Rates of Change
Learning Objectives
By the end of this lesson, you will be able to:- Find the average rate of change of a function.
[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 |
[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 |
[latex]\begin{array}{l}\text{Average rate of change}=\frac{\text{Change in output}}{\text{Change in input}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{\Delta y}{\Delta x}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}\end{array}[/latex]
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[latex]\frac{\Delta y}{\Delta x}=\frac{{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 (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.[latex]\frac{\Delta y}{\Delta x}=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}[/latex]
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: 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 |
Answer: In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is
[latex]\begin{array}{l}\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\\,\,\,\,\,\,\,\,\,\,=\frac{2.41-2.84}{2009 - 2007}\\\,\,\,\,\,\,\,\,\,\,=\frac{-0.43}{2\text{ years}}\\\,\,\,\,\,\,\,\,\,\,={-0.22}\text{ per year}\end{array}[/latex]
Analysis of the Solution
Note that a decrease is expressed by a negative change or "negative increase." A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases. 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 |
Answer: [latex]\frac{$2.84-$2.31}{5\text{ years}}=\frac{$0.53}{5\text{ years}}=$0.106[/latex] per year.
Example: Computing Average Rate of Change from a Graph
Given the function [latex]g\left(t\right)[/latex], find the average rate of change on the interval [latex]\left[-1,2\right][/latex].Answer: At [latex]t=-1[/latex], the graph shows [latex]g\left(-1\right)=4[/latex]. At [latex]t=2[/latex], the graph shows [latex]g\left(2\right)=1[/latex]. The horizontal change [latex]\Delta t=3[/latex] is shown by the red arrow, and the vertical change [latex]\Delta g\left(t\right)=-3[/latex] is shown by the turquoise arrow. The output changes by –3 while the input changes by 3, giving an average rate of change of
[latex]\frac{1 - 4}{2-\left(-1\right)}=\frac{-3}{3}=-1[/latex]
Analysis of the Solution
Note that the order we choose is very important. If, for example, we use [latex]\frac{{y}_{2}-{y}_{1}}{{x}_{1}-{x}_{2}}[/latex], we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex].Example: 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.Answer: We are computing the average rate of change of [latex]F\left(d\right)=\frac{2}{{d}^{2}}[/latex] on the interval [latex]\left[2,6\right][/latex].
[latex]\begin{array}{l}\text{Average rate of change.}=\frac{F\left(6\right)-F\left(2\right)}{6 - 2}\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{\frac{2}{{6}^{2}}-\frac{2}{{2}^{2}}}{6 - 2} \,\,\,\,\,\,\,\,\,\,\,\,\,\text{Simplify.} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{\frac{2}{36}-\frac{2}{4}}{4}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{-\frac{16}{36}}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Combine numerator terms.}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=-\frac{1}{9}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Simplify.}\end{array}[/latex]
The average rate of change is [latex]-\frac{1}{9}[/latex] newton per centimeter.Example: 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].Answer: We use the average rate of change formula.
[latex]\begin{array}{l}\text{Average rate of change}=\frac{g\left(a\right)-g\left(0\right)}{a - 0}\text{Evaluate.}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{\left({a}^{2}+3a+1\right)-\left({0}^{2}+3\left(0\right)+1\right)}{a - 0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Simplify.}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{{a}^{2}+3a+1 - 1}{a}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Simplify and factor.}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{a\left(a+3\right)}{a}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Divide by the common factor }a.\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=a+3\end{array}[/latex]
This result tells us the average rate of change in terms of [latex]a[/latex] between [latex]t=0[/latex] and any other point [latex]t=a[/latex]. For example, on the interval [latex]\left[0,5\right][/latex], the average rate of change would be [latex]5+3=8[/latex].Try It
Find the average rate of change of [latex]f\left(x\right)={x}^{2}+2x - 8[/latex] on the interval [latex]\left[5,a\right][/latex].Answer: [latex]a+7[/latex]
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