{"id":13,"date":"2011-11-25T21:05:08","date_gmt":"2011-11-25T12:05:08","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=13"},"modified":"2021-03-12T17:15:20","modified_gmt":"2021-03-12T08:15:20","slug":"tkr201106","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201106\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2011\uff1a\u7b2c6\u554f"},"content":{"rendered":"
    \r\n

  1. (1)<\/strong>\u3000\\(x , y\\) \u3092\u5b9f\u6570\u3068\u3057, \\(x \\gt 0\\) \u3068\u3059\u308b.\r\n\\(t\\) \u3092\u5909\u6570\u3068\u3059\u308b \\(2\\) \u6b21\u95a2\u6570 \\(f(t) = xt^2 +yt\\) \u306e \\(0 \\leqq t \\leqq 1\\) \u306b\u304a\u3051\u308b\u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u306e\u5dee\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u70b9 \\(( x , y )\\) \u5168\u4f53\u304b\u3089\u306a\u308b\u5ea7\u6a19\u5e73\u9762\u5185\u306e\u9818\u57df\u3092 \\(S\\) \u3068\u3059\u308b.
    \r\n\u300c \\(x \\gt 0\\) \u304b\u3064, \u5b9f\u6570 \\(z\\) \u3067 \\(0 \\leqq t \\leqq 1\\) \u306e\u7bc4\u56f2\u306e\u5168\u3066\u306e\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\u3066,\r\n\\[\r\n0 \\leqq xt^2 +yt +z \\leqq 1\r\n\\]\r\n\u3000\u3092\u6e80\u305f\u3059\u3088\u3046\u306a\u3082\u306e\u304c\u5b58\u5728\u3059\u308b. \u300d
    \r\n\\(S\\) \u306e\u6982\u5f62\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u70b9 \\(( x , y , z )\\) \u5168\u4f53\u304b\u3089\u306a\u308b\u5ea7\u6a19\u7a7a\u9593\u5185\u306e\u9818\u57df\u3092 \\(V\\) \u3068\u3059\u308b.
    \r\n\u300c \\(0 \\leqq x \\leqq 1\\) \u304b\u3064, \\(0 \\leqq t \\leqq 1\\) \u306e\u7bc4\u56f2\u306e\u5168\u3066\u306e\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\u3066,\r\n\\[\r\n0 \\leqq x t^2 +yt +z \\leqq 1\r\n\\]\r\n\u3000\u304c\u6210\u308a\u7acb\u3064. \u300d
    \r\n\\(V\\) \u306e\u4f53\u7a4d\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

    \\[\r\nf(t) = xt^2 +yt = x \\left( t -\\dfrac{y}{2x} \\right)^2 -\\dfrac{y^2}{4x}\r\n\\]\r\n\\(0 \\leqq t \\leqq 1\\) \u306b\u304a\u3044\u3066\u6700\u5927\u5024\u3068\u306a\u308a\u3046\u308b\u306e\u306f, \u4ee5\u4e0b\u306e \\(2\\) \u3064.\r\n\\[\r\nf(0) = 0 , \\quad f(1) = x+y\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\u6700\u5927\u5024\u306f, \u5927\u5c0f\u3092\u6bd4\u8f03\u3057\u3066<\/p>\r\n

      \r\n

    • \\(x+y \\geqq 0\\) \u3059\u306a\u308f\u3061 \\(y \\geqq -x\\) \uff08\u4e0b\u56f3\u9818\u57df(A)\uff09\u306e\u3068\u304d, \\(x+y\\)<\/p><\/li>\r\n

    • \\(x+y \\lt 0\\) \u3059\u306a\u308f\u3061 \\(y \\lt -x\\) \uff08\u4e0b\u56f3\u9818\u57df(B)\uff09\u306e\u3068\u304d, \\(0\\)<\/p><\/li>\r\n<\/ul>\r\n\"\"\r\n

      \\(0 \\leqq t \\leqq 1\\) \u306b\u304a\u3044\u3066\u6700\u5c0f\u5024\u3068\u306a\u308a\u3046\u308b\u306e\u306f, \u4ee5\u4e0b\u306e \\(3\\) \u3064.\r\n\\[\r\nf(0) = 0 , \\quad f(1) = x+y , \\quad f \\left( -\\dfrac{y}{2x} \\right) = -\\dfrac{y^2}{4x}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\u6700\u5c0f\u5024\u306f, \\(f(t)\\) \u306e\u8ef8\u306e\u4f4d\u7f6e\u306b\u6ce8\u610f\u3057\u3066<\/p>\r\n

        \r\n

      • \\(-\\dfrac{y}{2x} \\lt 0\\) \u3059\u306a\u308f\u3061 \\(y \\gt 0\\) \uff08\u4e0b\u56f3\u9818\u57df(\u30a2)\uff09\u306e\u3068\u304d, \\(0\\)<\/p><\/li>\r\n

      • \\(0 \\leqq -\\dfrac{y}{2x} \\leqq 1\\) \u3059\u306a\u308f\u3061 \\(-2x \\leqq y \\leqq 0\\) \uff08\u4e0b\u56f3\u9818\u57df(\u30a4)\uff09\u306e\u3068\u304d, \\(-\\dfrac{y^2}{4x}\\)<\/p><\/li>\r\n

      • \\(-\\dfrac{y}{2x} \\gt 1\\) \u3059\u306a\u308f\u3061 \\(y \\lt -2x\\) \uff08\u4e0b\u56f3\u9818\u57df(\u30a6)\uff09\u306e\u3068\u304d, \\(x+y\\)<\/p><\/li>\r\n<\/ul>\r\n\"\"\r\n

        \u4ee5\u4e0a\u3088\u308a, \u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u306e\u5dee\u306f\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} x+y & ( \\ y>0 \\text{\u306e\u3068\u304d}\\ ) \\\\ x+y+\\dfrac{y^2}{4x} & ( \\ -x \\leqq y \\leqq 0 \\text{\u306e\u3068\u304d}\\ ) \\\\ \\dfrac{y^2}{4x} & ( \\ -2x \\leqq y \\lt -x \\text{\u306e\u3068\u304d}\\ ) \\\\ -( x+y ) & ( \\ y \\lt -2x \\text{\u306e\u3068\u304d}\\ ) \\end{array} \\right.}\r\n\\]\r\n

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

        \u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\n-z \\leqq f(t) \\leqq 1-z \\quad ... [1]\r\n\\]\r\n\\(0 \\leqq t \\leqq 1\\) \u306b\u304a\u3044\u3066\u5e38\u306b [1] \u3092\u307f\u305f\u3059 \\(z\\) \u304c\u5b58\u5728\u3059\u308b\u6761\u4ef6\u306f, \\(f(t)\\) \u306e\u5024\u57df\u306e\u5e45,\r\n\u3059\u306a\u308f\u3061\u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u306e\u5dee\u304c \\(1\\) \u4ee5\u4e0b\u3067\u3042\u308b\u6761\u4ef6\u3067\u3042\u308b.
        \r\n(1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u3066\u8003\u3048\u308b.<\/p>\r\n

          \r\n

        1. 1*<\/strong>\u3000\\(y \\gt 0\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n0 \\leqq x+y \\leqq 1 \\\\\r\n\\text{\u2234} \\quad 0 \\leqq y \\leqq -x+1\r\n\\end{align}\\]<\/li>\r\n

        2. 2*<\/strong>\u3000\\(-x \\leqq y \\leqq 0\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n0 \\leqq x+y+\\dfrac{y^2}{4x} & \\leqq 1 \\\\\r\n0 \\leqq 4x^2 +4xy +y^2 & \\leqq 4x \\\\\r\n0 \\leqq ( 2x+y )^2 & \\leqq 4x \\\\\r\n0 \\leqq 2x+y & \\leqq 2\\sqrt{x} \\quad ( \\ \\text{\u2235} \\ 2x+y \\gt 0 \\ ) \\\\\r\n\\text{\u2234} \\quad -2x \\leqq y & \\leqq -2x +2\\sqrt{x}\r\n\\end{align}\\]<\/li>\r\n

        3. 3*<\/strong>\u3000\\(-2x \\leqq y \\lt -x\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n0 \\leqq \\dfrac{y^2}{4x} & \\leqq 1 \\\\\r\n0 \\leqq y^2 & \\leqq 4x \\\\\r\n-y & \\leqq 2\\sqrt{x} \\quad ( \\ \\text{\u2235} \\ y \\lt 0 \\ ) \\\\\r\n\\text{\u2234} \\quad -2 \\sqrt{x} \\leqq y & \\lt -x\r\n\\end{align}\\]<\/li>\r\n

        4. 4*<\/strong>\u3000\\(y \\lt -2x\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n0 \\leqq -( x+y ) \\leqq 1 \\\\\r\n\\text{\u2234} \\quad -x-1 \\leqq y \\leqq -2x\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

          1*<\/strong>\uff5e4*<\/strong> \u3088\u308a \\(S\\) \u306e\u6982\u5f62\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u305f\u3060\u3057, \u5b9f\u7dda\u5883\u754c\u306f\u542b\u307f, \u70b9\u7dda\u5883\u754c\u3068\u767d\u70b9\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n\"\"\r\n

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

          \\(g(t) = xt^2 +yt +z\\) \u3068\u304a\u304f.
          \r\n\\(x=k\\) \uff08 \\(0 \\leqq k \\leqq 1\\) \uff09\u3068\u5b9a\u6570\u3068\u307f\u306a\u3057, \u4e0e\u3048\u3089\u308c\u305f\u6761\u4ef6\u3092\u307f\u305f\u3059 \\(y\\) , \\(z\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u308b.
          \r\n\\(g(t) = f(t) +z\\) \u306a\u306e\u3067, (1)<\/strong> \u306e\u7d50\u679c\u3092\u5229\u7528\u3057\u3066, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

            \r\n

          1. 1*<\/strong>\u3000\\(y \\gt 0\\) \u306e\u3068\u304d
            \r\n\u6700\u5927\u5024\uff1a \\(y+z+k\\) , \u6700\u5c0f\u5024\uff1a \\(z\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n0 \\leqq z , \\quad y+z+k \\leqq 1 \\\\\r\n\\text{\u2234} \\quad 0 \\leqq z \\leqq -y+1-k\r\n\\end{align}\\]<\/li>\r\n

          2. 2*<\/strong>\u3000\\(-k \\leqq y \\leqq 0\\) \u306e\u3068\u304d
            \r\n\u6700\u5927\u5024\uff1a \\(y+z+k\\) , \u6700\u5c0f\u5024\uff1a \\(-\\dfrac{y^2}{4k} +z\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n0 \\leqq -\\dfrac{y^2}{4k} +z , \\quad y+z+k \\leqq 1 \\\\\r\n\\text{\u2234} \\quad \\dfrac{y^2}{4k} \\leqq z \\leqq -y+1-k\r\n\\end{align}\\]<\/li>\r\n

          3. 3*<\/strong>\u3000\\(-2k \\leqq y \\leqq -k\\) \u306e\u3068\u304d
            \r\n\u6700\u5927\u5024\uff1a \\(z\\) , \u6700\u5c0f\u5024\uff1a \\(-\\dfrac{y^2}{4k} +z\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n0 \\leqq -\\dfrac{y^2}{4k} +z , \\quad z \\leqq 1 \\\\\r\n\\text{\u2234} \\quad \\dfrac{y^2}{4k} \\leqq z \\leqq 1\r\n\\end{align}\\]<\/li>\r\n

          4. 4*<\/strong>\u3000\\(y \\lt -2k\\) \u306e\u3068\u304d
            \r\n\u6700\u5927\u5024\uff1a \\(z\\) , \u6700\u5c0f\u5024\uff1a \\(y+z+k\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n0 \\leqq y+z+k , \\quad z \\leqq 1 \\\\\r\n\\text{\u2234} \\quad -y-k \\leqq z \\leqq 1\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

            1*<\/strong>\uff5e4*<\/strong> \u3088\u308a\u6761\u4ef6\u3092\u307f\u305f\u3059\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u306f\u542b\u3080\uff09\u3067\u3042\u308b.<\/p>\r\n\"\"\r\n

            \u3053\u306e\u9762\u7a4d\u3092 \\(T _ k\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nT _ k & = 1 \\cdot 1 +\\dfrac{1}{2} k^2 - \\displaystyle\\int _ 0^{2k} \\dfrac{y^2}{4k} \\, dy \\\\\r\n& = 1 +\\dfrac{k^2}{2} -\\dfrac{1}{4k} \\left[ \\dfrac{y^3}{3} \\right] _ 0^{2k} \\\\\r\n& = 1 +\\dfrac{k^2}{2} -\\dfrac{2k^2}{3} = 1 +\\dfrac{k^2}{6}\r\n\\end{align}\\]\r\n\u3053\u306e\u9818\u57df\u306f, \u9818\u57df \\(V\\) \u306e\u5e73\u9762 \\(x=k\\) \u306b\u304a\u3051\u308b\u65ad\u9762\u306a\u306e\u3067, \u6c42\u3081\u308b\u4f53\u7a4d\u306f\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 0^1 T _ k \\, dk & = \\left[ k +\\dfrac{k^3}{18} \\right] _ 0^1 \\\\\r\n& = \\underline{\\dfrac{17}{18}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"(1)\u3000\\(x , y\\) \u3092\u5b9f\u6570\u3068\u3057, \\(x \\gt 0\\) \u3068\u3059\u308b. \\(t\\) \u3092\u5909\u6570\u3068\u3059\u308b \\(2\\) \u6b21\u95a2\u6570 \\(f(t) = xt^2 +yt\\) \u306e \\(0 \\leqq t \\leqq 1\\) \u306b\u304a\u3051\u308b\u6700 … \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":[20],"tags":[139,13],"class_list":["post-13","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2011","tag-tokyo_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/13","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=13"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/13\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=13"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=13"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=13"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}