Consider a continuous positive function f: R → R.  The graph of f(x) is the set of points in the (x,y) cartesian plane such that y = f(x) (see figure 1).

a) Consider that area A under the graph of f(x) for a < x < b (see figure 2.) 

Define: m = min{f(x): a < x < b} and M = max{f(x): a < x < b}.

Argue that m(b-a) < A < M(b-a)   (*)

b) Use (*) along with the intermediate value theorem to argue that there must exist an x*e[a,b] such that A = (b-a) f(x*) (**)

c) Consider now the function A(x) defined as the area under the graph of f(t) for a < t < x, as in Figure 3. Use Part b to argue that, for every h > 0 and every x > a, there is t_h in

[x; x+h] such that

A(x + h) - A(x) = f(t_h)h.

d) Use Part c to prove that A'(x) = f(x)