{"id":18,"date":"2011-11-25T21:15:43","date_gmt":"2011-11-25T12:15:43","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=18"},"modified":"2021-03-23T18:50:05","modified_gmt":"2021-03-23T09:50:05","slug":"kyr201101","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr201101\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb2011\uff1a\u7b2c1\u554f"},"content":{"rendered":"
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\u6b21\u306e\u5404\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u7bb1\u306e\u4e2d\u306b, \\(1\\) \u304b\u3089 \\(9\\) \u307e\u3067\u306e\u756a\u53f7\u3092 \\(1\\) \u3064\u305a\u3064\u66f8\u3044\u305f \\(9\\) \u679a\u306e\u30ab\u30fc\u30c9\u304c\u5165\u3063\u3066\u3044\u308b.\r\n\u305f\u3060\u3057, \u7570\u306a\u308b\u30ab\u30fc\u30c9\u306b\u306f\u7570\u306a\u308b\u756a\u53f7\u304c\u66f8\u304b\u308c\u3066\u3044\u308b\u3082\u306e\u3068\u3059\u308b.\r\n\u3053\u306e\u7bb1\u304b\u3089 \\(2\\) \u679a\u306e\u30ab\u30fc\u30c9\u3092\u540c\u6642\u306b\u9078\u3073, \u5c0f\u3055\u3044\u307b\u3046\u306e\u6570\u3092 \\(X\\) \u3068\u3059\u308b.\r\n\u3053\u308c\u3089\u306e\u30ab\u30fc\u30c9\u3092\u7bb1\u306b\u623b\u3057\u3066, \u518d\u3073 \\(2\\) \u679a\u306e\u30ab\u30fc\u30c9\u3092\u540c\u6642\u306b\u9078\u3073, \u5c0f\u3055\u3044\u307b\u3046\u306e\u6570\u3092 \\(Y\\) \u3068\u3059\u308b.\r\n\\(X=Y\\) \u3067\u3042\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ 0^{\\frac{1}{2}} ( x+1 ) \\sqrt{1 -2x^2} \\, dx\\) \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

    \\(X=k \\quad ( k = 1 , 2 , \\cdots , 8 )\\) \u3068\u306a\u308b\u78ba\u7387\u3092 \\(p _ k\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\np _ k & = \\dfrac{{} _ {10-k} \\text{C} {} _ 2 -{} _ {9-k} \\text{C} {} _ 2}{{} _ 9 \\text{C} {} _ 2} \\\\\r\n& = \\dfrac{( 10-k )( 9-k ) -( 9-k )( 8-k )}{9 \\cdot 8} \\\\\r\n& = \\dfrac{9-k}{36}\r\n\\end{align}\\]\r\n\\(Y=k \\quad ( k = 1 , 2 , \\cdots , 8 )\\) \u3068\u306a\u308b\u78ba\u7387\u3082 \\(p _ k\\) \u3068\u306a\u308b\u306e\u3067, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits_ {k=1}^8 {p _ k}^2 & = \\dfrac{1}{6^4} \\textstyle\\sum\\limits _ {k=1}^8 ( 9-k )^2 \\\\\r\n& = \\dfrac{1}{6^4} \\textstyle\\sum\\limits _ {k=1}^8 k^2 \\\\\r\n& = \\dfrac{1}{6^4} \\cdot \\dfrac{1}{6} \\cdot 8 \\cdot 9 \\cdot 17 \\\\\r\n& = \\underline{\\dfrac{17}{108}}\r\n\\end{align}\\]\r\n

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

    \u6c42\u3081\u308b\u5b9a\u7a4d\u5206\u3092 \\(I\\) \u3068\u304a\u304f.\r\n\\[\r\nI = \\underline{\\displaystyle\\int _ 0^{\\frac{1}{2}} x \\sqrt{1 -2x^2} \\, dx} _ {[ \\text{A} ]} + \\underline{\\displaystyle\\int _ 0^{\\frac{1}{2}} \\sqrt{1 -2x^2} \\, dx} _ {[ \\text{B} ]}\r\n\\]\r\n[A] \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n[ \\text{A} ] & = \\dfrac{1}{4} \\displaystyle\\int _ 0^{\\frac{1}{2}} \\sqrt{1 -2x^2} \\, ( 2x^2 )' \\, dx \\\\\r\n& = \\dfrac{1}{4} \\left[ -\\dfrac{2}{3} ( 1-2x^2 )^{\\frac{3}{2}} \\right] _ 0^{\\frac{1}{2}} \\\\\r\n& =\\dfrac{1}{6} \\left\\{ 1 -\\left( \\dfrac{1}{2} \\right)^{\\frac{3}{2}} \\right\\} \\\\\r\n& = \\dfrac{1}{6} -\\dfrac{\\sqrt{2}}{24}\r\n\\end{align}\\]\r\n[B] \u306b\u3064\u3044\u3066, \\(u = \\sqrt{2} x\\) \u3068\u7f6e\u63db\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n& du = \\sqrt{2} dx , \\\\\r\n& \\begin{array}{c|ccc} x & 0 & \\rightarrow & \\dfrac{1}{2} \\\\ \\hline u & 0 & \\rightarrow & \\dfrac{\\sqrt{2}}{2} \\end{array}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n[ \\text{B} ] = \\dfrac{\\sqrt{2}}{2} \\underline{\\displaystyle\\int _ 0^{\\frac{\\sqrt{2}}{2}} \\sqrt{1-u^2} \\, du} _ {[ \\text{C} ]}\r\n\\]\r\n[C] \u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u9762\u7a4d\u3092\u8868\u3059\u306e\u3067<\/p>\r\n\"\"\r\n

    \\[\\begin{align}\r\n[ \\text{B} ] & = \\dfrac{\\sqrt{2}}{2} \\left\\{ \\dfrac{1}{2} \\cdot 1^2 \\cdot \\dfrac{\\pi}{4} +\\dfrac{1}{2} \\left( \\dfrac{\\sqrt{2}}{2} \\right)^2 \\right\\} \\\\\r\n& = \\dfrac{\\sqrt{2}}{2} \\left( \\dfrac{\\pi}{8} +\\dfrac{1}{4} \\right)\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nI & = \\dfrac{1}{6} -\\dfrac{\\sqrt{2}}{24} +\\dfrac{\\sqrt{2}}{2} \\left( \\dfrac{\\pi}{8} +\\dfrac{1}{4} \\right) \\\\\r\n& = \\underline{\\dfrac{1}{6} +\\dfrac{\\sqrt{2}}{12} +\\dfrac{\\sqrt{2} \\pi}{16}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u5404\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\u7bb1\u306e\u4e2d\u306b, \\(1\\) \u304b\u3089 \\(9\\) \u307e\u3067\u306e\u756a\u53f7\u3092 \\(1\\) \u3064\u305a\u3064\u66f8\u3044\u305f \\(9\\) \u679a\u306e\u30ab\u30fc\u30c9\u304c\u5165\u3063\u3066\u3044\u308b. \u305f\u3060\u3057, \u7570\u306a\u308b\u30ab\u30fc\u30c9\u306b\u306f\u7570\u306a\u308b\u756a\u53f7\u304c\u66f8\u304b\u308c\u3066\u3044\u308b\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u7bb1\u304b … \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":[23],"tags":[140,13],"class_list":["post-18","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2011","tag-kyoto_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/18","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=18"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/18\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=18"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=18"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=18"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}