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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 Mean of a function ： ウィキペディア英語版 Mean of a function In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function ''f''(''x'') over the interval (''a,b'') is defined by : $\backslash bar=\backslash frac\backslash int\_a^bf(x)\backslash ,dx.$ Recall that a defining property of the average value $\backslash bar$ of finitely many numbers $y\_1,\; y\_2,\; \backslash dots,\; y\_n$ is that $n\backslash bar\; =\; y\_1\; +\; y\_2\; +\; \backslash cdots\; +\; y\_n$. In other words, $\backslash bar$ is the ''constant'' value which when ''added'' to itself $n$ times equals the result of adding the $n$ terms of $y\_i$. By analogy, a defining property of the average value $\backslash bar$ of a function over the interval $()$ is that : $\backslash int\_a^b\backslash bar\backslash ,dx\; =\; \backslash int\_a^bf(x)\backslash ,dx$ In other words, $\backslash bar$ is the ''constant'' value which when ''integrated'' over $()$ equals the result of integrating $f(x)$ over $()$. But by the second fundamental theorem of calculus, the integral of a constant $\backslash bar$ is just : $\backslash int\_a^b\backslash bar\backslash ,dx\; =\; \backslash barx\backslash bigr|\_a^b\; =\; \backslash barb\; -\; \backslash bara\; =\; (b\; -\; a)\backslash bar$ See also the first mean value theorem for integration, which guarantees that if $f$ is continuous then there exists a point $c\; \backslash in\; (a,\; b)$ such that : $\backslash int\_a^bf(x)\backslash ,dx\; =\; f(c)(b\; -\; a)$ The point $f(c)$ is called the mean value of $f(x)$ on $()$. So we write $\backslash bar\; =\; f(c)$ and rearrange the preceding equation to get the above definition. In several variables, the mean over a relatively compact domain ''U'' in a Euclidean space is defined by :$\backslash bar=\backslash frac\backslash int\_U\; f.$ This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of ''f'' to be :$\backslash exp\backslash left(\backslash frac\backslash int\_U\; \backslash log\; f\backslash right).$ More generally, in measure theory and probability theory, either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function. There is also a ''harmonic average'' of functions and a ''quadratic average'' (or ''root mean square'') of functions. ==See also== *Mean 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「Mean 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.