\u95a2\u6570 \\(f(x) = e^{-\\frac{x^2}{2}}\\) \u3092 \\(x \\gt 0\\) \u3067\u8003\u3048\u308b.\r\n\\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u70b9 \\(\\left( a , f(a) \\right)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092 \\(\\ell _ a\\) \u3068\u3057, \\(\\ell _ a\\) \u3068 \\(y\\) \u8ef8\u3068\u306e\u4ea4\u70b9\u3092 \\(( 0 , Y(a) )\\) \u3068\u3059\u308b.\r\n\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057, \u5b9f\u6570 \\(k\\) \u306b\u5bfe\u3057\u3066 \\(\\displaystyle\\lim _ {t \\rightarrow \\infty} t^k e^{-t} = 0\\) \u3067\u3042\u308b\u3053\u3068\u306f\u8a3c\u660e\u306a\u3057\u3067\u7528\u3044\u3066\u3082\u3088\u3044.<\/p>\r\n
(1)<\/strong>\u3000\\(Y(a)\\) \u304c\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(0 \\lt a \\lt b\\) \u3067\u3042\u308b \\(a , b\\) \u306b\u5bfe\u3057\u3066, \\(\\ell _ a\\) \u3068 \\(\\ell _ b\\) \u304c \\(x\\) \u8ef8\u4e0a\u3067\u4ea4\u308f\u308b\u3068\u304d, \\(a\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081, \\(b\\) \u3092 \\(a\\) \u3067\u8868\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000(2)<\/strong> \u306e \\(a , b\\) \u306b\u5bfe\u3057\u3066, \\(Z(a) = Y(a) -Y(b)\\) \u3068\u304a\u304f. \\(\\displaystyle\\lim _ {a \\rightarrow +0} Z(a)\\) \u304a\u3088\u3073 \\(\\displaystyle\\lim _ {a \\rightarrow +0} \\dfrac{Z'(a)}{a}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\r\nf'(x) = -x e^{-\\frac{x^2}{2}}\r\n\\]\r\n\u306a\u306e\u3067, \\(\\ell _ a\\) \u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = -a e^{-\\frac{a^2}{2}} ( x-a ) +e^{-\\frac{a^2}{2}} \\\\\r\n& = -a e^{-\\frac{a^2}{2}} x +\\left( a^2 +1 \\right) e^{-\\frac{a^2}{2}}\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\r\nY(a) = \\left( a^2 +1 \\right) e^{-\\frac{a^2}{2}}\r\n\\]\r\n\u3053\u308c\u3092 \\(a\\) \u306b\u3064\u3044\u3066\u5fae\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nY'(a) & = 2a e^{-\\frac{a^2}{2}} -\\left( a^2 +1 \\right) \\cdot a e^{-\\frac{a^2}{2}} \\\\\r\n& = a \\left( 1 -a^2 \\right) e^{-\\frac{a^2}{2}}\r\n\\end{align}\\]\r\n\\(Y'(a) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nx = 1\r\n\\]\r\n\u307e\u305f, \\(k = \\dfrac{a^2}{2}\\) ... [1] \u3068\u304a\u3051\u3070\r\n\\[\r\n\\displaystyle\\lim _ {a \\rightarrow +0} Y(a) = \\displaystyle\\lim _ {k \\rightarrow \\infty} = (2k+1) e^{-k} = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(Y(a)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} a & (0) & \\cdots & 1 & \\cdots & ( \\infty ) \\\\ \\hline Y'(a) & & + & 0 & - & \\\\ \\hline Y(a) & (1) & \\nearrow & \\text{\u6700\u5927} & \\searrow & (0) \\end{array}\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\r\nY(1) = 2 e^{-\\frac{1}{2}} = \\dfrac{2}{\\sqrt{e}}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u5024\u306e\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{0 \\lt Y(a) \\leqq \\dfrac{2}{\\sqrt{e}}}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\(\\ell _ a , \\ell _ b\\) \u3068 \\(x\\) \u8ef8\u3068\u306e\u4ea4\u70b9\u306e \\(x\\) \u5ea7\u6a19\u3092\u305d\u308c\u305e\u308c \\(x _ a , x _ b\\) \u3068\u304a\u304f. (3)<\/strong><\/p>\r\n (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nY(b) = \\left( \\dfrac{1}{a^2} +1 \\right) e^{-\\frac{1}{2 a^2}}\r\n\\]\r\n[1] \u3068\u540c\u69d8\u306b\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {a \\rightarrow +0} Y(a) & = ( 0+1 ) \\cdot 1 = 1 , \\\\\r\n\\displaystyle\\lim _ {a \\rightarrow +0} Y(b) & = \\displaystyle\\lim _ {k \\rightarrow \\infty} ( 2k-1 ) e^{-k} = 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {a \\rightarrow +0} Z(a) = 1-0 = \\underline{1}\r\n\\]\r\n\u7d9a\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{d Y(b)}{da} & = -\\dfrac{2}{a^3} e^{-\\frac{1}{2 a^2}} +\\left( \\dfrac{1}{a^2} +1 \\right) \\dfrac{1}{a^3} e^{-\\frac{1}{2 a^2}} \\\\\r\n& = \\dfrac{1}{a^3} \\left( \\dfrac{1}{a^2} -1 \\right) e^{-\\frac{1}{2 a^2}}\r\n\\end{align}\\]\r\n[1] \u3068\u540c\u69d8\u306b\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {a \\rightarrow +0} \\dfrac{Y'(a)}{a} & = ( 1-0 ) \\cdot 1 = 1 , \\\\\r\n\\displaystyle\\lim _ {a \\rightarrow +0} \\dfrac{1}{a} \\dfrac{d Y(b)}{da} & = \\displaystyle\\lim _ {k \\rightarrow \\infty} ( 2k )^2 ( 2k-1 ) e^{-k} = 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {a \\rightarrow +0} \\dfrac{Z'(a)}{a} = 1-0 = \\underline{1}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\(f(x) = e^{-\\frac{x^2}{2}}\\) \u3092 \\(x \\gt 0\\) \u3067\u8003\u3048\u308b. \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u70b9 \\(\\left( a , f(a) \\right)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092 \\ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\\(\\ell _ 1\\) \u306b\u3064\u3044\u3066, \\(y = 0\\) \u3092\u3068\u3051\u3070, \\(e^{-\\frac{a^2}{2}} \\neq 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n-a x _ a & +\\left( a^2 +1 \\right) = 0 \\\\\r\n\\text{\u2234} \\quad x _ a & = a +\\dfrac{1}{a}\r\n\\end{align}\\]\r\n\\(\\ell _ b\\) \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[\r\nx _ b = b +\\dfrac{1}{b}\r\n\\]\r\n\\(x _ a = x _ b\\) \u306a\u306e\u3067, \\(a \\neq b\\) \u306b\u6ce8\u610f\u3059\u308c\u3070\r\n\\[\\begin{align}\r\na +\\dfrac{1}{a} & = b +\\dfrac{1}{b} \\\\\r\na -b -\\dfrac{a-b}{ab} & = 0 \\\\\r\n( a-b ) ( ab -1 ) & = 0 \\\\\r\n\\text{\u2234} \\quad ab -1 & = 0 \\quad ( \\ \\text{\u2235} \\ a \\neq b \\ ) \\\\\r\n\\text{\u2234} \\quad b & = \\underline{\\dfrac{1}{a}}\r\n\\end{align}\\]\r\n\u307e\u305f, \\(0 \\lt a \\lt b\\) \u306a\u306e\u3067\r\n\\[\\begin{gather}\r\n0 \\lt a \\lt \\dfrac{1}{a} \\\\\r\n\\underline{0 \\lt a \\lt 1}\r\n\\end{gather}\\]\r\n