{"id":499,"date":"2012-12-03T14:13:23","date_gmt":"2012-12-03T05:13:23","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=499"},"modified":"2021-09-27T21:08:26","modified_gmt":"2021-09-27T12:08:26","slug":"thr201005","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201005\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2010\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(0 \\lt t \\lt 3\\) \u306e\u3068\u304d, \u9023\u7acb\u65b9\u7a0b\u5f0f\r\n\\[\r\n\\left\\{ \\begin{array}{l} 0 \\leqq y \\leqq \\sin x \\\\ 0 \\leqq x \\leqq t-y \\end{array} \\right.\r\n\\]\r\n\u306e\u8868\u3059\u9818\u57df\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b\u56de\u8ee2\u3057\u3066\u5f97\u3089\u308c\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092 \\(V(t)\\) \u3068\u3059\u308b.\r\n\\(\\dfrac{d}{dt} V(t) = \\dfrac{\\pi}{4}\\) \u3068\u306a\u308b \\(t\\) \u3068, \u305d\u306e\u3068\u304d\u306e \\(V(t)\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n

\r\n\r\n

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

\\(f(x) = \\sin x -(t-x)\\) \u3068\u304a\u304f\u3068, \\(0 \\lt x\\lt 3\\) \u306b\u304a\u3044\u3066\r\n\\[\r\nf'(x) = \\cos x +x \\gt 0\r\n\\]\r\n\u307e\u305f\r\n\\[\r\nf(0) = -t \\lt 0 , \\ f(3) = \\cos 3 +3 \\gt -1+3 \\gt 0\r\n\\]\r\n\u306a\u306e\u3067, \\(f(x)= 0\\) \u306f\u3053\u306e\u533a\u9593\u306b\u552f\u4e00\u306e\u89e3\u3092\u3082\u3064.
\r\n\u3053\u308c\u3092 \\(\\alpha\\) \u3068\u304a\u304f\u3068, \u4e0e\u3048\u3089\u308c\u305f\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u8868\u3059\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"\"\r\n

\u307e\u305f\r\n\\[\\begin{align}\r\nt -\\alpha & = \\sin \\alpha \\\\\r\n\\text{\u2234} \\quad t & = \\alpha +\\sin \\alpha \\quad ... [1]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nV(t) & = \\pi \\displaystyle\\int _ 0^{\\alpha} \\sin^2 x \\, dx +\\dfrac{\\pi}{3} (t -\\alpha)^3 \\\\\r\n& = \\dfrac{\\pi}{2} \\displaystyle\\int _ 0^{\\alpha} ( 1 -\\cos 2x ) \\, dx +\\dfrac{\\pi}{3} \\sin^3 \\alpha \\\\\r\n& = \\dfrac{\\pi}{2} \\left[ x -\\dfrac{\\sin 2x}{2} \\right] _ 0^{\\alpha} +\\dfrac{\\pi}{3} \\sin^3 \\alpha \\\\\r\n& = \\dfrac{\\pi \\alpha}{2} -\\dfrac{\\pi \\sin 2 \\alpha}{4} +\\dfrac{\\pi}{3} \\sin^3 \\alpha\r\n\\end{align}\\]\r\n[1] \u3088\u308a\r\n\\[\\begin{align}\r\ndt & = ( 1 +\\cos \\alpha ) \\, d \\alpha \\\\\r\n\\text{\u2234} \\quad \\dfrac{d \\alpha}{dt} & = \\dfrac{1}{1 +\\cos \\alpha}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{d V(t)}{dt} & = \\left( \\dfrac{\\pi}{2} -\\dfrac{\\pi \\cos 2 \\alpha}{2} +\\pi \\sin^2 x \\cos x \\right) \\dfrac{d \\alpha}{dt} \\\\\r\n& = \\dfrac{\\pi \\left\\{ 1 -( 2 \\cos^2 \\alpha -1 ) +( 1 -\\cos^2 \\alpha ) \\cos \\alpha \\right\\}}{2 ( 1 +\\cos \\alpha )} \\\\\r\n& = \\pi ( 1 -\\cos^2 \\alpha ) \\\\\r\n& = \\pi \\sin^2 \\alpha\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\dfrac{d}{dt} V(t) = \\dfrac{\\pi}{4}\\) \u3092\u89e3\u304f\u3068\r\n\\[\\begin{align}\r\n\\pi \\sin^2 \\alpha & = \\dfrac{\\pi}{4} \\\\\r\n\\sin^2 \\alpha & = \\dfrac{1}{4} \\\\\r\n\\text{\u2234} \\quad \\alpha & = \\dfrac{\\pi}{6} , \\dfrac{5 \\pi}{6}\r\n\\end{align}\\]\r\n\u3053\u306e\u3046\u3061, \\(\\pi \\gt 3\\) \u3088\u308a\r\n\\[\r\nt = \\dfrac{5 \\pi}{6} + \\dfrac{1}{2} \\gt \\dfrac{5}{2} + \\dfrac{1}{2} \\gt 3\r\n\\]\r\n\u306a\u306e\u3067, \u6c42\u3081\u308b \\(t\\) \u306e\u5024\u306f\r\n\\[\r\nt = \\underline{\\dfrac{\\pi}{6} +\\dfrac{1}{2}}\r\n\\]\r\n\u3053\u306e\u3068\u304d\r\n\\[\\begin{align}\r\nV \\left( \\dfrac{\\pi}{6} +\\dfrac{1}{2} \\right) & = \\dfrac{\\pi}{2} \\cdot \\dfrac{\\pi}{6} +\\dfrac{\\pi}{4} \\cdot \\dfrac{\\sqrt{3}}{2} +\\dfrac{\\pi}{3} \\left( \\dfrac{1}{2} \\right)^3 \\\\\r\n& = \\underline{\\dfrac{\\pi^2}{12} +\\dfrac{\\sqrt{3} \\pi}{8} +\\dfrac{\\pi }{24}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(0 \\lt t \\lt 3\\) \u306e\u3068\u304d, \u9023\u7acb\u65b9\u7a0b\u5f0f \\[ \\left\\{ \\begin{array}{l} 0 \\leqq y \\leqq \\sin x \\\\ 0 \\leqq x \\leqq t-y \\end{ar … \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":[74],"tags":[148,14],"class_list":["post-499","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2010","tag-tohoku_r","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/499","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=499"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/499\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=499"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=499"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=499"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}