1.3 Rates of change and behavior of graphs  (Page 2/15)

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Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values $\text{\hspace{0.17em}}{x}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{x}_{2}.$

1. Calculate the difference ${y}_{2}-{y}_{1}=\text{Δ}y.$
2. Calculate the difference ${x}_{2}-{x}_{1}=\text{Δ}x.$
3. Find the ratio $\text{\hspace{0.17em}}\frac{\text{Δ}y}{\text{Δ}x}.$

Computing an average rate of change

Using the data in [link] , find the average rate of change of the price of gasoline between 2007 and 2009.

In 2007, the price of gasoline was $2.84. In 2009, the cost was$2.41. The average rate of change is

Using the data in [link] , find the average rate of change between 2005 and 2010.

per year.

Computing average rate of change from a graph

Given the function $\text{\hspace{0.17em}}g\left(t\right)\text{\hspace{0.17em}}$ shown in [link] , find the average rate of change on the interval $\text{\hspace{0.17em}}\left[-1,2\right].$

At $t=-1,$ [link] shows $g\left(-1\right)=4.$ At $\text{\hspace{0.17em}}t=2,$ the graph shows $g\left(2\right)=1.$

The horizontal change $\text{\hspace{0.17em}}\text{Δ}t=3\text{\hspace{0.17em}}$ is shown by the red arrow, and the vertical change $\text{Δ}g\left(t\right)=-3$ is shown by the turquoise arrow. The output changes by –3 while the input changes by 3, giving an average rate of change of

$\frac{1-4}{2-\left(-1\right)}=\frac{-3}{3}=-1$

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 [link] . 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

Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours, for an average speed of

$\begin{array}{l}\begin{array}{l}\hfill \\ \frac{292-10}{6-0}=\frac{282}{6}\hfill \end{array}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=47\hfill \end{array}$

The average speed is 47 miles per hour.

Computing average rate of change for a function expressed as a formula

Compute the average rate of change of $f\left(x\right)={x}^{2}-\frac{1}{x}$ on the interval $\text{[2,}\text{\hspace{0.17em}}\text{4].}$

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

$\begin{array}{ll}f\left(2\right)={2}^{2}-\frac{1}{2}\begin{array}{cccc}& & & \end{array}\hfill & f\left(4\right)={4}^{2}-\frac{1}{4}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=4-\frac{1}{2}\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=16-\frac{1}{4}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{7}{2}\hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{63}{4}\hfill \end{array}$

Now we compute the average rate of change.

Find the average rate of change of $f\left(x\right)=x-2\sqrt{x}$ on the interval $\left[1,\text{\hspace{0.17em}}9\right].$

$\frac{1}{2}$

Finding the average rate of change of a force

The electrostatic force $\text{\hspace{0.17em}}F,$ measured in newtons, between two charged particles can be related to the distance between the particles $\text{\hspace{0.17em}}d,$ in centimeters, by the formula $\text{\hspace{0.17em}}F\left(d\right)=\frac{2}{{d}^{2}}.$ 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 $\text{\hspace{0.17em}}F\left(d\right)=\frac{2}{{d}^{2}}\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}\left[2,6\right].$

The average rate of change is $\text{\hspace{0.17em}}-\frac{1}{9}\text{\hspace{0.17em}}$ newton per centimeter.

Finding an average rate of change as an expression

Find the average rate of change of $g\left(t\right)={t}^{2}+3t+1$ on the interval $\left[0,\text{\hspace{0.17em}}a\right].$ The answer will be an expression involving $a.$

We use the average rate of change formula.

This result tells us the average rate of change in terms of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}t=0\text{\hspace{0.17em}}$ and any other point $\text{\hspace{0.17em}}t=a.\text{\hspace{0.17em}}$ For example, on the interval $\text{\hspace{0.17em}}\left[0,5\right],\text{\hspace{0.17em}}$ the average rate of change would be $\text{\hspace{0.17em}}5+3=8.$

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