Polynomial Approximations of Elementary Functions



We begin this section by deriving several polynomial approximations  of the function  near the point . Such approximations are said to be centered at .


As a first step, it seems reasonable to require that both  and . Use these requirements to find a first-degree polynomial approximation (of the form  ) of  at .


Now impose an additional requirement that  and find a second-degree polynomial approximation (of the form  ) of  at .


Can you find a third-degree approximation ?



Taylor and Maclaurin Polynomials



In the event that the approximating polynomial is expanded about some number c that is different than zero, it is convenient to write the polynomial in the form



Using this form makes it easier to determine what the coefficients  should be. We arrive at the following definition:


If f has n derivatives at c, then the polynomial



is called the nth Taylor polynomial for f at c. If , then



is also called the nth Maclaurin polynomial for f.



As an example, find the Taylor polynomials  through  for  centered at .


Find the Maclaurin polynomials  and  for .


Use a fourth Maclaurin polynomial to approximate .



Remainder of a Taylor Polynomial



To measure the accuracy of the approximations we have been discussing, we introduce the concept of a remainder  as follows:



The maximum value of  is called the error associated with the approximation.



Taylor’s Theorem.  If a function f is has  derivatives in an interval I containing c, then for each x in I there exists z between x and c such that




where .


A useful consequence of this is that .


Can you spot the oh-so-obvious connection to a result from Calculus 1?



As an example, compute the third Maclaurin polynomial for . Now use Taylor’s Theorem to approximate  by  and determine the accuracy of the approximation.


Determine the degree of the Taylor polynomial  centered at  that should be used to approximate  so that the error is less than 0.001.