(1)<\/strong>\u3000\\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u3092\u63cf\u3051.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(t \\gt 0\\) \u3092\u5a92\u4ecb\u5909\u6570\u3068\u3057\u3066, \\(x = f'(t)\\) , \\(y = f(t) -t f'(t)\\) \u3067\u8868\u3055\u308c\u308b\u66f2\u7dda\u306e\u6982\u5f62\u3092\u63cf\u3051.<\/p><\/li>\r\n (3)<\/strong>\u3000(2)<\/strong> \u306e\u66f2\u7dda\u306e\u63a5\u7dda\u304c \\(x\\) \u8ef8\u3068 \\(y\\) \u8ef8\u306b\u3088\u3063\u3066\u5207\u308a\u53d6\u3089\u308c\u3066\u3067\u304d\u308b\u7dda\u5206\u306e\u9577\u3055\u306f\u4e00\u5b9a\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\nf'(x) & = \\dfrac{1 \\cdot \\sqrt{1 +x^2} -x \\cdot \\frac{2x}{2 \\sqrt{1 +x^2}}}{1 +x^2} \\\\\r\n& = \\dfrac{( 1 +x^2 ) -x^2}{( 1 +x^2 )^{\\frac{3}{2}}} \\\\\r\n& = \\dfrac{1}{( 1 +x^2 )^{\\frac{3}{2}}} \\gt 0\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3059\u308b. (2)<\/strong><\/p>\r\n \\(x = x(t)\\) , \\(y = y(t)\\) \u3068\u3042\u3089\u308f\u3059. (3)<\/strong><\/p>\r\n \u70b9 \\(( x(t) , y(t) )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u5f0f\u306f, [1] \u3088\u308a\r\n\\[\\begin{align}\r\ny & = -t \\left( x -f'(t) \\right) +f(t) -t f'(t) \\\\\r\n& = -tx +f(t)\r\n\\end{align}\\]\r\n\u3053\u306e\u5f0f\u306b\u3064\u3044\u3066<\/p>\r\n \\(x = 0\\) \u306e\u3068\u304d\r\n\\[\r\ny = f(t)\r\n\\]<\/li>\r\n \\(y = 0\\) \u306e\u3068\u304d\r\n\\[\r\nx = \\dfrac{f(t)}{t}\r\n\\]<\/li>\r\n<\/ul>\r\n \u3057\u305f\u304c\u3063\u3066, \u63a5\u7dda\u304c \\(x\\) \u8ef8\u3068 \\(y\\) \u8ef8\u306b\u5207\u308a\u53d6\u3089\u308c\u3066\u3067\u304d\u308b\u7dda\u5206\u306e\u9577\u3055 \\(P\\) \u306f\r\n\\[\\begin{align}\r\nP & = \\left\\{ f(t) \\right\\}^2 +\\left\\{ \\dfrac{f(t)}{t} \\right\\}^2 \\\\\r\n& = \\dfrac{t^2}{1 +t^2} +\\dfrac{1}{1 +t^2} \\\\\r\n& = 1\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\(f(x) = \\dfrac{x}{\\sqrt{1 +x^2}}\\) \u306b\u3064\u3044\u3066, \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u3092\u63cf\u3051. (2)\u3000\\(t \\gt 0\\) \u3092\u5a92\u4ecb\u5909\u6570\u3068\u3057\u3066, \\ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\u307e\u305f\r\n\\[\r\nf( -x ) = -f(x) , \\quad f(0) = 0\r\n\\]\r\n\u3055\u3089\u306b\r\n\\[\r\nf(x) = \\pm \\dfrac{1}{\\sqrt{1 +\\frac{1}{x^2}}} \\rightarrow \\pm 1 \\quad ( \\ x \\rightarrow \\pm \\infty \\ \\text{\u306e\u3068\u304d} )\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u306f\u4e0b\u56f3.<\/p>\r\n![\"wsr20150101\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/wsr20150101.svg\")
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\r\n\u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\nx(t) & = ( 1 +t^2 )^{-\\frac{3}{2}} , \\\\\r\ny(t) & = t ( 1 +t^2 )^{-\\frac{1}{2}} -t ( 1 +t^2 )^{-\\frac{3}{2}} \\\\\r\n& = \\{ t ( 1 +x^2 ) -t \\} ( 1 +t^2 )^{-\\frac{3}{2}} \\\\\r\n& = t^3 ( 1 +t^2 )^{-\\frac{3}{2}}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nx'(t) & = -\\dfrac{3}{2} ( 1 +t^2 )^{-\\frac{5}{2}} \\cdot 2t \\\\\r\n& = -3t ( 1 +t^2 )^{-\\frac{5}{2}} , \\\\\r\ny'(t) & = 3t^2 ( 1 +t^2 )^{-\\frac{3}{2}} -3 t^4 ( 1 +t^2 )^{-\\frac{5}{2}} \\\\\r\n& = 3t^2 \\{ ( 1 +t^2 ) -t^2 \\} ( 1 +t^2 )^{-\\frac{5}{2}} \\\\\r\n& = 3t^2 ( 1 +t^2 )^{-\\frac{5}{2}}\r\n\\end{align}\\]\r\n\u3053\u308c\u3089\u3088\u308a\r\n\\[\\begin{align}\r\ny'(t) & = -t x'(t) \\\\\r\n\\text{\u2234} \\quad \\dfrac{dy}{dx} & = -t \\quad ... [1]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b\u66f2\u7dda\u306f\u5358\u8abf\u6e1b\u5c11\u3059\u308b.
\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {t \\rightarrow +0} x(t) & = 1 , \\ \\displaystyle\\lim _ {t \\rightarrow +0} y(t) = 0 , \\\r\n\\displaystyle\\lim _ {t \\rightarrow +0} \\dfrac{dy}{dx} = 0 , \\\\\r\n\\displaystyle\\lim _ {t \\rightarrow \\infty} x(t) & = 0 , \\ \\displaystyle\\lim _ {t \\rightarrow \\infty} y(t) = \\displaystyle\\lim _ {t \\rightarrow \\infty} \\dfrac{1}{\\left( 1 +\\frac{1}{t^2} \\right)^{\\frac{3}{2}}} = 1 , \\\\\r\n\\displaystyle\\lim _ {t \\rightarrow \\infty} \\dfrac{dy}{dx} & = -\\infty\r\n\\end{align}\\]\r\n\u306b\u6ce8\u610f\u3059\u308c\u3070, \u6c42\u3081\u308b\u66f2\u7dda\u306f\u4e0b\u56f3\u306e\u901a\u308a\u3067\u3042\u308b\uff08\u25cb\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n![\"wsr20150102\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/wsr20150102.svg\")
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