{"id":1144,"date":"2015-07-02T10:18:07","date_gmt":"2015-07-02T01:18:07","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1144"},"modified":"2021-09-15T07:21:29","modified_gmt":"2021-09-14T22:21:29","slug":"ngr201402","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201402\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2014\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\u3066, \\(2\\) \u70b9 P \\(( t , t^2 )\\) , Q \\(( t+1 , (t+1)^2 )\\) \u3092\u8003\u3048\u308b. \\(t\\) \u304c \\(-1 \\leqq t \\leqq 0\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \u7dda\u5206 PQ \u304c\u901a\u904e\u3057\u3066\u3067\u304d\u308b\u56f3\u5f62\u3092\u56f3\u793a\u3057, \u305d\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p>\r\n

\r\n\r\n

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

\u7dda\u5206 PQ \u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = \\left\\{ (t+1)^2 -t^2 \\right\\} (x-t) +t^2 \\\\\r\n& = (2t+1) x -t^2 -t \\quad ( t \\leqq x \\leqq t+1 ) \\quad ... [1] \\ .\r\n\\end{align}\\]\r\n\u7dda\u5206 PQ \u3068 \u76f4\u7dda \\(x = k \\ ( -1 \\leqq k \\leqq 1 )\\) \u306e\u4ea4\u70b9\u306f \\(\\left( k , f(k) \\right)\\) \u3068\u8868\u305b\u308b.
\r\n\\(f(k)\\) \u306e\u6700\u5927\u5024\u3092 \\(M(k)\\) , \u6700\u5c0f\u5024\u3092 \\(m(k)\\) \u3068\u304a\u3051\u3070, \\(x = k\\) \u306b\u304a\u3051\u308b\u7dda\u5206 PQ \u306e\u901a\u904e\u9818\u57df\u306f\r\n\\[\r\nm(k) \\leqq y \\leqq M(k) \\ .\r\n\\]\r\n\u3068\u8868\u305b\u308b.
\r\n[1] \u3092 \\(t\\) \u306e\u95a2\u6570\u3068\u307f\u306a\u3057\u3066, \\(y = f(t)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf(t) & = -t^2 +(2k-1) t +k \\\\\r\n& = - \\left\\{ t -\\left( k -\\dfrac{1}{2} \\right) \\right\\}^2 +k^2 +\\dfrac{1}{4} \\quad ( k-1 \\leqq t \\leqq k ) \\quad ... [2] \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf \\left( k -\\dfrac{1}{2} \\right) & = k^2 +\\dfrac{1}{4} , \\\\\r\nf(x) & = f( x-1 ) = k^2 \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(t\\) \u306e\u3068\u308a\u5f97\u308b\u7bc4\u56f2\u306b\u6ce8\u610f\u3057\u3066, \\(M(k)\\) \u306e\u5019\u88dc\u306f\r\n\\[\\begin{gather}\r\nf(-1) = -k , \\quad f(0) = k , \\\\\r\nf \\left( k -\\dfrac{1}{2} \\right) = k^2 +\\dfrac{1}{4} \\\\\r\n\\left( \\text{\u305f\u3060\u3057} \\ -1 \\leqq k -\\dfrac{1}{2} \\leqq 0 \\ \\text{\u3059\u306a\u308f\u3061} \\ -\\dfrac{1}{2} \\leqq k \\leqq \\dfrac{1}{2} \\text{\u306e\u3068\u304d} \\right) \\ .\r\n\\end{gather}\\]\r\n\\(m(k)\\) \u306e\u5019\u88dc\u306f\r\n\\[\\begin{gather}\r\nf(-1) = -k , \\quad f(0) = k , \\\\\r\nf(k) = f( k-1 ) = k^2 \\ .\r\n\\end{gather}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u305d\u308c\u305e\u308c\u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308c\u3070, \u6c42\u3081\u308b\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u542b\u3080\uff09\u3068\u306a\u308b.<\/p>\r\n\"ngr20140201\"\r\n

\u307e\u305f, \u3053\u306e\u9818\u57df\u306e\u9762\u7a4d \\(S\\) \u306f, \u5bfe\u79f0\u6027\u3082\u5229\u7528\u3057\u3066\r\n\\[\\begin{align}\r\nS & = 2 \\displaystyle\\int _ 0^{\\frac{1}{2}} \\left\\{\\left( x^2 +\\dfrac{1}{4} \\right) -x^2 \\right\\} \\, dx +2 \\displaystyle\\int _ {\\frac{1}{2}}^1 ( x -x^2 ) \\, dx \\\\\r\n& = 2 \\cdot \\dfrac{1}{4} \\cdot \\dfrac{1}{2} +\\left[ x^2 -\\dfrac{2 x^3}{3} \\right] _ {\\frac{1}{2}}^1 \\\\\r\n& = \\dfrac{1}{4} +\\dfrac{1}{3} -\\dfrac{1}{6} \\\\\r\n& = \\underline{\\dfrac{5}{12}} \\ .\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\u3066, \\(2\\) \u70b9 P \\(( t , t^2 )\\) , Q \\(( t+1 , (t+1)^2 )\\) \u3092\u8003\u3048\u308b. \\(t\\) \u304c \\(-1 \\leqq t \\leqq 0\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f … \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":[119],"tags":[143,112],"class_list":["post-1144","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2014","tag-nagoya_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1144","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=1144"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1144\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1144"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1144"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}