{"id":739,"date":"2013-04-25T01:24:39","date_gmt":"2013-04-24T16:24:39","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=739"},"modified":"2021-03-10T16:12:30","modified_gmt":"2021-03-10T07:12:30","slug":"tkr201302","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201302\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2013\uff1a\u7b2c2\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(a\\) \u3092\u5b9f\u6570\u3068\u3057, \\(x \\gt 0\\) \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 \\(f(x)\\) , \\(g(x)\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b.\r\n\\[\\begin{align}\r\nf(x) & = \\dfrac{\\cos x}{x} \\\\\r\ng(x) & = \\sin x +ax\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u3068 \\(y = g(x)\\) \u306e\u30b0\u30e9\u30d5\u304c \\(x \\gt 0\\) \u306b\u304a\u3044\u3066\u5171\u6709\u70b9\u3092\u3061\u3087\u3046\u3069 \\(3\\) \u3064\u6301\u3064\u3088\u3046\u306a \\(a\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p>\r\n<hr>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p>\\(f(x) = g(x)\\) \u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\n\\dfrac{\\cos x}{x^2} -\\dfrac{\\sin x}{x} = a \\quad ... [\\text{A}]\r\n\\]\r\n\u306a\u306e\u3067, [A] \u306e\u5de6\u8fba\u3092 \\(h(x)\\) \u3068\u304a\u3044\u3066, \\(y = h(x)\\) \u306e\u30b0\u30e9\u30d5\u3068\u76f4\u7dda \\(y=a\\) \u304c\u5171\u6709\u70b9\u3092 \\(3\\) \u3064\u6301\u3064\u305f\u3081\u306e \\(a\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.<br \/>\r\n\\(h(x)\\) \u3092\u5fae\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nh'(x) & = \\dfrac{-x^2 \\sin x -2x \\cos x}{x^4} -\\dfrac{x \\cos x -\\sin x}{x^2} \\\\\r\n& = -\\dfrac{( x^2+2 ) \\cos x}{x^3}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(h'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nx = \\dfrac{2n-1}{2} \\pi \\quad \\left( n = 1, 2, \\cdots \\right)\r\n\\]\r\n\u3053\u306e\u3068\u304d\r\n\\[\\begin{align}\r\nh \\left( \\dfrac{2n-1}{2} \\pi \\right) & = -\\dfrac{2}{(2n-1) \\pi} \\sin \\dfrac{2n-1}{2} \\pi \\\\\r\n& = \\dfrac{2 (-1)^{n-1}}{(2n-1) \\pi}\r\n\\end{align}\\]\r\n\u307e\u305f, \u5b9a\u7fa9\u57df \\(x \\gt 0\\) \u3067\u306e\u6975\u9650\u3092\u8003\u3048\u308b\u3068<\/p>\r\n<ul>\r\n<li><p>\\(x \\rightarrow +\\infty\\) \u306e\u3068\u304d<br \/>\r\n\\[\r\nh(x) \\rightarrow 0\r\n\\]<\/li>\r\n<li><p>\\(x \\rightarrow +0\\) \u306e\u3068\u304d<br \/>\r\n\\(\\cos x \\rightarrow 1\\) , \\(\\dfrac{\\sin x}{x} \\rightarrow 1\\) \u306a\u306e\u3067\r\n\\[\r\nh(x) \\rightarrow \\infty\r\n\\]<\/li>\r\n<\/ul>\r\n<p>\u4ee5\u4e0a\u3088\u308a, \\(x \\gt 0\\) \u306b\u304a\u3051\u308b \\(h(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccccccccc} x & (0) & \\cdots & \\dfrac{\\pi}{2} & \\cdots & \\dfrac{3 \\pi}{2} & \\cdots & \\dfrac{5 \\pi}{2} & \\cdots & \\dfrac{7 \\pi}{2} & \\cdots & ( \\infty )\\\\ \\hline h'(x) & & - & 0 & + & 0 & - & 0 & + & 0 & - & \\\\ \\hline h(x) & ( \\infty ) & \\searrow & -\\dfrac{2}{\\pi} & \\nearrow & \\dfrac{2}{3 \\pi} & \\searrow & -\\dfrac{2}{5 \\pi} & \\nearrow & \\dfrac{2}{7 \\pi} & \\searrow & (0) \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(y = h(x)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b. <\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tokyo_r_2013_01_012.png\" alt=\"tokyo_r_2013_01_01\" class=\"aligncenter size-full\" \/>\r\n<p>\u3088\u3063\u3066, \u6761\u4ef6\u3092\u307f\u305f\u3059 \\(a\\) \u306e\u5024\u306f\r\n\\[\r\n\\underline{a = -\\dfrac{2}{5 \\pi} , \\, \\dfrac{2}{7 \\pi} \\lt a \\lt \\dfrac{2}{3 \\pi}}\r\n\\]","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u5b9f\u6570\u3068\u3057, \\(x \\gt 0\\) \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 \\(f(x)\\) , \\(g(x)\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \\[\\begin{align} f(x) &#038; = \\dfrac{\\cos x}{x} \\\\ &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tkr201302\/\">\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":[86],"tags":[139,111],"class_list":["post-739","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2013","tag-tokyo_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/739","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=739"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/739\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=739"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=739"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=739"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}