{"id":635,"date":"2013-02-15T00:32:31","date_gmt":"2013-02-14T15:32:31","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=635"},"modified":"2021-09-10T21:30:19","modified_gmt":"2021-09-10T12:30:19","slug":"tok200702","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok200702\/","title":{"rendered":"\u6771\u5de5\u59272007\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u6b63\u6570 \\(a\\) \u306b\u5bfe\u3057\u3066, \u653e\u7269\u7dda \\(y = x^2\\) \u4e0a\u306e\u70b9 \\(A \\ (a,a^2)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092, \\(A\\) \u3092\u4e2d\u5fc3\u306b \\(-30^{\\circ}\\) \u56de\u8ee2\u3057\u305f\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u3059\u308b.\r\n\\(\\ell\\) \u3068 \\(y = x^2\\) \u306e\u4ea4\u70b9\u3067 \\(A\\) \u3067\u306a\u3044\u65b9\u3092 \\(B\\) \u3068\u3059\u308b. \u3055\u3089\u306b, \u70b9 \\((a,0)\\) \u3092 \\(C\\) , \u539f\u70b9\u3092 \\(O\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\ell\\) \u306e\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u7dda\u5206 \\(OC , CA\\) \u3068 \\(y = x^2\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S(a)\\) , \u7dda\u5206 \\(AB\\) \u3068 \\(y = x^2\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(T(a)\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d\r\n\\[\r\n\\displaystyle\\lim _ {a \\rightarrow \\infty} \\dfrac{T(a)}{S(a)}\r\n\\]\r\n\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

    \u70b9 \\(A\\) \u306b\u304a\u3051\u308b \\(y = x^2\\) \u306e\u63a5\u7dda\u3092 \\(m\\) \u3068\u3057, \\(m , \\ell\\) \u304c \\(x\\) \u8ef8\u6b63\u65b9\u5411\u3068\u306a\u3059\u89d2\u3092\u305d\u308c\u305e\u308c \\(\\theta _ {m} , \\theta _ {\\ell}\\) \u3068\u304a\u304f.
    \r\n\\(y = x^2\\) \u3088\u308a, \\(y'=2x\\) \u306a\u306e\u3067\r\n\\[\r\n\\tan \\theta _ {m} = 2a\r\n\\]\r\n\u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n\\tan \\theta _ {\\ell} & = \\tan ( \\theta _ {m} -30^{\\circ} ) \\\\\r\n& = \\dfrac{\\tan \\theta _ {m} - \\tan 30^{\\circ}}{1 +\\tan \\theta _ {m} \\tan 30^{\\circ}} \\\\\r\n& = \\dfrac{2a -\\frac{1}{\\sqrt{3}}}{1 +2a \\cdot \\frac{1}{\\sqrt{3}}} \\\\\r\n& = \\dfrac{2 \\sqrt{3} a -1}{2a +\\sqrt{3}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(\\ell\\) \u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\r\n\\underline{y = \\dfrac{2 \\sqrt{3} a -1}{2a +\\sqrt{3}} (x-a) +a^2}\r\n\\]\r\n

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

    \\[\r\nS(a) = \\displaystyle\\int _ 0^a x^2 \\, dx = \\left[ \\dfrac{x^3}{3} \\right] _ 0^a = \\dfrac{a^3}{3}\r\n\\]\r\n

    (1)<\/strong> \u306e\u7d50\u679c\u3068 \\(y = x^2\\) \u3088\u308a, \\(y\\) \u3092\u6d88\u53bb\u3057\u3066\r\n\\[\\begin{gather}\r\n\\left( 2a +\\sqrt{3} \\right) x^2 = \\left( 2 \\sqrt{3} a -1 \\right) (x-a) +\\left( 2a +\\sqrt{3} \\right) a^2 \\\\\r\n\\left( 2a +\\sqrt{3} \\right) x^2 -\\left( 2 \\sqrt{3} a -1 \\right) x -a \\left( 2a^2 -\\sqrt{3} a +1 \\right) \\\\\r\n\\left\\{ \\left( 2a +\\sqrt{3} \\right) x +2a^2 -\\sqrt{3} a +1 \\right\\} (x-a) = 0 \\\\\r\n\\text{\u2234} \\quad x = a , \\ -a -\\sqrt{3} +\\dfrac{4}{2a +\\sqrt{3}}\r\n\\end{gather}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nT(a) & = \\dfrac{1}{6} \\left\\{ a -\\left( -a -\\sqrt{3} +\\dfrac{4}{2a +\\sqrt{3}} \\right) \\right\\}^3 \\\\\r\n& = \\dfrac{1}{6} \\left( 2a -\\sqrt{3} +\\dfrac{4}{2a +\\sqrt{3}} \\right)^3\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\dfrac{T(a)}{S(a)} & = \\dfrac{1}{2} \\left\\{ 2 -\\dfrac{\\sqrt{3}}{a} +\\dfrac{4}{a \\left( 2a +\\sqrt{3} \\right)} \\right\\}^3 \\\\\r\n& \\rightarrow \\dfrac{1}{2} \\cdot 2^3 = 4 \\quad ( n \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {a \\rightarrow \\infty} \\dfrac{T(a)}{S(a)} = \\underline{4}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b63\u6570 \\(a\\) \u306b\u5bfe\u3057\u3066, \u653e\u7269\u7dda \\(y = x^2\\) \u4e0a\u306e\u70b9 \\(A \\ (a,a^2)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092, \\(A\\) \u3092\u4e2d\u5fc3\u306b \\(-30^{\\circ}\\) \u56de\u8ee2\u3057\u305f\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u3059\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":[91],"tags":[141,109],"class_list":["post-635","post","type-post","status-publish","format-standard","hentry","category-toko_2007","tag-toko","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/635","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=635"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/635\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=635"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=635"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=635"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}