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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Elasticity (mathematics) ： ウィキペディア英語版 Elasticity of a function In mathematics, the elasticity or point elasticity of a positive differentiable function ''f'' of a positive variable (positive input, positive output)〔The elasticity can also be defined if the input and/or output is consistently negative, or simply away from any points where the input or output is zero, but in practice the elasticity is used for positive quantities.〕 at point ''a'' is defined as :$Ef(a)\; =\; \backslash fracf\text{'}(a)$ :$=\backslash lim\_\backslash frac\backslash frac=\backslash lim\_\backslash frac\backslash frac=\backslash lim\_\backslash frac\}\}\backslash approx\; \backslash frac$ or equivalently :$Ef(x)\; =\; \backslash frac.$ It is thus the ratio of the relative (percentage) change in the function's output $f(x)$ with respect to the relative change in its input $x$, for infinitesimal changes from a point $(a,\; f(a))$. Equivalently, it is the ratio of the infinitesimal change of the logarithm of a function with respect to the infinitesimal change of the logarithm of the argument. The elasticity of a function is a constant $\backslash alpha$ if and only if the function has the form $f(x)\; =\; C\; x\; ^\; \backslash alpha$ for a constant $C>0$. The elasticity at a point is the limit of the arc elasticity between two points as the separation between those two points approaches zero. The concept of elasticity is widely used in economics; see elasticity (economics) for details. 〔• Hanoch, G. (1975) “The elasticity of scale and the shape of average costs,” American Economic Review 65, pp. 492-497.〕 〔• Panzar, J.C. and R.D. Willig (1977) “Economies of scale in multi-output production, Quarterly Journal of Economics 91, 481-493.〕〔• (Zelenyuk, V. (2013) “A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation.” European Journal of Operational Research 228:3, pp 592–600 )〕 ==Rules== Rules for finding the elasticity of products and quotients are simpler than those for derivatives. Let ''f, g'' be differentiable. Then〔 :$E\; (\; f(x)\; \backslash cdot\; g(x)\; )\; =\; E\; f(x)\; +\; E\; g(x)$ :$E\; \backslash frac\; =\; E\; f(x)\; -\; E\; g(x)$ :$E\; (\; f(x)\; +\; g(x)\; )\; =\; \backslash frac$ :$E\; (\; f(x)\; -\; g(x)\; )\; =\; \backslash frac$ The derivative can be expressed in terms of elasticity as :$D\; f(x)\; =\; \backslash frac$ Let ''a'' and ''b'' be constants. Then :$E\; (\; a\; )\; =\; 0\; \backslash$ :$E\; (\; a\; \backslash cdot\; f(x)\; )\; =\; E\; f(x)$, :$E\; (b\; x^a)\; =\; a\; \backslash$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Elasticity of a function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.