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?