Learning Outcomes
- Find the average rate of change of a function.
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.
[latex]y[/latex] | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |
[latex]C\left(y\right)[/latex] | 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |
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.
Finding the Average Rate of Change of a Function
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.
The Greek letter [latex]\Delta [/latex] (delta) signifies the change in a quantity; we read the ratio as “delta-y over delta-x” or “the change in [latex]y[/latex] divided by the change in [latex]x[/latex].” Occasionally we write [latex]\Delta f[/latex] instead of [latex]\Delta y[/latex], which still represents the change in the function’s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.
In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was
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 (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
A General Note: Rate of Change
A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are “output units per input units.”
The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.
How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[/latex] and [latex]{x}_{2}[/latex].
- Calculate the difference [latex]{y}_{2}-{y}_{1}=\Delta y[/latex].
- Calculate the difference [latex]{x}_{2}-{x}_{1}=\Delta x[/latex].
- Find the ratio [latex]\frac{\Delta y}{\Delta x}[/latex].
Example 1: Computing an Average Rate of Change
Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.
[latex]y[/latex] | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |
[latex]C\left(y\right)[/latex] | 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |
The following video provides another example of how to find the average rate of change between two points from a table of values.
Try It
Using the data in the table below, find the average rate of change between 2005 and 2010.
[latex]y[/latex] | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |
[latex]C\left(y\right)[/latex] | 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |
Example 2: Computing Average Rate of Change from a Graph
Given the function [latex]g\left(t\right)[/latex] shown in Figure 1, find the average rate of change on the interval [latex]\left[-1,2\right][/latex].
Example 3: Computing Average Rate of Change from a Table
After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in the table below. Find her average speed over the first 6 hours.
t (hours) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
D(t) (miles) | 10 | 55 | 90 | 153 | 214 | 240 | 282 | 300 |
Example 4: Computing Average Rate of Change for a Function Expressed as a Formula
Compute the average rate of change of [latex]f\left(x\right)={x}^{2}-\frac{1}{x}[/latex] on the interval [latex]\text{[2,}\text{4].}[/latex]
The following video provides another example of finding the average rate of change of a function given a formula and an interval.
Try It
Find the average rate of change of [latex]f\left(x\right)=x - 2\sqrt{x}[/latex] on the interval [latex]\left[1,9\right][/latex].
Try It
Example 5: Finding the Average Rate of Change of a Force
The electrostatic force [latex]F[/latex], measured in newtons, between two charged particles can be related to the distance between the particles [latex]d[/latex], in centimeters, by the formula [latex]F\left(d\right)=\frac{2}{{d}^{2}}[/latex]. Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.
Example 6: Finding an Average Rate of Change as an Expression
Find the average rate of change of [latex]g\left(t\right)={t}^{2}+3t+1[/latex] on the interval [latex]\left[0,a\right][/latex]. The answer will be an expression involving [latex]a[/latex].
Key Equations
Average rate of change | [latex]\dfrac{\Delta y}{\Delta x}=\dfrac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}[/latex] |
Key Concepts
- A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data.
- Identifying points that mark the interval on a graph can be used to find the average rate of change.
- Comparing pairs of input and output values in a table can also be used to find the average rate of change.
- An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.
- The average rate of change can sometimes be determined as an expression.
Glossary
- average rate of change
- the difference in the output values of a function found for two values of the input divided by the difference between the inputs
- rate of change
- the change of an output quantity relative to the change of the input quantity
- http://www.eia.gov/totalenergy/data/annual/showtext.cfm?t=ptb0524. Accessed 3/5/2014. ↵