TITLE: The role of the mean value theorem (MVT) in first-year calculus
QUESTION [25 upvotes]: Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that interval. Most of the related exercises that books list are either about examples related to the necessity of the hypothesis of the theorem (i.e. just checking it) or about proving theoretical facts (e.g. general properties of functions, or derivatives of functions, inequalities, Taylor, ...) I can imagine a text that doesn't require the student to understand logical rigor, but that instructs them on the role of the MVT in the theory.
It is a fact that many times we fail in making students able to use MVT in these kind problems. 
Assume that we are considering a first introduction to calculus for students that mostly will use it in application, students that will work in Biology, Engineering, Chemistry,... 
Is it possible to remove MVT from the program and get a consistent exposition of the rest of the results and techniques in Calculus?
How does eliminating MVT from the curriculum affect students from specialties exemplified above (i.e. In what ways is MVT used in further studies that they have to take or that they need)?
Can the role of MVT be replaced by a more easy to use/easy to grasp result?
Are there other uses (exercises) more suitable for the uses that MVT has for students of this specialties?  
Note added later: "Franklin" appears to have altered the meaning of the question with his later edits.  I'll say more on this soon.......

REPLY [4 votes]: 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 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.