Teaching calculus with infinitesimals and differentials

Abstract

Several new approaches to calculus in the U.S. have been studied recently that are grounded in infinitesimals or differentials rather than limits. These approaches seek to restore to differential notation the direct referential power it had during the first century after calculus was developed. In these approaches, a differential equation like dy = 2x·dx is a relationship between increments of x and y, making dy/dx an actual quotient rather than code language for limh→0f(x+h)-f(x)h. An integral ∫ab2xdx is a sum of pieces of the form 2x·dx, not the limit of a sequence of Riemann sums. One goal is for students to develop understandings of calculus notation that are imbued with more direct referential meaning, enabling them to better interpret and model situations by means of this notation. In this article I motivate and describe some key elements of differentials-based calculus courses, and I summarize research indicating that students in such courses develop robust quantitative meanings for notations in single- and multi-variable calculus.