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. This method is used in fields such as engineering, physics, economics, and ecology.

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 ${\displaystyle y=f(x)}$ at any ${\displaystyle x=a}$ based on the value and slope of the function at ${\displaystyle x=b}$, given that ${\displaystyle f(x)}$ is differentiable on ${\displaystyle [a,b]}$ (or ${\displaystyle [b,a]}$) and that ${\displaystyle a}$ is close to ${\displaystyle b}$. In short, linearization approximates the output of a function near ${\displaystyle x=a}$.

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