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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 Linearization ： ウィキペディア英語版 Linearization In mathematics linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems.〔(The linearization problem in complex dimension one dynamical systems at Scholarpedia )〕 This method is used in fields such as engineering, physics, economics, and ecology. ==Linearization of a function== Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function $y\; =\; f(x)$ at any $x\; =\; a$ based on the value and slope of the function at $x\; =\; b$, given that $f(x)$ is differentiable on $(b)$ (or $(a)$) and that $a$ is close to $b$. In short, linearization approximates the output of a function near $x\; =\; a$. For example, $\backslash sqrt\; =\; 2$. However, what would be a good approximation of $\backslash sqrt\; =\; \backslash sqrt$? For any given function $y\; =\; f(x)$, $f(x)$ can be approximated if it is near a known differentiable point. The most basic requisite is that $L\_a(a)\; =\; f(a)$, where $L\_a(x)$ is the linearization of $f(x)$ at $x\; =\; a$. The point-slope form of an equation forms an equation of a line, given a point $(H,\; K)$ and slope $M$. The general form of this equation is: $y\; -\; K\; =\; M(x\; -\; H)$. Using the point $(a,\; f(a))$, $L\_a(x)$ becomes $y\; =\; f(a)\; +\; M(x\; -\; a)$. Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to $f(x)$ at $x\; =\; a$. While the concept of local linearity applies the most to points arbitrarily close to $x\; =\; a$, those relatively close work relatively well for linear approximations. The slope $M$ should be, most accurately, the slope of the tangent line at $x\; =\; a$. Visually, the accompanying diagram shows the tangent line of $f(x)$ at $x$. At $f(x+h)$, where $h$ is any small positive or negative value, $f(x+h)$ is very nearly the value of the tangent line at the point $(x+h,\; L(x+h))$. The final equation for the linearization of a function at $x\; =\; a$ is: $y\; =\; f(a)\; +\; f\text{'}(a)(x\; -\; a)\backslash ,$ For $x\; =\; a$, $f(a)\; =\; f(x)$. The derivative of $f(x)$ is $f\text{'}(x)$, and the slope of $f(x)$ at $a$ is $f\text{'}(a)$. 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「Linearization」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.