The average rate of change of a function f(x)from x = a to x = b
= The slope of the secant line joining the points (a,f(a)) and (b,f(b)) on the graph of f
=$\frac{f(b)-f(a)}{b-a}$
= $\frac{f(x+\Delta x)-f(x)}{\Delta x}$
where $\Delta x = b-a$.
The following applets demonstrate the slope of a secant line and how the secant lines converge to the tangent line as $\Delta x$ goes to zero. The function x is not that interesting, so you should change it to one of the other functions available.
Slope of Secant Line Applet
Slope of Secant Line Applet: Go to Differential Calculus - Definition of the Derivative