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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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    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.2012-07-24

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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.2012-07-24
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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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    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...2012-07-24
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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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    I strongly agree with Robert Israel in principle. Howe$v$er, when I was learning calculus for the first time, I distinctly remember finding Rolle's theorem simple and the Mean Value $T$heorem confusing. Kind of odd, but that was indeed my first impression.2012-07-26
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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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    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 MVT2012-07-30
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You may want to consider the Race Track Principle. For example, see the Wikipedia page for it.

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    @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.2012-07-24