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Graph of a cubic function.

Estimate the intervals where the function is increasing or decreasing.

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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 )

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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.

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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 )

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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
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[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

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

f ( x ) = x 2 on [ 1 ,   5 ]

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h ( x ) = 5 2 x 2 on [ −2 , 4 ]

-4

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q ( x ) = x 3 on [ −4 , 2 ]

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g ( x ) = 3 x 3 1 on [ −3 , 3 ]

27

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y = 1 x on [ 1 ,  3 ]

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

–0.167

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k ( t ) = 6 t 2 + 4 t 3 on [ −1 , 3 ]

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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 ,   )

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

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g ( t ) = t t + 3

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

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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 ,   )

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

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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

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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 .

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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

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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?

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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

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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.

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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

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Questions & Answers

what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
Practice Key Terms 9

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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