sean-fitzpatrick
commented
2 years ago I think these would be a good addition: my perspective is that the MVT is primarily a theoretical result, and the main reasons to learn it in Calc 1 are to nail down some of the facts we use later on:
- positive/negative derivative implies increasing/decreasing
- positive/negative second derivative implies concave up/down
- Fundamental Theorem of Calculus (part II)
- the arc length formula
I don't assign the exercises currently in the book that ask students to "find the value of c" whose existence is predicted by the MVT. The one application that we discuss in class is aircraft patrol for cars speeding on the highway.
Ultimately, this one is up to @APEXCalculus. Greg, what do you think of these changes?
APEXCalculus
@efrenruiz and I have some modifications/additions to the MVT section to propose.
First, we have written up the results of Sean's two videos on the corollaries of the MVT as a theorem, and placed the videos and write ups into proofs: Theorem 3.2.11: Corollaries of the Mean Value Theorem
Second, we have added a new example that applies the corollary about equivalent derivatives. Example 3.2.14: Applying a Mean Value Theorem Corollary The idea here is that showing that g(x)-h(x)=C directly would require plenty of Fun with Trig Substitutions. Conversely, the algebra showing equivalence of derivatives is pretty straightforward.
Sean's final example proving that sin(x) is Lipschitz seemed a bit abstract to us, so we wanted to do something more applied here, and "objects with the same velocity function have positions which differ by a constant" seems like a good one to us.
The second corollary is basically the same result as Theorem 5.1.4: Antiderivative Forms, so it may be worth adding something there mentioning that fact.
Any thoughts, suggestions, corrections about this new content? I can prepare a pull request if we like these changes.