John H. Mathews

Using a computer algebra system to teach double integration

Evaluating double integrals is an important topic in calculus. The mechanics of evaluating interated integrals can be assisted with a computer algebra system. Methods using the computer algebra software MAPLE are presented.

Using MATLAB as a programming language for numerical analysis

Todays software offers more for numerical analysis than just programming. The software MATLAB can be used to do things the traditional way; writing loops; branching using logical decisions and invoking subroutines. Now a larger programming environment is available; graphics and built in subroutine libraries. These features are influencing the way numerical analysis is taught. MATLAB is based on lists and many algorithms can be streamlined by taking advantage of this structure. Graphical output for interpolation, curve fitting and the solution of differential equations is easily produced by manipulating these data structures. This article illustrates how MATLAB can be used in a numerical analysis course to enhance the teaching of: Newtons method, Gaussian elimination, Chebyshev approximation, least squares polynomials, error analysis for numerical differentiation, adaptive quadrature, Runge‐Kutta methods, and the solution of Laplaces equation. Our students have enjoyed MATLAB, and had a better experienc...

Mathematica, a new tool for studying optimization

An important topic in calculus is optimization. Historically, it has been presented as an application involving curve sketching and differentiation, and is used to show how the theory is put to practical use. From the pedagogical side, we want students to understand the first and second derivative tests and their relationship to a graph. But, we have the unhappy situation that many students are weak in both these areas. We show how a computer algebra system (CAS) is used to assist the teaching of optimization. Computation of derivatives are handled with the symbolic capabilities of a CAS. Topics from analytic geometry, such as the inflection point, are illuminated with the high‐resolution graphics capabilities of CAS.

Using Mathematicato teach Bessel functions

By use of a computer and Mathematicathe elementary function cos x is visualized as a limit of the graphs of Taylor polynomials. A slight modification of the Taylor polynomials for cos xleads to a computer visualization of the graph of the Bessel function J0 (x).By similar modifications the student can develop and study the students own analogues of trigonometric functions and of Bessel functions. A computer model of a vibrating elastic string, which can be animated, is produced by use of Fourier series and Mathematica.Similarly, a model of a vibrating elastic membrane is developed by use of J 0(x).The student can perform numerous computer experiments involving vibrating membranes with various initial conditions and involving study of the period of the motion. In addition, the student can study the motion of ‘pseudo‐membranes’ by use of the students own analogues of Bessel functions.

Finding Least Squares Lines with Mathematica.

ABSTRACT A detailed study of least squares lines reveals that there are several cases to consider. There are three “close fitting lines”, they minimize of the sum of the squares of the vertical distances, horizontal distances, and orthogonal distances, respectively. Students are interested in exploring these three cases as a computer laboratory exercise. The software Mathematica can be used to program the numerical computations and obtain the graph of the lines and points. This demonstrates that Mathematica is more than a computer algebra system; it is also a programming language which can be used for teaching numerical analysis.

Rational approximation in an educational mathematics laboratory

Abstract This paper discusses rational approximations for functions, and shows how this is a generalization of the polynomial approximation. Two techniques are presented, Pade approximation and an interpolation method. The techniques are instructive for students in an undergraduate computing laboratory environment and can be incorporated in either a calculus or numerical methods course.

Using a Computer Algebra System to Solve for Maxima and Minima

The availability of computer algebra systems (CASs) such as Derive, MACSYMA, Maple, Mathematica, and SMP are influencing the way we teach calculus. New approaches to old problems are being initiated at colleges and universities world? wide. In this note we use the CAS software Mathematica to solve some typical optimization problems from freshman calculus. Our purpose is to demonstrate how the availability of CAS alters our approach and treatment of computations. As a first example, consider minimizing the distance from the point (6,3) to the parabola y = x2. This requires minimizing the objective function fix) =