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First I'll apologize for necrobumping this thread after well over a year, but, it seemed to me that there is a simple answer to the OP's question which was never directly addressed.

Students are interested in existence and uniqueness of solutions to equations. They like the fact that $x^2 = -1$ has no solutions (in the real numbers) but $x^2=1$ has two solutions. They like the fact that $2x + 5y = 1$ has infinitely many solutions (in the $(x,y)$-coordinate plane).

I like to ask students: what are the solutions of the equation $y'=1$? The format of the answer must be a function $y=f(x)$. And of course they can all come up with infinitely many different solutions.

Then I ask them: are there any other solutions? If not, why not? They don't have to be "interested" in abstract mathematics and proofs and rigor in order to appreciate this question and to take interest when I show them why constant the functions $x + C$ are, in fact, all the solutions. And then one easily continues on to solutions of equations like $y' =$ any other of your favorite functions.

So yes, MVT can and should be taught in calculus.

Students are interested in existence and uniqueness of solutions to equations. They like the fact that $x^2 = -1$ has no solutions (in the real numbers) but $x^2=1$ has two solutions. They like the fact that $2x + 5y = 1$ has infinitely many solutions (in the $(x,y)$-coordinate plane).
I like to ask students: what are the solutions of the equation $y'=1$? The format of the answer must be a function $y=f(x)$. And of course they can all come up with infinitely many different solutions.
Then I ask them: are there any other solutions? If not, why not? They don't have to be "interested" in abstract mathematics and proofs and rigor in order to appreciate this question and to take interest when I show them why constant functions are, in fact, all the solutions. And then one easily continues on to solutions of equations like $y' =$ any other of your favorite functions.