3.3 Rates of change and behavior of graphs  (Page 6/15)

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Estimate the intervals where the function is increasing or decreasing.

Estimate the point(s) at which the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ has a local maximum or a local minimum.

local maximum: local minimum:

For the following exercises, consider the graph in [link] .

If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.

If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.

absolute maximum at approximately absolute minimum at approximately

Numeric

[link] gives the annual sales (in millions of dollars) of a product from 1998 to 2006. What was the average rate of change of annual sales (a) between 2001 and 2002, and (b) between 2001 and 2004?

Year Sales
(millions of dollars)
1998 201
1999 219
2000 233
2001 243
2002 249
2003 251
2004 249
2005 243
2006 233

[link] gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?

Year Population
(thousands)
2000 87
2001 84
2002 83
2003 80
2004 77
2005 76
2006 78
2007 81
2008 85

a. –3000; b. –1250

For the following exercises, find the average rate of change of each function on the interval specified.

$f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ on

$h\left(x\right)=5-2{x}^{2}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[-2,\text{4}\right]$

-4

$q\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[-4,\text{2}\right]$

$g\left(x\right)=3{x}^{3}-1\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[-3,\text{3}\right]$

27

$y=\frac{1}{x}\text{\hspace{0.17em}}$ on

$p\left(t\right)=\frac{\left({t}^{2}-4\right)\left(t+1\right)}{{t}^{2}+3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[-3,\text{1}\right]$

–0.167

$k\left(t\right)=6{t}^{2}+\frac{4}{{t}^{3}}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[-1,3\right]$

Technology

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

$f\left(x\right)={x}^{4}-4{x}^{3}+5$

Local minimum at $\text{\hspace{0.17em}}\left(3,-22\right),\text{\hspace{0.17em}}$ decreasing on increasing on

$h\left(x\right)={x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}-1$

$g\left(t\right)=t\sqrt{t+3}$

Local minimum at $\text{\hspace{0.17em}}\left(-2,-2\right),\text{\hspace{0.17em}}$ decreasing on $\text{\hspace{0.17em}}\left(-3,-2\right),\text{\hspace{0.17em}}$ increasing on

$k\left(t\right)=3{t}^{\frac{2}{3}}-t$

$m\left(x\right)={x}^{4}+2{x}^{3}-12{x}^{2}-10x+4$

Local maximum at local minima at $\text{\hspace{0.17em}}\left(-3.25,-47\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(2.1,-32\right),\text{\hspace{0.17em}}$ decreasing on $\text{\hspace{0.17em}}\left(-\infty ,-3.25\right)\text{\hspace{0.17em}}$ and increasing on and

$n\left(x\right)={x}^{4}-8{x}^{3}+18{x}^{2}-6x+2$

Extension

The graph of the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is shown in [link] .

Based on the calculator screen shot, the point is which of the following?

1. a relative (local) maximum of the function
2. the vertex of the function
3. the absolute maximum of the function
4. a zero of the function

A

Let $f\left(x\right)=\frac{1}{x}.$ Find a number $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ such that the average rate of change of the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}\left(1,c\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}-\frac{1}{4}.$

Let $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}$ . Find the number $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ such that the average rate of change of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}\left(2,b\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}-\frac{1}{10}.$

$b=5$

Real-world applications

At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?

A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3:40 p.m. At this time, he started pumping gas into the tank. At exactly 3:44, the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

2.7 gallons per minute

Near the surface of the moon, the distance that an object falls is a function of time. It is given by $\text{\hspace{0.17em}}d\left(t\right)=2.6667{t}^{2},\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in seconds and $\text{\hspace{0.17em}}d\left(t\right)\text{\hspace{0.17em}}$ is in feet. If an object is dropped from a certain height, find the average velocity of the object from $\text{\hspace{0.17em}}t=1\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}t=2.$

The graph in [link] illustrates the decay of a radioactive substance over $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ days.

Use the graph to estimate the average decay rate from $\text{\hspace{0.17em}}t=5\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}t=15.$

approximately –0.6 milligrams per day

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