Average rate of change of a function

Let f be a function de fined on the interval [a; b]. The average rate of change of f over [a; b] is defined as:

 

Geometrically, if we have the graph of y = f(x), the average rate of change over [a; b] is the slope of the line which connects (a; f(a)) and (b; f(b)). This is called the secant line through these points.

The graph of y = f(x) and its secant line through (a; f(a)) and (b; f(b))

Example

Suppose represents the costs, in hundreds, to produce x thousand pens.
Find and interpret the average rate of change as production is increased from making 3000 to 5000 pens.

Solution

 The average rate of change as production is increased from making 3000 to 5000 pens is

As production is increased from 3000 to 5000 pens, the cost decreases at an average rate of $200 per 1000 pens produced (20¢ per pen).