{"id":44,"date":"2011-11-25T22:17:07","date_gmt":"2011-11-25T13:17:07","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=44"},"modified":"2021-09-14T09:55:29","modified_gmt":"2021-09-14T00:55:29","slug":"iks201102","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201102\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272011\uff1a\u7b2c2\u554f"},"content":{"rendered":"<hr \/>\n<p>\u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066, \u539f\u70b9\u3092 O \u3068\u3057, \u6b21\u306e\u3088\u3046\u306a3\u70b9 P , Q , R \u3092\u8003\u3048\u308b.<\/p>\r\n<ol>\r\n<li><p>(a)\u3000\u70b9 P \u306f \\(x\\) \u8ef8\u4e0a\u306b\u3042\u308a, \u305d\u306e \\(x\\) \u5ea7\u6a19\u306f\u6b63\u3067\u3042\u308b.<\/p><\/li>\r\n<li><p>(b)\u3000\u70b9 Q \u306f\u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u3063\u3066, \\(\\text{OQ} = \\text{QP} = 1\\) \u3092\u6e80\u305f\u3059.<\/p><\/li>\r\n<li><p>(c)\u3000\u70b9 R \u306f\u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u3063\u3066, \\(\\text{OR} +\\text{RP} = 2\\) \u3092\u6e80\u305f\u3057, \u304b\u3064\u7dda\u5206 RP \u304c \\(x\\) \u8ef8\u306b\u5782\u76f4\u3068\u306a\u308b.<\/p><\/li>\r\n<\/ol>\r\n<p>\u305f\u3060\u3057, \u5ea7\u6a19\u8ef8\u306f\u7b2c \\(1\\) \u8c61\u9650\u306b\u542b\u3081\u306a\u3044\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\u4e0a\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059 \\(2\\) \u70b9 Q , R \u304c\u5b58\u5728\u3059\u308b\u3088\u3046\u306a, \u70b9 P \u306e \\(x\\) \u5ea7\u6a19\u304c\u53d6\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000<strong>(1)<\/strong> \u306e\u7bc4\u56f2\u3092\u70b9 P \u304c\u52d5\u304f\u3068\u304d, \u7dda\u5206 QR \u304c\u901a\u904e\u3059\u308b\u9818\u57df\u3092\u56f3\u793a\u3057, \u305d\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\u7dda\u5206 OP \u306e\u4e2d\u70b9\u3092 M \u3068\u3059\u308b. <strong>(1)<\/strong> \u306e\u7bc4\u56f2\u3092\u70b9 P \u304c\u52d5\u304f\u3068\u304d, \u56db\u89d2\u5f62 MPRQ \u306e\u9762\u7a4d\u3092\u6700\u5927\u306b\u3059\u308b\u70b9 P \u306e \\(x\\) \u5ea7\u6a19\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/ikashika_201102_01.png\" alt=\"\" title=\"ikashika_201102_01\" class=\"aligncenter size-full\" \/>\r\n<p>\u25b3OPQ \u306e\u8fba\u306e\u9577\u3055\u306b\u7740\u76ee\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n1+1 & \\gt x , \\ 1+x \\gt 1 \\\\\r\n\\text{\u2234} \\quad 0 & \\lt x \\lt 2\n\\end{align}\\]\r\n\u3053\u306e\u7bc4\u56f2\u3067 \u25b3OPR \u3082\u6210\u7acb\u3059\u308b\u306e\u3067, \u6c42\u3081\u308b\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{0 \\lt x \\lt 2}\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>Q \\(( \\cos \\theta , \\sin \\theta )\\) \u3068\u304a\u304f\u3068, P \\(( 2\\cos \\theta , 0 )\\) \u3068\u8868\u305b\u308b.<br \/>\r\nQ \u306f\u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u308b\u3053\u3068\u3068, <strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(0 \\lt \\theta \\lt \\dfrac{\\pi}{2} \\quad ... [1]\\) .<br \/>\r\nR \\(( 2\\cos \\theta , r )\\) \u3068\u304a\u304f\u3068, \u25b3OPR \u306b\u7740\u76ee\u3057\u3066\r\n\\[\\begin{align}\r\n( 2\\cos \\theta )^2 +r^2 & = (2-r)^2 \\\\\r\n4\\cos^2 \\theta & = 4-4r \\\\\r\n\\text{\u2234} \\quad r = 1-\\cos^2 \\theta & = \\sin^2 \\theta\n\\end{align}\\]\r\n[1] \u306e\u7bc4\u56f2\u306b\u304a\u3044\u3066, \\(\\sin \\theta , \\cos \\theta\\) \u304c\u5358\u8abf\u306b\u5909\u5316\u3059\u308b\u3053\u3068\u304b\u3089, \\(\\theta\\) \u3092 \\(0 \\rightarrow \\dfrac{\\pi}{2}\\) \u3068\u52d5\u304b\u3059\u3068<\/p>\r\n<ul>\r\n<li><p>Q \u306f \\(y = \\sqrt{1-x^2} \\ ... [2]\\) \u4e0a\u3092, \\(( 1 , 0 ) \\rightarrow ( 0 , 1 )\\)<\/p><\/li>\r\n<li><p>R \u306f \\(y = 1 -\\dfrac{x^2}{4} \\ ...[3]\\) \u4e0a\u3092 \\(( 2 , 0 ) \\rightarrow ( 0 , 1 )\\)<\/p><\/li>\r\n<\/ul>\r\n<p>\u3068\u52d5\u304f.<br \/>\r\n\u307e\u305f, \u66f2\u7dda [2] [3] \u306e\u4f4d\u7f6e\u95a2\u4fc2\u306f, \u3069\u3061\u3089\u3082\u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u308a\r\n\\[\\begin{align}\r\n\\left( 1 -\\dfrac{x^2}{4} \\right)^2 -\\left( \\sqrt{1-x^2} \\right)^2 & = 1 -\\dfrac{x^2}{2} +\\dfrac{x^4}{16} -\\left( 1-x^2 \\right) \\\\\r\n& = \\dfrac{x^4}{16} +\\dfrac{x^2}{2} \\gt 0\n\\end{align}\\]\r\n\u306a\u306e\u3067, [3] \u304c [2] \u3088\u308a\u4e0a\u5074\u306b\u3042\u308b.<br \/>\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u767d\u70b9, \u70b9\u7dda\u5883\u754c\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/ikashika_201102_02.png\" alt=\"\" title=\"ikashika_201102_02\" class=\"aligncenter size-full\" \/>\r\n<p>\u307e\u305f, \u3053\u306e\u9818\u57df\u306e\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ 0^2 \\left( 1 -\\dfrac{x^2}{4} \\right) \\, dx -\\dfrac{1}{2} \\cdot \\dfrac{\\pi}{2} \\cdot 1^2 \\\\\r\n& = \\left[ x -\\dfrac{x^3}{12} \\right] _ 0^2 -\\dfrac{\\pi}{4} \\\\\r\n& = \\underline{\\dfrac{4}{3} -\\dfrac{\\pi}{4}}\n\\end{align}\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p>M \\(( \\cos \\theta , 0 )\\) \u306a\u306e\u3067, \u56db\u89d2\u5f62 MPRQ \u306f\u53f0\u5f62\u3068\u306a\u308b.<br \/>\r\n\u3053\u306e\u9762\u7a4d\u3092 \\(T(\\theta)\\) \u3068\u304a\u304f\u3068,\r\n\\[\r\nT(\\theta) = \\dfrac{1}{2} \\left( \\sin \\theta +\\sin^2 \\theta \\right) \\cdot \\cos \\theta = \\dfrac{1}{4} \\sin 2\\theta ( \\sin \\theta +1 )\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n4 T'(\\theta) & = 2\\cos 2\\theta ( \\sin \\theta +1 ) +\\sin 2\\theta \\cos \\theta \\\\\r\n& = 2( 1 -2\\sin^2 \\theta )( \\sin \\theta +1 ) +2\\sin \\theta( 1- \\sin^2 \\theta) \\\\\r\n& = -2( \\sin \\theta +1 )( 3 \\sin^2 \\theta -\\sin \\theta -1) \\\\\r\n& = -2( \\sin \\theta +1 ) \\left( \\sin \\theta -\\dfrac{1+\\sqrt{13}}{6} \\right) \\left( \\sin \\theta -\\dfrac{1-\\sqrt{13}}{6} \\right)\n\\end{align}\\]\r\n[1] \u3088\u308a, \\(0 \\lt \\sin \\theta \\lt 1\\) \u306b\u6ce8\u610f\u3057\u3066, \\(T'(\\theta) =0\\) \u3092\u89e3\u304f\u3068\r\n\\[\r\n\\sin \\theta = \\dfrac{1+\\sqrt{13}}{6}\n\\]\r\n\u3053\u306e\u89e3\u3092 \\(\\alpha\\) \u3068\u304a\u304f\u3068, \\(T(\\theta)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} \\theta & 0 & \\cdots & \\alpha & \\cdots & \\dfrac{\\pi}{2} \\\\ \\hline T'( \\theta ) & & + & 0 & - & \\\\ \\hline T( \\theta ) & 0 & \\nearrow & \\text{\u6700\u5927} & \\searrow & 0 \\end{array}\r\n\\]\r\n\u3059\u306a\u308f\u3061, \\(\\theta = \\alpha\\) \u306e\u3068\u304d, \\(T(\\theta)\\) \u306f\u6700\u5927\u3068\u306a\u308b.<br \/>\r\n\u3053\u306e\u3068\u304d\u306e P \u306e \\(x\\) \u5ea7\u6a19\u306f,\r\n\\[\\begin{align}\r\n2\\cos \\alpha & = 2\\sqrt{1-\\sin^2 \\alpha} =2\\sqrt{1-\\dfrac{\\sin \\alpha +1}{3}} \\\\\r\n& = 2 \\sqrt{1 -\\dfrac{7+\\sqrt{13}}{18}} \\\\\r\n& = \\underline{\\dfrac{\\sqrt{22-2\\sqrt{13}}}{3}}\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066, \u539f\u70b9\u3092 O \u3068\u3057, \u6b21\u306e\u3088\u3046\u306a3\u70b9 P , Q , R \u3092\u8003\u3048\u308b. (a)\u3000\u70b9 P \u306f \\(x\\) \u8ef8\u4e0a\u306b\u3042\u308a, \u305d\u306e \\(x\\) \u5ea7\u6a19\u306f\u6b63\u3067\u3042\u308b. (b)\u3000\u70b9 Q \u306f\u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u3063\u3066,  &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/iks201102\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[29],"tags":[145,13],"class_list":["post-44","post","type-post","status-publish","format-standard","hentry","category-ikashika_2011","tag-ikashika","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/44","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=44"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/44\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=44"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=44"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=44"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}