{"id":1157,"date":"2015-07-08T23:14:07","date_gmt":"2015-07-08T14:14:07","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1157"},"modified":"2021-09-13T20:54:35","modified_gmt":"2021-09-13T11:54:35","slug":"iks201403","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201403\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272014\uff1a\u7b2c3\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570, \\(k\\) \u3092\u81ea\u7136\u6570\u3068\u3057, \\(x \\gt 0\\) \u3067\u5b9a\u7fa9\u3055\u308c\u308b\u95a2\u6570\r\n\\[\r\nf(x) = \\displaystyle\\int _ a^{ax} \\dfrac{k +\\sqrt[k]{u}}{ku} \\, du \\ .\r\n\\]\r\n\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(f(x)\\) \u306e\u5897\u6e1b\u304a\u3088\u3073\u51f9\u51f8\u3092\u8abf\u3079, \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u3092\u304b\u3051.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(S\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b\u3068\u304d, \\(f(p) = S\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570 \\(p\\) \u304c\u305f\u3060 \\(1\\) \u3064\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\\(b = \\dfrac{k}{k +\\sqrt[k]{a}}\\) \u3068\u304a\u304f\u3068\u304d, <strong>(2)<\/strong> \u306e \\(S , p\\) \u306b\u3064\u3044\u3066, \u6b21\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b.\r\n\\[\r\n1 +bS \\lt p \\lt e^{bS}\r\n\\]<\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\(F(u) = \\displaystyle\\int \\dfrac{k +\\sqrt[k]{u}}{ku} \\, du\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nF(u) & = \\displaystyle\\int \\left( \\dfrac{1}{u} +\\dfrac{u^{\\frac{1}{k} -1}}{k} \\right) \\, du \\\\\r\n& = \\log u +\\sqrt[k]{u} +C \\quad ( \\ C : \\text{\u7a4d\u5206\u5b9a\u6570} \\ ) \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf(x) & = F(ax) -F(a) \\\\\r\n& = \\log ax +\\sqrt[k]{ax} -\\log a -\\sqrt[k]{a} \\\\\r\n& = \\log x +\\sqrt[k]{a} \\left( \\sqrt[k]{x} -1 \\right) \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u9806\u6b21\u5fae\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nf'(x) & = \\dfrac{1}{x} +\\sqrt[k]{a} \\cdot \\dfrac{\\sqrt[k]{x}}{kx} \\\\\r\n& = \\dfrac{k +\\sqrt[k]{ax}}{kx} \\gt 0 , \\\\\r\nf''(x) & = -\\dfrac{1}{x^2} -\\dfrac{\\sqrt[k]{a}}{k} \\cdot \\dfrac{k-1}{k} x^{\\frac{1}{k} -2} \\\\\r\n& = -\\dfrac{1}{x^2} -\\dfrac{(k-1) \\sqrt[k]{ax}}{k^2 x^2} \\lt 0 \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306f \\(x \\gt 0\\) \u306b\u304a\u3044\u3066, \u5358\u8abf\u5897\u52a0\u3067\u3042\u308a, \u4e0a\u306b\u51f8\u3067\u3042\u308b.<br \/>\r\n\u3055\u3089\u306b\r\n\\[\r\nf(1) = 0 , \\\r\n\\displaystyle\\lim _\r\n{x \\rightarrow +0} f(x) = -\\infty , \\ \\displaystyle\\lim _ {x \\rightarrow +\\infty} f(x) = \\infty \\ .\r\n\\]\r\n\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070, \\(y = f(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\/iks20140301.svg\" alt=\"iks20140301\" class=\"aligncenter size-full\" \/>\r\n<p><strong>(2)<\/strong><\/p>\r\n<p><strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u3068, \u76f4\u7dda \\(y = S\\) \u306f\u305f\u3060 \\(1\\) \u3064\u306e\u5171\u6709\u70b9\u3092\u3082\u3064.<br \/>\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n<p><strong>(3)<\/strong><\/p>\r\n<p><strong>(2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\np \\gt 1 \\quad ... [1] \\ .\r\n\\]\r\n<ul>\r\n<li><p>\\(1 +bS \\lt p\\) \u306e\u8a3c\u660e<br \/>\r\n\\(f(1) = 0\\) , \\(\\dfrac{1}{b} = \\dfrac{k +\\sqrt[k]{a}}{k} = f'(1)\\) \u3067\u3042\u308b\u3053\u3068\u306b\u7528\u3044\u308b.<br \/>\r\n\\(f(x)\\) \u306f\u4e0a\u306b\u51f8\u3067\u3042\u308b\u304b\u3089\r\n\\[\\begin{align}\r\n\\dfrac{f(p) -f(1)}{p-1} & \\lt f'(1) \\\\\r\n\\dfrac{S}{p-1} & \\lt \\dfrac{1}{b} \\\\\r\n\\text{\u2234} \\quad 1 +bS & \\lt p \\ .\r\n\\end{align}\\]<\/li>\r\n<li><p>\\(p \\lt e^{bS}\\) \u306e\u8a3c\u660e<br \/>\r\n\\[\\begin{align}\r\n\\dfrac{f(p)}{\\log p} -\\dfrac{1}{b} & = \\left\\{ 1 +\\dfrac{\\sqrt[k]{a} \\left( \\sqrt[k]{p} -1 \\right)}{\\log p} \\right\\} -\\left( 1 +\\dfrac{\\sqrt[k]{a}}{k} \\right) \\\\\r\n& = \\sqrt[k]{a} \\left( \\dfrac{\\sqrt[k]{p} -1}{\\log p} -\\dfrac{1}{k} \\right) \\\\\r\n& = \\dfrac{\\sqrt[k]{a}}{\\log p} \\underline{\\left( \\sqrt[k]{p} -1 -\\log \\sqrt[k]{p} \\right)} _ {[2]} \\ .\r\n\\end{align}\\]\r\n[2]\u306b\u3064\u3044\u3066, \u4e00\u822c\u306b \\(x \\gt 1\\) \u306b\u3064\u3044\u3066 \\(x -1 \\gt \\log x\\) \u306a\u306e\u3067, [1]\u3088\u308a\r\n\\[\r\n[2] \\gt 0 \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\dfrac{f(p)}{\\log p} -\\dfrac{1}{b} & \\gt 0 \\\\\r\n\\log p & \\lt bS \\\\\r\n\\text{\u2234} \\quad p & \\lt e^{bS} \\ .\r\n\\end{align}\\]<\/li>\r\n<\/ul>\r\n<p>\u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570, \\(k\\) \u3092\u81ea\u7136\u6570\u3068\u3057, \\(x \\gt 0\\) \u3067\u5b9a\u7fa9\u3055\u308c\u308b\u95a2\u6570 \\[ f(x) = \\displaystyle\\int _ a^{ax} \\dfrac{k +\\sqrt[k]{u}}{ku &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/iks201403\/\">\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":[117],"tags":[145,112],"class_list":["post-1157","post","type-post","status-publish","format-standard","hentry","category-ikashika_2014","tag-ikashika","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1157","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=1157"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1157\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}