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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.

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    $\begingroup$ 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. $\endgroup$ – Thoth19 Jun 24 '14 at 16:10
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    $\begingroup$ 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. $\endgroup$ – Steven Gubkin Jun 24 '14 at 16:25
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    $\begingroup$ Might also be a better fit on the math educators stack exchange. $\endgroup$ – Steven Gubkin Jun 24 '14 at 16:25
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    $\begingroup$ 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. $\endgroup$ – Ben Crowell Jun 26 '14 at 16:47
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    $\begingroup$ 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. $\endgroup$ – Alexander Gruber Jun 28 '14 at 13:58
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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.

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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.

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    $\begingroup$ 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. $\endgroup$ – Alexander Gruber Jun 28 '14 at 13:51
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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/

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