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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#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives

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(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[gauravnainwal]

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[monochrm]

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[johnsu]

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 😢

[aaravjayalwal]

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

[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.

[VictorHugo-xnjz]

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.

[rafaelschipiura]

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.