ARC and IRC

Average rate of change: the average rate of change of a function f(x) over the interval [a, b] is the change in f(x) over the interval divided by the change in x over the interval, in other words, the rise over the run or the average slope of the function on the interval. Since linear functions (y = mx+b) have constant slopes, the average rate of change of a linear function over any interval is m, its slope. To find the average rate of change of a non-linear function over [a, b], you must first find the points (a, f(a)) and (b, f(b)). (These are the points on f(x) when x=a and x=b, respectively.) Next you find the slope between these two points. Voila, you have found the average rate of change over the interval.

Instantaneous rate of change: the instantaneous rate of change of a function f(x) at a point x=c, is the slope of the line tangent to f(x) at x=c. To find this slope, you will need to first find the derivative of f(x) since the derivative tells you the slope of f(x) at any point. Once you have f '(x), plug in c for x to get f '(c). This value is the instantaneous rate of change of f(x) at c.

Put more simply, if you drove your car for 2 hours and covered 80 miles the average rate of change of your position (which in this example would be your average velocity) would be easy to find since you would need only divide 80 by 2 to obtain an average rate of change of 40 m/h. The instantaneous rate of change, on the other hand, would be different depending upon whether you wanted the instantaneous rate of change after 40 minutes, or after 50 minutes, or at any other time since your velocity at any given instant would be constantly changing. If you could model the distance you had traveled as a function of time, however, you could find your instantaneous speed at any given time by finding the derivative of the function.

Further reading if you are still having trouble:
http://www.mathematicshelpcentral.com/
http://www.ugrad.math.ubc.ca/

I will be on later tonight. If you have any questions, leave a comment.