The Fundamental Theorem of Calculus part 1 (part iii)
So, we are now ready to prove the FTC part 1. We’re going to follow the proof in Stewart and add in some discussion as we go along to motivate what we are doing. What we are going to prove is that:
for when is continuous on .
we define and we want to find the derivative of . We will do this by using the fundamental definition of the derivative, so let’s look at calculating this function at and – ie. how much does it change when we change by a little bit?
But remember that the definite integral is just the area, so this difference is the area between a and x+h minus the area between a and x. Which is just the area between x and x+h. Using the properties of integrals, we can write this formally as:
and we can write, for :
Restated, we can think of this as the area between x and x+h divided by h.…