Chapter 12: Approximating Functions and Error

In Chapter 10 you learned and applied many tests to determine convergence of series. You have also learned about power series. In this chapter, you will discover some special power series, called Taylor series and Maclaurin series.

The chapter begins with polynomials that approximate well-known functions and you will determine the accuracy of these approximations. A focus on the size of the error will help you develop a formula to bound this error. 

Chapter Goals

Write Taylor and Maclaurin
series to approximate common functions.

Approximate error of Taylor
polynomials.

Determine the radius and
interval of convergence for
Taylor series.

Manipulate Taylor polynomials.

Chapter Outline

Section 12.1

You will write nth-degree Taylor polynomials that closely resembles a given function, and use them to make approximations. Then, you will write Taylor series that converge with a non-polynomial function on a set interval.

Section 12.2

You will determine the interval of convergence for Taylor series.

Section 12.3

You will bound the error of Taylor polynomials.

Section 12.4

Finally, you will use Taylor series to demonstrate l’Hôpital’s Rule.