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

 Page 2 / 15

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 $\text{\hspace{0.17em}}5+3=8.$

how to solve the Identity ?
what type of identity
Jeffrey
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
how do you get the (1.4427)^t in the carp problem?
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
hello
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
rotation by 80 of (x^2/9)-(y^2/16)=1
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
what is the standard form if the focus is at (0,2) ?