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

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
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how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations