{"id":1212,"date":"2015-07-31T23:37:57","date_gmt":"2015-07-31T14:37:57","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1212"},"modified":"2015-07-31T23:37:57","modified_gmt":"2015-07-31T14:37:57","slug":"htb201402","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb201402\/","title":{"rendered":"\u4e00\u6a4b\u59272014\uff1a\u7b2c2\u554f"},"content":{"rendered":"\\(0 \\lt t \\lt 1\\) \u3068\u3057, \u653e\u7269\u7dda \\(C : \\ y = x^2\\) \u4e0a\u306e\u70b9 \\(( t , t^2 )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092 \\(l\\) \u3068\u3059\u308b. \\(C\\) \u3068 \\(l\\) \u3068 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S_1\\) \u3068\u3057, \\(C\\) \u3068 \\(l\\) \u3068\u76f4\u7dda \\(x = 1\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S_2\\) \u3068\u3059\u308b. \\(S_1 +S_2\\) \u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088. \r\n
\r\n\r\n

\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n\r\n\\(C\\) \u306b\u3064\u3044\u3066, \\(y' = 2x\\) \u306a\u306e\u3067\r\n\\[ \\begin{align}\r\nl : \\ y & = 2t (x-t) +t^2 \\\\\r\n& = 2tx -t^2\r\n\\end{align} \\]\r\n\u3053\u306e\u76f4\u7dda\u306b\u3064\u3044\u3066, \\(y = 0\\) \u3068\u3059\u308c\u3070\r\n\\[ \\begin{align}\r\n2tx -t^2 & = 0 \\\\\r\n\\text{\u2234} \\quad x & = \\dfrac{t}{2}\r\n\\end{align} \\]\r\n\u307e\u305f, \\(x = 1\\) \u3068\u3059\u308c\u3070\r\n\\[\r\ny = 2t -t^2 = t (2-t)\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[ \\begin{align}\r\nS_1 +S_2 & = \\displaystyle\\int_0^1 x^2 \\, dx -\\dfrac{1}{2} \\left( 1 -\\dfrac{t}{2} \\right) t (2-t) \\\\\r\n& = \\left[ \\dfrac{x^3}{3} \\right]_0^1 -\\dfrac{t}{4} (2-t)^2 \\\\\r\n& = -\\dfrac{t^3}{4} +2t^2 -t +\\dfrac{1}{3}\r\n\\end{align} \\]\r\n\u3053\u308c\u3092 \\(f(t)\\) \u3068\u304a\u3051\u3070\r\n\\[ \\begin{align}\r\nf'(t) & = -\\dfrac{3 t^2}{4} +2t -1 \\\\\r\n& = -\\dfrac{1}{4} ( 3t -2 ) ( t -2 )\r\n\\end{align} \\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(0 \\lt t \\lt 1\\) \u306b\u304a\u3051\u308b \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b. \r\n\\[\r\n\\begin{array}{c|ccc} t & (0) & \\cdots & \\dfrac{2}{3} & \\cdots & (1) \\\\ \\hline f'(t) & & - & 0 & + & \\\\ \\hline f(t) & & \\searrow & \\text{\u6700\u5c0f} & \\nearrow & \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5c0f\u5024\u306f\r\n\\[ \\begin{align}\r\nf \\left( \\dfrac{2}{3} \\right) & = -\\dfrac{1}{4} \\cdot \\dfrac{8}{27} +\\dfrac{4}{9} -\\dfrac{2}{3} +\\dfrac{1}{3} \\\\\r\n& = \\underline{\\dfrac{1}{27}}\r\n\\end{align} \\]\r\n","protected":false},"excerpt":{"rendered":"\\(0 \\lt t \\lt 1\\) \u3068\u3057, \u653e\u7269\u7dda \\(C : \\ y = x^2\\) \u4e0a\u306e\u70b9 \\(( t , t^2 )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092 \\(l\\) \u3068\u3059\u308b. \\(C\\) \u3068 \\(l\\) \u3068 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c … \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":[124],"tags":[146,112],"class_list":["post-1212","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2014","tag-hitotsubashi","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1212","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=1212"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1212\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1212"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1212"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1212"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}