Real Analysis 56 | Proof of the Fundamental Theorem of Calculus

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.
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[@gauravnainwal5026]

I want to say this to you-- your channel is a goldmine. You make concise videos on essential topics of a subject. This saves so much time for those who want to learn the essentials without having to watch hours long lectures. I am so so grateful for this service of yours. Request: Please make series on Linear Algebra (with advanced topics), and Multivariable Analysis (with manifold stuff, and classical theorems of vector calculus). That will be so awesome! Thank you again.

[@brightsideofmaths]

Thank you very much!
Linear Algebra and multivarible calculus are in production, see here: tbsom.de/s/la
Advance topics will come :)

[@monochr0m]

Ich mache grade meine Promotion in Physik und bin längst mit allen Mathematik Kursen durch, aber die Videos machen trotzdem immer Spaß :D

[@johnsu9949]

I've got queries in the proof of the second fundamental theorem of calculus:
1) Fo(a) = 0 implies the theorem holds for Fo because on the LHS the integral is indeed just 0 and on the RHS we have F(a) - F(a) = 0 = LHS right?
2) In the last step Fo(b) - Fo(a) = integral from a to b f(t) dt because we already proved that the theorem works for Fo? I got quite confused here because we proved it works purely by using Fo(a) = 0 and nothing about Fo(b) :(
3) Lastly, how can we relate area to slope through FTC I can't seem to draw any conlcusion 😢

[@brightsideofmaths]

Such long questions are best discussed in my community forum.

[@synaestheziac]

What is the justification for the step “there exists mu such that…” around @5:30? It’s intuitively clear, but at the moment I can’t think of which specific theorem allows us to make that step.

[@brightsideofmaths]

Just define mu by the quotient of both integrals. That's all :)

[@mastershooter64]

another awesome video :D

[@brightsideofmaths]

Thanks again!

[@VictorHugo-xn9jz]

I can't understand the part where you transform [ int (x) to (x + h) of f(t) dt ] into [ f(x hat) times h ] with the mean value theorem of integration. Shouldn't it be [ f(t hat) times h ] ? It's at 8:05.

[@brightsideofmaths]

It's the means value theorem. How you call the intermediate point does not matter. It's just a name.

[@aaravjayalwal3416]

i am studying in iitb, and my teacher failed to teach a topic, that could be taught so easily

[@rafaelschipiura9865]

I think giving an example of what the fundamental theorem of calculus doesn't prove would be beneficial here. I'm thinking about the integral of the Normal density.

[@brightsideofmaths]

This video is about the proof of the theorem :)

[@khalidmohammadjama]

Ander great video

[@brightsideofmaths]

Thanks!