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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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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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