A computational point of view on teaching derivatives

Abstract

The strong connection between mathematics and informatics highlights the important role of scientific computing in many applications. Unfortunately, mathematics is traditionally taught without investigating possible connections between abstract problem solving and the use of algorithms capable of being implemented on a computer. Since mathematical theory and computing practice are taught separately, many students fail to appreciate the utility of mathematics. In this paper, we briefly explain how a typical lecture on obtaining derivatives in differential calculus can benefit from examples implemented in high-level languages like MATLAB, Python or R. Such examples can help to guide the students to a better understanding of the theoretical concepts and limits of finite precision floating-point arithmetic. We argue that, while the historical findings of Leibniz and Newton are good starting points for introducing finite difference approximation methods, this does not preclude new approaches to numerically computing derivatives using Infinite Computer Arithmetic.