Linearization - Linearization of A Function

Linearization of A Function

Linearizations of a function are lines — ones that are usually 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 continuous 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 continuous 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 continuous 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 .