\u66f2\u7dda \\(C\\) , \\(x\\) \u8ef8\u304a\u3088\u3073 \\(2\\) \u3064\u306e\u76f4\u7dda \\(x = \\alpha\\) , \\(x = \\beta\\) \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S _ 1\\) ,<\/p><\/li>\r\n
\u66f2\u7dda \\(C\\) \u304a\u3088\u3073 \\(2\\) \u3064\u306e\u76f4\u7dda PR , QR \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S _ 2\\)<\/p><\/li>\r\n<\/ul>\r\n
\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n
- \r\n
(1)<\/strong>\u3000\\(\\dfrac{S _ 2}{S _ 1}\\) \u3092 \\(\\alpha\\) \u3068 \\(\\beta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n
(2)<\/strong>\u3000\\(\\alpha \\lt \\dfrac{e}{t}\\) , \\(\\beta \\lt 2 \\log t\\) \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3057, \\(\\displaystyle\\lim _ {t \\rightarrow \\infty} \\dfrac{S _ 2}{S _ 1}\\) \u3092\u6c42\u3081\u3088.\r\n\u5fc5\u8981\u306a\u3089\u3070, \\(x \\gt 0\\) \u306e\u3068\u304d \\(e^x \\gt x^2\\) \u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u306a\u3057\u306b\u7528\u3044\u3066\u3088\u3044.<\/p><\/li>\r\n<\/ol>\r\n
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
(1)<\/strong><\/p>\r\n
P , Q \u306f \\(C\\) \u3068\u76f4\u7dda \\(y = tx\\) \u306e\u4ea4\u70b9\u306a\u306e\u3067\r\n\\[\r\ne^{\\alpha} = t \\alpha , \\ e^{\\beta} = t \\beta \\ .\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nS _ 1 & = \\displaystyle\\int _ {\\alpha}^{\\beta} e^x \\, dx \\\\\r\n& = e^{\\beta} -e^{\\alpha} = t ( \\beta -\\alpha ) \\quad ... [1] \\ .\r\n\\end{align}\\]\r\n\\(C\\) \u306e\u5f0f\u3088\u308a, \\(y' = e^x\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\text{PR} : \\ y & = e^{\\alpha} ( x -\\alpha ) +e^{\\alpha} \\\\\r\n& = e^{\\alpha} x +( 1 -\\alpha ) e^{\\alpha} \\\\\r\n& = t \\alpha ( x -\\alpha +1 ) , \\\\\r\n\\text{QR} : \\ y & = t \\beta ( x -\\beta +1 ) \\ .\r\n\\end{align}\\]\r\n\\(2\\) \u5f0f\u3088\u308a \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068, \\(t \\neq 0\\) , \\(\\alpha \\neq \\beta\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\alpha x -\\alpha ( & \\alpha -1 ) = \\beta x -\\beta ( \\beta -1 ) \\\\\r\n\\text{\u2234} \\quad x & = \\dfrac{{\\beta}^2 -{\\alpha}^2 -( \\beta -\\alpha )}{\\beta -\\alpha} \\\\\r\n& = \\alpha +\\beta -1 \\ .\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d\r\n\\[\r\ny = t \\alpha \\beta \\ .\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n\\text{Q} \\ ( \\alpha +\\beta -1 , t \\alpha \\beta ) \\ .\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\triangle \\text{PQR} & = \\dfrac{1}{2} ( \\beta -\\alpha ) \\left\\{ t ( \\alpha +\\beta -1 ) -t \\alpha \\beta \\right\\} \\\\\r\n& = \\dfrac{1}{2} ( \\beta -\\alpha ) ( 1 -\\alpha ) ( \\beta -1 ) \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS _ 2 & = \\triangle \\text{PQR} -\\left\\{ \\dfrac{1}{2} ( \\beta -\\alpha ) ( t \\alpha +t \\beta ) -S _ 3 \\right\\} \\\\\r\n& = \\dfrac{t}{2} ( \\beta -\\alpha ) \\left\\{ \\alpha +\\beta -( 1 -\\alpha ) ( \\beta -1 ) +2 \\right\\} \\\\\r\n& = \\dfrac{t}{2} ( \\beta -\\alpha ) ( 1 -\\alpha \\beta ) \\quad ... [2] \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, [1] [2] \u3088\u308a\r\n\\[\r\n\\dfrac{S _ 2}{S _ 1} = \\underline{\\dfrac{1 -\\alpha \\beta}{2}} \\ .\r\n\\]\r\n
(2)<\/strong><\/p>\r\n
\\(f(x) = e^x -tx\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(x) = e^x -t \\ .\r\n\\]\r\n\\(f'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nx = \\log t \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306e \\(x \\gt 0\\) \u306b\u304a\u3051\u308b\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} x & (0) & \\cdots & \\log t & \\cdots & ( \\infty ) \\\\ \\hline f'(x) & & - & 0 & + & \\\\ \\hline f(x) & (1) & \\searrow & -t ( \\log t -1 ) & \\nearrow & ( \\infty ) \\end{array}\r\n\\]\r\n\\(\\alpha , \\beta\\) \u304c \\(f(x) = 0\\) \u306e\u89e3\u3067\u3042\u308b\u306e\u3067, \\(f(x)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n
![\"ngr20150301\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/ngr20150301.svg\")
\r\n\\(t \\gt e\\) \u306b\u6ce8\u610f\u3059\u308c\u3070\r\n\\[\r\nf \\left( \\dfrac{e}{t} \\right) = e^{\\frac{e}{t}} -e \\lt 0 \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u30b0\u30e9\u30d5\u304b\u3089\r\n\\[\r\n0 \\lt \\alpha \\lt \\dfrac{e}{t} \\quad ... [3] \\ .\r\n\\]\r\n\u307e\u305f, \\(e^x \\gt x^2\\) \u3088\u308a \\(x \\gt 2 \\log x\\) \u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nf( 2 \\log t ) & = t^2 -2t \\log t \\\\\r\n& = t ( t -2 \\log t ) \\gt 0 \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u30b0\u30e9\u30d5\u3088\u308a\r\n\\[\r\n\\log t \\lt \\beta \\lt 2 \\log t \\quad ... [4] \\ .\r\n\\]\r\n[3] [4] \u3088\u308a\r\n\\[\r\n0 \\lt \\alpha \\beta \\lt \\dfrac{2e \\log t}{t} \\quad ... [5] \\ .\r\n\\]\r\n\u3053\u3053\u3067, \\(e^x \\gt x^2\\) \u306b\u3064\u3044\u3066, \\(x = \\log t\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nt \\gt ( \\log t )^2 \\ \\text{\u3059\u306a\u308f\u3061} \\ \\sqrt{t} \\gt \\log t \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n\\dfrac{\\log t}{t} \\lt \\dfrac{1}{\\sqrt{t}} \\rightarrow 0 \\quad ( \\ t \\rightarrow \\infty \\ \\text{\u306e\u3068\u304d} ) \\ .\r\n\\]\r\n\u3086\u3048\u306b, [5] \u3068\u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u304b\u3089\r\n\\[\r\n\\displaystyle\\lim _ {t \\rightarrow \\infty} \\alpha \\beta = 0 \\ .\r\n\\]\r\n\u3088\u3063\u3066, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {t \\rightarrow \\infty} \\dfrac{S _ 2}{S _ 1} = \\displaystyle\\lim _ {t \\rightarrow \\infty} \\dfrac{1 -\\alpha \\beta}{2} = \\underline{\\dfrac{1}{2}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(e\\) \u3092\u81ea\u7136\u5bfe\u6570\u306e\u5e95\u3068\u3057, \\(t\\) \u3092 \\(t \\gt e\\) \u3068\u306a\u308b\u5b9f\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u66f2\u7dda \\(C : \\ y = e^x\\) \u3068 \u76f4\u7dda \\(y = tx\\) \u306f\u76f8\u7570\u306a\u308b \\(2\\) \u70b9\u3067\u4ea4\u308f\u308b\u306e\u3067, … \u7d9a\u304d\u3092\u8aad\u3080