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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 is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
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
Jhon Reply
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
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
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?
Diddy Reply
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
Practice Key Terms 9

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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