{"id":701,"date":"2013-03-26T21:23:03","date_gmt":"2013-03-26T12:23:03","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=701"},"modified":"2021-10-04T18:13:06","modified_gmt":"2021-10-04T09:13:06","slug":"kbr200702","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr200702\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2007\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u95a2\u6570 \\(f(x) = b +\\dfrac{1}{b} -e^{ax} -e^{-ax}\\) \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.\r\n\u305f\u3060\u3057, \\(a \\gt 0\\) , \\(b \\gt 1\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(f(x) \\geqq 0\\) \u3092\u6e80\u305f\u3059 \\(x\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u66f2\u7dda \\(y = \\sqrt{f(x)}\\) \u3068 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b1\u56de\u8ee2\u3055\u305b\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d \\(V\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a = b \\log b\\) \u306e\u3068\u304d, (2)<\/strong> \u3067\u6c42\u3081\u305f\u4f53\u7a4d \\(V\\) \u3092 \\(V(b)\\) \u3067\u8868\u3059. \u3053\u306e\u3068\u304d, \\(\\displaystyle\\lim _ {b \\rightarrow \\infty} V(b) = 2 \\pi\\) \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

    (1)<\/strong><\/p>\r\n

    \\[\r\nf(-x) = b+\\dfrac{1}{b} -e^{-ax} -e^{ax} = f(x)\r\n\\]\r\n\u306a\u306e\u3067, \\(x \\geqq 0\\) \u306b\u3064\u3044\u3066\u8003\u3048\u308b.\r\n\\[\\begin{align}\r\nf'(x) & = -a e^{ax} +a e^{-ax} \\\\\r\n& = -a e^{-ax} \\left( e^{2ax} -1 \\right) \\\\\r\n& \\leqq 0 \\quad ( \\ \\text{\u2235} \\ x \\geqq 0 \\ )\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(x \\geqq 0\\) \u306b\u304a\u3044\u3066 \\(f(x)\\) \u306f\u5358\u8abf\u6e1b\u5c11\u3059\u308b.
    \r\n\u307e\u305f, \\(b \\gt 1\\) \u306a\u306e\u3067, \u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\\begin{align}\r\nf(0) & = b+\\dfrac{1}{b} -2 \\\\\r\n& \\gt 2 \\sqrt{b \\cdot \\dfrac{1}{b}} -2 = 0\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)=0\\) \u306f \\(x \\geqq 0\\) \u306b\u9ad8\u3005 \\(1\\) \u3064\u306e\u89e3\u3092\u3082\u3064.
    \r\n\u3053\u3053\u3067, \\(e^ax = b\\) \u3088\u308a \\(x = \\dfrac{\\log b}{a}\\) \u3067\u3042\u308b\u3053\u3068\u304b\u3089\r\n\\[\r\nf \\left( \\dfrac{\\log b}{a} \\right) = b +\\dfrac{1}{b} -b -\\dfrac{1}{b} = 0\r\n\\]\r\n\u306a\u306e\u3067, \\(x = \\dfrac{\\log b}{a}\\) \u306f, \\(f(x)=0\\) \u306e \\(x \\geqq 0\\) \u306b\u304a\u3051\u308b\u552f\u4e00\u306e\u89e3\u3067\u3042\u308b.
    \r\n\u3088\u3063\u3066, \\(f(x) \\geqq 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\n\\underline{-\\dfrac{\\log b}{a} \\leqq x \\leqq \\dfrac{\\log b}{a}}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nV & = 2 \\pi \\displaystyle\\int _ 0^{\\frac{\\log b}{a}} f(x) \\, dx \\\\\r\n& = 2 \\pi \\left[ \\left( b +\\dfrac{1}{b} \\right) x -\\dfrac{e^{ax}}{a} +\\dfrac{e^{-ax}}{a} \\right] _ 0^{\\frac{\\log b}{a}} \\\\\r\n& = \\dfrac{2 \\pi}{a} \\left\\{ \\left( b +\\dfrac{1}{b} \\right) \\log b -b +\\dfrac{1}{b} \\right\\} \\\\\r\n& = \\underline{\\dfrac{2\\pi \\left\\{ (b^2+1) \\log b -b^2 +1 \\right\\}}{ab}}\r\n\\end{align}\\]\r\n

    (3)<\/strong><\/p>\r\n

    \\(a = b\\log b\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nV(b) & = 2 \\pi \\left( 1 +\\dfrac{1}{b^2} -\\dfrac{1}{\\log b} +\\dfrac{1}{b^2 \\log b} \\right) \\\\\r\n& \\rightarrow 2 \\pi \\quad ( \\ b \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {b \\rightarrow \\infty} V(b) = 2 \\pi\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\(f(x) = b +\\dfrac{1}{b} -e^{ax} -e^{-ax}\\) \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057, \\(a \\gt 0\\) , \\(b \\gt 1\\) \u3068\u3059\u308b. (1)\u3000\\(f(x) … \u7d9a\u304d\u3092\u8aad\u3080 →<\/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":[101],"tags":[144,109],"class_list":["post-701","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2007","tag-tsukuba_r","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/701","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=701"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/701\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=701"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=701"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}