## The Fundamental Theorem of Calculus

Truly a theorem that was waiting to happen, the "FTOC" was discovered independently in several forms by a handful of mathematicians of the seventeenth century. Most notably, Newton and Leibniz were able to capitalize on the theorem, essentially inventing a whole new field of mathematics: calculus. The familiar modern notation you see below is that of Leibniz.

The idea:

The processes of differentiation and integration are inverses: what one does, the other undoes.

The statements. The analytic version contains the necessary mathematical precision. The other statements are analogies.

Part 1

 D If you start with a continuous function, integrate it to get a new function, then differentiate the new function, the result is the function you started with. A If f is continuous on [a,b] and if F(x) = , then F'(x) = f(x). N Take a sequence of numbers y1, y2, y3, ..., yn and let Y0 be a "starting number." Form a new sequence Yi by successively adding the old sequence to Y0. Thus, Y1 = Y0 + y1, Y2 = Y1 + y2, ..., etc. Using the new sequence, find consecutive differences. You get the original sequence back. See the example. G If you use a continuous function to define a region, form a new function which computes the area of that region, then find the slope of the area function, the result is the original function.

Part 2

 D The net change in a function is the integral of its derivative. A If f is continuous on [a,b] and if F is any antiderivative of f, then = F(b) - F(a). N Take a sequence of numbers y1, y2, y3, ..., yn and a starting number Y0, and form the new sequence Yi as above. Then the sum of the first sequence is the net change Yn - Y0 in the new sequence. See the example. G The area under a function is the net change in its antiderivative.