{"id":1888,"date":"2021-09-18T15:41:02","date_gmt":"2021-09-18T06:41:02","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1888"},"modified":"2021-09-18T19:27:40","modified_gmt":"2021-09-18T10:27:40","slug":"thr201606","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201606\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2016\uff1a\u7b2c6\u554f"},"content":{"rendered":"<hr \/>\n<p>\u95a2\u6570\r\n\\[\r\nf(x) = \\displaystyle\\int _ {0}^{\\pi} \\left| \\sin (t-x) -\\sin 2t \\right| \\, dt\r\n\\]\r\n\u306e\u533a\u9593 \\(0 \\leqq x \\leqq \\pi\\) \u306b\u304a\u3051\u308b\u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p>\\(g(x) = \\sin (t-x) -\\sin 2t\\) \u3068\u304a\u304f.<br \/>\r\n\u548c\u30fb\u7a4d\u306e\u516c\u5f0f\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\ng(x) & = 2 \\cos \\dfrac{(t-x) +2t}{2} \\, \\sin \\dfrac{(t-x) -2t}{2} \\\\\r\n& = -2 \\underline{\\cos \\dfrac{3t -x}{3}} _ {[1]} \\, \\underline{\\sin \\dfrac{t+x}{2}} _ {[2]} \\ .\r\n\\end{align}\\]\r\n\\(0 \\leqq x \\leqq \\pi\\) , \\(0 \\leqq t \\leqq \\pi\\) \u306b\u6ce8\u610f\u3057\u3066, [1] [2] \u306e\u7b26\u53f7\u306b\u3064\u3044\u3066\u8003\u3048\u308b.<br \/>\r\n[1] \u306b\u3064\u3044\u3066\u306f<\/p>\r\n<ul>\r\n<li><p>\\(-\\dfrac{\\pi}{2} \\leqq \\dfrac{3t -x}{2} \\leqq \\dfrac{\\pi}{2}\\) \u3059\u306a\u308f\u3061 \\(\\dfrac{x -\\pi}{3} \\leqq 0 \\leqq t \\leqq \\dfrac{x +\\pi}{3}\\) \u306e\u3068\u304d\r\n\\[\r\n[1] \\geqq 0 \\ .\r\n\\]<\/li>\r\n<li><p>\\(\\dfrac{\\pi}{2} \\leqq \\dfrac{3t -x}{2} \\leqq \\dfrac{3 \\pi}{2}\\) \u3059\u306a\u308f\u3061 \\(\\dfrac{x +\\pi}{3} \\leqq t \\leqq \\pi \\leqq x +\\dfrac{\\pi}{3}\\) \u306e\u3068\u304d\r\n\\[\r\n[1] \\leqq 0 \\ .\r\n\\]<\/li>\r\n<\/ul>\r\n[2] \u306b\u3064\u3044\u3066\u306f, \u5e38\u306b \\(0 \\leqq \\dfrac{t+x}{2} \\leqq \\pi\\) \u306a\u306e\u3067\r\n\\[\r\n[2] \\geqq 0 \\ .\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \\(\\alpha = \\dfrac{x +\\pi}{3}\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nf(x) = \\displaystyle\\int _ {0}^{\\alpha} g(t) \\, dt - \\displaystyle\\int _ {\\alpha}^{\\pi} g(t) \\, dt \\ .\r\n\\]\r\n\u3053\u3053\u3067, \\(G(t) = \\displaystyle\\int g(t) \\, dt\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nG(t) = \\cos (t-x) -\\dfrac{1}{2} \\cos 2t +C \\quad ( \\ C \\ \\text{\u306f\u7a4d\u5206\u5b9a\u6570})\r\n\\]\r\n\u3067\u3042\u308a\r\n\\[\\begin{align}\r\nG( \\alpha ) & = \\cos ( \\alpha -x ) -\\dfrac{1}{2} \\cos 2 \\alpha +C \\\\\r\n& = \\cos \\dfrac{\\pi -2x}{3} -\\dfrac{1}{2} \\cos \\dfrac{2 ( \\pi +x )}{3} \\ , \\\\\r\nG(0) & = \\cos x -\\dfrac{1}{2} +C \\ , \\\\\r\nG( \\pi ) & = -\\cos x -\\dfrac{1}{2} +C \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nf(x) & = 2 G( \\alpha ) -G(0) -G( \\pi ) \\\\\r\n& = 2 \\left( \\dfrac{1}{2} \\cos \\dfrac{2x}{3} +\\dfrac{\\sqrt{3}}{2} \\sin \\dfrac{2x}{3} \\right) \\\\\r\n& \\qquad -\\left( -\\dfrac{1}{2} \\cos \\dfrac{2x}{3} -\\dfrac{\\sqrt{3}}{2} \\sin \\dfrac{2x}{3} \\right) \\\\\r\n& \\qquad \\qquad -\\cos x +\\dfrac{1}{2} +\\cos x +\\dfrac{1}{2} \\\\\r\n& = \\dfrac{3}{2} \\cos \\dfrac{2x}{3} +\\dfrac{3 \\sqrt{3}}{2} \\sin \\dfrac{2x}{3} +1 \\\\\r\n& = \\dfrac{3}{4} \\left( \\dfrac{\\sqrt{3}}{2} \\sin \\dfrac{2x}{3} +\\dfrac{1}{2} \\cos \\dfrac{2x}{3} \\right) +1 \\\\\r\n& = 3 \\sin \\underline{\\left( \\dfrac{2x}{3} +\\dfrac{\\pi}{6} \\right)} _ {[3]} +1 \\ .\r\n\\end{align}\\]\r\n\\(\\dfrac{\\pi}{6} \\leqq [3] \\leqq \\dfrac{5 \\pi}{6}\\) \u306a\u306e\u3067<\/p>\r\n<ul>\r\n<li><p>\u6700\u5927\u5024\u306f, \\(f \\left( \\dfrac{\\pi}{2} \\right) = 3 \\cdot 1 +1 = \\underline{4}\\) .<\/p><\/li>\r\n<li><p>\u6700\u5c0f\u5024\u306f, \\(f \\left( \\dfrac{\\pi}{6} \\right) = f \\left( \\dfrac{5 \\pi}{6} \\right) = 3 \\cdot \\dfrac{1}{2} +1 = \\underline{\\dfrac{5}{2}}\\) .<\/p><\/li>\r\n<\/ul>\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\[ f(x) = \\displaystyle\\int _ {0}^{\\pi} \\left| \\sin (t-x) -\\sin 2t \\right| \\, dt \\] \u306e\u533a\u9593 \\(0 \\leqq x \\leqq \\ &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/thr201606\/\">\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":[155],"tags":[148,162],"class_list":["post-1888","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2016","tag-tohoku_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1888","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=1888"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1888\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1888"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1888"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1888"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}