1. Introduction
When I first learned the Mean Value Theorem and the Intermediate Value Theorem, I thought they were both intuitively obvious and utterly useless. In one of my courses in analysis, I was struck when, after proving the Mean Value Theorem, my instructor said that all of calculus was downhill from there. But it was a case of not being able to see the forest for the trees, and I missed the big picture.
I have since come to realize that almost every major (and often, minor) result of calculus is a direct and immediate consequence of the Mean Value Theorem and the Intermediate Value Theorem. In this talk, we will focus on the forest, the big picture, and see the Mean Value Theorem for what it really is: the true Fundamental Theorem of Calculus.
2. Warming up with standard results
Let's remind ourselves of the major players. Suppose
Then there exists a
Then there exists a
Both are best understood by drawing pictures. The IVT is visualized by accepting that we can't get from one point to another without crossing a line. The MVT is visualized by accepting that we can't get from one point to another without going in a particular direction... as long as our movement makes sense. An aside for readers: notice that we don't actually require the function to be continuously differentiable, but instead only differentiable. It is intuition defying that the MVT works even with noncontinuously differentiable functions, so long as they can be differentiated.
In standard calculus courses, what sort of problems might we be asked to solve that has to do with these theorems? It is peculiar that the IVT seems relegated to showing that there are roots, and the MVT is relegated to demonstrating speed traps and, coincidentally, guaranteeing that there are no roots. For example, prototypical question number one:
Example 1 Show that the functionhas exactly one root, and that it occurs in .
Proof: Notice that
This is perhaps somewhat useful, but totally uninspired. It feels more like a trick then a monumentally important pair of theorems from calculus. You'll notice that we actually used the following, based on the MVT.
If
Similarly, we see that if
These are technically useful, but utterly uninspired. It seems obvious that having positive derivative means that the function is increasing. So what have we gained here? Answer: not a whole lot
There is another technically useful, classical application. If
In other words, this says that any two functions with the same derivative are the same, up to a constant. Thought of differently, it says that antidifferentiation is unique up to addition by constants. This is actually very important, but it still doesn't feel particularly interesting.
There is something that all of these examples do have that hints at a sort of power behind the MVT. We have a sort of 'local-global duality,' in a manner of speaking. Just as in a piece of music, there are certain motifs that recur throughout mathematics. One of these is an interplay between global properties and local properties.
For example, being an extremum is global property. But having zero derivative is a local property. Within the Mean Value Theorem, we have a guarantee of a local point satisfying a non-local condition, a local point of a certain derivative that happens to be the slope a secant line from other points.
The last pair of lemmas demonstrate a striking interplay between local and global. Having the same derivative at each point is global (or lots and lots of local properties). Having the same value, up to a constant, is the same. But the proof works by showing that there would be a single local point that breaks the rule. Let's make this more striking by mentioning a near-miss, the Devil's Staircase.
Take the unit interval
Aside: This function is a great counterexample and can be constructed in better ways that explicitly demonstrate some of its great properties. For instance, it's possible to show it is the uniform limit of a sequence of continuous functions, and is therefore continuous. This was a deep type of problem that affected a lot of the classical mathematicians we associate with the formation of calculus, but this type of problem is completely excluded from introductory calculus courses. Perhaps for good reason, as it's intuition breaking? Or perhaps that's the point? It's certainly quite complicated and demands much higher mathematical maturity, for better or worse.
Consider this the end of the first third of the talk.
3. Fundamental Theorem of Calculus
The Fundamental Theorem(s) of calculus are extreme examples of local-global duality. You might recall that they are actually two statements. Right now, let us define the symbol
For
- If
, then . - If
is any function such that , then .
Let's parse these claims.
I also claim that the first is the IVT, and the second is the MVT. Let's prove it, right here, right now, no holds barred.
Our method of proof will be the "just do it'' method. For the first, let's differentiate
Further, as
For the second, we have to be a little bit more careful. We call a function integrable if its Riemann sums converge, in which case we say that the limit of the Riemann sums is the value of the integral. By this, I mean
Here's the bit of magic. For ease, suppose
(For a longer explanation of this, including pictures, see my other note)
Aside: to those unconvinced of the actual convergence here, you're right. We didn't show it. But this is what the additional written note is for. In the next subsection, not included in the talk, we actually handle the convergence
3.1. Convergence of the Riemann sum
When we have
Then on each partition segment, we know that
4. Taylor's Theorem
I hope to have convinced you that the MVT is deceptively useful, and that calculus is deceptively straightforward. It's just that so much of the signal gets lost in the noise. Before I go on, I'd like to mention another note I wrote, also with beautiful gifs that I'm exceptionally proud of. See An Intuitive Overview of Taylor Series.
Before we move on to the other heavy hitting theorems, let's refresh ourselves with a question. We know that on
The answer is no. Considering
To end, there are many possibilities. Nowhere near all of them fit into the talk, and I'll choose to talk about whichever one seems of greatest interest at the time. In no particular order, the MVT also gives
- l'Hopital's Rule
- Cauchy's Mean Value Theorem
- Taylor's Theorem
- A higher order MVT
In fact, these four things are incredibly interconnected.
Suppose
Aside: It is an interesting question to ask why we demand the limit on the right to exist. This has to do with the partial converse above, which doesn't hold. The problem is that even as
With l'Hopital's rule, verifying one form of Taylor's Theorem is very easy.
In particular, call the
Some would find this sort of direction strange, as a different, common approach is to prove l'Hopital's Rule from Taylor's Theorem. And one way of proving Taylor's Theorem is from a stronger Mean Value Theorem, called Cauchy's Mean Value Theorem.
I must take an aside and mention that I have always disliked Cauchy's Mean Value Theorem. It is not intuitive to me. It is very powerful, and can do everything of the Mean Value Theorem and more. But while the regular Mean Value Theorem is so incredibly clear and geometric, the Cauchy Mean Value Theorem is not. In some sense, it is the Mean Value Theorem as in Riemann-Stieltjes integration. I will mention something about this at the end, in the references.
Returning to the topic at hand.
If
As is so often, the proof is to choose a clever function and apply the normal Mean Value Theorem. For us right now, the clever function is
Let us provide an alternative proof of Taylor's Theorem. In order to apply Cauchy's Mean Value Theorem, you must cleverly choose the functions
Fix an
For that matter, we could immediately use Cauchy's Mean Value Theorem to provide another proof of l'Hopital's Rule. We'll prove it in the
There is a little known generalization of the Mean Value Theorem to higher derivatives (Gowers's Blog is the only reference the author has seen). A reference for the rest of this talk is a currently unpublished note of mine (which is in the process of being submitted to an expository journal), which I largely steal this from.
Similar to the ordinary Mean Value Theorem, we begin with a generalized Rolle's Theorem:
Let
The proof is almost immediate from Rolle's Theorem. The idea is to use Rolle's Theorem iteratively, each time moving up in the order of the derivative. Since
With Rolle's Theorem, one proves the Mean Value Theorem for a general differentiable
We will follow the same idea. Take the degree
The degree
But we can say more. As
So just as the Mean Value Theorem is the first case of Taylor's Theorem, the general form of Taylor's Theorem is a general order Mean Value Theorem.
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Comments (2)
2014-11-06 Tim McGrath
As much as I appreciate the connections and the insights, it would take me weeks or months to fully grasp the concepts. A lot of calculus requires a great deal of effort. Granted, it was worth the investment to learn the expansion of sin (x), but other curious items in math, such as Pascal's Triangle, Euler's Identity, and the Basel Problem are just as mysterious and intriguing but far more accessible to the layman,
2016-09-27 Deepak Suwalka
Thanks, It's a good blog. I like the way you have described the mean value theorem. it's really appreciable. but if you proved some examples and problems than it will be a great post.