(1)<\/strong>\u3000\\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u3092\u6b21\u306e\u70b9\u306b\u6ce8\u610f\u3057\u3066\u63cf\u3051\uff1a \\(f(x)\\) \u306e\u5897\u6e1b, \u30b0\u30e9\u30d5\u306e\u51f9\u51f8, \\(x \\rightarrow +0\\) , \\(x \\rightarrow \\infty\\) \u306e\u3068\u304d\u306e \\(f(x)\\) \u306e\u6319\u52d5.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \\(k = 1 , 2 , \\cdots , n\\) \u306b\u5bfe\u3057\u3066 \\(x\\) \u304c \\(e^{\\frac{k-1}{n}} \\leqq x \\leqq e^{\\frac{k}{n}}\\) \u3092\u52d5\u304f\u3068\u304d\u306e \\(f(x)\\) \u306e\u6700\u5927\u5024\u3092 \\(M _ k\\) , \u6700\u5c0f\u5024\u3092 \\(m _ k\\) \u3068\u3057,\r\n\\[\\begin{align}\r\nA _ n & = \\textstyle\\sum\\limits _ {k=1}^n M _ k \\left( e^{\\frac{k}{n}} -e^{\\frac{k-1}{n}} \\right) \\\\\r\nB _ n & = \\textstyle\\sum\\limits _ {k=1}^n m _ k \\left( e^{\\frac{k}{n}} -e^{\\frac{k-1}{n}} \\right)\r\n\\end{align}\\]\r\n\u3068\u304a\u304f. \\(A _ n , B _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(\\displaystyle\\lim _ {n \\rightarrow \\infty} A _ n , \\ \\displaystyle\\lim _ {n \\rightarrow \\infty} B _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (4)<\/strong>\u3000\u5404 \\(n\\) \u306b\u5bfe\u3057\u3066 \\(B _ n \\lt \\displaystyle\\int _ 1^e f(x) \\, dx \\lt A _ n\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(f(x) =\\dfrac{\\log x}{x}\\) \u306e\u5b9a\u7fa9\u57df\u306f, \\(x \\gt 0\\) .\r\n\\[\\begin{align}\r\nf'(x) & = \\dfrac{\\dfrac{1}{x} \\cdot x -1 \\cdot \\log x}{x^2} = \\dfrac{1 -\\log x}{x^2} , \\\\\r\nf''(x) & = \\dfrac{-\\dfrac{1}{x} \\cdot x^2 -2x( 1-\\log x )}{x^4} \\\\\r\n& = \\dfrac{2 \\log x -3}{x^3}\r\n\\end{align}\\]\r\n\\(f'(x)=0\\) \u3092\u89e3\u304f\u3068, \\(x = e\\) . (2)<\/strong><\/p>\r\n \\(1 \\lt x \\lt e\\) \u306b\u304a\u3044\u3066, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nM _ k & = f \\left( e^{\\frac{k}{n}} \\right) = \\dfrac{k}{n} e^{-\\frac{k}{n}} , \\\\\r\nm _ k & = f \\left( e^{\\frac{k-1}{n}} \\right) = \\dfrac{k-1}{n} e^{-\\frac{k-1}{n}}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nA _ n & = \\textstyle\\sum\\limits _ {k=1}^n \\dfrac{k}{n} e^{-\\frac{k}{n}} \\left( e^{\\frac{k}{n}} -e^{\\frac{k-1}{n}} \\right) \\\\\r\n& = \\textstyle\\sum\\limits _ {k=1}^n \\dfrac{k}{n} \\left( 1 -e^{-\\frac{1}{n}} \\right) = \\dfrac{n(n+1)}{2} \\cdot \\dfrac{1 -e^{-\\frac{1}{n}}}{n} \\\\\r\n& = \\underline{\\dfrac{n+1}{2} \\left( 1 -e^{-\\frac{1}{n}} \\right)} , \\\\\r\nB _ n & = \\textstyle\\sum\\limits _ {k=1}^n \\dfrac{k-1}{n} e^{-\\frac{k-1}{n}} \\left( e^{\\frac{k}{n}} -e^{\\frac{k-1}{n}} \\right) \\\\\r\n& = \\textstyle\\sum\\limits _ {k=1}^n \\dfrac{k-1}{n} \\left( e^{\\frac{1}{n}} -1 \\right) = \\dfrac{n(n-1)}{2} \\cdot \\dfrac{e^{\\frac{1}{n}} -1}{n} \\\\\r\n& = \\underline{\\dfrac{n-1}{2} \\left( e^{\\frac{1}{n}} -1 \\right)}\r\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n \\(N=\\dfrac{1}{n}\\) \u3068\u304a\u304f\u3068, \\(n \\rightarrow \\infty\\) \u306e\u3068\u304d, \\(N \\rightarrow 0\\) .\r\n\\[\\begin{align}\r\nA _ n & = \\dfrac{1+N}{2N} \\left( 1 -e^{-N} \\right) = \\dfrac{1}{2} \\cdot \\dfrac{1 -e^{-N}}{N} +\\dfrac{1 -e^{-N}}{2} \\\\\r\n& \\rightarrow \\dfrac{1}{2} \\left( e^{-0} \\right) +\\dfrac{1-1}{2} \\quad ( \\ N \\rightarrow 0 \\text{\u306e\u3068\u304d} ) \\\\\r\n& = \\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nB _ n & = \\dfrac{1-N}{2N} \\left( e^{N} -1 \\right) = -\\dfrac{1}{2} \\cdot \\dfrac{1-e^{N}}{N} +\\dfrac{e^{N}-1}{2} \\\\\r\n& \\rightarrow -\\dfrac{1}{2} \\left( e^{0} \\right) +\\dfrac{1-1}{2} \\quad ( \\ N \\rightarrow 0 \\text{\u306e\u3068\u304d} ) \\\\\r\n& = \\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} A _ n = \\displaystyle\\lim _ {n \\rightarrow \\infty} B _ n = \\underline{\\dfrac{1}{2}}\r\n\\]\r\n (4)<\/strong><\/p>\r\n \\(1 \\lt x \\lt e\\) \u306b\u304a\u3044\u3066, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u306a\u306e\u3067, \\(e^{\\frac{k-1}{n}} \\leqq x \\leqq e^{\\frac{k}{n}} \\ ( k = 1 , \\cdots , n )\\) \u306b\u7740\u76ee\u3059\u308c\u3070\r\n\\[\r\nm _ k \\left( e^{\\frac{k}{n}} -e^{\\frac{k-1}{n}} \\right) \\lt \\displaystyle\\int _ {e^{\\frac{k-1}{n}}}^{e^{\\frac{k}{n}}} f(x) \\, dx \\lt M _ k \\left( e^{\\frac{k}{n}} -e^{\\frac{k-1}{n}} \\right)\r\n\\]\r\n\u3053\u308c\u306b \\(k = 1 , \\cdots , n\\) \u3092\u4ee3\u5165\u3057\u3066, \u8fba\u3005\u52a0\u3048\u308c\u3070\r\n\\[\r\nB _ n \\lt \\displaystyle\\int _ 1^e ! f(x) \\, dx \\lt A _ n\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(f(x) = \\dfrac{\\log x}{x}\\) \u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u3092\u6b21\u306e\u70b9\u306b\u6ce8\u610f\u3057\u3066\u63cf\u3051\uff1a \\(f(x)\\) \u306e\u5897\u6e1b, \u30b0\u30e9\u30d5\u306e\u51f9\u51f8, \\(x \\ri … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\\(f''(x)=0\\) \u3092\u89e3\u304f\u3068, \\(x = e^{\\frac{3}{2}}\\) .
\r\n\\(t = \\dfrac{1}{x}\\) \u3068\u304a\u304f\u3068, \\(x \\rightarrow +0\\) \u306e\u3068\u304d, \\(t \\rightarrow +\\infty\\) \u306a\u306e\u3067\r\n\\[\r\nf(x) = t \\log \\dfrac{1}{t} = -t \\log t \\rightarrow -\\infty \\quad ( \\ t \\rightarrow +\\infty \\text{\u306e\u3068\u304d} )\r\n\\]\r\n\u306a\u306e\u3067, \\(\\displaystyle\\lim _ {x \\rightarrow +0} f(x) = -\\infty\\) .
\r\n\\(X = \\log x\\) \u3068\u304a\u304f\u3068, \\(x \\rightarrow +\\infty\\) \u306e\u3068\u304d, \\(X \\rightarrow +\\infty\\) \u306a\u306e\u3067\r\n\\[\r\nf(x) = \\dfrac{X}{e^X} \\rightarrow 0 \\quad ( \\ X \\rightarrow +\\infty \\text{\u306e\u3068\u304d} )\r\n\\]\r\n\u306a\u306e\u3067, \\(\\displaystyle\\lim _ {x \\rightarrow +\\infty} f(x) = 0\\) .
\r\n\u4ee5\u4e0a\u3088\u308a, \\(f(x)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccccc} x & (0) & \\cdots & e & \\cdots & e^{\\frac{3}{2}} & \\cdots & ( +\\infty ) \\\\ \\hline f'(x) & & + & 0 & - & & - & \\\\ \\hline f''(x) & & - & & - & 0 & + & \\\\ \\hline f(a) & (-\\infty) & \\nearrow (\\cap) & \\dfrac{1}{e} & \\searrow (\\cap) & \\dfrac{3}{2 e^{\\frac{3}{2}}} & \\searrow (\\cup) & ( 0 ) \\end{array}\r\n\\]\r\n\u3086\u3048\u306b, \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u306f\u4e0b\u56f3\u306e\u3068\u304a\u308a.<\/p>\r\n![\"waseda_2011_03_01\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/waseda_2011_03_01.png\")
<\/p>\r\n