Is the Mean Value Theorem interesting to engineers, scientists, and others? 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!
 A: 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. 
A: 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.
A: 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.   
A: 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. 
A: 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. 
A: You may want to consider the Race Track Principle. For example, see the Wikipedia page for it.
