As no one seems to have mentioned it, students in the specialties listed in the OP frequently need to understand the behaviour of solutions of ordinary differential equations. I haven't taught such a course recently enough to crank out a list of places where it comes up, but abstractly I feel like there are a "zillion" situations where you end up using MVT to justify some basic intuition about solutions of ODE. For example, "The vector field points down and left over here, so once a solution enters region A, it can never leave", or certain kinds of stability analysis (e.g. when linear stability analysis fails).