**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
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