{"id":854,"date":"2013-08-03T13:57:46","date_gmt":"2013-08-03T04:57:46","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=854"},"modified":"2021-09-15T09:37:11","modified_gmt":"2021-09-15T00:37:11","slug":"ngr201304","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201304\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2013\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\u534a\u5f84 \\(1\\) \u306e\u5186\u76e4 \\(C_1\\) \u304c\u534a\u5f84 \\(2\\) \u306e\u5186\u76e4 \\(C_2\\) \u306b\u8cbc\u308a\u4ed8\u3051\u3089\u308c\u3066\u304a\u308a, \\(2\\) \u3064\u306e\u5186\u76e4\u306e\u4e2d\u5fc3\u306f\u4e00\u81f4\u3059\u308b.\r\n\\(C_2\\) \u306e\u5468\u4e0a\u306b\u3042\u308b\u5b9a\u70b9\u3092 A \u3068\u3059\u308b. \u56f3\u306e\u3088\u3046\u306b, \u6642\u523b \\(t=0\\) \u306b\u304a\u3044\u3066 \\(C_1\\) \u306fO \\((0,0)\\) \u3067 \\(x\\) \u8ef8\u306b\u63a5\u3057, A \u306f\u5ea7\u6a19 \\((0,-1)\\) \u306e\u4f4d\u7f6e\u306b\u3042\u308b.\r\n\\(2\\) \u3064\u306e\u5186\u76e4\u306f\u4e00\u4f53\u3068\u306a\u308a, \\(C_1\\) \u306f \\(x\\) \u8ef8\u4e0a\u3092\u3059\u3079\u308b\u3053\u3068\u306a\u304f\u8ee2\u304c\u3063\u3066\u3044\u304f.\r\n\u6642\u523b \\(t\\) \u3067 \\(C_1\\) \u306e\u4e2d\u5fc3\u304c \\((t,1)\\) \u306b\u3042\u308b\u3088\u3046\u306b\u8ee2\u304c\u308b\u3068\u304d, \\(0 \\leqq t \\leqq 2 \\pi\\) \u306b\u304a\u3044\u3066 A \u304c\u63cf\u304f\u66f2\u7dda\u3092 \\(C\\) \u3068\u3059\u308b.<\/p>\r\n<p><strong>(1)<\/strong>\u3000\u6642\u523b \\(t\\) \u306b\u304a\u3051\u308b A \u306e\u5ea7\u6a19\u3092 \\(\\left( x(t) , y(t) \\right)\\) \u3067\u8868\u3059. \\(\\left( x(t) , y(t) \\right)\\) \u3092\u6c42\u3081\u3088.<\/p>\r\n<p><strong>(2)<\/strong>\u3000\\(x(t)\\) \u3068 \\(y(t)\\) \u306e \\(t\\) \u306b\u95a2\u3059\u308b\u5897\u6e1b\u3092\u8abf\u3079, \\(x(t)\\) \u3042\u308b\u3044\u306f \\(y(t)\\) \u304c\u6700\u5927\u5024\u307e\u305f\u306f\u6700\u5c0f\u5024\u3092\u3068\u308b\u3068\u304d\u306e A \u306e\u5ea7\u6a19\u3092\u5168\u3066\u6c42\u3081\u3088.<\/p>\r\n<p><strong>(3)<\/strong>\u3000\\(C\\) \u3068 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p>\r\n<img decoding=\"async\" src=\"\/nyushi\/wp-content\/uploads\/nagoya_r_2013_04_01.png\" alt=\"nagoya_r_2013_04_01\" class=\"aligncenter size-full\" \/>\r\n<hr \/>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\u6642\u523b \\(t\\) \u306b\u304a\u3051\u308b \\(C_1 , C_2\\) \u306e\u4e2d\u5fc3\u3092 P, \\(C_1\\) \u3068 \\(x\\) \u8ef8\u306e\u63a5\u70b9\u3092 T \u3068\u304a\u304f\u3068\r\n\\[\r\n\\text{P} ( t , 1 ) , \\ \\text{T} ( t , 0 ) , \\ \\angle\\text{TPA} = t \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u6c42\u3081\u308b A \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\left( x(t) , y(t) \\right) = \\underline{\\left( t -2 \\sin t , 1 -2\\cos t \\right)} \\ .\r\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p><strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[ \\begin{align}\r\nx'(t) & = 1 -2\\cos t \\ , \\\\\r\ny'(t) & = 2 \\sin t \\ .\r\n\\end{align} \\]\r\n\\(x'(t) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nt = \\dfrac{\\pi}{3} , \\dfrac{5 \\pi}{3} \\ .\r\n\\]\r\n\\(y'(t) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nt = 0 , \\pi , 2 \\pi \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(x , y\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccccccc} t & 0 & \\cdots & \\frac{\\pi}{3} & \\cdots & \\pi & \\cdots & \\frac{5 \\pi}{3} & \\cdots & 2 \\pi\\\\ \\hline x'(t) & & - & 0 & + & & + & 0 & - & \\\\ \\hline y'(t) & 0 & + & & + & 0 & - & & - & 0 \\\\ \\hline x(t) & 0 & \\leftarrow & \\frac{\\pi}{3} -\\sqrt{3} & \\rightarrow & \\pi & \\rightarrow & \\frac{5 \\pi}{3} +\\sqrt{3} & \\leftarrow & 2 \\pi \\\\ \\hline y(t) & -1 & \\uparrow & 0 & \\uparrow & 3 & \\downarrow & 0 & \\downarrow & -1 \\end{array} \\\\\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b A \u306e\u5ea7\u6a19\u306f<\/p>\r\n<ul>\r\n<li><p>\\(x(t)\\) \u304c\u6700\u5c0f\u306b\u306a\u308b\u3068\u304d, \\(\\underline{\\left( \\dfrac{\\pi}{3} -\\sqrt{3} , 0 \\right)}\\)<\/p><\/li>\r\n<li><p>\\(x(t)\\) \u304c\u6700\u5927\u306b\u306a\u308b\u3068\u304d, \\(\\underline{\\left( \\dfrac{5 \\pi}{3} +\\sqrt{3} , 0 \\right)}\\)<\/p><\/li>\r\n<li><p>\\(y(t)\\) \u304c\u6700\u5c0f\u306b\u306a\u308b\u3068\u304d, \\(\\underline{( 0 , -1 ) , \\ ( 2 \\pi , -1 )}\\)<\/p><\/li>\r\n<li><p>\\(y(t)\\) \u304c\u6700\u5927\u306b\u306a\u308b\u3068\u304d, \\(\\underline{( \\pi , 3 )}\\)<\/p><\/li>\r\n<\/ul>\r\n<p><strong>(3)<\/strong><\/p>\r\n<p><strong>(2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(C\\) \u3068 \\(x\\) \u8ef8\u306b\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n<img decoding=\"async\" src=\"\/nyushi\/wp-content\/uploads\/nagoya_r_2013_04_02.png\" alt=\"nagoya_r_2013_04_02\" class=\"aligncenter size-full\" \/>\r\n<p>\u3088\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int_{\\frac{\\pi}{3} -\\sqrt{3}}^{\\frac{5 \\pi}{3} +\\sqrt{3}} y(t) \\, dx \\\\\r\n& = \\displaystyle\\int_{\\frac{\\pi}{3}}^{\\frac{5 \\pi}{3}} y(t) x'(t) \\, dt \\\\\r\n& = \\displaystyle\\int_{\\frac{\\pi}{3}}^{\\frac{5 \\pi}{3}} \\left( 1 -2 \\cos t \\right)^2 \\, dt \\\\\r\n& = \\displaystyle\\int_{\\frac{\\pi}{3}}^{\\frac{5 \\pi}{3}} \\left( 1 -4 \\cos t +4 \\cdot \\dfrac{1 +\\cos 2t}{2}\\right) \\, dt \\\\\r\n& = \\left[ 3t -4 \\sin t +\\sin 2t \\right]_{\\frac{\\pi}{3}}^{\\frac{5 \\pi}{3}} \\\\\r\n& = \\underline{4 \\pi +3 \\sqrt{3}} \\ .\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u534a\u5f84 \\(1\\) \u306e\u5186\u76e4 \\(C_1\\) \u304c\u534a\u5f84 \\(2\\) \u306e\u5186\u76e4 \\(C_2\\) \u306b\u8cbc\u308a\u4ed8\u3051\u3089\u308c\u3066\u304a\u308a, \\(2\\) \u3064\u306e\u5186\u76e4\u306e\u4e2d\u5fc3\u306f\u4e00\u81f4\u3059\u308b. \\(C_2\\) \u306e\u5468\u4e0a\u306b\u3042\u308b\u5b9a\u70b9\u3092 A \u3068\u3059\u308b. \u56f3\u306e\u3088\u3046\u306b, \u6642\u523b  &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/ngr201304\/\">\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":[98],"tags":[143,111],"class_list":["post-854","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2013","tag-nagoya_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/854","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=854"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/854\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=854"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=854"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=854"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}