How Do You Calculate and Interpret an Average Rate of Change?

I posted a worksheet to help you work on calculating average rates of change. Let’s look at what students did in class on the problem in this worksheet.

Let’s start with the data table.

Year

Wireless Revenue (billions $)

Wireless Connections (millions)

2006

38.0

59.1

2007

43.9

65.7

2008

49.3

72.1

2009

60.3

96.5

2010

63.4

102.2

2011

70.2

107.8

These numbers indicate how fast the revenue is increasing over each period. Comparing these average rates of change, we see that the rate is getting larger.

The numbers above indicate how fast the number of wireless connections are increasing over each period. Like the revenue, the average rate over 2008 through 2011 is larger than the average rate over 2006 through 2008. YES! Time for me to invest!

Not so fast…let’s look at another set of average rates. Calculate the average rate of change of wireless revenue with respect to wireless connections over the same periods.

\displaystyle \text{Ave Rate over 2006 to 2008}=\frac{49.3-38.0}{72.1-59.1}=0.869 \frac{\text{billion dollars}}{\text{million connections}}

Since one billion divided by one million is one thousand, this is equivalent to 0.869 thousand dollars per connection or 869 dollars per connection. This means that increasing the number of connections by one increases the wireless revenue by $869. Now look at the same rate over 2008 through 2011:

\displaystyle \text{Ave Rate over 2008 to 2011}=\frac{70.2-49.3}{107.8-72.1}=0.585 \frac{\text{thousand dollars}}{\text{connections}}

Over this period, increasing the number of connections by one raises the wireless revenue by $585. Even though revenue still rises, it is doing so at a lower rate. Depending on the cost of operating the business and the turnover rate for existing customers, this may be a reason for concern. If nothing else, it warrants more questions.

The numbers above (and the rates calculated by students) may be represented using line segments on a scatter plot.

The slope of a line segment corresponding to the years 2006 (59.1, 38.0) and 2008 (72.1, 49.3) is 0.869. Similarly, the slope of a line segment corresponding to the years 2008 (72.1, 49.3) and 2011 (107.8, 70.2) is 0.585. As you can see from the graph, the less steep segment extends from 2008 to 2011 so the average rate is lower.