Real Analysis | The Fundamental Theorem of Calculus

We prove both parts of the Fundamental Theorem of Calculus.

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[goodplacetostop]

22:53

Michael: *uploads a video outside of regular schedule*
Me: _I’ve been tricked, I’ve been backstabbed and I’ve been quite possibly, bamboozled_

[hybmnzz]

18:10
I think you should define M as the supremum of the function on the interval [a, b] instead of [x, y].
Because for Lipschitz continuity you need to show there exists an M such that for any x, y the inequality follows.
Not that for any x, y there exists an M. Our choice of M should NOT depend on x, y.

(In fact you could erroneously prove any function is Lipschitz continuous if you do this subtle error of choosing M afterwards)

[stefanveskovic]

I feel very happy that YouTube recommended your video. Thank you for the best content

[samisiddiqi]

I really love this course. We should learn this before Calculus, or at least combine this, Precalculus, and Calculus, into one subject.

[mathwithjanine]

Awesome video! You are a great teacher!

[AltinoSantos]

Thank you very much for your all videos. They help teachers and students a lot. That is an elegant proof of FTC based on MVT. I enjoy your videos, mainly those about calculus. Thanks again.

[jonaskoelker]

At 12:20 Michael constructs a sequence of partitions P_n such that U(f, P_n) - L(f, P_n) -> 0 as n -> infinity.

I'm not sure it follows from this that U(f, P_n) is convergent on its own, and likewise for L(f, P_n).

To fix this, let Q_n = P_1 u ... u P_n. Then U(f, Q_{n+1}) <= U(f, Q_n) for all n since every refinement is an improvement, and the upper limits are bounded below by every lower limit. So the sequence is monotonic and bounded and thus convergent. Likewise for L.

[luissaybe]

Great content, best channel to learn about analysis

[arvindsrinivasan]

I’ve been wanting to see the proof for the generalization of this (Stokes theorem) using differential forms... I know it’s not really real analysis

[JDMathematicsAndDataScience]

I appreciate these videos. Do you plan on continuing through to Stone-Weierstrass and other topics?

[arturwiadrowski]

Professor, I really enjoy your videos. Could you please do one on some identities involving the trilogarithm, for instance about the value it takes at 1/2 or at 1/goldenratio^2?

[punditgi]

Capstone video. We have arrived at the Promised Land! 😇

[TechToppers]

22:53
When you are earlier than the Guy named, *Good Place to Stop* !
Hey, Michael Sir! Did you screw up again on scheduling stuff?? Or, this was planned so that *Good Place to Stop* comments after me.

[MsOops]

I think part 1 of FTC is supposed to be part 2 of FTC and vice versa. At least that's how it's shown on other sites

[ahorribleperson]

Is this meant to be visable? (11/19/2020)

[CM_France]

Hi,

For fun:

1 "all the way up to",

4 "great",
1 "ok, great",

1 "next what we want to do is",

3 "so let's go ahead and",
1 "now let's go ahead and",
3 "so let's may be go ahead and".

[aaronhunt]

Yeah, the timing of this video is almost spooky. I was reviewing this earlier today 😆... What are the odds???? .... hmm 🤔

[solcarzemog]

NOBODY on Youtube proves the fundamental theorem sequentially. They all prove it backwards, meaning, they use the result to prove the hypothesis.

[grzechu]

can you put your also Real Analysis videos in one playlist already?