Using Integral Transforms to Estimate Higher Order Derivatives

When doing error analysis for numerical quadrature, achieving good uniform bounds on higher order derivatives of the integrand is paramount. As undergraduates become increasingly adept with programmable calculators, numerical integration schemes such as Simpson’s Rule and the Trapezoidal Rule take on a new relevance. Although it may be a rare calculus class that dwells overmuch on error bounds for such schemes, this may be due as much to the perceived paucity of interesting examples for which decent error bounds are readily achievable as to the general weakness in algebraic skills necessary for the requisite understanding of inequalities. The purpose of this article, therefore, is to offer some interesting, non-trivial examples for which the error analysis, if not elegant, is at least simple enough to carry out in the classroom.