Use Mathematica to teach calculus

I am interested in coming up with a few lectures from first-semester calculus that I can incorporate Mathematica into in a natural way. I have already written some code for a lecture on the $\delta\mbox{-}\epsilon$ definition of a limit, and I am looking for suggestions about things other people have done in their classrooms.

I would also be interested in knowing if anyone has done examples that "break" Mathematica. That is, examples that require human ingenuity that a computer can't simply compute the answer to.

• Do your students have access to Mathematica? They might want to try examples or something for themselves. If not, Sage is a useful, and somewhat similar tool that is free online. I know my HS calculus teacher wanted to use that to help his classes. For delta-epsilon, you could got for some function that is logarithmic so that the limit appears more easily. I would say to write a function that just computes the function, and then probes around for the limit bounds, using some sort of algorithm that tries to check order of magnitude before actual value.
– Thoth19
Jun 24 '14 at 16:10
• There is actually a book "Calculus and Mathematica" which is pretty good (although it contains some curse words). It may or may not be out of print, but I recommend trying to find a copy, at least for inspiration. Jun 24 '14 at 16:25
• Might also be a better fit on the math educators stack exchange. Jun 24 '14 at 16:25
• IMO there's always something wrong when one of the main goals of a college course is to teach students to use a specific piece of proprietary software.
– user507
Jun 26 '14 at 16:47
• When I took calculus at University of Cincinnati, we had a 1 credit hour co-requisite called "Calculus Lab" where a TA would lead us through constructing Mathematica notebooks to demonstrate the stuff we were doing in lecture. It honestly couldn't have been more helpful to my understanding, and I would support adding something like this in just about any first year calculus course. Having students meet in a computer lab solves the problem of requiring them to all buy student copies. Jun 28 '14 at 13:58

For several years I taught I course (which no longer exists) making heavy use of Maple. All the material is at http://neil-strickland.staff.shef.ac.uk/courses/MAS100/. Obviously technical details would change if you wanted to use Mathematica instead, but the general approach might still be useful.

Here is an example that "breaks" Mathematica, for now:

$$\int_{-\infty}^\infty \text{erf}(x+1)\,e^{-x^2} dx = \sqrt{\pi}\ \text{erf}(\,1/\sqrt{2}\,)$$

Mathematica doesn't find this answer, even though you can find the exact result with human ingenuity, or with the right table of integrals. However....

I have submitted this comment to the Mathematica developers, and I expect that they will teach Mathematica some techniques for integrating this soon enough.

Also, this is not an example at the level of a first-semester calculus course. Mathematica has been trained to solve all the problems in various textbooks, so I doubt you'll find a simple example of a break.

• Not sure why you're getting downvoted. OP did ask for examples of breaking Mathematica, and didn't say they necessarily had to have anything to do with first year calculus. Jun 28 '14 at 13:51

Here's some courseware that uses a custom version of Mathematica to teach Calculus, but it looks like it costs about \$99/student. On the other hand, that's probably cheaper than buying Mathematica: http://www.makingmath.com/