Blank B.E., Krantz S.G. Calculus: Single Variable
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Answers to Selected
Exercises
Section 1.1
1. 213/100 3. 23/99
5. 4996/999 7. 0.025
9. 1:
6 (The overline identifies the repeating block.)
11. 0.72 13. [1, 3]
15. (1,N) 17. [214, 6]
19.
010 5105
21.
010 5105
23.
010 5105
25. {x :|x 2 1| # 2}
27. {x :|x 2 1| , π 1 1}
31. Both approximations are rational numbers, but π is not.
33. [34.155, 34.845]; {x :|x 2 34.5| # 0.345}
35.
010 5105
37.
010 5104 2
39.
010 5105
41.
010 5105
43.
010 5105
45. {23}
47. fx : 1 , x # 2
ffiffiffi
2
p
g
49. {s : 213 , s ,25/3}
51. {w :|w 1 1/2| , 1/2}
59. [0.4485, 0.4495]
61. [0.000499, 0.001499]
63. 4.001
65. Calculated values of α β using N 5 12, 14, 16, 18, and 20
digits are tabulated:
N α β
12 34562960.8
14 2345629.6
16 0.000000
18 0.000000
20 1.0369
67.
x Relative Error
10
10
0.5
10
12
9.901 3 10
23
10
14
9.999 3 10
25
10
16
9.99999 3 10
27
10
18
9.9999999 3 10
29
A-1
x Relative Error
210
12
0.0101
210
14
1.0001 3 10
24
210
16
1.000001 3 10
26
210
18
1.00000001 3 10
28
210
20
1.000000000 3 10
210
69.
x Relative Error
10
3
1.031
10
6
0.0345
10
9
0.00003
10
12
0.3 3 10
27
10
15
0.3 3 10
210
x Relative Error
210
3
0.9709
210
6
0.0322
210
9
0.00003
210
12
0.3 3 10
27
210
15
0.3 3 10
210
Section 1.2
1.
y
x
2108
(p 3, p 3)
(7, 3)
(2, 4)
64
3
2
1
2
3
4
2
p
,
8
3
2
, p
,
3
6
3.
y
x
(5, 3)
(4, 2)
(2, 0)
(1, 1)
(0, 2)
(1, 3)
(2, 4)
(3, 5)
(4, 6)
(5, 7)
442
4
6
4
5. jABj5 2
ffiffiffiffiffi
13
p
; jACj5
ffiffiffiffiffiffiffiffi
130
p
; jBCj5
ffiffiffiffiffiffiffiffi
170
p
7. Center: (1, 3); radius: 3
9. Center: (0, 25); radius:
ffiffiffi
2
p
11. Center: (0, 1/2); radius: 1/2
13. Center: (22, 1); radius:
ffiffiffiffiffiffiffiffiffiffi
17=3
p
15. Center: (4, 21); radius:
ffiffiffiffiffi
17
p
17. (x 1 3)
2
1 ( y 2 5)
2
5 36
19. (x 1 4)
2
1 ( y 2 π)
2
5 25
21. Vertex: (0, 23); axis of symmetry: x 5 0
23. Vertex: (1, 1); axis of symmetry: x 5 1
25. Vertex: (23, 9/2); axis of symmetry: x 523
27. Ellipse; standard form: x
2
/(1/2)
2
1 y
2
5 1; center: (0, 0)
29. Ellipse; standard form: (x 1 1/2)
2
/2
2
1 y
2
/(2/3)
2
5 1; cen-
ter: (21/2, 0)
31. Ellipse; standard form: (x 1 1)
2
/7
2
1 (y 1 3)
2
/
(7/2)
2
5 1; center: (21,23)
33. Hyperbola; standard form: x
2
=ð10=
ffiffiffi
2
p
Þ
2
2 ðy 1 1Þ
2
=
ð10=
ffiffiffi
3
p
Þ
2
5 1; center: (0, 21)
35.
4
4
2
2
424 2
{(x, y) : x 3}
x
y
A-2 Answers to Selected Exercises
37.
{(x, y) : x 7, y 4 1}
8
8
4
4
848 4
x
y
39.
{
(
x, y
)
: x
2
y
2
16}
x
y
2
2
5
41.
x
52312 1
{
(
x, y
)
:
(
x 2
)
2
y
2
4}
y
4
4
2
2
43.
8104622
y x 1
y x 1
x
y
10
5
{(x, y) : x y 1, x 4}
45.
y
1
3
3
3
{(
x, y
)
: x
2
y
2
9, x 0, y 1
}
x
47. (3, 21) 49. (0, 0)
51. Yes: It is a circle centered at (1/5, 23/10) with radius
ffiffiffiffiffiffiffiffi
133
p
=10.
53. No: It is the equation of a hyperbola.
55. 81/2 57. 1/2
59. a 527, b 5 1
61.
{(x, y) : xⴢ y 1}
x
y
Section 1.2 A-3
63.
x
3
2
1
2
3
1
y
2112
{(x, y) : x y 1}
65.
1
2
2
6
4 3 1
y 2x 5
x 13
{
(
x, y
)
: 3x 1 0 y 2x 5}
x
y
67.
{
(
x, y
)
: x y
2
x
2
}
y x
x y
2
y x
x
y
8
6
4
2
2
4
6
8
6824
69.
x
2246
y
1
2
3
4
5
(65, 125)
x 3y 6
{(
x, y
)
: y 2x, x 3y 6
}
73. (x 1 1)
2
1 y
2
5 5
2
75.
x
y
2.5
22
11
2.0
1.5
1.0
2
1
2
x
2
y
77.
2341
6
4
2
y
x
y x
2
2.2x
y 2.3 1.1x 1.7x
2
(1.149, 1.208)
(0.741,
2.181)
2
4
6
79.
1
01
x
1 1.3x
2
1.7x y
2
0.9y
y
x
2
0.5x 2.1y
2
1
A-4 Answers to Selected Exercises
81.
y 2x
y 2 x x
2
y 4x
y 3.82x
y
x
5
0
1
Section 1.3
1.
1
1
x
y
Slope not defined
Slope
1兾4
Slope
1兾4
Slope
1兾3
Slope
1兾3
Slope
2
Slope
2
Slope 0
Slope 0
3.
20
20
30
x
y
22
5. y 5 5(x 1 3) 1 7
7. y 52ðx 1
ffiffiffi
2
p
Þ
9. y 5 (211/4)(x 2 2) 1 7ory 5 (211/4)(x 2 6) 2 4
11. y 5 (1/4)(x 1 7) 2 4ory 5 (1/4)(x 29)
13. y 524x 1 9
15. y 5
ffiffiffi
2
p
x 2
ffiffiffi
3
p
17. y 5 3x 1 12
19. y 5 3x 1 6
21. y 5 3x 1 1
23. y 524x 2 1
25. x/(22) 1 y/6 5 1
27. x/(21) 1 y/3 5 1
29. y 5 (21/2)x 23/2
31. y 5 2x 2 3
33. y 5 2x 2 4
35. y 5 (3/2)x 2 1/2
37. x-intercept:
2/3; y-intercept: 1/2; slope: 23/4
39. x-intercept: 3; y-intercept: 21; slope: 1/3
41.
y
x
8
4
0
13
13 2
8
4
y
3
(
x1
)
2
x/2 1
43. The figure for 41 has this plot.
45.
y
x
3 2
2
4
6
8
112
2x
3y 6
y
2x 4
47. The figure for 45 has this plot.
49. No; no
51. y 5 3x 229
53. All points (a,2a 217) with a 6¼5; for example, (0, 217)
55. All points (a,12 3a) with a 6¼22; for example, (1, 22)
57. One example is (a, b) 5 (1, 0) and (c, d) 5 (3, 4).
59. x 5 1, y 5 2
61. x 524, y 5 6
Section 1.3 A-5
63. (3, 1)
67. 21999997 and 2000003
69. Equation of budget line: x p
X
1 y p
Y
5 C,0# x, and
0 # y; x-intercept: C/p
X
; y-intercept: C/p
Y
; slope: 2p
X
/p
Y
.
The budget line Lu that corresponds to Cu is parallel to
the budget line L that corresponds to C. Lu lies above L if
Cu .C; Lu lies below L if Cu , C.
75. Slope (1.002001 2 .998001)/(1.001 2 .999), or 2.0 (See
accompanying figure.)
1.00100.9990 0.9995 1.0000 1.0005
0.998
0.999
1.000
1.001
1.002
(0.999, 0.998001)
(1.001, 1.002001)
y x
2
x
y
77. Slope (0.001999998 2 (20.001999998))/(0.001 2
(20.001)), or 1.999998 (See accompanying figure.)
x
y
0.002
0.002
0.001
(0.001,
0.001999998)
(
0.001, 0.001999998
)
0.001
x
2
1
2x
y
79. Slope 0 (See accompanying figure.)
0.999999999 1 1.000000001
2
1
x
y
81.
y 2.30945x
65.1552
y
x
190
180
170
160
150
36 40 44 48 52
83. Q 5 ð3=4;
ffiffiffi
3
p
=2Þ; line ‘ has equation y 5 ð1=
ffiffiffi
3
p
Þ
ðx 2 3=4Þ1
ffiffiffi
3
p
=2
2.0
0.5
12
x
ᐉ
y
3
1.0
1.5
P
y x
Q
y
1
(x 34)
3
2
3
85. m 4.76923 minimizes the sum of the squared
errors; m 4.6666667 minimizes the sum of the abso-
lute errors
1.0
0.5
4.6
Min.
SSE
4.76923
4.9
m
SSE
5.0
4.6666666 4.6666667
4.66666667
m
SAE
4.6666668
Min.
SAE
A-6 Answers to Selected Exercises
y
x
15
10
5
1234
Line that minimizes SAE
Regression line (minimizes SSE)
Section 1.4
1. {x 2R : x 6¼21}
3. fx 2 R : x #2
ffiffiffi
2
p
or x $
ffiffiffi
2
p
g (or, equivalently,
fx 2 R :
ffiffiffi
2
p
# jxjg )
5. fx 2 R : x 6¼ 21 and x 6¼ 1g
7. R
9.
y
x
22
5
10
5
y 3 x
2
2
y x
2
1
y 1 x
2
11. The figure for Exercise 9 has this plot.
13. The figure for Exercise 9 has this plot.
15.
y
x
1212
y x
2
17.
y
x
642 2
4
y 2x 4
19.
x
y
y (x 4)
12
85 6 7
21.
y
x
842 62
y 1 x 1
23.
x
y
2112
x
2
2 x
2
if x 1
if x 1
25. s(α) 5 (π /180) rα
27. A(r, α) 5 (π/360) r
2
α
29. A 52(r 1 s); B 5 rs
31. f ðxÞ5
1 1 2x if 0 # x # 1
3ð3 2 xÞ=2if1, x # 3
x 2 3if3, x # 4
1if4, x # 5
8
>
>
<
>
>
:
Section 1.4 A-7
33. mðxÞ5
0:10 if 0 , x , 8025
0:15 if 8025 , x , 32550;
0:25 if 32550 , x , 50000
AðxÞ5 TðxÞ
8
<
:
35. f
1
5 2 f
n
5 2 f
n21
for 2 # n
37. f
1
5 1 f
n
5 n f
n21
for 2 # n
39. f
1
5 1 f
n
5 n 1 f
n21
for 2 # n
41. bxc5
IntðxÞ if x . 0orx 2 Z
IntðxÞ2 1ifx , 0
y
x
3 2 1312
2
3
2
3
1
1
y Int
(
x
)
(
also called y trunc
(
x
))
y
x
3 2 1312
2
3
2
3
1
y x
43. I(P, m, n) 5 12mn 2 P
45. 1, 1, 2, 5, 14, 42, 132, 429, 1430
47. R(x) 5 x(0 , x)(x # 1) 1 (1 , x)
y
x
0.2
0.4
0.6
0.8
1
112
y R(x)
49. b(x) 5 x(0 , x)(x # 1) 1 (2 2 x), (1 , x)(x # 2)
y
x
2 14123
1
y b(x)
top of right column
51.
p
1
1.0
y
0.80.60.40.2
y
0.98p
0.97p 0.01
p
0.3
0.005
y
0.0040.0030.0020.001
0.2
0.1
The probability of a true positive becomes very small as x
approaches 0. Routine screening is not advisable if the
disease is rare. y 5 R(x)
53. a. x
0
5 4;
b.
h 10
23
10
24
10
25
F(h, 3.99) 1.5031 3 10
23
1.437 3 10
24
1.44 3 10
25
F(h, x
0
) 7.1972 3 10
25
7.1997 3 10
27
7.200 3 10
29
A-8 Answers to Selected Exercises
c.
y
x
0.0015
0.001
0.0005
0
0.0002 0.0004 0.0006 0.0008 0.001
y F(3.99, h)
d.
y
x
6e-05
4e-05
2e-05
0
0.0002 0.0004 0.0006 0.0008 0.001
y F(x
0
, h)
57. p
1
5 2.828427124, p
2
5 3.061467458
p
3
5 3.121445152, p
4
5 3.136548492
p
5
5 3.140331158, p
6
5 3.141277254
p
7
5 3.141513803, p
8
5 3.141572945
Section 1.5
1. (x
3
2 x
2
1 6x 2 4)/(x 2 1),
3. (x 2 2)/(x 2 3)
5. 2x
3
2 7x
2
1 10x 2 30
7. (x 2 2)
2
/(x 2 3)
2
1 5
9. 2x
11. f (x) 5 x 2 2, g(x) 5 x
2
13. f (x) 5 x
3
1 3x, g(x) 5 x
4
15. g(x) 5 3x
2
1 1
17. g(x) 5 x/(x
2
1 2)
19. 3
21. 1/2
23. (x 1 5)(x 2 1)
25. (x 1 2)(x 2 2)(x
2
1 2x 1 2)
27. f
21
ðtÞ5
ffiffiffiffiffiffiffiffiffiffi
t 2 1
p
29. f
21
ðtÞ5 ð21 1
ffiffiffiffiffiffiffiffiffiffiffiffi
1 1 4t
p
Þ=2
31. f
21
(t) 5 (t 1 27)
1/3
33. f
21
ðtÞ5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t=ðt 2 1Þ
p
35. f
21
(t) 5 t
2
2 3
37. Not onto
39. a. Invertible
0.2
0.2 0.4 0.6 0.8 1.0
0.4
0.6
0.8
1.0
b. Not invertible c. Not invertible d. Not invertible
e. Invertible
0.2
0.2 0.4 0.6 0.8 1.0
0.4
0.6
0.8
1.0
f. Invertible
0.2
0.2 0.4 0.6 0.8 1.0
0.4
0.6
0.8
1.0
41. h 522; horizontal translation 2 units to the right
43. h 521; horizontal translation 1 unit to the right
Section 1.5 A-9
45. h 5 2; k 5212; The graph of g is obtained by translating
the graph of f 2 units to the left and 12 units down.
47. h 5 3; k 522; The graph of g is obtained by translating
the graph of f 3 units to the left and 2 units down.
49. The line y 5 3x 2 7
51. The parabola x 5 3y
2
1 1
53. deg( p q) 5 deg( p) 1 deg(q); deg( p 3 q) 5 deg( p) deg(q);
deg( p 3 q) 5 deg(q 3 p); deg( p6q) # max(deg( p),deg(q))
55. deg( f ) 5 n
2
; every root of p is a root of f;
q(x) 5 x
2
2 x 2 6.
61. g(x) 5 x
2
1 2
63. g(x) 5 2x 1 18
65. f (x) 5 x 2 4
67. f (x) 5 (x
2
2 1)
1/3
69. f (x) 5 (x 2 3)
2
71. f (x) 5 (1 2 x
3
)/(x
2
1 1)
73. True for p 5 0 and p 5 2
75. φðxÞ5 π; ψðxÞ5 π
1=5
; μðxÞ5 x
1=5
1 π;
λðxÞ5 ðx 1 πÞ
1=5
77.
x
y
y f (x)
10864022
6
4
(0, 0)
(3, 6)
(6, 0)
2
2
0
x
y
y f ( f (x))
10864022
6
4
(0, 0) (3, 0)
(32, 6)
(92, 6)
(6, 0)
2
2
0
79. G(x) 5 x
p
, F(x) 5 px, H(x) 5 log
10
( p) 1 x
81. x 5 y
2/3
83. y 5 3 2
x
85. f 3 S 5 f; g 3 S 52g
87.
6
y
x
y
x
2
2x2
4
2
5
05
y1x
2
2x2
89.
1
1
x
22
y
y
x
3
1
x
2
1
y
1 x
3
x
2
1
91.
x
1.00.5
0.6
y
0.4
0.2
x
0.5450.525 0.530 0.535 0.540
0.388
y
0.384
0.376
0.380
t 0.63068
(0.52753, 0.37982)
(0.53456, 0.38488)
t 0.57399
( f(t), g(t))
y
18x
25
93.
y
x
y f
1
(x)
2
1
5 5 10 15
A-10 Answers to Selected Exercises