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I am trying to decide whether to include whether to include the Mean Value Theorem in a calculus course I will be teaching. I am sort of leaning away from it, in light of the interesting discussion found here on MathOverflow (see especially the answer from Jeff Strom). I think it is very possible to teach what the Mean Value Theorem says, and assign some canned problems (for the function $f(x) = x^2 - x$, find a point $a$ that satisfies the conclusion of the Mean Value Theorem on the interval $[1, 4]$), but I question how interesting this is. The real interest of MVT is that it allows you to turn geometric intuition into proofs, and my course will unfortunately not do proofs.

However, my specific question: Is MVT also interesting for other reasons? Do courses in engineering, economics, science, or any other discipline use it (other than to prove mathematical theorems)? Is the canned problem above more interesting than I have given it credit for?

Essentially -- is there any reason to include it, other than those I can anticipate as a mathematician?

Thank you!

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The Mean Value Theorem is very relevant to error estimates. However, the formulas can be mentioned without detailed MVT justification. – André Nicolas Jul 24 '12 at 19:37
I think that although the formulas can be introduced without proof, the proof of MVT really communicates the concept in a way that just staring at formulas cannot. – Emily Jul 24 '12 at 19:40

The Mean Value Theorem is the starting point for a chain of results leading to Taylor's theorem, with the associated estimates of various remainder terms. As far as I am concerned, the most important part of Taylor's theorem is being able to obtain accurate error estimates. This should be of interest to users of mathematics in a ``practical" context. The Mean Value theorem is also relevant to estimates for the rate of convergence of Newton's method. So there are numerous reasons why it could/should be of interest to an audience other than one of mathematical specialists, and its later uses could at least be outlined without going through a completely rigorous development.

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That is a very good answer (which I'm a little embarrassed not to have remembered!) +1 sir. – Frank Thorne Jul 24 '12 at 21:19

Yes, there are many applications. For instance, Broyden's method is an alternative root finding algorithm to Newton's method when it is unfeasible or impossible to compute the derivative of a function at a single point. Broyden's method works because of the mean value theorem.

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Note that Broyden's method is in fact a multivariate generalization of the usual secant method, which is in itself a modification of Newton's method where one uses secants ;) instead of tangents... – J. M. Jul 24 '12 at 21:46
I've always considered the secant method to just be, in essence, Broyden's method in one variable. To-may-to, to-mah-to? – Emily Jul 24 '12 at 21:48
Clearly. :) I was thinking that for most people, it might be easier for them to imagine the secant method first (since one dimension is "easy"), and then generalize accordingly... – J. M. Jul 24 '12 at 21:52

The MVT is what tells you that a function whose derivative is positive on an interval is increasing there. As such, it is extremely important e.g. for curve-sketching. Of course, the instructor in a non-rigourous calculus course can easily hand-wave past this point without mentioning the MVT. And the engineers, scientists and economists who use this all the time will probably not realize the connection to the MVT.

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My textbook made a big deal of this. In my mind (and as you suggest) this is one reason I was considering skipping MVT. The derivative is the slope, and if the slope is positive, then the graph is increasing. This is "obvious", and yet the book makes a big deal about this. I imagine the reason for this is lost on all but the strongest students. – Frank Thorne Jul 24 '12 at 21:13
What is a theorem good for? It can be used to prove other theorems --- but your students aren't interested in doing that. It might tell you how to calculate something --- but the MVT is an existence theorem, which doesn't tell you how to find $\xi$ where $f'(\xi) = (f(a)-f(b))/(b-a)$. It might tell you something that you wouldn't have guessed otherwise --- but the MVT is pretty much intuitively obvious. – Robert Israel Jul 24 '12 at 22:53
@FrankThorne: in the MO-post you mentioned, Roy Smith argued that you can do this using only Rolle's theorem. – wildildildlife Jul 24 '12 at 23:03
@wildildildlife: If you're going to talk about Rolle's theorem, you might as well talk about the MVT. I don't see the point of only doing a special case, when the more general result is no harder to state or understand. – Robert Israel Jul 24 '12 at 23:49
I think I agree, although I don't have much teaching experience... I'd say the statement of Rolle is slightly easier to understand and remember than MVT. – wildildildlife Jul 25 '12 at 11:39

Every physicist's favourite theorem is - or should be - Taylor's theorem. I would guess that not every physicist knows that this is a theorem: it's just that useful method that allows you to introduce simplifying formulas and e.g. solve perturbation problems. But Taylor's theorem is just a neat inductive application of the MVT. So maybe this is a good reason for teaching the MVT to physicists and engineers - it provides justification (and perhaps some intuition) for one of their favourite mathematical tools.

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What about the fact that if a student has taken a basic calculus course, he/she is expected to know the MVT? I think the MVT is a great theorem to introduce geometric intuition as well.

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Who expects them to know the MVT, and more importantly: for what purpose? – wildildildlife Jul 24 '12 at 22:56
It was a selection question when I was being interviewed for my master's program (long story). What I meant by my comment was that, there are some theorems which are considered basic in Calculus and anyone who has taken a Calculus course should know this theorem AND the name of the theorem! – Anu Jul 25 '12 at 19:50
This was a selection question in an interview for a non-mathematics master's program?! – wildildildlife Jul 27 '12 at 21:09
Applied mathematics master's program (well PhD program really). I have a degree in Mathematics, but now work in Geophysics and a few of the algorithms I studied used the concept of MVT or a concept analogous to MVT – Anu Jul 30 '12 at 15:29

You may want to consider the Race Track Principle. For example, see the Wikipedia page for it.

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"If a horse named Franky Fleetfeet always runs faster than a horse named Greg Gooseleg, then if Frank and Greg start a race from the same place and the same time, then Frank will win." Unfortunately, I don't think my students would appreciate why one would make a substantial effort to justify this. – Frank Thorne Jul 24 '12 at 21:18
@Frank Thorne: My intent was to consider the possibility of letting this principle replace MVT in classroom justifications where one would ordinarily cite MVT, not that one attempt to prove this principle. I have not personally done this, but I've heard others swear by it. For example, I believe it played a prominent role in Jerry Uhl's calculus program at University of Illinois. – Dave L. Renfro Jul 24 '12 at 21:59

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