Blank B.E., Krantz S.G. Calculus: Single Variable
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33. y 52(1/20)(x 2 5) 1 50
35. y 52(1/12) (x 1 2) 2 7
37. y 5 x/6 1 19/6
39. y 52x/28 1 85/14
41. 6 43. 0.05 cents
45. 5 47. 16 49. 25/4
51. y 5 12(x 2 2) 1 13
53. y 5 12(x 1 3) 1 38
55. A : 10; B :3; C : 21; D : 23; E :0; F :20
57. 64π in
59. Positive: 21/2 , t; negative: t ,21/2
61. y 5 18 x/2
1/3
63. 1, 5
65. (1, 26), (1/3, 4/27)
67. 2
ffiffiffiffiffiffiffiffi
2=3
p
;
ffiffiffiffiffiffiffiffi
2=3
p
; 2
ffiffiffi
2
p
;
ffiffiffi
2
p
69. 2
71. a and b have the same sign.
77. a. i.28 ii. 6.4 iii. 7.84 iv. 7.984 b. 8
Section 3.2
1. cos(3) 2 sin(3)
3. 2 5. 592
7. 3 cos(c) 24 sin(c)
9. 36 2 sin(2)
11. 2π 2 1/(πx
2
)
13. 12x
2
1 12x
15. 3 17. 2
ffiffiffi
2
p
and
ffiffiffi
2
p
19.
y
y 6x
2
x
1221
5
5
20
10
15
21.
y
x
y 2x 1
y 2x 1
2 121
2
4
2
4
23. y 5
1
2
x 2
π
3
1
1
2
ffiffiffi
3
p
25. y 5
ffiffiffi
2
p
ðx 2 3π=4Þ2 3
ffiffiffi
2
p
27. 1=ð2
ffiffiffi
x
p
Þ
29. 3=ð2
ffiffiffiffiffiffiffiffiffiffiffiffiffi
3x 1 7
p
Þ
33. 0 35. 232
37. 25x 1 16
39. 21/(1 1 x)
2
41. 2 x
21=2
ð1 1
ffiffiffi
x
p
Þ
22
=2
43. No 45. Yes
47. f ðxÞ5
ffiffiffi
x
p
; c 5 4
49. f (x) 5 1/x, c 5 5
55. (1 2 c
2
)/(1 1 c
2
)
2
59. E( p) 52pqu( p)/q( p)
67.
0.2
1.0
0.8
0.6
0.4
0.6
0.8 1.00.2 0.4
x
y
y x
2
2x
y x
1
4
69.
y
x
2.00.8 1.2 1.6
0.5
1.5
1.0
2.0
y x 2
y
1
x
71. f u(c) 0.22727273
y
x
2.01.20.8 1.6
0.4
0.6
0.5
0.7
y 0.2272727275(x 1)
y
x
x 1
Section 3.2 A-21
73. f u(c) 0.7019192
y
x
2.00.5 1.0 1.5
0.4
0.8
1.2
y sin(x)
y 0.701919 x
p
4
2
2
79. f u(1) does not exist
1.41.0 1.20.80.6
x
1.5
1.0
0.5
1.5
1.0
0.5
y
y
x
3
3x 2
x 1
81. f u(π/2) does not exist
2
10
14
6
2.0
2.5 3.00.5 1.0 1.5
x
y
cos(x)
13
y
x
p
2
83. f u(0) 5 0
y
x
0.4 0.2 0.4
0.1
0.2
0.1
0.2
y
x
sin(x)/x 1
85.
y
x
3.02.5
0.5
0.4
0.8
y (cos(x)
4
)
15
, 0
p
2
87.
y
x
Q
1
1.0
1
P
(0, 0)
1.0 2.0
y
x cos(x
2
)
13
, 0
p
2
89. f u(x) 5 2 cos(2x)
y
x
2
2
p
y 10000
sin 2x sin(2x)
1
5000
Section 3.3
1. 15x
2
2 6x
3. 3x
2
=π 2 π sinðxÞ
5.
4
5
cosðxÞ1
3
5
sinðxÞ1 5
7. 2x(x 1 3) 1 x
2
,or3x
2
1 6x
9. 3x
2
cos(x) 2 x
3
sin(x)
11. 2 sin(x) cos(x)
13. cos
2
(x) 2 sin
2
(x)
15. 7(2x 1 1) sin(x) 1 7(x
2
1 x) cos(x)
17. 2sin(x)/x
2
1 cos(x)/x
19. 21/(1 1 x)
2
21. 212x/(3x
2
1 4)
2
23. 2(1 2 3 sin(x))/(x 1 3 cos(x))
2
25. 227(x
2
2 sin(x))/(3 cos(x) 1 x
3
)
2
A-22 Answers to Selected Exercises
27. 1/(1 1 x)
2
29. 2/(1 1 x)
2
31. 4/x
3
33.
x cosðxÞ2 sinðxÞ
x
2
35.
2x cosðxÞ1 ðx
2
1 7ÞsinðxÞ
cos
2
ðxÞ
37. 2 sin(x)/(1 1 cos(x))
2
39.
xðx 1 1ÞcosðxÞ1 sinðxÞ
ðx 1 1Þ
2
41. 6x
5
1 6x
2
1 4x
3
1 2
43. 2 sin(x) 1 sin
2
(x) sec(x) tan(x)
45. y 5 (5 2 x)/2
47. y 5 6 2 3x
49. y 5 x
51. sec(x)(11 x tan(x))
53. 22 sec(x) tan(x)/(1 1 2 sec(x))
2
55. 5.0 57. 20
59. ( f g)u(2) 5 22, ( f =g)u(2) 5 29/32
61. ( f g)u(21) 5232, ( f
2
)u(21) 5 20, (g
2
)u(21) 5264
63. 2x 65. 26/x
7
67. sin(x) cos(x) 1 x(cos
2
(x) 2 sin
2
(x))
69. 6x(1 1 x
2
)
2
71. a. 2/(c 1 1)
2
b. y 5 2(x 1 c
2
)/(c 1 1)
2
c. 4 5 2(2 1 c
2
)/(c 1 1)
2
d. y 5 2x and y 5 2x/9 1 32/9
73. A 5 4, B 521
81. b. 0.62% decrease
c. E(70) 7/3, E(90) 39/4, E(80) 102/25
83.
n Forward Backward Center
1 .48943 2.48943 0.00000
2 .04934 2.04934 0.00000
3 .00493 2.00493 0.00000
4 .00049 2.00049 0.00000
5 .00005 2.00005 0.00000
85.
n Forward Backward Center
1 .24279 .24401 .24342
2 .24337 .24349 .24343
3 .24342 .24343 .24343
4 .24344 .24343 .24344
5 .24340 .24340 .24350
87. 0.86603, h 5 1/1000 suffices.
89. 21.41421, h 5 1/1000 suffices.
91.
2 4 6
y
1
x
1
93. L(2.70) 0.99322, L(2.73) 1.00435.
a
a
0
L(a)
2.7186
1.0002
2.7190
1.0001
1.0000
0.9999
a
0
2:718 and
d
dx
a
x
0
j
x50
L ð 2:718Þ5 1:
Section 3.4
1. 80x
9
1 30x
26
3. 10x
2/3
2 15x
22/5
5. 2 1 e
x
7.
x 2
3
2
x
25=2
e
x
9. 29x
210
cot (x) 2x
29
csc
2
(x)
11.
1
2
x
23=2
secðxÞ
2xtanðxÞ2 1
13. 2csc(x) cot(x) 2 csc
2
(x)
15. 2cot
2
(x) csc (x) 2 csc
3
(x)
17. tan
2
(x) sec(x) 1 sec
3
(x)
19. e
x
(sin(x) 1 cos(x))
21. 221e
x
/(5 1 7e
x
)
2
23. 2
secðxÞ
1 1 secðxÞ
tan
2
ðxÞ
25. x(x 1 2e
x
2 xe
x
)/(x 1 e
x
)
2
27.
x
22=3
2 1
3ð1 1 x
2=3
Þ
2
Section 3.4 A-23
29. y 5 x 1 4
31. y 52(x 2 π)/π
3
33. y 5 14ðx 2 π=3Þ1 2
ffiffiffi
3
p
35. y 5 8x 1 4πx 2 π
2
37. 2e
2x
39. 22 csc
2
(x) cot(x)
41. x
7
22x
2
1 6x 1 C, where C is any constant
43. x
9
/9 1 x
6
23x
2
/2 1 C, where C is any constant
45. 2
ffiffiffi
x
p
1 C, where C is any constant
47. 8 cot(x) 1 C, where C is any constant
51. tan(x) 2 x 1 C, where C is any constant
53. 27 55. 63 59. 1 63. 0 and 1
65.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p=ð2 2 pÞ
p
71. a. bp /(a 2 bp) b. b
73. tanhu(x) 5 sech
2
(x), sechu(x) 52tanh(x)sech(x)
75. 1 77.
Interval
where f increases
Interval
where
f decreases
Point at which
f has a horizontal
tangent
75. [20.4805, 0.5] [21.3, 20.4805] 20.4805
77. [0, 4.9936] [4.9936, 7] 4.9936
Interval where
f u . 0
Interval where
f u,0
Point(s) at
which f u 5 0
75. [20.4805, 0.5] [21.3, 2.4805] 20.4805
77. [0, 4.9936] [4.9936, 7] 4.9936
79. g
k
u(0) 5 k
y
k
2 121
1
2
1
2
h
y
e
kh2
e
kh2
h
1
10000
81. a. (μ
4
τ
0
p
0
2 τ
0
p
0
1 2μ
2
Δ)/( p 1 μ
2
p
0
)
2
e.
5
4
3
2
1
0
p
123456
(4.520, 1.287)
(2.1, 1.7)
(1.249, 2.875)
0.425p 4.57
4.6003 1.3811p
2.0587 0.1708p
p 0.525
Section 3.5
1. 3 cos(3x)
3. 5e
5x
5. (1/x
2
) sin(1/x)
7. (2x 1 3) cos(x
2
1 3x)
9. 2sin(x) sec(cos(x)) tan(cos(x))
11. 27 sec
2
(1 2 7x)
13. (1 1 x/2)/(1 1 x)
3/2
15. 22x csc(x
2
1 4)(csc (x
2
1 4) 1 cot(x
2
1 4))
17. (2/x
3
) exp(21/x
2
)
19. 3(2 1 cos(x))(2x 1 sin(x))
2
21. x=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
4 1 x
2
p
23. 3 sin
2
(x) cos( x)
25. 6e
x
=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 e
x
p
27. ðf 3 gÞ uðxÞ5 2x 1 7; ðg 3 f ÞuðxÞ5 ð2x
3
1 7xÞ=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
4
1 7x
2
p
29. ðf 3 gÞ uðxÞ5 3x
2
=ðx
3
1 1Þ
2
; ðg 3 f ÞuðxÞ5 3x
2
=ð1 1 xÞ
4
31. 2
x
ln(2) 33. 2(3
2x
) ln(3)
35. 8
x
ln(8) 2 8x
7
37. 2(1/x
2
)(11 e)
1/x
ln(1 1 e)
39. (3
x
ln(3) 2 (3
x
1 2)ln(5))/5
x
41. 32 43. 4
45. 5ð1 1 2xÞ
4
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 ð1 1 2xÞ
5
q
47. 26 cos
2
(2x) sin(2x)
49. 3 sec
2
ð
ffiffiffiffiffiffiffiffiffiffiffiffiffi
3x 1 2
p
Þ=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
3x 1 2
p
51. 5/36 53. 9
ffiffiffiffiffi
10
p
=40
A-24 Answers to Selected Exercises
59. 20.001213 yr
21
,0,0
61. 8
ffiffiffi
2
p
in
2
=min
63. ð1 2 2tÞf uðt
2
Þ; f uðtÞ2 guð
ffiffi
t
p
Þ5 ð1 2 2
ffiffi
t
p
Þf uðtÞ
65. 10
ffiffiffiffiffiffiffiffi
7=x
p
sec
5
ð
ffiffiffiffiffi
7x
p
Þtanð
ffiffiffiffiffi
7x
p
Þ
67. sec
2
ð
ffiffiffi
x
p
Þ=
ffiffiffi
x
p
1 3 sec
2
ðxÞ=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
tanðxÞ
p
69. xð5x
3
1 1Þð25x
3
1 30x 2 1Þ=ðx
2
1 1Þ
3=2
71. 4π
2
x sin
π sinðπx
2
Þ
cos
π sinðπx
2
Þ
cosðπx
2
Þ
73. 3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
ffiffiffiffiffiffiffiffiffiffiffiffiffi
2x 2 1
p
1
ffiffiffiffiffiffiffiffiffiffiffiffiffi
2x 1 1
p
Þ=ð4x
2
2 1Þ
q
75. Yes: c 5 1/3
77. a. P
0
; M
b. P
0
M
2
ðM P
0
Þke
kMt
=
P
0
þðM P
0
Þe
kMt
2
c. 0
79. a.
ffiffiffiffiffiffiffiffi
g=κ
p
b. 4g expð2t
ffiffiffiffiffiffi
gκ
p
Þ=
1 þ expð2t
ffiffiffiffiffiffi
gκ
p
Þ
2
c. 0
81. coshu(ax) 5 a sinh(ax), sinhu(ax) 5 a cosh(ax)
85. 0.38677, 0.26795, 1
87. 21.921 3 10
211
g/yr
89.
y
200
120
140
160
180
y T(t)
t
180100 120 140 16060 8020 40
(25, 175)
(32.425, 155)
(45.1175, 135)
(57.81, 125)
(112.67, 115.5)
t
180
4.0
1.5
2.0
2.5
3.0
3.5
1.0
0.5
y T(t)
y
140 160
(112.67, 0.0273)
(57.81, 0.5461)
(45.1175, 1.092)
(32.425, 2.184)
(25, 3.277)
100 12060 80
20 40
91. v is everywhere continuous; v is differentiable at all
points except at t 5 8. vu is continuous and differentiable
at all points of its domain, which does not include the
point t 5 8.
y
t
20
160
120
80
40
y v(t)
161284
y
200
100
y v(t)
t
2010
Section 3.6
1. 1/2 3. 1/3 5. 1/6 7. 4/5
9. 1/3 11. 22/t
3
13. (t22)
24/5
/5
15. 1=ð2
ffiffiffiffiffiffiffiffiffiffi
1 1 t
p
Þ
17. t=
ffiffiffiffiffiffiffiffiffiffiffiffi
t
2
2 9
p
19. 1=2 1 t=ð2
ffiffiffiffiffiffiffiffiffiffiffiffi
t
2
2 4
p
Þ,or
ðt
2
1 t
ffiffiffiffiffiffiffiffiffiffiffiffi
t
2
2 4
p
2 2Þ=ðt
2
1 t
ffiffiffiffiffiffiffiffiffiffiffiffi
t
2
2 4
p
2 4Þ
21. 2 ln(t)/t
23. 2
t
ln(2)
25. 1=
3t ln(2)(log
2
(t))
2/3
27. 1/x 29. ln(x)
31. (1 2 ln(x))/x
2
33. 3=
x
ffiffiffiffiffiffiffiffiffiffiffi
lnðxÞ
p
35. tan(x)
37. 1/(x ln(x))
39.
1 1 exp(x)
=
x 1 exp(x)
41.
lnð3Þ
lnð2Þ
3
log
2
ðxÞ
x
43. 5=
(5x 1 3)ln(10)
45. 2
x
ln(1 1 4x) 1 4=
(1 1 4x)ln(2)
47. 3x
2
log
10
(6 2 x)2x
3
=
(6 2 x)ln(10)
49. 1=
x(5)ln(x)
Section 3.6 A-25
51.
4ð2x 1 1Þðx
2
1 xÞ
3
ðx
3
1 x
2
2 1Þ
3
ðx
2
1 2Þ
2
1
3ð3x
2
1 2xÞðx
2
1 xÞ
4
ðx
3
1 x
2
2 1Þ
2
ðx
2
1 2Þ
2
2
4xðx
2
1 xÞ
4
ðx
3
1 x
2
2 1Þ
3
ðx
2
1 2Þ
3
53. 3x
3x
lnðxÞ1 1
55.
2lnðxÞ1 1
x
x
2
11
57. 2/x
59. 21/(2x)
61. s 5254(t 2 1/3) 1 6
63. s 5 (t 2 3)/5 1 1
65. f
21
(5) 5 2, ( f
21
)u(5) 5 1/14
67. f
21
(3/2) 5 1, ( f
21
)u(3/2) 5 1
69. f
21
(0) 5 1, ( f
21
)u(0) 5 1
71. f
21
(9) 5 64, ( f
21
)u(9) 5 128 ln(2)/3
73. 1/7
75. 2x
ln(x)
ln(x)/x
77. (log
2
(x))
x
(ln(log
2
(x)) 1 1/(ln(2)log
2
(x)))
79. 2 ln(x)/x
81.
ðg 3 f Þ
21
uðuÞ5 1=
guðtÞ f uðsÞ
where t 5 f ðsÞ
and u 5 gðtÞ:
83. c 5 e, γ 5 1
85. a. length / time c. y 5 C d. domain: [0,N), image:
(C, c(0)]; e. domain: (C, c(0)], image: ½0; NÞ; c
1
ðsÞ5
2
V
kA
ln
s C
cð0ÞC
f : cuðtÞ5ðCkA=VÞexpð2kAt=VÞ;
ðc
1
ÞuðsÞ52V ðkAðs CÞÞ
1
87. p 5 p
0
2
p
0
q
0
Eðp
0
Þ
ðq 2 q
0
Þ
89. 0.3065
91. 0.02701
93.
x
x 0.4602 (y2)0.8526
y
32.521.51
Graph of f
1
for f
(
x
)
1xexp
(
x
)
/2
1
0.5
0
0.5
0.8526
1
(2, f
1
(2))
x f
1
(y)
95.
1.2
0.5352
0.8
0.4
0
1.6 2 2.4 2.8
x
y
Graph of f
1
for f (x)
sin
2
(
x
)
cos
(
x
)
2.6
(2, f
1
(2))
x f
1
(y)
x 0.7207 (y 2) 0.5352
Section 3.7
1. 10x 2 6, 10, 0
3. 21/x
2
,2/x
3
, 26/x
4
5. 1/x, 21/x
2
,2/x
3
7. 3=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
6x 1 5
p
; 29=ð6x 1 5Þ
3=2
; 81=ð6x 1 5Þ
5=2
9. 23 sin(3x),29 cos(3 x), 27 sin(3x)
11. 2e
2x
,4e
2x
,8e
2x
13. 1/(x 1 1), 21/(x 1 1)
2
, 2/(x 1 1)
3
15. 22/(x 21)
2
, 4/(x 21)
3
, 212/(x 21)
4
17. 90x(x
2
1 1)
2/3
, 30(x
2
1 1)
21/3
(7x
2
1 3),
40x(x
2
1 1)
24/3
(7x
2
1 9)
19. cos(2x) 2 2x sin(2x), 2 4 sin(2x) 2 4x cos(2x),
212 cos(2x) 1 8x sin(2x)
21. 2
x
ln(2), 2
x
ln
2
(2), 2
x
ln
3
(2)
23. e
x
sin(x) 1 cos(x)
,2e
x
cos(x), 2e
x
cos(x)2 sin(x)
25. 1=
xln(3)
, 21=
x
2
ln(3)
,2=
x
3
ln(3)
27. cos(x)
29. 696
31. 2360x
2
1 840x
33. 64 sin(4x 1 3)
35. π/2 37. 0 m/s
2
39. 216 m/s
2
41. 0 m/s
2
43. 6 m/s
2
45. 38 m/s
2
, 74 m/s
2
47. 128 2 32t ft/s, 232 ft/s
2
49. 324 ft 51. 2176 ft/s
53. (26 2 3t) ft/s
2
59. x
3
2 4x
2
1 2x 1 2
61. 2 cos(x) 2 4x sin(x) 2 x
2
cos(x)
A-26 Answers to Selected Exercises
63. (2 ln(x) 2 3)/x
3
65. x
23/2
(x
2
1 x21/4)e
x
69. P
0
ðxÞ5 1; P
1
ðxÞ5 x; P
2
ðxÞ5 ð3x
2
2 1Þ=2;
P
3
ðxÞ5 ð5x
3
2 3xÞ=2; P
4
ðxÞ5 ð35x
4
2 30x
2
1 3Þ=8
71. H
0
ðxÞ5 1; H
1
ðxÞ5 2x; H
2
ðxÞ5 4x
2
2 2;
H
3
ðxÞ5 8x
3
2 12x; H
4
ðxÞ5 16x
4
2 48x
2
1 12
79.
0
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
y
x
P
f(x) x
g(x)
h(x)
k(x)
81.
f(x) sin
2
(x)
g(x)
h(x)
k(x)
0
0.5 1 1.5 2
0.5
1
x
y
P
83. 20.0061 85. 20.0069
Section 3.8
1. 6y
1/2
(dy/dx)
3. 2y
21/2
2 xy
23/2
(dy/dx)
5. e
y
1 xe
y
(dy/dx)
7. 28/5 9. 21/16 11. 25/12 13. 21/5
15. y 5 13/7 2 3x/14
17. y 5
ffiffiffi
3
p
ðx 2 π=3 Þ=2 1 1=4
19. y 5 x/2
21. y 5 x/4 2 1
23. y 5 12/5 2 2x/5
25. y 527(x 2 2) 1 2
27. y 52(x 2 π/4) 1 π/4
29. 21/4; 23/32
31. 29/8; 135/128
33. 21; 22
35. y 5 3
x/2 2 1
37. y 5 x/7 2 32/63
39. y 5 ð3 1 2
ffiffi
ffi
2
p
Þðx 2 π=4 2
ffiffiffi
2
p
=2Þ1 π=4 1
ffiffiffi
2
p
=2
41. y 5 124x/45 2 16/15
43. y 52
ffiffiffi
3
p
e
ð12
ffiffi
3
p
Þ=2
ðx 2 e
ffiffi
3
p
=2
Þ1 e
1=2
45. y 5 ln(27)(x 2 5)/ln(4) 1 5
51. 1
53. 4(ln(2) 2 1)/(2 ln(2) 2 1)
55. a
2
57. 2Ax
27=2
y
3
ðb 1 3 x=2Þðy 2 2Þ
21
expðb=xÞ
59. a. 20.289 b. 0.503
61. 21.0006
63. 22.862
65. y
0
5210.6657; y 521.00018x 2 0.6639
67. y
0
5 4.4599; y 523.9162x 2 3.4214
69.
1
0.5
1 0.5 0
0.5
0.5 1
1
x
P
0
y
x
3
2xy y
3
0
y
0.69558 x
0.58261
71.
0
1234
1.5
2
2.5
3
3.5
4
x
P
0
y
x
3
x
2
y
2
y
3
0
y
0.38516 x
2.74256
Section 3.8 A-27
73.
5
055
5
x
P
0
(4,3 )
y
Tear Drop Curve x
8 cos(t),
y 8 sin
(
t
)
(
sin
(
t/2
))
2
y
3 x 73
33
75.
0.4 0.6 0.8
1
0
1
P
0
(1/2,1)
y
y 2x
x
x
(
1t
2
)
/
(
13t
2
)
, y 4t/
(
13t
2
)
77.
10
0.4
0.2
0.2
0.4
x
y
Cubic of Tschirnhaus x
1 3t
2
, y t 3t
3
P
0
(1/4,1/8)
y
5 x 1
12 48
79. (4
1/3
,0)
81. quð4Þ520:4317; qð4:2Þ5 4:0680; q ð 3:8Þ5 4: 2408;
quð4ÞD
0
qð4; 0:4Þ520:432
Section 3.9
1. 1.975 3. 0.4985417 5. 4.05
7. 3.825 9. 0.48891
11. 2.6 2 π /2 1.0292
13. 1.138968
15. 2 1 20/exp(3) 2.995741
17. 0.83
19. 1 1 0.13 ln(16) 1.3604
21. 0.9627
23. 1.2 2 π /4 0.4146
25. c 5 25; estimate: 49/10; error: 5 1.02 3 10
23
27. c 5 9; estimate: 2.02917; error: 5 7.5 3 10
24
29. c 5 60; estimate: 1=2 1
ffiffiffi
3
p
π=72; error : 5 0:002
31. 1/3 2 2x/9 33. x/8 35. 1 2 x
37. 1=2 2
ffiffiffi
3
p
ðx 2 π=3Þ=2
39.
10
3
ln
5
3
x 2
1
2
1
5
3
41. 1
43. 1 1 p(a 2 b)x
47. 2.6 49. 3/2
51. Slope: 20.26124; 3235 units at $6.80; 3287 units at $6.60
53. 365987
55. One-step approximation: 3.03333; three-step approx-
imation: 3.03309; exact: 3.03296 . . .
57. One-step approximation: 1.10364; three-step approx-
imation: 1.10109; exact: 1.09861 . . .
59. y
1
5 2.6; z
1
5 2.68; exact solution: 4 exp(0.2) 2 2.2(about
2.6856)
61. y
1
5 1.5; z
1
5 1.8828125; exact solution: 5=32 1
11=ð4
ffiffiffi
e
p
Þðabout 1:82421Þ
63. Approximate demand: 9989; exact demand: 10000; rela-
tive error: 0.11%
65. Approximate demand: 3072693; exact demand: 3200000;
relative error: 3.98%
Section 3.10
1. π/2 3. 2π/2 5. π/3 7. π/4
9. 2π/3 11. π/4 13. 2π/4
15. 2π/4 17. 5π/6
19. 21=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2x 2 x
2
p
21. arcsinðxÞ1 x=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2 x
2
p
23. 21=
lnð2Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2 x
2
p
arccosðxÞ
25. e
x
/(1 1 e
2x
)
27. arctan(x) 1 x/(x
2
1 1)
29. 2x arcsecðxÞ1 x=
ffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
2 1
p
31. 22=ðx
ffiffiffiffiffiffiffiffiffiffiffiffiffi
x
4
2 1
p
Þ
A-28 Answers to Selected Exercises
33. 21=
2
ffiffiffi
x
p
ðx 1 1Þ
35. 3 cosh(3x)
37. coth(x)
39.
x sinh(x) 2 cosh(x)
x
2
41. sech
2
(x)/tanh(x)
43. sech
2
(tan x) sec
2
(x)
45. 2sech ð2
ffiffiffi
x
p
Þtanhð
ffiffiffi
x
p
Þ=ð2
ffiffiffi
x
p
Þ
47. 5=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 25x
2
p
49. coshðxÞ=
1 1 sinh
2
ðxÞ
51. 2csc(x)
53. 2csch(x)
55. arcsinðxÞ5 π=6 1 ð2=
ffiffiffi
3
p
Þðx 2 1=2Þ
57. π=3 1 ðx 2
ffiffiffi
3
p
Þ=4
77. pðu; vÞ5 u
2
1 v
4
2 6v
2
2 2uv
2
2 6u 1 9
79. pðu; vÞ5 u
2
1 v
6
2 2uv
3
2 u
81. f u(2.1) 0.0887
1.5
1
0.5
0
12345
y
x
y 0.0887x 1.3356
y arcsec(1x
4
)
83. f u(4.5) 0.47213
x
y
4
2
0246810
y 0.47213x 0.12892
y sinh( )
1
x
x
85. f u(2.5) 0.34815
y
x
2
0
123456
2
y arcsinh (log
2
(x))
y 0.34815x 0.22138
87.
y
x
y gd(x)
y gd
1
(x)
y
p
2
x
p
2
y
p
2
x
p
2
REVIEW EXERCIS ES FOR CHAPTER 3
1. y 5 12x 1 16
3. y 5 (23/32)x 1 9/2
5. y 5 ð6 1 π
ffiffiffi
3
p
Þx 2 π
ffiffiffi
3
p
=6
7. y 5 (7/3)x 2 2
9. y 5 (3/e
2
)x 2 2/e
11. y 523 ln(2)x 1 3
13.
1
2
x
21=2
15. 2(x 1 6)/ x
3
17. 3x
2
2 3/x
4
19. 1 1 2/x
2
21. 29x
2
/(2 1 x
3
)
4
23. cos(x)
25. 22 sin(2x 1 π/4)
27. 2 sec(2x)tan(2x)
29. x
22
csc
2
(1/x)
31. 30x sin
2
(5x
2
)cos(5x
2
)
33. 3x
2
sin(x
2
) 1 2x
4
cos(x
2
)
35. (1 1 sin(x) 1 cos(x))/(1 1 cos(x))
2
37. 23x sinð3x
2
Þ=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 cosð3x
2
Þ
p
39. (1 1 3x) e
3x
41. (x
2
1 x 1 1) e
x
/(x 1 1)
2
43. 22
2x
ln(2)
45. 1/x 2 1/x
2
47. x
2
(1 1 3 ln(x))
49. 1/(x ln(3x))
51. 2x log
2
(x 1 2) 1 x
2
/((x 1 2) ln(2)
53. ln(30)2
x
3
x
5
x
Review Exercises for Chapter 3 A-29
55. 22/(x
2
1 4) 57. 21
59. 2x sinh(x
2
)
61. e
x
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 e
2x
p
63. 60 232t; 232
65. 240t/(1 1 t
2
)
2
; 40(3t
2
2 1)/(1 1 t
2
)
3
67. 5e
25t
; 225e
25t
69. 1/15 71. 1 73. 1/(2e)
75. 1/(1 1 exp(y))
77. 2x/(1 22y)
79. y
2
/(x(y
2
2 1))
81. 1/(1 1 3y
2
); 26y/(1 1 3y
2
)
3
83. 2 sin(x)/cos(y); 2 sec
2
(y) (cos(y) cos(x) 1 2 sin
2
(x)tan(y))
85. y 5 4x 2 6
87. y 5
ffiffiffi
3
p
x 1 ð2 2 π
ffiffiffi
3
p
=3Þ
89. y 52x 1 4
91. 5.825 93. 21.3
95. f uðcÞ2:214; f uuðcÞ4:144
97. f uðcÞ27:605; f uuðcÞ30:5
Section 4.1
1. 60 3. 1 5. 3 7. 22 9. 23
11. s
0
524
13. v
0
522
15. s
0
5224
17. s
0
5 1/2; tangent line: y 5 ðx 1 13Þ=6
19. v
0
521/2; tangent line: y 5 4x 2 2
21. s
0
5232/23; tangent line: y 5 ð16x 2 2Þ=23
23. 10 in
2
/min
25. 27 cm
2
/s
27. 3
29. 0.48 ft/s
31. 1/(384π) cm/min
33. 81π/8 35. 115 ft/s
37. 1 39. 28
41. 120/π cm/min
43. 0.15 Ω/s
45. (1/4, 1/2)
47. (1/2) cm/min
51. 405π cm
3
/min
53. 2.349 cm/s
Section 4.2
1. Local minimum for c 5 2
3. Local maximum for c 5 0
5. Local minimum for c 5 (4n 1 3)π/8, integer n; local
maximum for c 5 (4n 1 1)π/8, integer n
7. 6 9. 21, 0, 1 11. 25, 1 13. 1
15. 0 17. 1 19. e 21. 3
23. 1 25. 4/9 27. ð24 1
ffiffiffiffiffi
13
p
Þ=3 29. 0
31. 1/31 33. 5 35. 5x 1 C
37. x
3
/3 1 πx 1 C
39. sin(x)1 C
41. 15 43. 11 45. 2 1 e
3
47. 3/2
49. 3π/4 51. π/4 53. 1 1
ffiffiffi
3
p
55.
ffiffiffiffiffi
19
p
57. f u(5) does not exist.
59. f u(0) does not exist.
91.
y
x
1
2
4
y 3x
2
2 sin(x)
y x
3
2x cos(x)
Local
maximum
Local
minimum
x 0.6767 x 0.9704
93.
x
y 1 4xe
x
2
y x 2 exp(x
2
)
Local
maximum
Local minimum
x
1.277 x 0.2687
A-30 Answers to Selected Exercises