200806\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2008\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\\(k \\gt 1\\) \u3068\u3057\u3066, \\(f(x) = x^2+2kx\\) \u3068\u304a\u304f.\r\n\u66f2\u7dda \\(y=f(x)\\) \u3068\u5186 \\(C : \\ x^2+y^2 = 1\\) \u306e \\(2\\) \u3064\u306e\u4ea4\u70b9\u306e\u5185\u3067, \u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u308b\u3082\u306e\u3092 P \u3068\u3057, \u7b2c \\(3\\) \u8c61\u9650\u306b\u3042\u308b\u3082\u306e\u3092 Q \u3068\u3059\u308b.\r\n\u70b9 O \\((0, 0)\\) , A \\((1, 0)\\) , B \\((-1, 0)\\) \u306b\u5bfe\u3057\u3066, \\(\\alpha = \\angle \\text{AOP}\\) , \\(\\beta = \\angle \\text{BOQ}\\) \u3068\u304a\u304f\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(k\\) \u3092 \\(\\alpha\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u66f2\u7dda \\(y=f(x)\\) \u3068\u5186 \\(C\\) \u3067\u56f2\u307e\u308c\u308b \\(2\\) \u3064\u306e\u56f3\u5f62\u306e\u5185\u3067, \\(y = f(x)\\) \u306e\u4e0a\u5074\u306b\u3042\u308b\u3082\u306e\u306e\u9762\u7a4d \\(S(k)\\) \u3092 \\(\\alpha\\) \u3068 \\(\\beta\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(\\displaystyle\\lim _ {k \\rightarrow \\infty} S(k)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n\"tohoku_r_2008_06_01\"\r\n

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

    P \\(( \\cos \\alpha , \\sin \\alpha )\\) \u304c \\(y=f(x)\\) \u4e0a\u306b\u3042\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n\\sin \\alpha & = \\cos^2 \\alpha +2k \\cos \\alpha \\\\\r\n\\text{\u2234} \\quad k & = \\underline{\\dfrac{\\tan \\alpha -\\cos \\alpha}{2}} \\quad ( \\ \\text{\u2235} \\ \\cos \\alpha \\neq 0 \\ ) \\quad ... [1]\r\n\\end{align}\\]\r\n

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

    Q \\(( -\\cos \\beta , -\\sin \\beta )\\) \u3082 \\(y=f(x)\\) \u4e0a\u306b\u3042\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n-\\sin \\beta & = \\cos^2 \\beta -2k \\cos \\beta \\\\\r\n\\text{\u2234} \\quad k & = \\dfrac{\\tan \\beta +\\cos \\beta}{2} \\quad ( \\ \\text{\u2235} \\ \\cos \\beta \\neq 0 \\ ) \\quad ... [2]\r\n\\end{align}\\]\r\n\u304a\u3046\u304e\u5f62 OPQ \u306e\u9762\u7a4d\u306f\r\n\\[\r\n\\dfrac{1}{2} \\cdot 1^2 ( \\pi -\\alpha +\\beta ) = \\dfrac{\\pi -\\alpha +\\beta}{2}\r\n\\]\r\n\u7dda\u5206 OP \u3068 \\(y = f(x)\\) \u306b\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u306f\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 0^{\\cos \\alpha} & \\left\\{ ( \\tan \\alpha ) x -( x^2+2kx ) \\right\\} \\, dx \\\\\r\n& = \\displaystyle\\int _ 0^{\\cos \\alpha} \\left\\{ ( 2\\cos \\alpha ) x -x^2 \\right\\} \\, dx \\quad ( \\ \\text{\u2235} \\ [1] \\ ) \\\\\r\n& = \\left[ ( \\cos \\alpha ) x^2 -\\dfrac{x^3}{3} \\right] _ 0^{\\cos \\alpha} \\\\\r\n& = \\dfrac{\\cos^3 \\alpha}{6}\r\n\\end{align}\\]\r\n\u7dda\u5206 OQ \u3068 \\(y = f(x)\\) \u306b\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u306f\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ {-\\cos \\beta}^0 & \\left\\{ ( \\tan \\beta ) x -( x^2+2kx ) \\right\\} \\, dx \\\\\r\n& = \\displaystyle\\int _ {-\\cos \\beta}^0 \\left\\{ ( -2\\cos \\beta ) x -x^2 \\right\\} \\, dx \\quad ( \\ \\text{\u2235} \\ [2] \\ ) \\\\\r\n& = \\left[ ( -\\cos \\alpha ) x^2 -\\dfrac{x^3}{3} \\right] _ {-\\cos \\beta}^0 \\\\\r\n& = \\dfrac{\\cos^3 \\beta}{6}\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u9762\u7a4d \\(S(k)\\) \u306f\r\n\\[\r\nS(k) = \\underline{\\dfrac{\\pi -\\alpha +\\beta}{2} +\\dfrac{\\cos^3 \\alpha}{6} +\\dfrac{\\cos^3 \\beta}{6}}\r\n\\]\r\n

    (3)<\/strong><\/p>\r\n[1] [2] \u3088\u308a\r\n\\[\r\nk = \\dfrac{\\tan \\alpha -\\cos \\alpha}{2} = \\dfrac{\\tan \\beta +\\cos \\beta}{2}\r\n\\]\r\n\\(-1 \\leqq \\cos \\alpha \\leqq 1\\) , \\(-1 \\leqq \\cos \\beta \\leqq 1\\) \u306a\u306e\u3067, \\(k \\rightarrow \\infty\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n\\tan \\alpha \\rightarrow \\infty & , \\ \\tan \\beta \\rightarrow \\infty \\\\\r\n\\text{\u2234} \\quad \\alpha \\rightarrow \\dfrac{\\pi}{2} +0 & , \\ \\beta \\rightarrow \\dfrac{\\pi}{2} +0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {k \\rightarrow \\infty} S(k) & = \\dfrac{\\pi -\\frac{\\pi}{2} +\\frac{\\pi}{2}}{2} +\\dfrac{0}{6} +\\dfrac{0}{6} \\\\\r\n& = \\underline{\\dfrac{\\pi}{2}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(k \\gt 1\\) \u3068\u3057\u3066, \\(f(x) = x^2+2kx\\) \u3068\u304a\u304f. \u66f2\u7dda \\(y=f(x)\\) \u3068\u5186 \\(C : \\ x^2+y^2 = 1\\) \u306e \\(2\\) \u3064\u306e\u4ea4\u70b9\u306e\u5185\u3067, \u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u308b … \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":[76],"tags":[148,16],"class_list":["post-523","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2008","tag-tohoku_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/523","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=523"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/523\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=523"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=523"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=523"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}