In the Derivatives as Rates of Change section, one of the topics you will explore is how to calculate the average rate of change and instantaneous rate of change. Here we will review how to calculate the average rate of change.

Finding the Average Rate of Change

Gasoline costs have experienced some wild fluctuations over the last several decades. The table below^[1] lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012. The cost of gasoline can be considered as a function of year.

If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed per year. In this section, we will investigate changes such as these.

The price change per year is a rate of change because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in the table above did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.

[latex]\begin{align}\text{Average rate of change} &=\frac{\text{Change in output}}{\text{Change in input}}\\[2mm] &=\frac{\Delta y}{\Delta x}\\[2mm] &=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\[2mm] &=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}\end{align}[/latex]

In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was

[latex]\dfrac{\Delta y}{\Delta x}=\dfrac{{1.37}}{\text{7 years}}\approx 0.196\text{ dollars per year}[/latex]

On average, the price of gas increased by about 19.6¢ each year.

Other examples of rates of change include:

• A population of rats increasing by 40 rats per week
• A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)
• A car driving 27 miles per gallon of gasoline (distance traveled changes by 27 miles for each gallon)
• The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage
• The amount of money in a college account decreasing by $4,000 per quarter

The following video provides another example of how to find the average rate of change between two points from a table of values.

You can view the transcript for “Ex: Find the Average Rate of Change From a Table – Temperatures” here (opens in new window).