Section Summary: 12.12

Applications of Taylor Polynomials

Definitions

Theorems

Properties, Hints, etc.

There are three possibilities for estimating the size of the error made when using a Taylor polynomial to approximate a function:

1. Graph the difference on the interval of interest and estimate the error.
2. If the series alternates, use the Alternating Series Estimation Theorem.
3. Use Taylor's Inequality: if , then

   

Summary

First of all, polynomials are wonderfully nice functions: infinitely differentiable, easily integrated, smooth, etc.

Taylor polynomials are useful for approximating more complicated functions: we can compute approximations using only the operations of addition, subtraction, multiplication, and division (which are perfect for computers). Your calculator makes supreme use of this: functions such as sine and log are computed this way.

Taylor polynomials match the function they're approximating on all derivatives up to degree n at the point about which the expansion is made. For example, the Taylor polynomial will get the function value (the zeroth derivative) as well as the first derivative right. So the higher the degree of the Taylor polynomial, the better the function will be approximated at x=a.

Taylor's Inequality may help us determine just how good an approximation we have. It is important when approximating to know just how bad things can get....