{"id":604,"date":"2013-02-02T07:51:42","date_gmt":"2013-02-01T22:51:42","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=604"},"modified":"2021-03-18T15:25:12","modified_gmt":"2021-03-18T06:25:12","slug":"tkr200706","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr200706\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2007\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(0 \\lt x \\lt a\\) \u3092\u307f\u305f\u3059\u5b9f\u6570 \\(x , a\\) \u306b\u5bfe\u3057, \u6b21\u3092\u793a\u305b.\r\n\\[\r\n\\dfrac{2x}{a} \\lt \\displaystyle\\int _ {a-x}^{a+x} \\dfrac{1}{t} \\, dt \\lt x \\left( \\dfrac{1}{a+x} +\\dfrac{1}{a-x} \\right)\r\n\\]<\/li>\r\n

  2. (2)<\/strong>\u3000(1)<\/strong> \u3092\u5229\u7528\u3057\u3066, \u6b21\u3092\u793a\u305b.\r\n\\[\r\n0.68 \\lt \\log 2 \\lt 0.71\r\n\\]\r\n\u305f\u3060\u3057, \\(\\log 2\\) \u306f \\(2\\) \u306e\u81ea\u7136\u5bfe\u6570\u3092\u8868\u3059.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    A \\(\\left( a , \\dfrac{1}{a}\\right)\\) , B \\(\\left( a-x , \\dfrac{1}{a-x}\\right)\\) , C \\(\\left( a+x , \\dfrac{1}{a+x}\\right)\\) \u3068\u304a\u304f.
    \r\n\u76f4\u7dda BC \u306e\u5f0f\u3092 \\(y = f(t)\\) , \u70b9 A \u306b\u304a\u3051\u308b \\(y =\\dfrac{1}{t}\\) \u306e\u63a5\u7dda\u306e\u5f0f\u3092 \\(y = g(t)\\) \u3068\u304a\u3051\u3070,\r\n\\(y =\\dfrac{1}{t}\\) \u306f\u4e0b\u306b\u51f8\u306a\u306e\u3067, \\(a-x \\lt t \\lt a+x\\) \u306b\u304a\u3044\u3066\r\n\\[\r\ng(t) \\lt \\dfrac{1}{t} \\lt f(t) \\quad ... [1]\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nf(t) & = \\dfrac{\\frac{1}{a+x} -\\frac{1}{a-x}}{2x} ( t -x-a ) +\\dfrac{1}{a+x} \\\\\r\n& = \\dfrac{-(t-x-a) +(a-x)}{(a+x)(a-x)} \\\\\r\n& = \\dfrac{2a-t}{(a+x)(a-x)}\r\n\\end{align}\\]\r\n\\[\\begin{align}\r\ng(t) & = -\\dfrac{1}{a^2} (t-a) +\\dfrac{1}{a} \\\\\r\n& = -\\dfrac{2a-t}{a^2}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u306e\u8fba\u3005\u3092 \\(t\\) \u306b\u3064\u3044\u3066 \\(a-x \\rightarrow a+x\\) \u3067\u7a4d\u5206\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ {a-x}^{a+x} (2a-t) \\, dt & = \\left[ 2at - \\dfrac{t^2}{2} \\right] _ {a-x}^{a+x} \\\\\r\n& = 4ax -\\dfrac{(a+x)^2 -(a-x)^2}{2} \\\\\r\n& = 2ax\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{2ax}{a^2} & \\lt \\displaystyle\\int _ {a-x}^{a+x} \\dfrac{1}{t} \\, dt \\lt \\dfrac{2ax}{(a+x)(a-x)} \\\\\r\n\\text{\u2234} \\quad \\dfrac{2x}{a} & \\lt \\displaystyle\\int _ {a-x}^{a+x} \\dfrac{1}{t} \\, dt \\lt x \\left( \\dfrac{1}{a-x} +\\dfrac{1}{a+x} \\right)\r\n\\end{align}\\]\r\n

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

    \\[\\begin{align}\r\n\\displaystyle\\int _ {a-x}^{a+x} \\dfrac{1}{t} \\, dt & = \\left[ \\log t \\right] _ {a-x}^{a+x} \\\\\r\n& = \\log \\dfrac{a+x}{a-x}\r\n\\end{align}\\]\r\n\\(h = \\dfrac{x}{a}\\) \u3068\u304a\u304f\u3068, (1)<\/strong> \u306e\u7d50\u679c\u306f\r\n\\[\r\n2h \\lt \\log \\dfrac{1+h}{1-h} \\lt \\dfrac{h}{1-h} +\\dfrac{h}{1+h} \\quad ... [2]\r\n\\]\r\n\u3053\u3053\u3067, \\(\\dfrac{a+x}{a-x} = \\sqrt{2}\\) \u3068\u306a\u308b\u5834\u5408\u3092\u8003\u3048\u308b\u3068\r\n\\[\\begin{align}\r\na+x & = \\sqrt{2} (a-x) \\\\\r\n\\text{\u2234} \\quad h & = \\dfrac{\\sqrt{2}-1}{\\sqrt{2}+1} = 3 -2 \\sqrt{2}\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, \\(1.414 \\lt \\sqrt{2} \\lt 1.415\\) \u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n2h & = 6 -4 \\sqrt{2} \\\\\r\n& \\gt 6 -4 \\cdot 1.415 = 0.34\r\n\\end{align}\\]\r\n\\[\\begin{align}\r\n\\dfrac{h}{1-h} +\\dfrac{h}{1+h} & = \\dfrac{3 -2 \\sqrt{2}}{2 \\sqrt{2} -2} +\\dfrac{3 -2 \\sqrt{2}}{4 -2 \\sqrt{2}} \\\\\r\n& = \\dfrac{1}{2} \\left\\{ \\left( 3 -2 \\sqrt{2} \\right) \\left( \\sqrt{2} +1 \\right) +\\dfrac{\\left( 3 -2 \\sqrt{2} \\right) \\left( 2 +\\sqrt{2} \\right)}{2} \\right\\} \\\\\r\n& = \\dfrac{2 \\left( \\sqrt{2} -1 \\right) +2 -\\sqrt{2}}{4} \\\\\r\n& = \\dfrac{\\sqrt{2}}{4} \\gt \\dfrac{1.414}{4} = 0.3535\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [2] \u3088\u308a\r\n\\[\\begin{align}\r\n0.34 & \\lt \\dfrac{1}{2} \\log 2 \\lt 0.3535 \\\\\r\n\\text{\u2234} \\quad 0.68 & \\lt \\log 2 \\lt 0.707\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(0 \\lt x \\lt a\\) \u3092\u307f\u305f\u3059\u5b9f\u6570 \\(x , a\\) \u306b\u5bfe\u3057, \u6b21\u3092\u793a\u305b. \\[ \\dfrac{2x}{a} \\lt \\displaystyle\\int _ {a-x}^{a … \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":[85],"tags":[139,109],"class_list":["post-604","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2007","tag-tokyo_r","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/604","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=604"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/604\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=604"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=604"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=604"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}