Taylor’s Theorem and Derivative Tests for Extrema：泰勒的定理和极值导数试验.doc

Taylor Approximations and Definite Integrals Sheldon P. Gordon Farmingdale State University of New York gordonsp@ Abstract We investigate the possibility of approximating the value of a definite integral by approximating the integrand rather than using numerical methods to approximate the value of the definite integral. Particular cases considered include examples where the integral is improper, such as an elliptic integral. Keywords Taylor polynomials, Taylor approximations, numerical integration, definite integrals Many of us tend to think of Taylor’s theorem as the capstone for the first year of calculus. But, if it is truly a capstone, then it should provide some broader perspectives on and deeper insights into topics that were previously encountered. Unfortunately, too few of us take the time to take advantage of some of these opportunities and not all of these opportunities are well known. For instance, in an earlier article [1], the author examined the use of Taylor approximations to provide insight into some of the standard limits that arise in calculus, not just to focus on techniques and tricks, such as l’H?pital’s rule, that find the values of such limits. In a more recent article [2], the author investigated ways in which one can look back at the derivative tests for extrema (as well as derivative tests for points of inflections) from the point of view of Taylor approximations. In the present article, we look at how the use of Taylor polynomial approximations can provide additional insights into, and tools for dealing with, definite integrals. Specifically, we consider two often troublesome cases, one where antiderivatives for the integrand do not exist in closed form, so that the fundamental theorem of calculus cannot be used, and the other where we need to evaluate improper integrals. Consider the definite integral , which is essentially the integral needed to evaluate probabilities involving a normal distribution, something that comes up in