Special features of this approach (will expand each into more elaborate prose at a later date).
Calculus addresses two foundational problems:
You know how fast a quantity
changes at every moment; you want to know how much of it there is
at every moment.
You know how much of a quantity
there is at every moment; you want to know how fast it is changing
at every moment.
Rooted in research on students’ mathematical thinking and understanding
Always refer to values of variables
Differentials are at the heart of this approach. A differential is a variable that varies through (small) intervals. We speak of differentials only in the context of constant rate of change. The differential dx is a varying change in x. The symbol $\Delta x$ represents the size of intervals through which dx varies. The differential dy is always a constant multiple of dx.
We avoid “slope”. Rather we emphasize rate of change.
The special meaning of “at a moment that $x=x_0$" is developed in Chapter 3 and is used throughout the book. By a moment of a variable we mean a small interval containing a particular value of the variable.
We develop the concept of a function that gives the rate of change of another function at a moment of the other function’s independent variable. This is the meaning of derivative that we develop.
An integral is not an area. It is an accumulation. If the quantity that is accumulating is area, then an integral is an area. But most times an integral is not an area.
Antiderivatives are functions.
We do not develop derivatives, then integrals, then bring them together with the FTC. Rather, like Newton, we
begin with the FTC in that rate of change and accumulation are two sides of a coin. We use the concept of rate of change to develop accumulation functions and we use the concept of accumulation to develop rate of change functions.
We state the FTC even before developing rate of change from accumulation (derivatives). In fact, we use the FTC as motivation for finding derivatives.
Uses of the textbook as a visual aid during class.