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).
![729_The Graph of function f.png](https://secure.tutorsglobe.com/CMSImages/729_The Graph of function f.png)
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) (*)
![1791_Area Under the graph.png](https://secure.tutorsglobe.com/CMSImages/1791_Area Under the graph.png)
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 th in
[x; x+h] such that
A(x + h) - A(x) = f(th)h.
![617_Area Under the graph of function f.png](https://secure.tutorsglobe.com/CMSImages/617_Area%20Under%20the%20graph%20of%20function%20f.png)
d) Use Part c to prove that A'(x) = f(x)