{"id":396,"date":"2012-05-18T23:26:18","date_gmt":"2012-05-18T14:26:18","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=396"},"modified":"2021-09-15T07:47:37","modified_gmt":"2021-09-14T22:47:37","slug":"ngr201201","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201201\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2012\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(a\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3057, \\(xy\\) \u5e73\u9762\u4e0a\u306e\u66f2\u7dda \\(C\\) \u306e\u65b9\u7a0b\u5f0f\u3092 \\(y = x^3 -a^2x\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(C\\) \u4e0a\u306e\u70b9 A \\(( t, t^3 -a^2t )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u63a5\u7dda\u3092 \\(\\ell\\) \u3068\u3059\u308b. \\(\\ell\\) \u3068 \\(C\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306e\u9762\u7a4d \\(S(t)\\) \u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(t\\) \u306f \\(0\\) \u3067\u306a\u3044\u3068\u3059\u308b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(b\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \\(C\\) \u306e\u63a5\u7dda\u306e\u3046\u3061 \\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 B \\(( 2a , b )\\) \u3092\u901a\u308b\u3082\u306e\u306e\u672c\u6570\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(C\\) \u306e\u63a5\u7dda\u306e\u3046\u3061\u70b9 B \\(( 2a , b )\\) \u3092\u901a\u308b\u3082\u306e\u304c \\(2\\) \u672c\u306e\u3082\u306e\u5834\u5408\u3092\u8003\u3048, \u305d\u308c\u3089\u306e\u63a5\u7dda\u3092 \\(\\ell _ 1 , \\ell _ 2\\) \u3068\u3059\u308b. \u305f\u3060\u3057, \\(\\ell _ 1\\) \u3068 \\(\\ell _ 2\\) \u306f\u3069\u3061\u3089\u3082\u539f\u70b9 \\((0,0)\\) \u306f\u901a\u3089\u306a\u3044\u3068\u3059\u308b. \\(\\ell _ 1\\) \u3068 \\(C\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306e\u9762\u7a4d\u3092 \\(S _ 1\\) \u3068\u3057, \\(\\ell _ 2\\) \u3068 \\(C\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306e\u9762\u7a4d\u3092 \\(S _ 2\\) \u3068\u3059\u308b. \\(S _ 1 \\geqq S _ 2\\) \u3068\u3057\u3066, \\(\\dfrac{S _ 1}{S _ 2}\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/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) = x^3-a^2x\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf(-x) = -x^3+a^2x = -f(x)\n\\]\r\n\u306a\u306e\u3067, \u66f2\u7dda \\(C\\) \u306f\u539f\u70b9\u306b\u3064\u3044\u3066\u5bfe\u79f0\u3067\u3042\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066, \\(t \\gt 0\\) \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.
    \r\n\\(f'(x) = 3x^2-a^2\\) \u306a\u306e\u3067 , \\(\\ell\\) \u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = (3t^2-a^2)(x-t) +t^3-a^2t \\\\\r\n& = (3t^2-a^2)x -2t^3\n\\end{align}\\]\r\n\u3053\u308c\u3068 \\(C\\) \u306e\u65b9\u7a0b\u5f0f\u3088\u308a, \\(y\\) \u3092\u6d88\u53bb\u3057\u3066\r\n\\[\\begin{align}\r\nx^3-a^2x & = (3t^2-a^2)x -2t^3 \\\\\r\n(x-t)^2 (x+2t) & = 0 \\\\\r\n\\text{\u2234} \\quad x & = t , -2t\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nS(t) & = \\displaystyle\\int _ {-2t}^t \\left[ x^3-a^2x -\\left\\{ (3t^2-a^2)x -2t^3 \\right\\} \\right] \\, dx \\\\\r\n& = \\displaystyle\\int _ {-2t}^t (x-t)^2(x+2t) \\, dx \\\\\r\n& = \\displaystyle\\int _ {-2t}^t \\left\\{ (x-t)^3 +3t(x-t)^2 \\right\\} \\, dx \\\\\r\n& = \\left[ \\dfrac{(x-t)^4}{4} +t(x-t)^3 \\right] _ {-2t}^t \\\\\r\n& = -\\dfrac{81 t^4}{4} +27 t^4 =\\underline{\\dfrac{27t^4}{4}}\n\\end{align}\\]\r\n

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

    \\(\\ell\\) \u304c \\((2a,b)\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nb & = 2a(3t^2-a^2) -2t^3 \\\\\r\n\\text{\u2234} \\quad b & = -2t^3 +6at^2 -2a^3 \\quad ... [1]\n\\end{align}\\]\r\n\u53f3\u8fba\u3092 \\(g(t)\\) \u3068\u304a\u3044\u3066, \u76f4\u7dda \\(y=b\\) \u3068\u66f2\u7dda \\(y=g(t)\\) \u306e\u5171\u6709\u70b9\u306e\u500b\u6570\u3092\u8003\u3048\u308c\u3070\u3088\u3044.\r\n\\[\r\ng'(t) =-6t^2 +12at =-6t (t-2a)\n\\]\r\n\u306a\u306e\u3067 \\(g(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} t & \\cdots & 0 & \\cdots & 2a & \\cdots \\\\ \\hline g'(t) & - & 0 & + & 0 & - \\\\ \\hline g(t) & \\searrow & -2a^3 & \\nearrow & 6a^3 & \\searrow \\end{array}\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u672c\u6570\u306f\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} 1 & \\left( \\ b \\lt -2a^3 , 6a^3 \\lt b \\text{\u306e\u3068\u304d} \\right) \\\\ 2 & \\left( \\ b = -2a^3 , 6a^3 \\text{\u306e\u3068\u304d} \\right) \\\\ 3 & \\left( \\ -2a^3 \\lt b \\lt 6a^3 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\n\\]\r\n

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

      \r\n

    1. 1*<\/strong>\u3000\\(b = -2a^3\\) \u306e\u3068\u304d
      \r\n[1] \u3088\u308a\r\n\\[\\begin{align}\r\n-2a^3 & = -2t^3 +6at^2 -2a^3 \\\\\r\nt^2 (t-3a) & = 0 \\\\\r\n\\text{\u2234} \\quad t & = 0 , 3a\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u63a5\u7dda\u306e \\(1\\) \u3064\u304c\u539f\u70b9\u3092\u901a\u308b\u306e\u3067, \u4e0d\u9069.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(b = 6a^3\\) \u306e\u3068\u304d
      \r\n[1] \u3088\u308a\r\n\\[\\begin{align}\r\n6a^3 = -2t^3 +6at^2 & -2a^3 \\\\\r\nt^3 -3at^2 +4a^3 & = 0 \\\\\r\n(t-2a)^2 (t+a) & = 0 \\\\\r\n\\text{\u2234} \\quad t = -a , 2a &\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      \u3088\u3063\u3066, (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n\\dfrac{S _ 1}{S _ 2} = \\dfrac{\\dfrac{27}{4} (2a)^4}{\\dfrac{27}{4} (-a)^4} = \\underline{16}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3057, \\(xy\\) \u5e73\u9762\u4e0a\u306e\u66f2\u7dda \\(C\\) \u306e\u65b9\u7a0b\u5f0f\u3092 \\(y = x^3 -a^2x\\) \u3068\u3059\u308b. (1)\u3000\\(C\\) \u4e0a\u306e\u70b9 A \\(( t, t^3 -a^2t )\\) \u306b\u304a\u3051\u308b \\(C\\ … \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":[63],"tags":[143,68],"class_list":["post-396","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2012","tag-nagoya_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/396","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=396"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/396\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=396"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=396"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=396"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}