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.
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 The numerator of this difference quotient is in miles and the denominator is in gallons yielding units of miles per gallon on the average rate of change. In this case, we would think of the distance traveled as being dependent on the number of gallons in the tank of the automobile. A general guideline to use is when we discuss the average rate of change of quantity 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 in all situations, and then adapt this definition to specific applications. Keep in mind that the numerator and denominator each describe changes in some quantities. The units on the average rate of change correspond to the units on the quantities in the numerator and denominator.
Example 1 Find the Average Rate of Change from a Table
The table below defines the relationship y = f (x). Use this table to compute the average rates of change below.
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 b. Find the average rate of change of from 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 Note that what you are doing is calculating the slope between the ordered pairs (2, 5) and (5, 26). In both parts, the average rate of change is negative indicating that 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. (Source: National Association of Theater Owners, www.natoonline.org) In each part, calculate the indicated average rate of change.
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 From the table we know the price of a ticket in 1987 was $3.91 and the price of a ticket in 1999 was $5.06. The average rate of change of P with respect to t from t = 1987 to t = 1999 is The numerator of this quotient is a difference in prices and corresponds to a change of 1.15 dollars. The denominator is a difference in years and corresponds to a change of 12 years. The difference quotient is If we round this average rate of change to the nearest cent, we get approximately 0.10 dollars per year. This tells us that each year from 1987 through 1999, the ticket prices rose by an average of about 0.10 dollars or 10 cents.
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 This means that ticket prices increased by an average of approximately 0.24 dollars or 24 cents each year from 1999 to 2009.
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 where we think of the symbol ∆ as indicating “change in”. The symbol ∆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: (Source: Insurance Research Council)
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 The years serve to reference the particular data we’ll use to compute the change, but are not otherwise involved in the calculation. Using the percent uninsured and unemployed in 2003 (6% unemployed and 14.9% uninsured) and 2007 (4.6% unemployed and 13.8 uninsured), we get The unit on the numerator of the difference quotient is percent uninsured and the unit on the denominator of the difference quotient is percent unemployed. This means the units on the average rate of change is percent uninsured per percent unemployed.
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 Then the rate can be interpreted as a 1 percent change in unemployment leads to a 0.79 percent change in the percent of motorists that are uninsured. Although this is not exactly a change of “three-fourths of 1 percent”, it is close enough to be consistent. In popular media, a phrase like “three-fourths of 1 percent” is more palatable than the number 0.79 percent. It is interesting to note that even though the percents are both decreasing, the average rate of change is interpreted in terms of increases. We could have also interpreted the average rate as In this case we would say that a 1 percent drop in the percent unemployed leads to a 0.79 percent drop in the percent of motorists uninsured.