Blank B.E., Krantz S.G. Calculus: Single Variable
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Section 1.6
1. sinðπ=6Þ5 1=2; cosðπ=6Þ5
ffiffiffi
3
p
=2; tanðπ=6Þ5
ffiffiffi
3
p
=3;
cscðπ=6Þ5 2; secðπ=6Þ5 2
ffiffiffi
3
p
=3; cotðπ=6Þ5
ffiffiffi
3
p
3. sinð2π=3Þ5
ffiffiffi
3
p
=2; cosð 2π=3Þ521=2;
tanð2π=3Þ52
ffiffi
ffi
3
p
; cscð2π=3Þ5 2
ffiffiffi
3
p
=3;
secð2π=3Þ522; cotð2π=3Þ52
ffiffiffi
3
p
=3
5.
ffiffiffi
3
p
=4 7. ð
ffiffiffi
3
p
1 1Þ=2 9. 3 11. 1
13. 21 15. 2
ffiffiffi
2
p
=3 17. 24/25 19.
ffiffiffiffiffiffiffiffiffiffiffiffi
25=26
p
21. sin, cos, tan, csc, sec, cot
23. tan, cot
25.
y
6
t
642
1
1
4 2
y sin(2t)
27.
y
6
t
642
1
1
4 2
y sin(2t)
29. x 5 cos(t), y 5 sin( t)
31. x 5 sin(t), y 5 cos( t)
33. rθ
35. The first equation holds only for θ 5 nπ/2 where n is an
integer. The second equation is an identity: It holds for
every value of θ.
45. sinð7π=12Þ5
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 1
ffiffiffi
3
p
q
; cosð7 π=12Þ52
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2
ffiffiffi
3
p
q
;
tanð7π=12Þ522 2
ffiffiffi
3
p
;
cscð7π=12Þ5 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2
ffiffiffi
3
p
p
; secð7π=12Þ522
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 1
ffiffiffi
3
p
p
;
cotð7π=12Þ522 1
ffiffiffi
3
p
51. C 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A
2
1 B
2
p
; φ satisfies sin(φ) 5 A/C and cos(φ) 5
B/C
55. The line segment x/a 1 y/b 5 1 traversed from (a,0) to
(0, b) and then back to (a, 0).
57. y 5 ab/x,0, x ,N
59. 2π 61. π 63. 2π
65. T(t) 5 27.5 1 23.5 sin(πt/6 2 π/2).
73. T(t) 5 59 1 19 sin( πt/6 2 7π/12)
T
t
12
30
80
T(t) 59 19 sin
pt
6
7p
12
75. ω 5 7; the frequency of the modulated amplitude is 1
x
y
77.
60
0.001
t
y
y 10
20
tan
t
3
t
3
2sin 10
20
t
y 10
20
t 3ⴢ 10
20
sin(t)
2 cos(t)
REVIEW EXERCIS ES FOR CHAPTER 1
1. (211, 1) 3. [0, 3]
5. (2N, 21/6),(1/2,N)
7. (2N, 22],(0, 3]
Review Exercises for Chapter 1 A-11
9. 13 11. (3,27/2)
13. circle; axes of symmetry: x 5 1, y 521; radius:
ffiffiffi
2
p
15. ellipse; axes of symmetry: x 521, y 5 2
17. y 5 3x 2 6
19. y 52x 2 4
21. y 526(x 1 2) 1 1
23. y 5 2(x 1 2) 1 1
25. x/4 1 y/(23) 5 1
27. x/(3/2) 1 y/3 5 1
29. (2, 3)
31. (21,28) and (4, 7)
33. (2N,N)
35. (2N,5),(5,N)
37. (2N, 22),(22,N)
39. [25, 23),(23,
3),(3, 5]
41. 1/2, 2/3, 3/4, 4/5, 5/6
43. 4, 8, 14, 24, 42
45. 7, 15, 31, 63, 127
47. 16 49. 41
51. 4x 1 12
ffiffiffi
x
p
1 9
53. 2x
2
/(2 1 x
2
)
55. 2x
4
1 4x
2
1 5
57. 2
ffiffiffiffiffiffiffiffiffiffiffi
1 1 x
p
1 3
59. 2(3x 1 4)/(2x 1 3)
61. T 5 [1/5, 1/2]; f
21
(t) 5 1/t 2 1
63. T 5 ½ 2; 5; f
21
ðtÞ521 1
ffiffiffiffiffiffiffiffiffiffi
t 2 1
p
65. Onto but not one-to-one
67. One-to-one but not onto
69. Translate left 2 units, down 1
71. Translate right 3 units, down 9
73. The line segment from (21/3, 0) to (0, 1/2)
75. The line segment from (1, 0) to (0, 1)
77. sinðAÞ5 3=
ffiffiffiffiffi
13
p
; cosðAÞ5 2=
ffiffiffiffiffi
13
p
; tanðAÞ5 3=2;
cotðAÞ5 2=3; secðAÞ5
ffiffiffiffiffi
13
p
=2; cscðAÞ5
ffiffiffiffiffi
13
p
=3
79. sinðAÞ5 1=
ffiffiffiffiffi
10
p
; cosðAÞ5 3=
ffiffiffiffi
ffi
10
p
;
tanðAÞ5 1=3; cotðAÞ5 3; secðAÞ5
ffiffiffiffiffi
10
p
=3; cscðAÞ5
ffiffiffiffiffi
10
p
81. sinðα 1 π=2Þ5 5=13; cosðα 1 π=2Þ5212=13;
tanðα 1 π=2Þ525=12; cotðα 1 π=2Þ5212=5;
secðα 1 π=2Þ5213=12; cscðα 1 π=2Þ5 13=5
83. 2 85. 22
ffiffi
ffi
2
p
87. 3
ffiffiffi
7
p
=8 89. 224/7
Section 2.1
1. 5 3. 57 5. 21
7. 24 9. 10 11. 21/14
13. Does not have a limit.
15. lim
x-2
f ðxÞ522
17. lim
x-5
f ðxÞ5 7
19. 0.01 21. 0.002 23. 0.2 25. 0.0019
27. lim
x-2
2
f ðxÞ5 0 6¼ 4 5 lim
x-2
1
f ðxÞ
29. a. The domains of g, f, h, and k are, respectively,
fx 2 R : x 6¼ 1g; R; R; and R: b. f and h
c. lim
x-1
gðxÞ5 lim
x-1
f ðxÞ5 lim
x-1
hðxÞ5 lim
x-1
kðxÞ5 2
31. a. The domains of f, g, k, and h are, respectively, R; R; R;
and fx 2 R : x 6¼ 0g: b. f and k
c. lim
x-0
gðxÞ¼lim
x-0
f ðxÞ¼lim
x-0
hðxÞ¼lim
x-0
kðxÞ¼0
33. lim
x-0
1
HðxÞ5 1; lim
x-0
2
HðxÞ5 0; lim
x-0
HðxÞ does not
exist
35. 3.28 3 10
23
37. a. Does not exist. b. 0 c. 1 d. 21
39. See Figure for Exercise 41, Section 1.4;
lim
x-n
1
bxc5 1 1 lim
x-n
2
bxc 6¼ lim
x-n
2
bxc
41. a. 3 b. 2.5 c. 2.1 d. 2 1 h e. 2
43. a. 4 b. 0.1 c. 0.01 d. ε
45. a. 8 b. 0.1 c. 0.01 d. ε
47. 6 1 h;6
49. h 1 4; 4
51. 0
53. |x 2 2| , 0.0095
1.010
1.005
1.000
0.995
0.990
0.985
2.0001.995 2.005
f (x) 4x
2
0.0095 0.0095
e 0.01
e 0.01
A-12 Answers to Selected Exercises
55. |x 2 10| , 0.00046
100.010
100.005
100.000
99.995
99.990
99.985
10.00009.9996 9.9998 10.0002 10.0004
0.00046 0.00046
e 0.01
e 0.01
57. δ 5 0.5166368178 (or smaller)
59. 7.82 3 10
24
mm
61. a.
xf(x)
π/22 1/10 20.99833
π/22 1/100 20.99998
π/2 2 1/1000 20.99999
b. 21 c. δ 5 0.24 will do
d.
0.990
0.995
1.000
1.005
1.010
1.61.4 1.5 1.7 1.8
0.24 0.24
p
2
63. a.
xf(x)
1/10 0.16658
1/100 0.16666
1/1000 0.16666
b. 1/6 c. δ 5 1.05 will do
d.
0.5 10.5
1.05
1
1
6
0.01
1
6
1
6
0.01
1.05
0.18
0.17
0.16
65. a. 4 b. h , 0.01662 will do.
c. h , 0.0016662 will do.
67. a. 3 b. h , 0.45187 will do.
c. h , 0.044205 will do.
69. a.
vðtÞ5
sinðt 1 0:0001Þ2 sinðt 2 0:0001Þ
=0:0002
642
x
1.0
0.5
1.0
0.5
y
x(t) sin(t)
y(t) (sin(t 0.0001)
sin(t 0.0001))0.0002
b. vðtÞ is the greatest near t 5 0 and 2π; vðtÞ is the most
negative near t 5 π
c.
vðtÞ is zero near t 5 π/2 and 3π/2
d.
vðtÞ, 0 for t in I 5 [π/2, 3π/2]; x(t) decreases on I;
vðtÞ, 0 suggests that x is decreasing.
e. cos ( t)
Section 2.2
1. 6 3. 4 5.
ffiffiffi
2
p
7. 0
9. lim
x-5
2
f ðxÞ5 15 6¼ 4 5 lim
x-5
1
f ðxÞ
11. lim
x-5
2
f ðxÞ5 0 6¼ 24 5 lim
x-5
1
f ðxÞ
13. lim
x-4
2
f ðxÞ5 17 5 lim
x-4
1
f ðxÞ
15. lim
x-4
2
f ðxÞ5 64 5 lim
x-4
1
f ðxÞ
17. 14 19. 25 21. 2 23. 24/5
25. 1/2 27. 6 29. 22
Section 2.2 A-13
31. ðπ 2
ffiffiffi
π
3
p
Þ=ð
ffiffiffi
π
p
2 3Þ
33. 0 35. 1 37. 3
39. lim
x-2
f ðxÞ5 2
41. lim
x-0
f ðxÞ5 25
43. lim
x-1
1
f ðxÞ5 0 and lim
x-2
2
f ðxÞ5 0
45. lim
x-ð22Þ
1
f ðxÞ5 2 and lim
x-2
2
f ðxÞ5 0
Neither lim
x-0
1
f ðxÞ nor lim
x-0
2
f ðxÞ exist
47. 21 49. 28 51. 36 53. 28
55. Does not exist.
57. 3 59. 2 61. 0 63. π/180
65. 6 67. 4 69. 1 71. 21/8
73. 3/2 75. 21 77. 3/2 79. 1
87. 0.00062 will do.
y
y
x
3
x
2
2.9x2
x
40
39.98
39.96
39.94
39.92
39.9
39.88
2 2.0002 2.0004 2.0006
0.1
89. 0.046 will do.
1.1
1
0.9
1.96 1.98 2 2.02 2.04
0.046 0.046
yx
3
3x
2
2x1
91. From the graph: approximately 0.01745 (See Exercise 63
for exact limit.)
0.0175
0.017
0.0165
0.016
40 20 0 20 40
y
g(x) sin(x°)/x
x
93. lim
x-1
f ðxÞ5 2
2.04
2.02
2
0.8 0.9 1 1.1 1.2
x
h(x) 2| x1|/4
f(x) 2(x1)
2
cos
2
(x/(x1))
g
(
x
)
2
95. lim
x-0
f ðxÞ5 0
1
1
0.5 0.5
y
x
h(x) 2|x|
f(x) cos (x1/x)cos
(x1/x)
g
(
x
)
2|x|
Section 2.3
1. 21, 1, 2 3. 1, 2
5. f is continuous at all values of its domain, which is R.
7. f is continuous at all values of its domain, which is {x 2R :
x 6¼21}.
A-14 Answers to Selected Exercises
9. f is continuous at all values of its domain, which is R.
11. f is continuous at all values of its domain, which is {x 2R :
x 6¼3, x 6¼24}.
13. The domain of f is R; f is continuous at each point of
{x 2R : x 6¼3}.
15. f is continuous at all values of its domain, which is R.
17. f is continuous at all values of its domain, which is R.
19. The domain of f is R; f is continuous at each point of
{x 2R : x 6¼21}.
21. The domain of f is R; f is continuous at each point of
{x 2R : x 6¼0}.
23. f is continuous at all values of its domain, which is R.
25. F(2) 5 11
27. F(2) 5 4
29. Left continuous on (2N,N); right continuous on {x 2R :
x 6¼5};
continuous on {x 2R : x 6¼5}
31. Left continuous on {x 2R : x 6¼3}; right continuous on
(2N,N); continuous on {x 2R : x 6¼3}
33.
2.02.4 2.22.8 2.63.0
6
4
2
5
3
1
(3,
6)
(2,
3)
Maximum
value
Minimum
value
Minimum
f(x) x
2
2x 3
Maximum
x
y
35.
10
6
8
4
2
1.0 1.4 1.6 1.81.2 2.0
(2,
11)
(1,
6)
Maximum
value
Minimum
value
Minimum Maximum
f(x) x
2
2x 3
x
y
37.
2.0
3.0
1.0
134526
(2p,
1)
(0,
1)
(p,
3)
Maximum
value
Minimum
value
Minimum Maximum
f(x) 2 cos(x)
Minimum
x
y
39. 21=2;
ffiffiffi
2
p
=2
41. 1/4, 49/96
43. a 527/2, b 5 9/2
45. a 5 25/6, b 527/2
47. Discontinuous at each odd integer
49. Discontinuous at each integer n 6¼k 1 6m for some inte-
ger m and k 561
53. lim
x-1
f ðxÞ; lim
x-1
gðxÞ; lim
x-1
hðxÞ; and lim
x-1
kðxÞ all
exist and equal 2; g and h are continuous at 1, while k is
discontinuous at 1; g and h are identical.
3
1
2
x
2.00.5 1.0 1.5
y
y f(x)
Not defined at 1
x
2.00.5 1.0 1.5
3
1
2
3
1
2
y
y h(x)
y g(x)
Continuous at 1
x
2.00.5 1.0 1.5
y
y k(x)
Discontinuous at 1
Section 2.3 A-15
TðxÞ5
0:08x if 0 , x # 8025
642 1 0:12ðx 2 8025 Þ if 8025 , x # 32550
3585 1 0:16ðx 2 32550 Þ if 32550 , x # 78850
10993 1 0:20ðx 2 78850Þ if 78850 , x # 164550
28133 1 0:25ðx 2 164550Þ if 164550 , x # 357700
76420:50 1 0:30ðx 2 357700Þ if 357700 , x
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
55. a. 2 b. 21; c. 21; d. 0
65. False
y
x
y p(x)
131
1
4
3
2
2
(1, 4)
(
3, 2
)
67. False (The figure for answer 65 illustrates this as well.)
69. False
x
y
1
1
f(x) 1 if x 0
f(x) 1 if x 0
f is not continuous
x
y
1
f
2
(x) 1
f(x)
1
f
2
and
f
are continuous
71. False (The figure for answer 69 illustrates this as well.)
73. True; otherwise, the Extreme Value Theorem would be
contradicted.
83. 4.3017
85. No continuous extension
x
y
212 1
1.0
0.5
0.5
f (x) xx
87. A continuous extension F is obtained by setting F(2) 528.
1.5 2.52.0 3.0
10
8
6
4
2
y
x
f(x) (x
4
6x
3
7x
2
4x 4)(x 2)
89. For each γ $ 10, the equation p(x) 5 γ has at least one
solution x; the equation has no solution when γ , 10.
When γ 5 20, the solutions are 0.7267 and 4.1132.
91. Minimum at x 521.8301; maximum at x 5 0.9369
Section 2.4
1. N 3. 0 5. 1 7. 1 9. N
11. 1/3 13. N 15. N 17. 2N 19. 2N
21. Vertical asymptote: x 5 7; horizontal asymptote: y 5 1
y
x
201010
15
10
15
5
15
10
5
x 7
y 1
y
x
(x 7)
A-16 Answers to Selected Exercises
23. Vertical asymptote: x 5 0; horizontal asymptote: y 5 0
y
x
5
10
10
5
y xx
25. Horizontal asymptote: y 5 1
5050
x
y 1
y
1
1
1
y
(x
2
8)
13
(x 1)
23
27. Horizontal asymptote: y 5 0
10
1
10
y 1(x
2
1)
x
y
29. Vertical asymptote: x 5 0; horizontal asymptote: y 5 0
y
x
20
2
2
y x
13
x
31. Vertical asymptotes: x 524, x 521, x 5 0; horizontal
asymptote: y 5 0
55
x
y
5
x 1
x 4
y (x 4)
13
(x(x 1)
33. Vertical asymptote: x 5 0
y
x
1
10
y cot(x)
41. Vertical asymptotes: x 5 nπ for every nonzero integer n
y
x
5p5p
20
20
20
20
y x/sin
(
x
)
43. Vertical asymptote: x 5 0; horizontal asymptote: y 5 0
y
x
55
5
5
f (x) cos
2
(x)x
Section 2.4 A-17
45. Vertical asymptotes: x 5 nπ for every integer n
y
x
4p2p4p 2p
4
2
y 1sin(x)
47. x 5 0isnot a vertical asymptote.
f(x) sin(1x)x
x
y
80
60
40
20
60
40
20
0.20.2
49. f ð10
80
Þ5 10
220
andlim
x-N
f ðxÞ5 N;
gð10
80
Þ5 10
20
andlim
x-N
gðxÞ5 0
51. Vertical asymptote:
V
b
P RT(V b) aV
2
P
57. y 5 x
2
23x 1 9 is a parabolic asymptote of F(x);
y 5 x
2
25x 1 25 is a parabolic asymptote of G( x);
y 5 x
2
1 1 is a parabolic asymptote of H(x).
x
y
20
20
424 2
H(x)
x
73
x
2
x
13
x
13
1
y x
2
1
59. x . ð500 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
249999
p
Þ
2
999997:999999
61.
f(x) (3x
2
x cos(x))x
2
1)
x
y
4
1
20101020
Graphed in the viewing window.
[25, 25] [0, 4]
f(x) (3x
2
x cos(x))x
2
1)
x
y
4
1
2
3
200150
Graphed in the viewing window.
[
100, 200
]
[
0, 4
]
A-18 Answers to Selected Exercises
63.
x
y
1510515 10 5
2
2
2
f(x)
x
4
x sin(x)x
2
2x 2)
Graphed in the viewing window.
[15, 15] [0, 2]
x
y
1.10
0.95
1.00
1.05
600520 540 560 580
f(x) x
4
x sin(x)x
2
2x 2)
Graphed in the viewing window.
[
500, 600
]
[
0.9, 1.1
]
1
65.
x
y
500
400
300
200
100
3.01.0 2.0 2.50.5
f(x) (4 cos(x))(sin(x) cos(2x))
p
2
Section 2.5
1. 0 3. 3/2 5. 0
7. Does not converge.
9. 0
11. Does not converge.
13. 21 15. 1 17. 26 19. 1/2
21. 6 23. 0 25. 1/2 27. 0
29. 3 31.
ffiffiffi
3
p
33. 1/4 35. 0
37. 1 39. 0 41. 1/3 43. 1/6
45. 3/2 47. 3=ð3 2
ffiffiffi
3
p
Þ 49.10/9 51. 0
53. 2 55. 1/2 57.16 59. 608911/4950
77. lim
n-N
n
1=n
5 1
y
x
4
3
2
1
0.5 1.0 1.5 2.0
y x
x
Section 2.6
1. 2
ffiffi
3
p
3. 2
π
5. 11 7. 9
9. 23 11. 3 13.
2
3
2
1
3
x 15. 1
17. 22x 19. 3x 21. 9 23. 23/2 25. log
6
ð2Þ
27.
3 124321
x
80
60
40
20
y
f(x) 3
x
1
(0, 2)
(1, 4)
(2, 10)
(3, 28)
(4, 82)
29.
x
y
5
10
15
3125
4 3
g(x) 3 e
x
2 1
Section 2.6 A-19
31.
7
(1/4, 2)
(1/2, 3)
(1, 4)
(2, 5)
(4, 6)
(8, 7)
6
5
4
3
2
12345678
x
y
y 3 log
2
(2x)
33. e 35. exp(e) 37. e
3/4
39. e
2π
41. e
2π/4
43. 8 45. 0 47. 1
49. e
3
51. 1
53. Vertical asymptote: x 5 0; horizontal asymptotes: y 5 1
and y 521
55. Vertical asymptote: x 5 3; horizontal asymptotes: y 5 0
and y 5 4
57. a. 1060 b. 1060.90 c. 1061.36 d. 1061.83
e. 1061.84
59. a. 5829.57 b. 5841.02 c. 5846.91
d. 5852.84 e. 5852.90
61. 17510 63. 7211 65. 3.707 g 67. 15.77 g
69. 40 ln(2)/ln(3/2) 68.38 yr
71. 100(ln(2))/ln(7/5) 206 yr
73.
2
2
2
4
6
y
x
y | e
x
1|
y 1
75.
10
y ln(| x2|)
x 2
2
2
y
x
77. e 79. e
6
83. $44693
85. $1000(e
22/25
1 e
23/25
1 e
24/25
), or about $2662.18
87. Exact doubling time: 100 ln(2)/r; 69 better approximates
100 ln(2)
89. y 5 0, y 5 M
91. lim
t-N
CðtÞ5 α=β; horizontal asymptote : y 5 α=β
93. α 95. 63.84% 97. 19996 yr
REVIEW EX ERCIS ES FOR CHAPTER 2
1. Does not exist
3. 22 5. 212 7. 21/10 9. 6
11. Does not exist
13. 21 15. 1 17. 2 19. 4
21. 0 23. 7/3 25. 0 27. 3
29. Does not exist
31. 32 33. 4 35. 1 37. 1=
ffiffiffi
2
p
39. 21 41. 9 43. N
45. Does not exist
47. 0 49. 1 51. 1/e 53. 0
55. 0.01 57. 0.000500024 59. 1 61. 22
63. Horizontal asymptote: y 5 1; vertical asymptote x 523
65. Horizontal asymptote: y 5 0
67. Horizontal asymptote: y 5 3; vertical asymptote x 522
69. 9 70. e
3
71. 64 73. x 1 2
75. 22 77. log
9=2
ð18Þ
79. f ðxÞ5
0ifx is not an integer
1ifx is an integer
80. 3 81. 1 83. 0 85. 0
87. 3.97 more weeks
89. 1620 ln(5)/ln(2) 3761.5 yr
91. a. 0.775 b. 0.802
93. 0.524 95. 5001 yr
Section 3.1
1. 0 3. 42 5. 6 7. 7
9. 22 11. 24 13. Forward 15. Backward
17. 9 19. 225/2 21. 6 23. 26
25. y 5 20(x25) 1 50
27. y 5 12(x 1 2) 27
29. y 5 20x 2 20
31. y 5 8x 2 3
A-20 Answers to Selected Exercises