<< Chapter < Page Chapter >> Page >
Graph of a cubic function.

Estimate the intervals where the function is increasing or decreasing.

Got questions? Get instant answers now!

Estimate the point(s) at which the graph of f has a local maximum or a local minimum.

local maximum: ( 3 ,   60 ) , local minimum: ( 3 ,   60 )

Got questions? Get instant answers now!

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

Graph of a cubic function.

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

Got questions? Get instant answers now!

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

absolute maximum at approximately ( 7 ,   150 ) , absolute minimum at approximately ( −7.5 ,   −220 )

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

[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

Got questions? Get instant answers now!

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

f ( x ) = x 2 on [ 1 ,   5 ]

Got questions? Get instant answers now!

h ( x ) = 5 2 x 2 on [ −2 , 4 ]

-4

Got questions? Get instant answers now!

q ( x ) = x 3 on [ −4 , 2 ]

Got questions? Get instant answers now!

g ( x ) = 3 x 3 1 on [ −3 , 3 ]

27

Got questions? Get instant answers now!

y = 1 x on [ 1 ,  3 ]

Got questions? Get instant answers now!

p ( t ) = ( t 2 4 ) ( t + 1 ) t 2 + 3 on [ −3 , 1 ]

–0.167

Got questions? Get instant answers now!

k ( t ) = 6 t 2 + 4 t 3 on [ −1 , 3 ]

Got questions? Get instant answers now!

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 ( x ) = x 4 4 x 3 + 5

Local minimum at ( 3 , 22 ) , decreasing on ( ,   3 ) , increasing on ( 3 ,   )

Got questions? Get instant answers now!

h ( x ) = x 5 + 5 x 4 + 10 x 3 + 10 x 2 1

Got questions? Get instant answers now!

g ( t ) = t t + 3

Local minimum at ( 2 , 2 ) , decreasing on ( 3 , 2 ) , increasing on ( 2 ,   )

Got questions? Get instant answers now!

m ( x ) = x 4 + 2 x 3 12 x 2 10 x + 4

Local maximum at ( 0.5 ,   6 ) , local minima at ( 3.25 , 47 ) and ( 2.1 , 32 ) , decreasing on ( , 3.25 ) and ( 0.5 ,   2.1 ) , increasing on ( 3.25 ,   0.5 ) and ( 2.1 ,   )

Got questions? Get instant answers now!

n ( x ) = x 4 8 x 3 + 18 x 2 6 x + 2

Got questions? Get instant answers now!

Extension

The graph of the function f is shown in [link] .

Graph of f(x) on a graphing calculator.

Based on the calculator screen shot, the point ( 1.333 ,   5.185 ) 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

Got questions? Get instant answers now!

Let f ( x ) = 1 x . Find a number c such that the average rate of change of the function f on the interval ( 1 , c ) is 1 4 .

Got questions? Get instant answers now!

Let f ( x ) = 1 x . Find the number b such that the average rate of change of f on the interval ( 2 , b ) is 1 10 .

b = 5

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Near the surface of the moon, the distance that an object falls is a function of time. It is given by d ( t ) = 2.6667 t 2 , where t is in seconds and d ( t ) is in feet. If an object is dropped from a certain height, find the average velocity of the object from t = 1 to t = 2.

Got questions? Get instant answers now!

The graph in [link] illustrates the decay of a radioactive substance over t days.

Graph of an exponential function.

Use the graph to estimate the average decay rate from t = 5 to t = 15.

approximately –0.6 milligrams per day

Got questions? Get instant answers now!

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 9

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask