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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.

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1

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.