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30 CHAPTER 2 The Derivative
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Exercises 2-1
1.
( ) ( ) (5) (3)
5 3
5 1
2
4
2
2
f b f a f f
b a
2.
( ) ( ) (1) ( 1)
1 ( )
1 (1)
2
2
2
1
v b v a v v
b a
3.
( ) ( ) (4) ( 3)
4 ( )
12 12
7
0
7
0
v b v a v v
b a
4.
ln(5) ln(1)
5 1
ln(5)0
5
( ) ( ) (5) (1)
5 1
4
4
ln(5)
20
0.0805
z b z a z z
b a
5.
( ) ( ) (6) (0)
6 0
8 2
6
2 2 2
6
2
6
0.2357
q b q a q q
b a
6. average rate of change
change in number of warehouses
change in years
608 512 warehouses=
2012 2008 years
=24 warehouses per year
7. average rate of change
change in number of Gold Star members
change in years
24,846 21,445 thousands
2011 2009
1700.5 thousand members per year
8. average rate of change
change in number of Gold Star members/warehouse
change in number of warehouses
26,736 20,181 thousands
608 512
68.28 thousand Gold Star members per warehouse
2-1 Average Rate of Change 31
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
9. average rate of change
change in Starbucks net revenue
change in years
11.7 9.4 billion dollars
2011 2007
0.575 billion dollars per year
10. average rate of change
number of Starbucks stores
change in years
16,858 16,680 stores
2010 2008
89 stores per year
11. ( ) 2xf x
-7
-2
3
8
13
18
0 1 2 3 4
x
y
(3) (1) 8 2
3 1 2
6
2
3
f f
12. ( ) 5f x
-7
-5
-3
-1
1
3
5
7
0 1 2 3 4 x
y
(3) (1) 5 5
3 1 2
0
2
0
f f
13. 2( ) ( 2)f x x
-2
-1
0
1
2
3
4
5
0 1 2 3 4
x
(3) (1) 1 1
3 1 2
0
2
0
f f
14.
(3) (1) 4 0
3 1 2
4
2
2
f f
32 CHAPTER 2 The Derivative
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
15.
(3) (1) 1 3
3 1 2
2
2
1
f f
16.
700 600 GS members
700 600 warehouses
31719 26116 GS members
100 warehouses
56.03 thousand Gold Star members
per warehouse
m m
Increasing the number of warehouses
from 600 to 700 is predicted to
increase the number of Gold Star
members at an average rate of 56.03
thousand members per warehouse.
17.
5 3 dollars per pound
5 3 years
2.75 2.0028 dollars per pound
2 years
0.374 dollars per pound per year
p p
Between 2010 and 2012, the price of
peanut butter increased at an average
rate of $0.374 per pound per year.
18.
5 1 dollars per pound
5 1 years
5.0025 4.0481 dollars per pound
4 years
0.24 dollars per pound per year
p p
Between 2008 and 2012, the price of
potato chips increased at an average
rate of $0.24 per pound per year.
19.
20 10 thousand jobs
20 10 years
347.8 334.7 thousand jobs
10 years
1.31 thousand jobs per year
j j
Between 2008 and 2018, the number
of jobs in the dry cleaning and
laundry industry is predicted to
increase at an average rate of 1.31
jobs per year.
20.
20 15 thousand jobs
20 15 years
140 150.925 thousand jobs
5 years
2.185 thousand jobs per year
j j
Between 2008 and 2018, the number
of jobs in the ship- and boat-building
industry is predicted to decrease at
an average rate of 2.185 thousand
jobs per year.
2-2 Limits and Instantaneous Rates of Change 33
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Exercises 2-2
1. 0.1
(2 0.1) (2)
0.1
4.41 4
0.1
0.41
0.1
4.1
0.01
(2 0.01) (2)
0.01
4.0401 4
0.01
0.0401
0.01
4.01
0.001
(2 0.001) (2)
0.001
4.004001 4
0.001
0.004001
0.001
4.001
inst. rate of change 4
h
f f
h
f f
h
f f
2. 0.1
(2 0.1) (2)
0.1
6.56 0
0.1
65.6
0.01
(2 0.01) (2)
0.01
0.6416 0
0.01
64.16
0.001
(2 0.001) (2)
0.001
0.064016 0
0.001
64.016
inst. rate of change 64
h
s s
h
s s
h
s s
34 CHAPTER 2 The Derivative
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
3. 0.1
(5 0.1) (5)
0.1
22.4 22
0.1
0.4
0.1
4
0.01
(5 0.01) (5)
0.01
22.04 22
0.01
0.04
0.01
4
0.001
(5 0.001) (5)
0.001
22.004 22
0.001
0.004
0.001
4
inst. rate of change 4
h
w w
h
w w
h
w w
4.
0.1
(5 0.1) (5)
0.1
5 5
0.1
0
0.1
0
0.01
(5 0.01) (5)
0.01
5 5
0.01
0
0.01
0
0.001
(5 0.001) (5)
0.001
5 5
0.001
0
0.001
0
inst. rate of change 0
h
P P
h
P P
h
P P
2-2 Limits and Instantaneous Rates of Change 35
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
5.
0.1
(0.07 0.1) (0.07)
0.1
684.45 572.45
0.1
112
0.1
1120
0.01
(0.07 0.01) (0.07)
0.01
583.2 572.45
0.01
10.75
0.01
1075
0.001
(0.07 0.001) (0.07)
0.001
573.5205 572.45
0.001
1.0705
0.001
1070.5
inst
h
P P
h
P P
h
P P
. rate of change 1070
6.
0
2 2
0
2
0
2
0
0
0
(2 ) (2)(2) lim
(2 ) (2)lim
4 4 4lim
4lim
(4 )lim
lim(4 )
4
h
h
h
h
h
h
f h ff
h
h
h
h h
h
h h
h
h h
h
h
7.
0
2 2
0
2
0
2
0
2
0
0
0
(2 ) (2)(2) lim
16(2 ) 64 16(2) 64lim
16(4 4 ) 64 16(4) 64lim
64 64 16 64 64 64lim
64 16 0lim
16 4lim
lim 16(4 )
64
h
h
h
h
h
h
h
s h ss
h
h
h
h h
h
h h
h
h h
h
h h
h
h
36 CHAPTER 2 The Derivative
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
8.
0
0
0
0
0
(5 ) (5)(5) lim
4(5 ) 2 4(5) 2lim
20 4 2 22lim
4lim
lim(4)
4
h
h
h
h
h
w h ww
h
h
h
h
h
h
h
9.
0
0
0
(25 ) (25)(25) lim
5 5lim
0lim
0
h
h
h
P h PP
h
h
h
10.
2
0
2 2
0
2 2
0
0
2
0
500(1.1449 2.14 ) 500(1.1449
(0.07 ) (0.07)(0.07) lim
500(1 0.07 ) 500(1 0.07)lim
500(1.07 ) 500(1.07)lim
lim
500 (1.1449 2.14 ) 1.1449lim
li
h
h
h
h
h
h h
P h PP
h
h
h
h
h
h
h h
h
2
0
0
0
500 2.14m
500 2.14lim
lim 500(2.14 )
1070
h
h
h
h h
h
h h
h
h
11. 32 ; 3f x x x
h diff quotient
0.1 0.06940
0.01 0.07358
0.001 0.07402
Inst. rate of change ≈ 0.74
12. ( ) 230(0.9) ; 25tP t t
h diff quotient
0.1 1.7305
0.01 1.7388
0.001 1.7396
Inst. rate of change ≈ -1.740
13. 2( ) 5 ; 1.2P r r r
h diff quotient
0.1 12.5
0.01 12.05
0.001 12.005
Inst. rate of change ≈ 12
14. ln( ); 2y x x
h diff quotient
0.1 0.4879
0.01 0.4988
0.001 0.4999
Inst. rate of change ≈ 5.0
15. 3( ) ; 1xg x e x
h diff quotient
0.1 70.271
0.01 61.170
0.001 60.347
Inst. rate of change ≈ 60.0
2-2 Limits and Instantaneous Rates of Change 37
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
16. Using the Calculating the Difference
Quotient for Different Values of h Tech
Card with 0.0001h , we calculate the
instantaneous rate of change.
10 361.94y
In 2007, yogurt production is expected to
be increasing by 361.94 million pounds
per year.
17. Using the Calculating the Difference
Quotient for Different Values of h Tech
Card with 0.0001h , we calculate the
instantaneous rate of change.
55 7.597R
In 2035, the death rate due to heart
disease is expected to be decreasing by
7.597 deaths per 100,000 people per year.
18. Using the Calculating the Difference
Quotient for Different Values of h Tech
Card with 0.0001h , we calculate the
instantaneous rate of change.
100 282.1D
When the price of a DVD player is $100,
the number of DVD players sold is
expected to be decreasing by 282.1
thousand DVD players per dollar of price.
That is, increasing the price by $1 is
predicted to reduce DVD player sales by
282,100 units.
19. Using the Calculating the Difference
Quotient for Different Values of h Tech
Card with 0.0001h , we calculate the
instantaneous rate of change.
4 15.48T
Increasing the number of days that the
ticket authorizes entrance into the park
from four days to five days is predicted to
increase the ticket price by about $15.48.
20. Using the Calculating the Difference
Quotient for Different Values of h Tech
Card with 0.0001h , we calculate the
instantaneous rate of change.
10 2922.36U
In 2015, the population of the United
States is predicted to be increasing by
2922.36 thousand people per year.
21. s
ftvh 0;40 00 . Thus
.40)0(16)( 2 ttts
Find time when the can hits the pool
bottom:
2
2
2
( ) 0
16 40 0
16 40
402.5
16
2.5 1.58, and then
fts (1.58) 50.6
(using the tech. tip)
ftspeed 50.6 s
s t
t
t
t
t
s
22. 0 04 ; 20ft
h ft vs
; Thus
216 20 4
32 20
(1) 12
s t t t
s t t
fts
s
At 1 second, the velocity of the ball is –
12 feet per second.
38 CHAPTER 2 The Derivative
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly
accessible website, in whole or in part.
Exercises 2-3
1. 2( ) 4 ; (1, 3)f x x x
0
2
0
2
0
2
0
2
0
0
0
(1 ) (1)(1) lim
(1 ) 4(1 ) ( 3)lim
1 2 4 4 ( 3)lim
3 2 3lim
2lim
( 2 )lim
lim( 2 )
2
Equation of line
3 ( 2)(1)
3 2
1
2
h
h
h
h
h
h
h
f h ff
h
h h
h
h h h
h
h h
h
h h
h
h h
h
h
y mx b
b
b
b
y
1x
-10
-5
0
5
10
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
x
y
2-3 The Derivative as a Slope: Graphical Methods 39
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
2. 2( ) 6;(2,2)f x x
0
2
0
2
0
2
0
2
0
2
0
0
0
(2 ) (2)(2) lim
(2 ) 6 (2)lim
(4 4 ) 6 2lim
4 4 6 2lim
2 4 2lim
4lim
( 4 )lim
lim( 4 ) 4
Equation of line
2 ( 4)(2)
2
h
h
h
h
h
h
h
h
f h ff
h
h
h
h h
h
h h
h
h h
h
h h
h
h h
h
h
y mx b
b
8
10
4 10
b
b
y x
-10
-5
0
5
10
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
y
3. 2( ) 2 1;(0,1)g x x x
0
2
0
2
0
0
0
(0 ) (0)(0) lim
2 1 (1)lim
2lim
( 2)lim
lim( 2)
2
Equation of line
1 (2)(0)
1 0
1
2 1
h
h
h
h
h
g h gg
h
h h
h
h h
h
h h
h
h
y mx b
b
b
b
y x
-10
-5
0
5
10
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
y
40 CHAPTER 2 The Derivative
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4. 2( ) 4;(3,5)g x x
0
2
0
2
0
2
0
2
0
0
0
(3 ) (3)(2) lim
(3 ) 4 (5)lim
(9 6 ) 4 5lim
5 6 5lim
6lim
(6 )lim
lim(6 )
6
Equation of line
5 (6)(3)
5 18
13
6 13
h
h
h
h
h
h
h
g h gg
h
h
h
h h
h
h h
h
h h
h
h h
h
h
y mx b
b
b
b
y x
-15
-10
-5
0
5
10
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
y
5. 2( ) 4 5;(4, 5)g x x x
0
2
0
2
0
2
0
2
0
0
0
(4 ) (4)(4) lim
(4 ) 4(4 ) 5 ( 5)lim
16 8 16 4 5 5lim
4 5 5lim
4lim
(4 )lim
lim(4 )
4
Equation of line
5 (4)(4)
5 16
21
4 2
h
h
h
h
h
h
h
g h gg
h
h h
h
h h h
h
h h
h
h h
h
h h
h
h
y mx b
b
b
b
y x
1
-30
-20
-10
0
10
20
30
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 x
y
2-3 The Derivative as a Slope: Graphical Methods 41
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
6. 3( ) ; (2,8)f x x
3 2
2
2
2 3
2 3
0
3
0
2 3
0
2 3
0
2
0
2
0
:(2 ) (2 )(2 )
(2 )(4 4 )
8 8 2
4 4
8 12 6
(2 ) (2)(2) lim
(2 ) (8)lim
(8 12 6 ) (8)lim
12 6lim
(12 6 )lim
lim(12 6 )
12
h
h
h
h
h
h
note h h h
h h h
h h
h h h
h h h
so now
f h ff
h
h
h
h h h
h
h h h
h
h h h
h
h h
Equation of line
8 (12)(2)
8 24
16
12 16
y mx b
b
b
b
y x
-20
-15
-10
-5
0
5
10
15
20
-3 -2 -1 0 1 2 3 x
y
7. 3( ) ; (0,0)f x x
0
3
0
3
0
2
0
(0 ) (0)(0) lim
(0 ) (0)lim
lim
lim( )
0
Equation of line (note horizontal line)
0 (0)(0)
0
0
h
h
h
h
f h ff
h
h
h
h
h
h
y mx b
b
b
y
-15
-10
-5
0
5
10
15
-3 -2 -1 0 1 2 3 x
y
42 CHAPTER 2 The Derivative
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
8. 3( ) ; (1,1)f x x
3 2
2
2
2 3
2 3
0
3
0
2 3
0
2 3
0
2
0
2
0
: (1 ) (1 )(1 )
(1 )(1 2 )
1 2
2
1 3 3
(1 ) (1)(1) lim
(1 ) (1)lim
(1 3 3 ) (1)lim
3 3lim
(3 3 )lim
lim(3 3 )
3
Equatio
h
h
h
h
h
h
note h h h
h h h
h h
h h h
h h h
and now
f h ff
h
h
h
h h h
h
h h h
h
h h h
h
h h
n of line
1 (3)(1)
1 3
2
3 2
y mx b
b
b
b
y x
-10
-5
0
5
10
-3 -2 -1 0 1 2 3 x
y
9. 2( ) ( 3) ; (3,0)f x x
0
2
0
2
0
2
0
0
(3 ) (3)(3) lim
(3 3) (0)lim
(0)lim
lim
lim( )
0
Equation of line (note horizontal line)
0 (0)(3)
0 0
0
0
h
h
h
h
h
f h ff
h
h
h
h
h
h
h
h
y mx b
b
b
b
y
-5
-3
-1
1
3
5
7
9
-3 -2 -1 0 1 2 3 4 5 6
x
y
2-3 The Derivative as a Slope: Graphical Methods 43
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
10. 2( ) ( 2) ; ( 1,1)f x x
0
2
0
2
0
2
0
2
0
0
0
( 1 ) ( 1)( 1) lim
( 1 2) (1)lim
( 1) 1lim
2 1 1lim
2lim
( 2)lim
lim( 2)
2
Equation of line
1 (2)( 1)
1 2
3
2 3
h
h
h
h
h
h
h
f h ff
h
h
h
h
h
h h
h
h h
h
h h
h
h
y mx b
b
b
b
y x
-5
-3
-1
1
3
5
7
9
-5 -4 -3 -2 -1 0 1 2 3
x
y
In Exercises 11 – 20, the slope of the
tangent line was computed using the
procedures on the Chapter 2 Tech Card.
11. 6 55.4W thousand dollars per
year. In 2009, the median sales price
of a new home was decreasing at a
rate of $55,400 dollars per year.
Since 6 251.8W , the point
6, 251.8 lies on the tangent line.
55.4
251.8 55.4(6)
584.2
55.4 584.2
y x b
b
b
y x
We evaluate the tangent line at
7x .
55.4 7 584.2
196.4
y
According to the model, the
estimated median sales price of a
new home in 2010 was $196.4
thousand.
12. In the function ( )H t , a decimal
point was mistakenly dropped. The
correct function is 2( ) 0.0023 34.47 2265H t t t .
The solution is based on this
corrected function.
(8) 34.43H square feet per year.
In 2008, the average number of
square feet of a new home was
increasing at a rate of 34.43 square
feet per year. Since (8) 2541H , the
point 8, 2541 lies on the tangent
line.
34.43
2541 34.43(8)
2266
34.43 2266
y x b
b
b
y x
We evaluate the tangent line at
9x .
34.43(9) 2266
2576
y
44 CHAPTER 2 The Derivative
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
According to the model, the
estimated size of a new home in
2009 was 2576 square feet.
13. 13 0.1179R students per year.
In 2008, the student-to-teacher ratio
was dropping at a rate of 0.1179
students per year. Since
(13) 13.09R , the point 13,13.09
lies on the tangent line.
0.1179
13.09 0.1179 13
14.62
0.1179 14.62
y x b
b
b
y x
We evaluate the tangent line at
14x .
0.1179 14 14.62
12.97
y
According to the model, the student-
to-teacher ratio in 2009 was
approximately 12.97.
14. million dollars
(250) 7.775million cassettes
V
When the number of cassette tapes
shipped is 250 million, the value of
the tapes is increasing at a rate of
7.775 million dollars per million
cassette tapes shipped.
(250) 2126.4 million dollars
Tangent at (250,2126.4)
( )
2126.4 7.775 250
182.65
( ) 7.775 182.65
(251) 7.775 251 182.65
2134
V
T s ms b
b
b
T s t
T
The estimated shipment value when
251 million cassette tapes are
shipped is about 2134 million
dollars.
15. Since this is a linear function the
slope of the tangent line
(instantaneous rate of change) will
be the same as the slope of the line.
That is,
0.340P x
Since the derivative does not depend
on the value of x, it will be the same
for all values of x. So the rate of
change in private college enrollment
when 12,752 thousand public
students are enrolled is 0.340
thousand private college enrollments
per thousand public college
enrollments. (In other words, 340
private enrollments per 1000 public
enrollments.)
Since the original function is linear,
it is its own tangent line. So to
estimate the number of private
college enrollments when the public
college enrollment is 12,753
thousand, we simply evaluate the
original function at 12,753.
0.340 12,753 457
3879
y
We estimate that when 12,753 public
college students are enrolled, 3879
private college students are enrolled.
16. We estimate 6W with an average
rate of change.
67, 055 62, 0516
7 5
2502
W
2-3 The Derivative as a Slope: Graphical Methods 45
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
In 2006, the average annual salary in
the motion picture and sound
recording industry was increasing by
approximately $2502 per year.
17. We estimate 3I with an average
rate of change.
4.28 3.903
4 2
0.19
I
In 2007, the retail price of a ½ gallon
of ice cream was increasing by
approximately $0.19 per year.
18. We estimate 4f and 5f with
an average rate of change.
13,344 12,106
4 6195 3
f
13,344 12,832
5 5125 4
f
In 2008-2009, full-time resident
tuition and fees were increasing at a
rate of approximately $619 per year.
In 2009-2010, the rate of increase
was approximately $512 per year.
19. We estimate 3S with an average
rate of change.
6.07 5.793
4 2
0.14
S
In 2007, the price per pound of
boneless sirloin steak was increasing
at a rate of $0.14 per year.
20. We estimate 2B with an average
rate of change.
1.81 1.292
3 1
0.26
B
In 2006, the price per loaf of whole
wheat bread was increasing at a rate
of approximately $0.26 per year.
46 CHAPTER 2 The Derivative
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Exercises 2-4
1.
0
2 2
0
2 2 2
0
2
0
0
0
( ) lim
( ) 4 4lim
2 4 4 4lim
2 4lim
(2 4)lim
lim(2 4)
2 4
h
h
h
h
h
h
f x h f xf x
h
x h x h x x
h
x xh h x h x x
h
xh h h
h
h x h
h
x h
x
2.
2 2
2 2 2
0
0
0
2
0
0
0
( ) 2 1 2 1
2 2 2 1 2 1
( ) lim
lim
lim
2 2lim
(2 2)lim
lim(2 2)
2 2
h
h
h
h
h
h
x h x h x x
x xh h x h x x
g x h g xg x
h
h
h
xh h h
h
h x h
h
x h
x
2-4 The Derivative as a Function: Algebraic Method 47
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
3.
0
2 2
0
2 2 2
0
2
0
0
0
( ) lim
( ) 4 5 4 5lim
2 4 4 5 4 5lim
2 4lim
(2 4)lim
lim(2 4)
2 4
h
h
h
h
h
h
g x h g xg x
h
x h x h x x
h
x xh h x h x x
h
xh h h
h
h x h
h
x h
x
4.
0
3 3
0
3 2 2 3 3
0
2 2 3
0
2 2
0
2 2
0
2
( ) lim
( ) 2 2lim
3 3 2 2lim
3 3lim
(3 3 )lim
lim(3 3 )
3
h
h
h
h
h
h
j x h j xj x
h
x h x
h
x x h xh h x
h
x h xh h
h
h x xh h
h
x xh h
x
48 CHAPTER 2 The Derivative
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
5.
2
0
2 2
0
2 2 2
0
2
0
0
0
( ) 6 9
( ) lim
( ) 6( ) 9 6 9lim
2 6 6 9 6 9lim
2 6lim
(2 6)lim
lim(2 6)
2 6
h
h
h
h
h
h
f t t t
f t h f tf t
h
t h t h t t
h
t th h t h t t
h
th h h
h
h t h
h
t h
t
6.
0
2 2
0
2 2 2
0
2 2 2
0
2
0
0
0
( ) lim
2( ) 1 2 1lim
2( 2 ) 1 2 1lim
2 4 2 1 2 1lim
4 2lim
(4 2 1)lim
lim(4 2 1)
4 1
h
h
h
h
h
h
h
g x h g xg x
h
x h x h x x
h
x xh h x h x x
h
x xh h x h x x
h
xh h h
h
h x h
h
x h
x
(1) 4(1) 1 5
(3) 4(3) 1 13
(5) 4(5) 1 21
g
g
g
2-4 The Derivative as a Function: Algebraic Method 49
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
7.
0
2 2
0
2 2 2
0
2
0
0
0
( ) lim
( ) 2 2lim
2 2 2 2lim
2 2lim
(2 2)lim
lim(2 2)
2 2
(1) 2(1) 2
0
(3) 2(3) 2
4
(5) 2(5) 2
8
h
h
h
h
h
h
f x h f xf x
h
x h x h x x
h
x xh h x h x x
h
xh h h
h
h x h
h
x h
x
f
f
f
8. Since j(x) is linear with a slope of 0, ( ) 0 for all -values.j x x
9. Since W(x) is linear with a slope of 4, ( ) 4 for all -values.W x x
50 CHAPTER 2 The Derivative
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10.
0
2 2
0
2 2 2
0
2 2 2
0
2
0
0
0
( ) lim
3( ) 2 1 3 2 1lim
3( 2 ) 2 2 1 3 2 1lim
3 6 3 2 2 1 3 2 1lim
6 3 2lim
(6 3 2)lim
lim(6 3 2)
6 2
(
h
h
h
h
h
h
h
S x h S xS x
h
x h x h x x
h
x xh h x h x x
h
x xh h x h x x
h
xh h h
h
h x h
h
x h
x
S
1) 6(1) 2 4
(3) 6(3) 2 18 2 16
(5) 6(5) 2 30 2 28
S
S
11.
0
0
0
0
0
0.001
( )( ) lim
3 3lim
(3 )(3 ) 3lim
3 [3 1]lim
(3 1)(3 ) lim
(3 1)(3 )
0.001
1.099(3 )
h
x h x
h
x h x
h
x h
h
hx
h
x
x
P x h P xP x
h
h
h
h
h
2-4 The Derivative as a Function: Algebraic Method 51
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
12.
0
0
0
0
0
0.001
( )( ) lim
3(4 ) [ 3(4 )]lim
3(4 )(4 ) 3(4 )lim
3(4 )[4 1]lim
(4 1)3(4 ) lim
(4 1)3(4 )
0.001
3(1.387)(4 )
4.161(4 )
h
x h x
h
x h x
h
x h
h
hx
h
x
x
x
C x h C xC x
h
h
h
h
h
13. ( ) 5.042 (0.98)xR x
0
0
0
0
0
0.001
( )( ) lim
5.042(0.98 ) [5.042(0.98 )]lim
5.042(0.98 )(0.98 ) 5.042(0.98 )lim
5.042(0.98 )[0.98 1]lim
5.042(0.98 )(0.98 1)5.042(0.98 ) lim
(0.98 1)5.042(0.98 )
0.001
h
x h x
h
x h x
h
x h
h
x hx
h
x
R x h R xR x
h
h
h
h
h
5.042 0.020 (0.98 )
0.1008(0.98 )
x
x
52 CHAPTER 2 The Derivative
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14. We find W t .
2
2 2
2 2
( ) 9.2 55 253
9.2 2 55 55 253
9.2 18.4 9.2 55 55 253
W t h t h t h
t ht h t h
t ht h t h
0
2 2 2
0
2
0
0
( ) ( )lim
9.2 18.4 9.2 55 55 253 9.2 55 253lim
18.4 9.2 55lim
lim 18.4 9.2 55
18.4 55
h
h
h
h
W t h W tW t
h
t ht h t h t t
h
ht h h
h
t h
t
3 18.4 3 55
0.2 thousand dollars per year
W
5 18.4 5 55
37 thousand dollars per year
W
The median sales price was changing more quickly at the end of 2008.
2-4 The Derivative as a Function: Algebraic Method 53
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
15. We find U t .
2
2 2
2 2
( ) 5.1 33 194
5.1 2 33 33 194
5.1 10.2 5.1 33 33 194
U t h t h t h
t ht h t h
t ht h t h
0
2 2 2
0
2
0
0
( ) ( )lim
5.1 10.2 5.1 33 33 194 5.1 33 194lim
10.2 5.1 33lim
lim 10.2 5.1 33
10.2 5.1 0 33
10.2 33
h
h
h
h
U t h U tU t
h
t ht h t h t t
h
ht h h
h
t h
t
t
4 10.2 4 33
7.8 thousand dollars per year
U
6 10.2 6 33
28.2 thousand dollars per year
U
The median sales price was changing more quickly at the end of 2009.
54 CHAPTER 2 The Derivative
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16. We find s t .
2
2 2
2 2
( ) 0.8636 14.39 84.72
0.8636 2 14.39 14.39 84.72
0.8636 1.7272 0.8636 14.39 14.39 84.72
s t h t h t h
t ht h t h
t ht h t h
0
2 2 2
0
2
0
0
( ) ( )lim
0.8636 1.7272 0.8636 14.39 14.39 84.72 0.8636 1.7272 84.72lim
1.7272 0.8636 14.39lim
lim 1.7272 0.8636 14.39
1.7272 0.8636 0 14.39
1.7272 14.39
h
h
h
h
s t h s ts t
h
t ht h t h t t
h
ht h h
h
t h
t
t
10 1.7272 10 14.39
31.66 billion dollars per year
s
9 1.7272 9 14.39
29.93 billion dollars per year
s
31.66 29.93 1.73
According to the model, Wal-Mart net sales were increasing 1.73 billion dollars per
year more rapidly in 2006 than in 2005.
2-4 The Derivative as a Function: Algebraic Method 55
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17. We have
2
2 2
2 2
2 2
2
2.889( ) 2.613( ) 158.7
2.889( 2 ) 2.613 2.613 158.7
2.889 5.778 2.889 2.613 2.613 158.7
5.778 2.889 2.613 2.889 2.613 158.7
5.778 2.889 2.613
P t h t h t h
t ht h t h
t ht h t h
ht h h t t
ht h h P t
0
2
0
2
0
0
0
( ) ( )( ) lim
5.778 2.889 2.613 ( )lim
5.778 2.889 2.613lim
5.778 2.889 2.613lim
lim 5.778 2.889 2.613
5.778 2.613
h
h
h
h
h
P t h P tP t
h
ht h h P t P t
h
ht h h
h
h t h
h
t h
t
10 5.778 10 2.613
55.167
P
According to the model, per capita prescription drug spending was increasing at a rate
of about $55.17 per year in 2000. We want to find out when it was increasing at twice
this rate.
2 55.167 5.778 2.613
110.334 5.778 2.613
112.947 5.778
19.5
t
t
t
t
Since 19t is the end of 2009, we estimate that in mid2010 per capita prescription
drug spending will be increasing at a rate twice that of the 2000 rate.
56 CHAPTER 2 The Derivative
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18. We find b t .
2
2 2
2 2
( ) 4.29 278 2250
4.29 2 278 278 2250
4.29 8.58 4.29 278 278 2250
b t h t h t h
t ht h t h
t ht h t h
0
2 2 2
0
2
0
0
( ) ( )lim
4.29 8.58 4.29 278 278 2250 4.29 278 2250lim
8.58 4.29 278lim
lim 8.58 4.29 278
8.58 4.29 0 278
8.58 278
h
h
h
h
b t h b tb t
h
t ht h t h t t
h
ht h h
h
t h
t
t
15 8.58 15 278
149.3 million barrels per year
b
20 8.58 20 278
106.4 million barrels per year
b
According to the model, the difference in U.S. oil field production and net oil imports
was changing at a rate of –149.3 million barrels per year in 2000 at a rate of –106.4
million barrels per year in 2005.
19.
( ) 2 ( )
2 3
5
g x x f x
x x
x
20. 2
2
2
2
( ) 3 3, so
( ) 0
3 3 0
3 3
1
1
f x x
f x
x
x
x
x
2-5 Interpreting the Derivative 57
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Exercises 2-5
1. (10) 76.64W => The weight of a 10-
year old girl is about 76.64 pounds.
(10) 8.24W => A 10-year-old girl is
gaining weight at about 8.24 pounds
per year.
11 (10) (10)
(76.64 8.24) pounds
84.88 pounds,
W W W
.
the weight of an 11-year old girl.
2. The boys weight gain between their
tenth and eleventh year is about 7.29
pounds, while the girls is about 8.24
pounds per year, so the girls can be
expected to gain about 0.95 more
pounds.
3. In 2005, carbon monoxide pollution in
the United States was 2.27 parts per
million and was decreasing at a rate of
0.248 parts per million per year.
4. The population of Kazakhstan was
predicted to be 13,819 thousand people
in 2015 and expected to be decreasing
at a rate of 97.08 thousand people per
year.
5. The population of India was predicted
to be 1,188,644 thousand people in
2010 and expected to be increasing at a
rate of 17,697 thousand people per
year.
6. In 2005, the 44.69% of highway
accidents in the United States resulted
in injuries. In 2005, the percentage of
accidents with injuries was decreasing
by 0.295 percentage points per year.
7. When there are 14,000 thousand
public college students, there are 4303
thousand private college students and
the number private college students is
increasing by 0.340 thousand private
college students per thousand public
college students. In other words, a
1000 student increase in the number of
public college students corresponds
with a 340 student increase in private
college students.
8. When 20% of people smoke, the heart
disease death rate is 228 deaths per
100,000 people and is increasing by
14.08 deaths per 100,000 people per
percentage point. In other words, a 1
percentage point increase in the
percentage of people who smoke
corresponds with a 14.08 deaths per
100,000 people increase in the heart
disease death rate.
9. When 500 billion dollars are spent on
farm foods at home, 356.9 billion
dollars are spent on farm foods away
from home. When 500 billion dollars
are spend on farm foods at home, the
amount of money spent on farm foods
away from home is increasing at a
rate of 0.7905 billion dollars per
billion dollars spent on farm foods at
home. In other words, a 1 billion
dollar increase in spending on farm
foods at home corresponds with a
0.7905 billion dollar increase in
spending on farm foods away from
home.
58 CHAPTER 2 The Derivative
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10. When 30 million cassette tapes are
shipped the value of the shipped
cassettes is 226 million dollars and is
increasing at a rate of 9.5 million
dollars per million cassettes. In other
words, when 30 million cassettes are
shipped, a 1 million cassette increase
in cassettes shipped corresponds with
a 9.5 million dollar increase in the
value of the shipped cassettes.
One comparatively easy way to
do a tangent line approximation is to
add the value of the derivative to the
value of the function.*
31 (30) 30
226 9.5
235.5
V V V
When 31 million cassette tapes are
shipped, the value of the shipment is
predicted to be 235.5 million dollars
*The strategy of adding the value of
the derivative to the function value is
based on the point-slope form of the
line as demonstrated below.
266 9.5 25
9.5( 25) 266
9.5(26 25) 266 Let 26
9.5 266
235.5
y x
y x
y x
y
y
11. In 2005 there were 42.0 million
Medicare enrollees and the number
of enrollees was increasing at a rate
of 0.427 million enrollees per year.
One comparatively easy way to
do a tangent line approximation is to
add the value of the derivative to the
value of the function.
26 (25) 25
42.0 0.427
42.4
M M M
12. When there are 42 million Medicare
enrollees there are 3331 million
prescriptions and the number of
prescriptions is increasing at a rate of
79.84 million prescriptions per
million Medicare enrollees.
43 (42) (42)
3331 79.84
3411 million prescriptions
p p p
When there are 43 million Medicare
enrollees the number of prescriptions
is predicted to be 3411 million.
13. In 2007, the population of the United
States was 302,366 thousand people
and was increasing at a rate of 2742
thousand people per year.
(18) (17) 17
302,366 2742
305,108
P P P
In 2008, the population of the United
States was predicted to be 305,108
thousand people.
14. In 2003, Wal-Mart net sales were
227.8 billion dollars and were
increasing at a rate of 26.48 billion
dollars per year.
(8) (7) 7
227.8 26.48
254.3
s s s
In 2004, Wal-Mart net sales were
approximately 254.3 billion dollars,
according to the model.
2-5 Interpreting the Derivative 59
© 2015 Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
15. When the amount of private money
spent on health care is 1500 billion
dollars the amount of public money
spent on health care is 1448 billion
dollars and is increasing by 1.228
billion dollars of public money per
billion dollars of private money.
(1501) (1500) 1500
1448 1.228
1449
G G G
When 1501 billion dollars of private
money are spent on health care, 1449
billion dollars of public money is
spent on health care.
16. In 2010, there were 4896 million
pounds of yogurt produced in the
United States and yogurt production
was increasing at a rate of 451.9
million pounds per year.
(14) (13) 13
4896 451.9
5348
y y y
In 2011, yogurt production was
predicted to be approximately 5348
million pounds.
17. When a 3-day ticket is purchased the
price is $157.10 and is increasing at
a rate of 26.90 per additional day of
park admittance.
(4) (3) 3
157.10 26.90
184
T T T
The price of a 4-day ticket is
predicted to be approximately $184.
18. When the life expectancy of an
American male is 75 years, the life
expectancy of an American female is
80.2 years and is increasing at a rate
of 1.22 years of female life per year
of male life.
(76) (75) 75
80.2 1.22
81.42
f f f
When the life expectancy of an
American male is 76 years, the life
expectancy of an American female is
predicted to be 81.42 years.
19. We are looking for where the graph
is increasing most rapidly and where
the graph is decreasing most rapidly.
The increasing portion of the graph
appears to be steepest at
approximately 8t and the
decreasing portion of the graph
appears to be steepest at 14t . In
1989, the number of AIDS deaths
was increasing most rapidly. In
1995, the number of AIDS deaths
were decreasing most rapidly.
20. Since ( )f a f x for all x, f a
is a maximum value for f. In addition
f is a continuous, smooth function on
the interval , , so the tangent
line at ,x f a must be horizontal;
hence 0f a .
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