{"id":1133,"date":"2015-06-23T01:45:04","date_gmt":"2015-06-22T16:45:04","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1133"},"modified":"2021-09-24T17:34:23","modified_gmt":"2021-09-24T08:34:23","slug":"tok201405","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201405\/","title":{"rendered":"\u6771\u5de5\u59272014\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\( xy\\) \u5e73\u9762\u4e0a\u306e\u66f2\u7dda \\(C : \\ y = x^3+x^2+1\\) \u3092\u8003\u3048, \\(C\\) \u4e0a\u306e\u70b9 \\((1,3)\\) \u3092 \\(\\text{P} {} _ {0}\\) \u3068\u3059\u308b.\r\n\\(k = 1 , 2 , 3 , \\cdots\\) \u306b\u5bfe\u3057\u3066, \u70b9 \\( \\text{P} {} _ {k-1} \\ ( x _ {k-1} , y _ {k-1} )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u63a5\u7dda\u3068 \\(C\\) \u306e\u4ea4\u70b9\u306e\u3046\u3061\u3067 \\( \\text{P} {} _ {k-1}\\) \u3068\u7570\u306a\u308b\u70b9\u3092 \\(\\text{P} {} _ k \\ ( x _ k , y _ k )\\) \u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d, \\( \\text{P} {} _ {k-1}\\) \u3068 \\(\\text{P} {} _ k\\) \u3092\u7d50\u3076\u7dda\u5206\u3068 \\(C\\) \u306b\u3088\u3063\u3066\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S _ k\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(S _ 1\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(x _ k\\) \u3092 \\(k\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(\\textstyle\\sum\\limits _ {k=1}^{\\infty} \\dfrac{1}{S _ k}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

    \\(C\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ny' = 3x^2 +2x\r\n\\]\r\n\u70b9 \\(P {} _ {k-1}\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u63a5\u7dda\u3092 \\(\\ell _ {k}\\) \u3068\u304a\u304f\u3068, \\(\\ell _ {k}\\) \u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = \\left( 3 {x _ {k-1}}^2 +2 x _ {k-1} \\right) \\left( x -x _ {k-1} \\right) +{x _ {k-1}}^3 +{x _ {k-1}}^2 +1 \\\\\r\n& = \\left( 3 {x _ {k-1}}^2 +2 x _ {k-1} \\right) x -2 {x _ {k-1}}^3 -{x _ {k-1}}^2 +1\r\n\\end{align}\\]\r\n\\(C\\) \u3068 \\(\\ell _ {k}\\) \u306e\u5f0f\u304b\u3089, \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\r\nx^3 +x^2 +1 = \\left( 3 {x _ {k-1}}^2 +2 x _ {k-1} \\right) x -2 {x _ {k-1}}^3 -{x _ {k-1}}^2 +1 \\\\\r\n\\left( x -x _ {k-1} \\right)^2 \\left( x +2x _ {k-1} +1 \\right) = 0 \\\\\r\n\\text{\u2234} \\quad x = x _ {k-1} , \\ -2x _ {k-1} -1\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\nx _ k = -2x _ {k-1} -1 \\quad ... [1]\r\n\\]\r\n\u3053\u306e\u3068\u304d\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ {x _ {k-1}}^{x _ k} & \\left( x -x _ {k-1} \\right)^2 \\left( x -x _ k \\right) \\, dx \\\\\r\n& = \\displaystyle\\int _ {x _ {k-1}}^{x _ k} \\left\\{ \\left( x -x _ {k-1} \\right)^3 + \\left( x _ k -x _ {k-1} \\right) \\left( x -x _ k \\right)^2 \\right\\} \\, dx \\\\\r\n& = \\left[ \\dfrac{1}{4} \\left( x -x _ {k-1} \\right)^4 + \\dfrac{x _ k -x _ {k-1}}{3} \\left( x -x _ {k-1} \\right)^3 \\right] _ {x _ {k-1}}^{x _ k} \\\\\r\n& = \\dfrac{\\left( x _ k -x _ {k-1} \\right)^4}{12} \\gt 0\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nS _ k = \\dfrac{\\left( x _ k -x _ {k-1} \\right)^4}{12} \\quad ... [2]\r\n\\]\r\n

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

    \\(x _ 0 = 1\\) \u306a\u306e\u3067, [1] \u3088\u308a\r\n\\[\r\nx _ 1 = -2 \\cdot 1 -1 = -3\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [2] \u3088\u308a\r\n\\[\r\nS _ 1 = \\dfrac{( -3-1 )^4}{12} = \\underline{\\dfrac{64}{3}}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n[1] \u3092\u5909\u5f62\u3059\u308c\u3070\r\n\\[\r\nx _ k +\\dfrac{1}{3} = -2 \\left( x _ {k-1} +\\dfrac{1}{3} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \u6570\u5217 \\(\\left\\{ x _ k +\\dfrac{1}{3} \\right\\}\\) \u306f, \u521d\u9805 \\(x _ 0 +\\dfrac{1}{3} = \\dfrac{4}{3}\\) , \u516c\u6bd4 \\(-2\\) \u306e\u7b49\u6bd4\u6570\u5217\u3067\u3042\u308b.
    \r\n\u3059\u306a\u308f\u3061\r\n\\[\r\nx _ k +\\dfrac{1}{3} = \\dfrac{4}{3} (-2)^k \\\\\r\n\\text{\u2234} \\quad x _ k = \\underline{\\dfrac{(-2)^{k+2} -1}{3}}\r\n\\]\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u3068 [2] \u3088\u308a\r\n\\[\\begin{align}\r\nS _ k & = \\dfrac{1}{12} \\left\\{ \\dfrac{(-2)^{k+2} -1}{3} -\\dfrac{(-2)^{k+1} -1}{3} \\right\\}^4 \\\\\r\n& = \\dfrac{2^{4(k+1)}}{12} \\\\\r\n& = \\dfrac{2^{4k+2}}{3}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=1}^{\\infty} \\dfrac{1}{S _ k} & = \\dfrac{3}{64} \\cdot \\dfrac{1}{1 -\\frac{1}{16}} \\\\\r\n& = \\dfrac{3}{64} \\cdot \\dfrac{16}{15} \\\\\r\n& = \\underline{\\dfrac{1}{20}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\( xy\\) \u5e73\u9762\u4e0a\u306e\u66f2\u7dda \\(C : \\ y = x^3+x^2+1\\) \u3092\u8003\u3048, \\(C\\) \u4e0a\u306e\u70b9 \\((1,3)\\) \u3092 \\(\\text{P} {} _ {0}\\) \u3068\u3059\u308b. \\(k = 1 , 2 , 3 , … \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":[116],"tags":[],"class_list":["post-1133","post","type-post","status-publish","format-standard","hentry","category-toko_2014"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1133","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=1133"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1133\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1133"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1133"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}