Question

I have a question about proof on FTC1, lecture 20, 36:48 (http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-3-the-definite-integral-and-its-applications/part-a-definition-of-the-definite-integral-and-first-fundamental-theorem/session-50-combining-the-fundamental-theorem-and-the-mean-value-theorem/)

Answer 1

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-3-the-definite-integral-and-its-applications/part-a-definition-of-the-definite-integral-and-first-fundamental-theorem/session-50-combining-the-fundamental-theorem-and-the-mean-value-theorem/

Answer 2

FTC1 states that if F'(x)=f(x)
\[\int\limits_{a}^{b}f(x)dx=F'(b)-F'(a)\]
Here is a summary of how the proof goes.
define
\[G(x)=\int\limits_{a}^{x}f(t)dt\]
According to FTC2, G'(x)=f(x)
Then, since F'(x) = f(x) by assumption, G'(x)=F'(x)
Then, it follows that F(x)=G(x)+c (step *)
Then, the proof goes on to substitute F(b)-F(a) with G(x)+c (and the rest is irrelevant).

My question is in the step *, Prof Jerison mentioned that it follows due to MVT. I completely fail to see how MVT is used here. I can understand that fact without MVT (pretty much obvious after going thru calculus this far). But how is MVT used?

Sorry for making fuss about small things here but I really want to have a grasp of MVT

Answer 3

By the way, the version of MVT I know (and the one covered in the course) is this
f(b)-f(a)/b-a = f'(c) for some c, a<=c<=b

Answer 4

In lecture 14, Prof. Jerison showed how the version of MVT you stated,\[\frac{ f(b)-f(a) }{ b-a }=f'(c)\]can be solved for f(b):\[f(b)=f(a)+f'(c)(b-a)\]and mentioned that this is another customary form of MVT. He then went on to show how this version can be used to prove some facts we've used in graphing: (1) if f' is positive, f is increasing; (2) if f' is negative, f is decreasing, and (3) if f' is 0, f is constant. As to the last one, we can easily see that if f' is 0, f'(c) is zero, so we get f(b) = f(a). Further, this will be true for any pair of points a and b that we choose, so f has to be constant.

As he wrote (3) on the board, Prof. Jerison remarked that it seems like the simplest one but is actually "the key to everything." He didn't explain what he meant by that, but it becomes clear later. In lecture 15 he uses this fact to establish that antiderivatives are unique up to a constant: If F'(x) = f(x) and G'(x) = f(x), then F(x) = G(x)+c. And this is the fact being used in step * of the proof of FTC1. FTC is the most important fact in all of calculus, and it rests on this corollary of MVT which is "the key to everything."

Answer 5

Yes, very clear and make perfect sense. So, the "key to everything" is the fact that if derivative is zero, function is a const. What amazes me is MVT is so trivial and yet so difficult to apply. The triviality is to such an extreme that it masks all important subtleties of the theorem. I did go thru both lecture 14 and 15 carefully but the fact completely failed to register in my mind.

Another unrelated question. Though MVT indeed proves f' zero imples f is const. It comes even without MVT. Slope zero can only mean horizontal lines which means constants. Do you have any comment on why we give so much credit to MVT for this fact?

Answer 6

In the Simmons calculus book used in at least some calculus classes at MIT, the author makes the following remark about MVT and two other bits of calculus theory (the extreme value theorem and the intermediate value theorem): "Part of the difficulty they cause for beginning students lies in the effort required to doubt them in the face of their compelling believability." Well said!

Your statement, "Slope zero can only mean horizontal lines which means constants," does not provide an alternate path to the conclusion we seek, eliminating the need for MVT. The reason is that without realizing it, you're *applying* MVT when you make that statement. MVT is so intuitively obvious that you're able to reach conclusions that flow from it without even realizing you're using it.

Answer 7

It is a great perspective. Thanks.