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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.