In my opinion (I have been a TA for 4 years), MVTs should be given in the following forms: MVT-Integrals: [Diestel] (Soft version) $f$ is B-integrable. Then for any $|E|>0,$ $\frac{\int_E fd\mu}{|E|}\in \bar{co}(f(E))$

MVT and MVT-Cauchy: As given by Apostol in his book Calc. vol I with convexity

Function theory heavily depends on these theorems+FTC. MVTs are related to, for instance, complex analysis which is a must for an engineer. Definitions of holomorphic functions and (sub)harmonic functions contain both volume and surface integral averages. Details can be found in [Krantz]. Also Taylor series(both real,complex analytic cases) uses these theorems. I should also mention the `''convexity'' which is very fundamental.