{"id":288,"date":"2012-02-02T20:54:50","date_gmt":"2012-02-02T11:54:50","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=288"},"modified":"2021-09-10T08:46:32","modified_gmt":"2021-09-09T23:46:32","slug":"osr200804","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr200804\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2008\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(t\\) \u3092\u8ca0\u306e\u5b9f\u6570\u3068\u3057, \\(xy\\) \u5e73\u9762\u4e0a\u3067\u66f2\u7dda \\(y = 2^{2x+2t}\\) \u3068\u66f2\u7dda \\(y = 2^{x+3t}\\) \u304a\u3088\u3073 \\(y\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u3092 \\(D\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(D\\) \u3092 \\(x\\) \u8ef8\u306e\u5468\u308a\u306b \\(1\\) \u56de\u8ee2\u3055\u305b\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d \\(V(t)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(t\\) \u304c\u8ca0\u306e\u5b9f\u6570\u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(V(t)\\) \u306e\u6700\u5927\u5024\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

    \\(2\\) \u3064\u306e\u66f2\u7dda\u306e\u5f0f\u3088\u308a\r\n\\[\r\n2^{2x+2t} -2^{x+3t} = 2^{x+2t} \\left( 2^x -2^t \\right)\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u9818\u57df \\(D\\) \u306e\u6982\u5f62\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"\"\r\n

    \u3086\u3048\u306b\r\n\\[\\begin{align}\r\nV(t) & = \\pi \\displaystyle\\int _ t^0 \\left( 2^{2 (2x+2t)} -2^{2 (x+3t)} \\right) \\, dx \\\\\r\n& = \\pi \\displaystyle\\int _ t^0 \\left( 2^{4t} 2^{4x} -2^{6t} 2^{2x} \\right) \\, dx \\\\\r\n& = 2^{4t} \\pi \\left[ \\dfrac{2^{4x}}{4 \\log 2} \\right] _ t^0 -2^{6t} \\pi \\left[ \\dfrac{2^{2x}}{2 \\log 2} \\right] _ t^0 \\\\\r\n& = \\dfrac{2^{4t} \\pi}{4 \\log 2} \\left( 1 -2^{4t}\\right) -\\dfrac{2^{6t} \\pi}{2 \\log 2} \\left( 1 -2^{2t}\\right) \\\\\r\n& = \\underline{\\dfrac{\\pi}{4 \\log 2} \\left( 2^{8t} -2^{6t+1} +2^{4t} \\right)}\n\\end{align}\\]\r\n

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

    \\(u = 2^{2t}\\) \u3068\u304a\u3051\u3070, \\(t \\lt 0\\) \u306e\u3068\u304d, \\(0 \\lt u \\lt 1 \\quad ... [1]\\) .
    \r\n\u307e\u305f\r\n\\[\r\n2^{8t} -2^{6t+1} +2^{4t} = u^4 -2u^3 +u^2\n\\]\r\n\u306a\u306e\u3067, \u3053\u308c\u3092 \\(f(u)\\) \u3068\u304a\u3044\u3066, [1] \u306b\u304a\u3051\u308b\u6700\u5927\u5024\u3092\u8003\u3048\u308b.\r\n\\[\r\nf'(u) = 4u^3 -6u^2 +2u =2u (u-1) (2u-1)\n\\]\r\n\\(f'(u) = 0\\) \u3092\u89e3\u304f\u3068\r\n\\[\r\nu =\\dfrac{1}{2}\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(u)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} u & (0) & \\cdots & \\dfrac{1}{2} & \\cdots & (1) \\\\ \\hline f'(u) & & + & 0 & - & \\\\ \\hline f(u) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(u)\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\nf \\left( \\dfrac{1}{2} \\right) =\\dfrac{1}{16} -2 \\cdot \\dfrac{1}{8} +\\dfrac{1}{4} =\\dfrac{1}{16}\n\\]\r\n\u3088\u3063\u3066, \\(V(t)\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{\\pi}{4 \\log 2} \\cdot \\dfrac{1}{16} =\\underline{\\dfrac{\\pi}{64 \\log 2} }\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(t\\) \u3092\u8ca0\u306e\u5b9f\u6570\u3068\u3057, \\(xy\\) \u5e73\u9762\u4e0a\u3067\u66f2\u7dda \\(y = 2^{2x+2t}\\) \u3068\u66f2\u7dda \\(y = 2^{x+3t}\\) \u304a\u3088\u3073 \\(y\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u3092 \\(D\\) \u3068\u3059\u308b. (1)\u3000\\(D\\) \u3092 … \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":[51],"tags":[142,16],"class_list":["post-288","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2008","tag-osaka_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/288","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=288"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/288\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=288"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=288"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=288"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}