September 14, 2011, 8:00 am

By Robert Talbert

Happy Hump Day! Here are some items of interest from the past week:

Read More

January 19, 2010, 9:22 pm

By Robert Talbert

This is probably the last of three articles on how piecewise-linear functions could be used as a helpful on-ramp to the big ideas in calculus. In the first article, we saw how it’s possible to develop some of the main conceptual ideas of the derivative, without much of the technical notation or jargon, by using piecewise-linear functions. In the second article, we saw how to use the piecewise-linear approach to develop an alternative limit-based definition of the derivative of a function at a point. To wrap things up, in this article I’ll discuss how this same sort of approach can help in students’ first contact with integration, again by way of a hypothetical classroom exercise.

When we took this approach with derivatives, we used the travels of three college students from their dorm rooms to the cafeteria. Each student had a different graph showing his position as a (piecewise-linear)…

Read More

May 5, 2008, 1:48 pm

By Robert Talbert

Asking the following question may make me less of a mathematician in some people’s eyes, and I’m fine with that, but: How do you explain the meaning of the differential *dx* inside an integral? And more importantly, how do you treat the dx in an integral so that, when you get to u-substitutions, all the substituting with du and dx and so on means more than just a mindless crunching of symbols?

Here’s how Stewart’s *Calculus* does it:

- In the section introducing the definite integral and its notation, it says: “The symbol dx has no official meaning by itself; \(\int_a^b f(x) \, dx\) is all one symbol.” (What kind of statement is that? If dx has “no official meaning”, then why is it there at all?)
- In the section on Indefinite Integrals and the Net Change Theorem, there is a note — almost an afterthought — on units at the very end, where there is an implied connection between…

Read More