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Concept
Linearization
Summary
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. 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 at any based on the value and slope of the function at , given that is differentiable on (or ) and that is close to . In short, linearization approximates the output of a function near .
For example, . However, what would be a good approximation of ?
For any given function , can be approximated if it is near a known differentiable point. The most basic requisite is that , where is the linearization of at . The point-slope form of an equation forms an equation of a line, given a point and slope . The general form of this equation is: .
Using the point , becomes . Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to at .
While the concept of local linearity applies the most to points arbitrarily close to , those relatively close work relatively well for linear approximations. The slope should be, most accurately, the slope of the tangent line at .
Visually, the accompanying diagram shows the tangent line of at . At , where is any small positive or negative value, is very nearly the value of the tangent line at the point .
The final equation for the linearization of a function at is:
For , . The derivative of is , and the slope of at is .
To find , we can use the fact that . The linearization of at is , because the function defines the slope of the function at . Substituting in , the linearization at 4 is .
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