Rate of change in tables and graphs

Analysis

Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.

Computing Average Rate of Change for a Function Expressed as a Formula

Compute the average rate of change of [latex]f\left(x\right)=^-\frac[/latex] on the interval [latex]\text<[2,>\,\text[/latex]

We can start by computing the function values at each endpoint of the interval.

[latex]\begin\hfill f\left(2\right)& =& ^-\frac\hfill & \hfill \phantom>f\left(4\right)& =& ^-\frac\hfill \\ & =& 4-\frac\hfill & & =& 16-\frac\hfill \\ & =& \frac\hfill & & =& \frac\hfill \end[/latex]

Now we compute the average rate of change.

Try It

Find the average rate of change of [latex]f\left(x\right)=x-2\sqrt[/latex] on the interval [latex]\left[1,\,9\right].[/latex]

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^>.[/latex]Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.

We are computing the average rate of change of[latex]\,F\left(d\right)=\frac^>\,[/latex]on the interval[latex]\,\left[2,6\right].[/latex]

The average rate of change is [latex]-\frac[/latex] newton per centimeter.

Finding an Average Rate of Change as an Expression

Find the average rate of change of [latex]g\left(t\right)=^+3t+1[/latex] on the interval [latex]\left[0,\,a\right].[/latex] The answer will be an expression involving [latex]a[/latex] in simplest form.

We use the average rate of change formula.

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)=^+2x-8\,[/latex] on the interval[latex]\,\left[5,a\right]\,[/latex]in simplest forms in terms

Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant

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. (Figure) shows examples of increasing and decreasing intervals on a function.

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

While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input 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 input 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.

Clearly, 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 whose graph is shown in (Figure), 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]

Figure 4.

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. (Figure) illustrates these ideas for a local maximum.

Figure 5. Definition of a local maximum

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

Local Minima and Local Maxima

A function[latex]\,f\,[/latex]is an increasing function on an open interval if[latex]\,f\left(b\right)>f\left(a\right)\,[/latex]for any two input values[latex]\,a\,[/latex]and[latex]\,b\,[/latex]in the given interval where[latex]\,b>a.[/latex]

Finding Increasing and Decreasing Intervals on a Graph

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

Figure 6.

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

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).

Finding Local Extrema from a Graph

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

Using technology, we find that the graph of the function looks like that in (Figure). 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]

Figure 7.

Analysis

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

Figure 8.

Based on these estimates, the function is increasing on the interval[latex]\,(-\infty \text-\text\text<.449)>\,[/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 location of the extrema is at[latex]\,±\sqrt,\,[/latex]but determining this requires calculus.)

Try It

Graph the function[latex]\,f\left(x\right)=^-6^-15x+20\,[/latex]to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.

The local maximum appears to occur at[latex]\,\left(-1,28\right),\,[/latex]and the local minimum occurs at[latex]\,\left(5,-80\right).\,[/latex]The function is increasing on[latex]\,\left(-\infty ,-1\right)\cup \left(5,\infty \right)\,[/latex]and decreasing on[latex]\,\left(-1,5\right).[/latex]

Finding Local Maxima and Minima from a Graph

For the function[latex]\,f\,[/latex]whose graph is shown in (Figure), find all local maxima and minima.

Figure 9

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 (Figure), (Figure), and (Figure).

Figure 10.

Figure 11.

Figure 12.

Use A Graph to Locate the Absolute Maximum and Absolute Minimum

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. See (Figure).

Figure 13.

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

Absolute Maxima and Minima

The absolute maximum of[latex]\,f\,[/latex]at[latex]\,x=c\,[/latex]is[latex]\,f\left(c\right)\,[/latex]where[latex]\,f\left(c\right)\ge f\left(x\right)\,[/latex]for all[latex]\,x\,[/latex]in the domain of[latex]\,f.[/latex]

The absolute minimum of[latex]\,f\,[/latex]at[latex]\,x=d\,[/latex]is[latex]\,f\left(d\right)\,[/latex]where[latex]\,f\left(d\right)\le f\left(x\right)\,[/latex]for all[latex]\,x\,[/latex]in the domain of[latex]\,f.[/latex]

Finding Absolute Maxima and Minima from a Graph

For the function[latex]\,f\,[/latex]shown in (Figure), find all absolute maxima and minima.

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]

Access this online resource for additional instruction and practice with rates of change.

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. See (Figure).
• Identifying points that mark the interval on a graph can be used to find the average rate of change. See (Figure).
• Comparing pairs of input and output values in a table can also be used to find the average rate of change. See (Figure).
• An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See (Figure) and (Figure).
• The average rate of change can sometimes be determined as an expression. See (Figure).
• A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See (Figure).
• 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. See (Figure) and (Figure).
• The highest and lowest points on a graph indicate the maxima and minima. See (Figure).

Section Exercises

Verbal

Can the average rate of change of a function be constant?

Yes, the average rate of change of all linear functions is constant.

If a function[latex]\,f\,[/latex]is increasing on[latex]\,\left(a,b\right)\,[/latex]and decreasing on[latex]\,\left(b,c\right),\,[/latex]then what can be said about the local extremum of[latex]\,f\,[/latex]on[latex]\,\left(a,c\right)?\,[/latex]

How are the absolute maximum and minimum similar to and different from the local extrema?

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

How does the graph of the absolute value function compare to the graph of the quadratic function,[latex]\,y=^,\,[/latex]in terms of increasing and decreasing intervals?

Algebraic

For the following exercises, find the average rate of change of each function on the interval specified for real numbers [latex]\,b\,[/latex]or[latex]\,h[/latex] in simplest form.