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Average Rate of Change

One way to describe how a quantity changes over some interval. is the average rate of change. The average rate of change of the function f(t) over the interval from  t= a to  t= b is given by

Consider, for example, the function

which can be used to model the population of Forsyth County, Georgia. The function describes the population in thousands in the year 2000 + t. In the examples below we will determine the average rate of change of the population of Forsyth County over two different intervals of time.

Example 1

What was the average rate of change of the population of  Forsyth County between 1990 and 2000?

Solution

Since t = -10 corresponds to the year 1990 and t = 0 corresponds to 2000, we first calculate the change in population between 1990 and 2000 as follows:

                                       

Next we divide by the length of time, 2000-1990 = 10 years:           

                                   

We see that the population of Forsyth County grew at an average rate of 4.098 thousand people per year between 1990 and 2000.

Example 2

What was the average rate of change of the population of Forsyth County between 1980 and 1990?

Solution

Since t = -20 corresponds to the year 1980 and t = -10 corresponds to 1990, we first calculate the change in population between 1980 and 1990 as follows:

            F(-10) - F(-20) = 22.0788 thousand.

Next we divide by the length of time, 1990 1980 = 10 years:

             

We see that the population of Forsyth County grew at an average rate of 2.208 thousand people per year between 1980 and 1990.

In the above examples, we computed the average rate of change of the function F(t) over the interval from t = -10 to t = 0 and then again over the interval from t = -20 to t = -10.    You may have noticed that we used the same formula for calculating the average rate of change as the one we use for calculating the slope of a line between two points. Figure 1 shows a graph of F(t) and the line through the points (-10, F(-10)) and (0, F(0)).

Figure 1

The slope of the line between the two points is

This is the same as the average annual population change between 1990 and 2000 computed in Example 1 above. Figure 2 shows the function F(t) and the line through the points (-20, F(-20)) and (-10, F(-10)).

Figure 2

The slope of the line between the two points is

           

                                         

Again, this is the same as the average annual population change between 1980 and 1990 computed in Example 2 above.

Terminology

A line connecting two points on a graph is called a secant line. The average rate of change in f(t) between t = a and t = b is the same as the slope of the secant line between the points (a, f(a)) and (b, f(b)) on the graph of f.