THIRTEEN - TAYLOR POLYNOMIALS AND APPROXIMATION

Abstract

It is a fact that many functions arising in applications are difficult to deal with. A continuous function, for example, may take a complicated form or it may take a simple form that, nevertheless, cannot be integrated. For this reason, mathematicians and physicists have developed methods for approximating certain functions by other functions that are much easier to handle. Some of the easiest functions to deal with are the polynomials because apart from having other useful properties, they can be differentiated and integrated any number of times and still remain polynomials. This chapter discusses how certain continuous functions can be approximated by polynomials. It discusses Taylor's theorem and Taylor polynomials. It discusses how a function can be approximated as closely as desired by a polynomial, provided that the function possesses a sufficient number of derivatives. The chapter presents a proof of Taylor's theorem and then discusses how it is possible to estimate the maximum size of the remainder term Rn(x). The chapter discusses how the bound on the magnitude of the remainder term can be used to find some interesting approximations.