Mistakes in Sage Calculus Tutorial on Differentiability

[fwchapman]

[Hello, everyone! I originally sent this as an email to Elliott Brossard, but since he's no longer maintaining content on the website, he suggested that I post my feedback here. -- Fred]

Hi Elliott,

Thanks for all your time and effort on the Sage Calculus Tutorial. The plots and the mathematical typesetting are beautifully done!

I'm writing to report some mathematical mistakes in the tricky proofs on the Differentiability page. You're trying to show that the absolute value function, denoted by g, is not differentiable at zero.

You argued that "since the limit of g'(x) as x approaches 0 from the left ≠ the limit of g'(x) as x approaches 0 from the right, g'(0) does not exist." The conclusion is incorrect. Since the left- and right-hand limits are different, you've proven that the two-sided limit of g'(x) as x approaches 0 does not exist. This shows that the derivative g' must be discontinuous at zero, if it exists there; however, this does not prove that g'(0) does not exist. In this counterexample, the derivative g' is discontinuous at zero, like in your example, but g'(0) actually exists.

To show that g'(0) does not exist, you have to appeal directly to the definition of the derivative as the limit of the difference quotient. You did this for g'(x) for general x and derived an expression that is undefined when x is 0, but that's not a valid proof of non-existence at 0. (Also, the 2x in the numerator immediately after cancellation should be 2x + h.)

You need to substitute x = 0 into the limit definition from the very beginning and then evaluate the two one-sided limits. You'll get the limit of |h|/h as h approaches 0. The limit from the left yields -h/h = -1, and the limit from the right yields h/h = 1. Since the one-sided limits disagree, the two-sided limit does not exist, and neither does the derivative g'(0).

If you want to update the web page, I'd be happy to give it another look. I taught university calculus for umpty-ump years and am currently copy editing a collection of calculus questions on the Khan Academy website.

All the best,

Fred Chapman Math Professor

[kcrisman]

Hi! Thanks for your feedback. Yeah, that entire collection of resources is not in such great shape, and (to be frank) never was. I believe it was a summer project by an undergraduate quite a few years ago. I'm not even exactly sure when it was incorporated into the website...

So far it wasn't maybe worth the effort to do anything with it, but if people are actually finding and/or using it that is a different matter. Do you think that the content is worth preserving? It was an experiment in trying to make a Sage-ified calc tutorial, as I recall, but Sage and a lot of other things have moved rapidly in the intervening time. (For instance, it's sort of silly that it doesn't use the Sage cell server technology.)

If so, I think we would welcome pull requests to improve it - most of us spend our limited time trying to improve core Sage functionality or "official" documentation, but would be happy to review changes to it. Or, especially given that there are other tutorials of similar material out there, perhaps it's time to retire this.

[fwchapman]

Hi Karl,

Thanks for your speedy reply! The online math articles I've seen elsewhere vary considerably in quality. What strikes me as unique about the Sage calculus tutorial is that it incorporates a computer algebra system into the presentation, which encourages students to experiment on their own. I think the approach has genuine value. The two mistakes I found pertain to some finer technical points of differential calculus (nonexistence of limits and derivatives). Otherwise, it seemed sound to me.

I'm new to both the Sage Math project and GitHub. I have no idea how GitHub works. What's a "pull request"?

If I may tell you a little about myself, I earned my Ph.D. in applied math with an emphasis in computer algebra when I was a member of the Symbolic Computation Group at the University of Waterloo. I'm especially interested in hybrid symbolic-numeric methods for the interpolation and approximation of multivariate functions, which has lots of practical applications. My thesis supervisor was Keith Geddes, one of the co-inventors of Maple. I've been an avid Maple user for over twenty years and am very keen on the potential of computer algebra to empower both mathematical research and teaching. Computer algebra deserves to be much more widely known and much more heavily used!

Nice to meet you,

Fred

[fwchapman]

P.S. In general, it's not easy to produce online math articles of high quality. I worked for Sapling Learning (a division of Macmillan New Ventures) as an author/reviewer of e-books on high school math, and I'm currently the math copy editor for the Khan Academy. I can tell you from my own experience that the process usually involves both peer review and copy editing.

I don't know whether the Sage Math project has a budget for such things, but if you do, I would be delighted to review and copy edit the calculus tutorial and any other material you feel could benefit from the attention of a professional writer/reviewer/editor. Please let me know if I can be of service.