ma016 - Graphs of Rational Functions for Computer Assisted...

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Graphs of Rational Functions for Computer Assisted Calculus Stan Byrd and Terry Walters, University of Tennesse at Chattanooga, Chattanooga, TN 37403 The College Mathematics Journal, September 1991, Volume 22, Number 4, pp. 332–334. I n this capsule, we suggest some calculus problems whose solutions involve pencil- and-paper techniques and some form of computer assistance. These are problems that can be used as calculus laboratory projects. We expect the computer to act as a strong and convenient number-cruncher, but we expect the student to supply the conceptual framework. For some of the problems below, finding the proper scaling so that one can see the extrema is a bit difficult, but we feel that a student will profit from this trial-and-error experience. The main computational difficulty of these problems is approximating all the roots of a polynomial, so your computer package should have a reliable polynomial root finder. Assuming a, b, c, and d are positive real numbers, we determine the important properties of the graphs of the family of rational functions, (1) (We encountered this family in a problem in the SIAM Review [ 3 ], where f is described as a mean-squared-error function for a class of regression models, and the author asks for conditions under which the minimum of f on is unique.) As we will show in the following discussion, this family of functions is a good place to make the transition from usual textbook rational function graphs to graphs that should be analyzed with the aid of a computer. (Students who hope to push buttons and get sufficient information will be disappointed.) In the following list we suggest some problems, following each with remarks about its
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ma016 - Graphs of Rational Functions for Computer Assisted...

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