{"id":58,"date":"2011-11-26T18:25:50","date_gmt":"2011-11-26T09:25:50","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=58"},"modified":"2021-10-23T04:12:38","modified_gmt":"2021-10-22T19:12:38","slug":"wsr201101","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201101\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52011\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u4e0a\u306e\u653e\u7269\u7dda \\(y = x^2\\) \u3092 \\(C\\) \u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(C\\) \u4e0a\u306e\u70b9 \\(( a , a^2 )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u6cd5\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u70b9 \\(( 1 , 2 )\\) \u3092\u901a\u308b \\(C\\) \u306e\u6cd5\u7dda\u306e\u6570\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u70b9 \\(\\left( t , t +\\dfrac{1}{2} \\right)\\) \u3092\u901a\u308b \\(C\\) \u306e\u6cd5\u7dda\u306e\u6570\u304c \\(2\\) \u3068\u306a\u308b\u305f\u3081\u306e \\(t\\) \u306b\u5bfe\u3059\u308b\u6761\u4ef6\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\(C : \\ y = x^2\\) \u3088\u308a, \\(y' =2x\\) .\r\n\u3057\u305f\u304c\u3063\u3066, \u70b9 \\(( a , a^2 )\\) \u3092\u901a\u308b\u6cd5\u7dda\u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\\begin{align}\r\n(x-a) \\cdot 1 +(y-a^2) \\cdot 2a = 0 & \\\\\r\n\\text{\u2234} \\quad \\underline{x+2ay-2a^3-a = 0} & \\quad ... [1]\r\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n[1] \u304c\u70b9 \\(( 1 , 2 )\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n1+4a-2a^3-a & = 0 \\\\\r\n2a^3-3a-1 & = 0 \\\\\r\n(a+1)(2a^2-2a-1) & =0 \\\\\r\n\\text{\u2234} \\quad a = -1 , & \\dfrac{1 \\pm \\sqrt{3}}{2}\r\n\\end{align}\\]\r\n\u7570\u306a\u308b\u5b9f\u6570\u89e3\u3092 \\(3\\) \u3064\u3082\u3064\u306e\u3067, \u6cd5\u7dda\u306e\u6570\u306f, \\(\\underline{3}\\) .<\/p>\r\n

    (3)<\/strong><\/p>\r\n[1] \u304c\u70b9 \\(\\left( t , t +\\dfrac{1}{2} \\right)\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nt +a(2t+1) -2a^3-a & = 0 \\\\\r\n2a^3-2ta+t & = 0 \\quad ... [2]\r\n\\end{align}\\]\r\n[2] \u304c \\(2\\) \u3064\u306e\u7570\u306a\u308b\u5b9f\u6570\u89e3\u3092\u3082\u3064\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.\r\n[2] \u306e\u5de6\u8fba\u3092 \\(f(a)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(a) = 6a^2-2t = 6 \\left( a^2-\\dfrac{t}{3} \\right)\r\n\\]\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(t \\leqq 0\\) \u306e\u3068\u304d
      \r\n\\(f'(a) \\geqq 0\\) \u306a\u306e\u3067, \\(f(a)\\) \u306f\u5358\u8abf\u5897\u52a0\u3068\u306a\u308a, \\(f(a) = 0\\) \u306f \\(1\\) \u3064\u3057\u304b\u5b9f\u6570\u89e3\u3092\u3082\u305f\u306a\u3044\u306e\u3067, \u4e0d\u9069.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(t \\gt 0\\) \u306e\u3068\u304d\r\n\\[\r\nf'(a) = 6\\left( a +\\sqrt{\\dfrac{t}{3}} \\right) \\left( a -\\sqrt{\\dfrac{t}{3}} \\right)\r\n\\]\r\n\\(f'(a) = 0\\) \u3092\u89e3\u304f\u3068\r\n\\[\r\na = \\pm \\sqrt{\\dfrac{t}{3}}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(a)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} a & \\cdots & -\\sqrt{\\dfrac{t}{3}} & \\cdots & \\sqrt{\\dfrac{t}{3}} & \\cdots \\\\ \\hline f'(a) & + & 0 & - & 0 & + \\\\ \\hline f(a) & \\nearrow & \\text{\u6975\u5927} & \\searrow & \\text{\u6975\u5c0f} & \\nearrow \\end{array}\r\n\\]\r\n\u3053\u3053\u3067,\r\n\\[\\begin{align}\r\nf \\left( \\pm \\sqrt{\\dfrac{t}{3}} \\right) & = \\pm \\dfrac{2t}{3}\\sqrt{\\dfrac{t}{3}} \\mp 2t\\sqrt{\\dfrac{t}{3}} +t \\\\\r\n& = t \\left( 1 \\mp \\dfrac{4}{3}\\sqrt{\\dfrac{t}{3}} \\right) \\quad ( \\text{\u8907\u53f7\u540c\u9806} )\r\n\\end{align}\\]\r\n\u6975\u5927\u5024\u306b\u3064\u3044\u3066, \\(t \\gt 0\\) \u306a\u306e\u3067,\r\n\\[\r\nf \\left( -\\sqrt{\\dfrac{t}{3}} \\right) = t \\left( 1 +\\dfrac{4}{3}\\sqrt{\\dfrac{t}{3}} \\right) \\gt 0\r\n\\]\r\n\u3068\u306a\u308b\u306e\u3067, [2] \u304c \\(2\\) \u3064\u306e\u7570\u306a\u308b\u5b9f\u6570\u89e3\u3092\u3082\u3064\u6761\u4ef6\u306f, \u6975\u5c0f\u5024\u306b\u7740\u76ee\u3057\u3066\r\n\\[\\begin{align}\r\nf \\left( \\sqrt{\\dfrac{t}{3}} \\right) = t \\left( 1 -\\dfrac{4}{3}\\sqrt{\\dfrac{t}{3}} \\right) & = 0 \\\\\r\n1 -\\dfrac{4}{3}\\sqrt{\\dfrac{t}{3}} & =0 \\\\\r\n\\sqrt{\\dfrac{t}{3}} & = \\dfrac{3}{4}\\\\\r\n\\text{\u2234} \\quad t & = \\dfrac{27}{16}\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      1*<\/strong>, 2*<\/strong>\u3088\u308a, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{t = \\dfrac{27}{16}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306e\u653e\u7269\u7dda \\(y = x^2\\) \u3092 \\(C\\) \u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\\(C\\) \u4e0a\u306e\u70b9 \\(( a , a^2 )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u6cd5\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088. (2)\u3000\u70b9 … \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":[38],"tags":[147,13],"class_list":["post-58","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2011","tag-waseda_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/58","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=58"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/58\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=58"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=58"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=58"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}