In this lecture we prove (with calculus 1 level rigor), and do examples surrounding, one of the most important facts in all of calculus, maybe in all of life -- The Fundamental Theorem of Calculus. This theorem ties together two areas of math that were thought of as separate for millennia: differential calculus and integral calculus (these are the modern names). The class was asked to read a proof showing that if a function that gave the area under a curve f exists, then the function must be an antiderivative of f. The Fundamental Theorem takes this further and guarantees the existence of an antiderivative for continuous functions. The theorem also says that differentiating an integral gives you back the original function; this shows that the derivative is the reverse of the integral. The integral is *almost* the reverse of the derivative, we've seen that when we take an indefinite integral, there is a +C term that arises that needs to be dealt with.
This fact is astounding. It says the area problem and the rate problem that we spoke about at the beginning of the semester are two sides of the same coin. Integration and differentiation are (very nearly) inverse operations--figuring out the rate of change of a function is the opposite of figuring out the area under the function. Who'd have thought?
We do examples after the theorem is proved.