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…….
Best Answer
My view is that there are essentially two strands in a first calculus course.
The first is not really concerned with a rigorous presentation; rather it tries to get the main ideas, their interrelations, and uses across.
The second is concerned with the technicalities, showing how abstract mathematics can lead to very useful, interesting, and important results.
This means that we are really working with two different definitions of the derivative. The first is to draw the tangent line and measure its slope. The second is to compute a certain limit. To be sure, the limit is motivated by the tangent approach, but no attempt is ever made to show that the two approaches give the same answer (indeed this can't be proved using the usual definitions, since ultimately the definition of tangent line is based on the derivative).
The MVT is the basis for all proofs that geometric intuition about slopes of tangent lines holds for the limit definition. That is, the metamathematical content of the MVT is that the intuition definition matches the formal definition.
When I realized this, I decided that this point was so subtle that I either have to make a big point of explaining the question or else drop it. This choice varies from class to class.
EDIT: You use the MVT to show that positive derivative corresponds to increasing function. This is obvious from the intuitive point of view, but not from the formal point of view.