A versatile approach to calculus and numerical methods

Abstract

Traditionally the calculus is the study of the symbolic algorithms for differentiation and integration, the relationship between them, and their use in solving problems. Only at the end of the course, when all else fails, are numerical methods introduced, such as the Newton-Raphson method of solving equations, or Simpson’s rule for calculating areas. The problem with such an approach is that it often produces students who are very well versed in the algorithms and can solve the most fiendish of symbolic problems, yet have no understanding of the meaning of what they are doing. Given the arrival of computer software which can carry out these algorithms mechanically, the question arises as to what parts of calculus need to be studied in the curriculum of the future. It is my contention that such a study can use the computer technology to produce a far more versatile approach to the subject, in which the numerical and graphical representations may be used from the outset to produce insights into the fundamental meanings, in which a wider understanding of the processes of change and growth will be possible than the narrow band of problems that can be solved by traditional symbolic methods of the calculus