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Real-world applications

For the following exercises, explain the notation in words. The volume f ( t ) of a tank of gasoline, in gallons, t minutes after noon.

f ' ( 30 ) = −20

At 12:30 p.m. , the rate of change of the number of gallons in the tank is –20 gallons per minute. That is, the tank is losing 20 gallons per minute.

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f ' ( 200 ) = 30

At 200 minutes after noon, the volume of gallons in the tank is changing at the rate of 30 gallons per minute.

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For the following exercises, explain the functions in words. The height, s , of a projectile after t seconds is given by s ( t ) = 16 t 2 + 80 t .

s ( 2 ) = 96

The height of the projectile after 2 seconds is 96 feet.

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s ( 3 ) = 96

The height of the projectile at t = 3 seconds is 96 feet.

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s ( 0 ) = 0 , s ( 5 ) = 0.

The height of the projectile is zero at t = 0 and again at t = 5. In other words, the projectile starts on the ground and falls to earth again after 5 seconds.

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For the following exercises, the volume V of a sphere with respect to its radius r is given by V = 4 3 π r 3 .

Find the average rate of change of V as r changes from 1 cm to 2 cm.

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Find the instantaneous rate of change of V when r = 3  cm .

36 π

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For the following exercises, the revenue generated by selling x items is given by R ( x ) = 2 x 2 + 10 x .

Find the average change of the revenue function as x changes from x = 10 to x = 20.

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Find R ' ( 10 ) and interpret.

$50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold.

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Find R ' ( 15 ) and interpret. Compare R ' ( 15 ) to R ' ( 10 ) , and explain the difference.

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For the following exercises, the cost of producing x cellphones is described by the function C ( x ) = x 2 4 x + 1000.

Find the average rate of change in the total cost as x changes from x = 10  to  x = 15.

$21 per unit

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Find the approximate marginal cost, when 15 cellphones have been produced, of producing the 16 th cellphone.

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Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21 st cellphone.

$36

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Extension

For the following exercises, use the definition for the derivative at a point x = a , lim x a f ( x ) f ( a ) x a , to find the derivative of the functions.

f ( x ) = 5 x 2 x + 4

f ' ( x ) = 10 a 1

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f ( x ) = x 2 + 4 x + 7

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f ( x ) = 4 3 x 2

4 ( 3 x ) 2

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Chapter review exercises

Finding Limits: A Numerical and Graphical Approach

For the following exercises, use [link] .

Graph of a piecewise function with two segments. The first segment goes from (-1, 2), a closed point, to (3, -6), a closed point, and the second segment goes from (3, 5), an open point, to (7, 9), a closed point.

lim x −1 + f ( x )

2

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lim x −1 f ( x )

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lim x 1 f ( x )

does not exist

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At what values of x is the function discontinuous? What condition of continuity is violated?

Discontinuous at  x = 1 ( lim x a f ( x )  does not exist ) , x = 3   ( jump discontinuity ) , and  x = 7   ( lim x a f ( x )  does not exist ) .

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Using [link] , estimate lim x 0 f ( x ) .

x F ( x )
−0.1 2.875
−0.01 2.92
−0.001 2.998
0 Undefined
0.001 2.9987
0.01 2.865
0.1 2.78145
0.15 2.678

3

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For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x approaches a .   If the function has limit as x approaches a , state it. If not, discuss why there is no limit.

f ( x ) = { | x | 1 , i f x 1 x 3 , i f x = 1    a = 1

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f ( x ) = { 1 x + 1 , i f x = 2 ( x + 1 ) 2 , i f x 2    a = 2

lim x 2 f ( x ) = 1

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f ( x ) = { x + 3 , i f x < 1 x 3 , i f x > 1    a = 1

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Practice Key Terms 7

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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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