You start by noticing that the Riemann sums (multiplication followed by addition) and the difference quotients (subtraction followed by division) undo each other. Their limits -- the integral and the ...

This is about the Riemann sums but the logic is there.

My home teaching setup. Still to be tested.

The debate seems to be where it was 20 years ago. The reason is that most universities have a "generic" mathematics curriculum, so if the engineers need calculus as it is now, everyone will also get ...

The context isn't entirely clear so I'll assume this is about teaching. Then, I support Pedro's answer but also want to add that doing both verbal and symbolic versions may be a good idea. For example:...

When I choose to closely follow the book, I use exercises as examples. For me, it's more work but also more freedom. Meanwhile, the students are exposed to both a polished presentation in the book and ...

The discussion seems to overlook this simple fact: the Lebesgue integral is not a generalization of and cannot serve as a substitute for the Riemann integral.

Property (3), i.e., the $\varepsilon-\delta$ definition of continuity, has numerous motivations/interpretations. For example, continuity can be interpreted as accuracy. Suppose we are shooting a ...

Excel comes to mind. This took just a few seconds to make:

I think both "the linear approximation of $y=f(x)$ at $x=a$" and "local linear approximation" are fine. Now, it should be made clear that the linear approximation is a function. Next, what kind of ...

My answer would be: "What if there is no formula?" Just as $x$ might stand for an unspecified number, $f$ might stand for an unspecified function.