## How do you calculate the average rate of change from a table?

We quantify how one quantity changes with respect to another using the average rate of change.

**Average Rate of Change**The average rate of change of

*f*with respect to

*x*from

*x*=

*a*to

*x*=

*b*is defined as

Since the numerator and denominator each contain a difference of two values, the quotient or the right side of the definition is often called a difference quotient. The variable *x* describes one of the quantities we are interested in comparing and the variable *f* describes the other quantity. In general, one of the quantities is thought to depend on the other. The quantity that depends on the other corresponds to the dependent variable *f* and the variable *x*, the independent variable, corresponds to the other. Often the choice of which quantity is which is not very clear cut. In many of those cases, we can use the units on the comparison to determine how the average rate of change should be computed. For instance, suppose the auto manufacturer is interested in the miles per gallon that its vehicles achieve. This means they wish to calculate how the miles change with respect to the change in the gallons in the tank. In this case, they would think of this average rate of change as *f* with respect to a quantity *x*, the change in the quantity *f* is in the numerator of the difference quotient. The change in the quantity *x* is in the denominator of the difference quotient. The average rate of change can be a positive number or a negative number. If the average rate of change is a positive number, the quantity corresponding to the numerator of the difference quotient increases as the quantity in the denominator increases. On the other hand, if the quantity in the numerator decreases as the quantity in the denominator increases, then the average rate of change is negative. The average rate of change can be viewed in many different ways. To make it as simple as possible, it is useful to work with the definition

**Example 1 Find the Average Rate of Change from a Table**

The table below defines the relationship *y* = *f *(*x*).

a. Find the average rate of change of *f* with respect to *x* over [0, 4].

**Solution** Apply the definition of the average rate of change to give *x* = 2 to *x* = 5.

**Solution** In this part, the interval is defined with slightly different phrasing. By saying, “from *x* = 2 to *x* = 5”, the interval over which the average rate of change is being found is being defined to be [2, 5]. Using the definition for average rate of change yields *f *(*x*) decreases as *x* increases.

**Example 2 Find the Average Rate of Change from a Table**

The average price for a ticket to a movie theater in North America for selected years is shown in the table below.

a. Find the average rate of change of ticket price with respect to time over the period 1987 to 1999.

**Solution** Use the definition of average rate of change to write as *P* with respect to *t* from *t* = 1987 to *t* = 1999 is

b. Find the average rate of change of ticket price with respect to time over the period 1999 to 2009.

**Solution** The price of a ticket in 1999 was $5.06 and $7.50 in 2009. The averate rate of change of *P* with respect to *t* from t = 1999 to t = 2009 is

c. Were ticket prices increasing faster during the period from 1987 to 1999 or during the period 1999 to 2009?

**Solution** The time period with the greater average rate of change corresponds to the period in which the prices are increasing faster. Since the average rate of change of price from 1999 to 2009 is approximately 0.24 dollars per year and the average rate of change of price from 1987 to 1999 is approximately 0.10 dollars, prices are rising faster from 1999 to 2009.

The numerator and denominator of the difference quotient is often symbolized using the greek symbol capital delta, ∆. For the average rate of change of price with respect to time, we could symbolize the difference quotient as *P* corresponds to a change in price since *P* represents price. This symbol helps us to economize on the amount of writing we need to do in order to indicate an average rate of change. In Example 1 and Example 2, there were only two rows of data in the table. It was fairly easy to decide what numbers go in the numerator of the difference quotient and which numbers go in the denominator of the difference quotient. In the next example, there are several columns of data and we’ll need to examine the average rate of change to determine how the difference quotient is formed.

**Example 3 Find the Average Rate of Change from a Table**

During the years 2003 through 2007, the percentage of Americans unemployed and the percentage of Americans driving without auto insurance both dropped according to the table below:

a. Find the average rate of change of uninsured with respect to unemployment over the period 2003 through 2007.

**Solution** We are interested in how the percent uninsured changes as the percent uninsured changes. The time period is not a part of the average rate of change except to reference the values for the percent unemployed and the percent of motorists that are unemployed. We could think of the values in the table as function values, however in this case we’ll simply think of the average rate as a ratio of changes. The average rate of change of the percent uninsured with respect to the percent unemployed is computed as

b. In a new release, an official with the Insurance Research Council was quoted as saying, “”If the unemployment rate goes up by 1 percent, we would anticipate that the percentage of people who are uninsured would go up by three-fourths of 1 percent.” Use the data from in 2003 and 2007 to support this statement.

**Solution** In part a, we calculated the average rate of change of the percent uninsured with respect to the percent unemployed as 0.79 over this time period. The units on this number are percent uninsured per percent unemployed. Think of the average rate as