Question 1

the first proof... thought a little about it and came up w/ an possible other way to solve it:

take it from the Riemann definition (right sum approx) by
b=x (upper bound)
d/dx(int[a;b]f(x)dx = d/dx(f(a+dx)dx+f(a+2dx)dx...n times...+f(b)dx)
as f(a+dx)dx+f(a+2dx)dx... etc are not x-dependent, the derivatives of all go to 0, so
d/dx(int[a;b]f(x)dx=d/dx(f(b)dx)=f(b)
seems reasonable for me, as it all relies on x being both, the upper bound (b) and the variable, as if i just put x=b, considering b as "the max{x}" in the defined interval [a;b]; besides it avoids all the f(t)dt mess that seems confusing.

so, i want to discuss this, even (probably), why is it wrong (didn't saw it anywhere), but seems quite intuitive for me, for dF/dx is just the last d(right area)=f(b)dx/dx + the local linearity stuff.

Accepted Answer

Looks like you did a good job there, but I still can't understand what you did because it's hard to understand it when it's written in that way. Can you, or anyone else, please write it down on a piece of paper and upload a photo of it on the internet so everybody can comprehend what's going on?
Thank you for your addition.

Question 2

In the mean value theorem for integrals proof Sal uses the fundamental theorem of calculus and here in the first part he uses the mean value theorem. Isn't that a circular argument because it says that MVT is true from FTC and FTC is true from MVT?

Accepted Answer

Yes.
But MVT can be proved independently from FTC. In fact, the most rigorous proofs are a bunch of 𝜖-𝛿 style computations, not a corollary from some other strong theorem.

Question 3

I understood the first part of the theorem this way. Integrating from a to x(between a and b) is like moving x-pivot from a to b. Along the way x -pivot moves, we add the y- value that we get from the function f(t) (Since F(t) is an integration.) So the F(t)'s derivative of the moment we enter the arbitrary x into f(t) must be +f(x), because F(x) is adding f(x) at that moment, and the derivative describes the rate of change at that moment. Is my intuition correct?

Accepted Answer

If i understood well, you might be thinking that the function itself pivotes like in this simulation: https://bit.ly/2JK8toj (Desmos)

Tell my if it wasn't usefull  :D

Question 4

I'm confused why we needed to prove that d/dx of F is f, when F was defined during creation as the area under f.  When we (or Leibnitz I suppose I should say) specified the "width" as dx, doesn't that in itself say that the derivative of F, (the rate at which the area, F, is changing at x) is given by f(x)? Stated another way, by specifying the width as dx, which is negligible in the area calculation by design, doesn't our area change by f(x)?  Perhaps this is not proof because it's recursive?

Accepted Answer

That's true, but it isn't a rigorous proof. It's good that you have the intuition for it, though!

Question 5

at 5:43 in the 1st video, the m.v.t. of def. integrals hasn't been taught so far.

Accepted Answer

You're right. I think it's taught in Unit 8. I'd recommend getting back to the proof once you learn the theorem. But, if you want a shortened version of it to better grasp the proof here, it basically says that for a function f(x) defined on the interval [a,b], there exists some c such that the value of the function at that point equals the average value of the function. This should make intuitive sense because if the function takes on a range of values, at one point, it must take on the average value too. Observe the striking similarity with the MVT for derivatives, where the derivative at a point was equal to the average rate of change of the function.

Question 6

What is that dt and why does it matter?

Accepted Answer

this is the formal explanation of dx
https://en.wikipedia.org/wiki/Integral#Meaning_of_the_symbol_dx

to be simple, it's the deferential of x and x + delta x

Question 7

Would there be any critical errors in this analogy? (I came up with this to understand those theorems more intuitively.)

f = F'
f = productivity (speed in producing value)
F = wealth (accumulated value) 
C = Inherited wealth
If productivity is static, wealth increases linearly
if productivity increases, wealth increases exponentially
-> INVEST IN EDUCATION!

Accepted Answer

The basic idea is correct. Wealth would not necessarily increase exponentially, but it would certainly increase at a much faster rate.

Question 8

What about indefinite integrals? Those don't correspond to any area of a curve, yet they equal the anti-derivative of the function. How does that work?

Accepted Answer

The indefinite integral is basically to have a general antidriative. Since the antiderivative is the integral. The problem is the derivative of f(x) is the same as the derivative of f(x)+1 or plus or minus any constant. So the general form as I called it was given a C to represent you could take the derivative of this where C is any real number, and you would get the original function.

Let me know if that didn't help.

Let me know if that didn't help.

Question 9

Where in the AP calc BC playlist is the video for the MVT of definite integrals? I've gone through all the material before this and I haven't seen it yet, but Sal talks about it in the first video here.

Accepted Answer

This is discussed in Unit 8 of AB.
https://www.khanacademy.org/math/ap-calculus-ab/ab-applications-of-integration-new/ab-8-1/v/mean-value-theorem-integrals

Question 10

is an integral basically an antiderivative?

Accepted Answer

yup cuz if you take the derivative of the integral of f(x), you get f(x)

Course: AP®︎/College Calculus AB > Unit 6

Proof of fundamental theorem of calculus

First, we prove the first part of the theorem.

Next, we offer some intuition into the correctness of the second part.

Finally, we prove the second part of the theorem based on the first part.

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