The teaching of calculus has changed a great deal over the past 300 years. Originally, calculus was introduced with differentials and was rigorously based on Taylor's theorem, with integration considered the inverse of differentiation. Cauchy changed all that by showing that calculus followed from the Mean Value Theorem and that integration is the limit of a sum. Weierstrass and Lesbesgue changed it again by making limits and integrals set-theoretic and by replacing the Mean Value theorem with results that followed from absolute continuity and uniform convergence. In this century, differential forms, operator theory, numerical analysis, and dynamical systems have continued the ongoing transformation of what calculus is and what it is used for.
How then to present calculus with a consistent interpretation while maintaining at least some semblance of rigor? After struggling for quite some time, we identified two important ideas which would allow us to do just that. First, we recognized that two themes have been central to calculus since the days of Newton to the present and will continue to be central for years to come. These two themes are differential equations and integration. That is, nearly all the theory and application of calculus is reflected in the study of differential equations and the theory of integration.
Second, we realized that the most modern realizations of calculus are also the best. That means using algebra to study the derivative and using simple functions to define integrals. It also means including applications which involve data, developing the idea of a mathematical model, and using sequences to study discrete dynamical systems. Thus, our "modern approach" is one that presents calculus as the foundation of modern mathematics, science, and engineering. <Back to home page>