Rates 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]

The Greek letter [latex]\Delta [/latex] (delta) signifies the change in a quantity; we read the ratio as “delta-[latex]y[/latex] over delta-[latex]x[/latex]” 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.

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.

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].

Show Solution

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]\dfrac{1 - 4}{2-\left(-1\right)}=\dfrac{-3}{3}=-1[/latex]

Analysis of the Solution

Note that the order we choose is very important. If, for example, we use [latex]\dfrac{{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].

The following video provides another example of finding the average rate of change of a function given a formula and an interval.

Behaviors of Functions

As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. The graph below shows examples of increasing and decreasing intervals on a function.

The function [latex]f\left(x\right)={x}^{3}-12x[/latex] is increasing on [latex]\left(-\infty \text{,}-\text{2}\right){{\cup }^{\text{ }}}^{\text{ }}\left(2,\infty \right)[/latex] and is decreasing on [latex]\left(-2\text{,}2\right)[/latex].

This video further explains how to find where a function is increasing or decreasing.

While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the output where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a local maximum. If a function has more than one, we say it has local maxima. Similarly, a value of the output where a function changes from decreasing to increasing as the input variable increases is called a local minimum. The plural form is “local minima.” Together, local maxima and minima are called local extrema, or local extreme values, of the function. (The singular form is “extremum.”) Often, the term local is replaced by the term relative. In this text, we will use the term local.

A function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.

For the function below, the local maximum is 16, and it occurs at [latex]x=-2[/latex]. The local minimum is [latex]-16[/latex] and it occurs at [latex]x=2[/latex].

To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. The graph below illustrates these ideas for a local maximum.

Definition of a local maximum.

These observations lead us to a formal definition of local extrema.

Example: Finding Increasing and Decreasing Intervals on a Graph

Given the function [latex]p\left(t\right)[/latex] in the graph below, identify the intervals on which the function appears to be increasing.

Show Solution

We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from [latex]t=1[/latex] to [latex]t=3[/latex] and from [latex]t=4[/latex] on.

In interval notation, we would say the function appears to be increasing on the interval (1,3) and the interval [latex]\left(4,\infty \right)[/latex].

Analysis of the Solution

Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1[/latex] , [latex]t=3[/latex] , and [latex]t=4[/latex] . These points are the local extrema (two minima and a maximum).

Example: Finding Local Extrema from a Graph

Graph the function [latex]f\left(x\right)=\dfrac{2}{x}+\dfrac{x}{3}[/latex]. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.

Show Solution

Using technology, we find that the graph of the function looks like that below. It appears there is a low point, or local minimum, between [latex]x=2[/latex] and [latex]x=3[/latex], and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[/latex] and [latex]x=-2[/latex].

Analysis of the Solution

Most graphing calculators and graphing utilities can estimate the location of maxima and minima. The graph below provides screen images from two different technologies, showing the estimate for the local maximum and minimum.

Based on these estimates, the function is increasing on the interval [latex]\left(-\infty ,-{2.449}\right)[/latex] and [latex]\left(2.449\text{,}\infty \right)[/latex]. Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact locations of the extrema are at [latex]\pm \sqrt{6}[/latex], but determining this requires calculus.)

Example: Finding Local Maxima and Minima from a Graph

For the function [latex]f[/latex] whose graph is shown below, find all local maxima and minima.

Show Solution

Observe the graph of [latex]f[/latex]. The graph attains a local maximum at [latex]x=1[/latex] because it is the highest point in an open interval around [latex]x=1[/latex]. The local maximum is the [latex]y[/latex] -coordinate at [latex]x=1[/latex], which is [latex]2[/latex].

The graph attains a local minimum at [latex]\text{ }x=-1\text{ }[/latex] because it is the lowest point in an open interval around [latex]x=-1[/latex]. The local minimum is the y-coordinate at [latex]x=-1[/latex], which is [latex]-2[/latex].

Analyzing the Toolkit Functions for Increasing or Decreasing Intervals

We will now return to our toolkit functions and discuss their graphical behavior in the table below.

Function	Increasing/Decreasing	Example
Constant Function [latex]f\left(x\right)={c}[/latex]	Neither increasing nor decreasing
Identity Function [latex]f\left(x\right)={x}[/latex]	Increasing
Quadratic Function [latex]f\left(x\right)={x}^{2}[/latex]	Increasing on [latex]\left(0,\infty\right)[/latex] Decreasing on [latex]\left(-\infty,0\right)[/latex] Minimum at [latex]x=0[/latex]
Cubic Function [latex]f\left(x\right)={x}^{3}[/latex]	Increasing
Reciprocal [latex]f\left(x\right)=\frac{1}{x}[/latex]	Decreasing [latex]\left(-\infty,0\right)\cup\left(0,\infty\right)[/latex]
Reciprocal Squared [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex]	Increasing on [latex]\left(-\infty,0\right)[/latex] Decreasing on [latex]\left(0,\infty\right)[/latex]
Cube Root [latex]f\left(x\right)=\sqrt[3]{x}[/latex]	Increasing
Square Root [latex]f\left(x\right)=\sqrt{x}[/latex]	Increasing on [latex]\left(0,\infty\right)[/latex]
Absolute Value [latex]f\left(x\right)=|x|[/latex]	Increasing on [latex]\left(0,\infty\right)[/latex] Decreasing on [latex]\left(-\infty,0\right)[/latex] Minimum at [latex]x=0[/latex]

Use a graph to locate the absolute maximum and absolute minimum of a function

There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\text{-}[/latex] coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively.

To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function.

Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\left(x\right)={x}^{3}[/latex] is one such function.

Example: Finding Absolute Maxima and Minima from a Graph

For the function [latex]f[/latex] shown below, find all absolute maxima and minima.

Show Solution

Observe the graph of [latex]f[/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[/latex] and [latex]x=2[/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y-coordinate at [latex]x=-2[/latex] and [latex]x=2[/latex], which is [latex]16[/latex].

The graph attains an absolute minimum at [latex]x=3[/latex], because it is the lowest point on the domain of the function’s graph. The absolute minimum is the y-coordinate at [latex]x=3[/latex], which is [latex]-10[/latex].

Key Equations

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.
• A function is increasing where its rate of change is positive and decreasing where its rate of change is negative.
• A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
• A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
• Minima and maxima are also called extrema.
• We can find local extrema from a graph.
• The highest and lowest points on a graph indicate the maxima and minima.