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# Proof of fundamental theorem of calculus

The first part of the fundamental theorem of calculus tells us that if we define 𝘍(𝘹) to be the definite integral of function ƒ from some constant 𝘢 to 𝘹, then 𝘍 is an antiderivative of ƒ. In other words, 𝘍'(𝘹)=ƒ(𝘹). See why this is so. Created by Sal Khan.

## Want to join the conversation?

• What is squeeze theorem?
I did read up, but I did not understand much of it....

<edited> I watched the video and I understand it now. I'm leaving this here (and not deleting it) just for the record.
Thanks for all the help everyone - particularly @TheCatofSauron ! •   I cannot explain this as well as the video can (It is in the limits section of calculus), but I will try anyway. Suppose I say that I have always been taller then or the sam height as my sister, but I've always been shorter than or the same height as my brother. If I then say that three months ago, my sister was 5' 11" and my brother was 5' 11", then my height must be in squeezed in between them, so my height was also 5' 11".
This helps in calculus if you know a variable, let's call it x, is less or equal to than y, but greater than or equal to z. Then if you find out that y and z are both equal to k, then x must be equal to k as well.
Of course, in calculus, it won't be that simple. It will involve sines and cosines and stuff, but I hope this helps! :)
• ; Is there a video on the mean value theorem for definite integrals? • At , Sal uses the Mean Value Theorem as a tool to prove the Fundamental Theorem of Calculus (F.T.C); but, isn't the F.T.C needed to demonstrate the MVT for integrals? If not, could someone please show me a proof of it that doesn't use the FTC?
Thank you very much! • The rough proof is something like this:
let g(t) be a continuous function on [a, b]
By the extreme value theorem we can write m <= g(t) <= M.
Therefore we can write m*(b-a) <= integral from a to b of g(t) <= M*(b-a). (There is a smaller box that has area less equal to the area under g(t) which is less equal to the area of some bigger box)
Then we can write m <= (integral from a to b of g(t))/(b-a) <= M.
From the intermediate value theorem there is some c in [a, b] such that g(c) = (integral from a to b of g(t))/(b-a). Then we are done. (The intermediate value theorem says for [a, b] if f(a) <= S <= f(b) then there is a c in [a, b] such that f(a) <= f(c) = S <= f(b) of course if f is continuous.)

I guess the proof was kind of long and Sal did not have videos on the intermediate value theorem is why he didn't prove it. This is 4 months late but maybe it can help someone else.
• We know that derivatives graph the slope of a function. I wonder, what is the "slope" of an area, and how is it related to the function under which this area is determined? • I think maybe it is clearer to think of `F(x)` as being a function that describes an area. Then `f(x)` is the rate of change in the area (i.e. how much the area changes for an increment of `x`).

In other words, `f(x)` does not give the slope of the area, it gives the slope of the graph that describes how area changes with `x`.
• When you take the derivative of a derivative (anti-derivative) using the Power Rule (not the reverse power rule), are you essentially applying the Fundamental Theorem of Calculus? • If I am not mistaken, the Fundamental Theorem of Calculus says that the indefinite integral of f(x) is the anti-derivative F(x). Can we say the reverse, that the derivative F'(x) is the anti-integral f(x)?

I hope I said this right. What I think that I am really asking is: is there a concept called "anti-integral" related to derivative similar to the relationship between integral and anti-derivative?
If you can figure out what I am trying to say, can you state it better?

Thanks for your help and insight. • where did the first definition come from .. i mean : how did we know that int(from a to x) f(x) = F(x) ? • In order to prove this theorem to begin with, we had to have a pretty good hunch that F'(x) = f(x). What would have been the thought process of earlier mathematicians to lead them to believe that F'(x) = f(x)? • It probably wasn't quite "out of the blue." I suspect Newton and Leibnitz would have realized that they could use the concept of infinitesimals, which they used to get to derivatives, to calculate area. Then, just trying a few functions, and plotting the areas as they varied the values of x (calculating area by brute force), it wouldn't take long to notice that the derivative of the area function was the original function.

Or, maybe one of them just noticed that the derivative of pi r^2 = 2pi r, and the rest is history.  