{"id":1236,"date":"2015-08-19T10:52:03","date_gmt":"2015-08-19T01:52:03","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1236"},"modified":"2021-03-22T20:06:56","modified_gmt":"2021-03-22T11:06:56","slug":"kyr201503","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr201503\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb2015\uff1a\u7b2c3\u554f"},"content":{"rendered":"<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(a\\) \u3092\u5b9f\u6570\u3068\u3059\u308b\u3068\u304d, \\(( a , 0 )\\) \u3092\u901a\u308a, \\(y = e^x +1\\) \u306b\u63a5\u3059\u308b\u76f4\u7dda\u304c\u305f\u3060 \\(1\\) \u3064\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(a _ 1 = 1\\) \u3068\u3057\u3066, \\(n = 1, 2, \\cdots\\) \u306b\u3064\u3044\u3066, \\(( a _ n , 0 )\\) \u3092\u901a\u308a, \\(y = e^x +1\\) \u306b\u63a5\u3059\u308b\u76f4\u7dda\u306e\u63a5\u70b9\u306e \\(x\\) \u5ea7\u6a19\u3092 \\(a _ {n+1}\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} ( a _ {n+1} -a _ n )\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\(C : \\ y = e^x +1\\) \u3068\u3059\u308c\u3070, \\(y' = e^x\\) \u306a\u306e\u3067, \u70b9 \\(( t , e^t +1 )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = e^t (x-t) +e^t +1 \\\\\r\n& = e^t x -(t-1) e^t +1\r\n\\end{align}\\]\r\n\u3053\u308c\u304c\u70b9 \\(( a , 0 )\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\ne^t a -(t-1) e^t +1 & = 0 \\\\\r\n\\text{\u2234} \\quad -e^{-t} +t-1 & = a \\quad ... [1]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u304c\u305f\u3060 \\(1\\) \u3064\u306e\u89e3 \\(t\\) \u3092\u3082\u3064\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044.<br \/>\r\n[1] \u306e\u5de6\u8fba\u3092 \\(f(t)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(t) = e^{-t} +1 \\gt 0\r\n\\]\r\n\u306a\u306e\u3067, \\(f(t)\\) \u306f\u5358\u8abf\u5897\u52a0\u95a2\u6570\u3067\u3042\u308b.<br \/>\r\n\u3055\u3089\u306b, \u6975\u9650\u306b\u3064\u3044\u3066\r\n\\[\r\n\\displaystyle\\lim _ {t \\rightarrow \\infty} f(t) = \\infty , \\ \\displaystyle\\lim _ {t \\rightarrow -\\infty} f(t) = -\\infty\r\n\\]\r\n\u306a\u306e\u3067, [1] \u306f \\(a\\) \u306e\u5024\u306b\u3088\u3089\u305a, \u305f\u3060 \\(1\\) \u3064\u306e\u89e3\u3092\u3082\u3064.<br \/>\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n<p><strong>(2)<\/strong><\/p>\r\n[1] \u3068 <strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n-e^{-a _ {n+1}} +a _ {n+1} -1 = a _ n \\quad ... [2]\r\n\\]\r\n\u3053\u308c\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\na _ {n+1} = a _ n +1 +e^{-a _ {n+1}} \\gt a _ n +1\r\n\\]\r\n\u3053\u308c\u3092\u7e70\u8fd4\u3057\u7528\u3044\u308c\u3070, \\(a _ 1 = 1\\) \u3088\u308a\r\n\\[\r\na _ n \\gt n\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} a _ n = \\infty\r\n\\]\r\n\u3088\u3063\u3066, \u518d\u5ea6 [2] \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} ( a _ {n+1} -a _ n ) & = \\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( 1 +e^{-a _ {n+1}} \\right) \\\\\r\n& = 1 +0 = \\underline{1}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"(1)\u3000\\(a\\) \u3092\u5b9f\u6570\u3068\u3059\u308b\u3068\u304d, \\(( a , 0 )\\) \u3092\u901a\u308a, \\(y = e^x +1\\) \u306b\u63a5\u3059\u308b\u76f4\u7dda\u304c\u305f\u3060 \\(1\\) \u3064\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b. (2)\u3000\\(a _ 1 = 1\\) \u3068\u3057\u3066, \\(n = &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/kyr201503\/\">\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":[126],"tags":[140,137],"class_list":["post-1236","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2015","tag-kyoto_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1236","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=1236"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1236\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1236"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1236"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}