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Engineering Electromagnetics 8th Edition - Solution Manual
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Solution manual for the Calculus Ninth Edition Textbook by Purcell
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CHAPTER 0 Preliminaries0.1 Concepts Review 1 ⎡ 2 1 ⎛ 1 1 ⎞⎤ 1
8. − ⎢ − ⎜ − ⎟ ⎥ = −
3 ⎣ 5 2 ⎝ 3 5 ⎠⎦ ⎡ 2 1 ⎛ 5 3 ⎞⎤
1. rational numbers 3 ⎢ − ⎜ − ⎟⎥
⎣ 5 2 ⎝ 15 15 ⎠ ⎦
2. dense 1 ⎡ 2 1 ⎛ 2 ⎞⎤ 1 ⎡2 1 ⎤
= − ⎢ − ⎜ ⎟⎥ = − ⎢ − ⎥
3. If not Q then not P. 3 ⎣ 5 2 ⎝ 15 ⎠ ⎦ 3 ⎣ 5 15 ⎦
1⎛ 6 1 ⎞ 1⎛ 5 ⎞ 1
4. theorems =− ⎜ − ⎟=− ⎜ ⎟=−
3 ⎝ 15 15 ⎠ 3 ⎝ 15 ⎠ 9
Problem Set 0.1 2 2
14 ⎛ 2 ⎞ 14 ⎛ 2 ⎞
2
⎟ = ⎜ ⎟ = ⎛⎜ ⎞⎟
14 6
9. ⎜
1. 4 − 2(8 − 11) + 6 = 4 − 2(−3) + 6 21 ⎜ 5 − 1 ⎟ 21 ⎜ 14 ⎟ 21 ⎝ 14 ⎠
⎝ 3⎠ ⎝ 3 ⎠
= 4 + 6 + 6 = 16 2
14 ⎛ 3 ⎞ 2⎛ 9 ⎞ 6
= ⎜ ⎟ = ⎜ ⎟=
21 ⎝ 7 ⎠ 3 ⎝ 49 ⎠ 49
2. 3 ⎡⎣ 2 − 4 ( 7 − 12 ) ⎤⎦ = 3[ 2 − 4(−5) ]
= 3[ 2 + 20] = 3(22) = 66 ⎛2 ⎞ ⎛ 2 35 ⎞ ⎛ 33 ⎞
⎜ − 5⎟ ⎜ − ⎟ ⎜ − ⎟
10. ⎝
7 ⎠ = ⎝ 7 7 ⎠ = ⎝ 7 ⎠ = − 33 = − 11
3. –4[5(–3 + 12 – 4) + 2(13 – 7)] ⎛ 1⎞ ⎛7 1⎞ ⎛6⎞ 6 2
= –4[5(5) + 2(6)] = –4[25 + 12] ⎜1 − ⎟ ⎜ − ⎟ ⎜ ⎟
⎝ 7⎠ ⎝7 7⎠ ⎝7⎠
= –4(37) = –148
11 – 12 11 – 4 7
7
4. 5 [ −1(7 + 12 − 16) + 4] + 2 11. 7 21 = 7 7 = 7 =
11 + 12 11 + 4 15 15
= 5 [ −1(3) + 4] + 2 = 5 ( −3 + 4 ) + 2 7 21 7 7 7
= 5 (1) + 2 = 5 + 2 = 7
1 3 7 4 6 7 5
− + − +
5
5 1 65 7 58 12. 2 4 8 = 8 8 8 = 8 =
5. – = – = 1 3 7 4 6 7 3 3
7 13 91 91 91 + − + −
2 4 8 8 8 8 8
3 3 1 3 3 1 1 1 2 3 2 1
6. + − = + − 13. 1 – =1– =1– = – =
4 − 7 21 6 −3 21 6 1
1+ 2 3 3 3 3 3
42 6 7 43 2
=− + − =−
42 42 42 42
3 3 3
14. 2 + = 2+ = 2+
5 2 5 7
1 ⎡1 ⎛ 1 1 ⎞ 1⎤ 1 ⎡1 ⎛ 3 – 4 ⎞ 1⎤ 1+ −
⎜ – ⎟+ = ⎜ ⎟+
3 ⎢⎣ 2 ⎝ 4 3 ⎠ 6 ⎥⎦ 3 ⎢⎣ 2 ⎝ 12 ⎠ 6 ⎥⎦
7. 2 2 2 2
6 14 6 20
1 ⎡1 ⎛ 1 ⎞ 1⎤ = 2+ = + =
= ⎢ ⎜– ⎟+ ⎥ 7 7 7 7
3 ⎣ 2 ⎝ 12 ⎠ 6 ⎦
( )( ) ( 5) – ( 3)
2 2
1⎡ 1 4⎤ 15. 5+ 3 5– 3 =
= ⎢– + ⎥
3 ⎣ 24 24 ⎦
=5–3= 2
1⎛ 3 ⎞ 1
= ⎜ ⎟=
3 ⎝ 24 ⎠ 24
Instructor’s Resource Manual Section 0.1 1
, ( ) = ( 5) ( 5 )( 3 ) + ( 3 )
2 2 2 12 4 2
16. 5− 3 −2 27. +
+
x + 2x x x + 2
2
= 5 − 2 15 + 3 = 8 − 2 15 12 4( x + 2) 2x
= + +
x( x + 2) x( x + 2) x( x + 2)
17. (3x − 4)( x + 1) = 3 x 2 + 3 x − 4 x − 4 12 + 4 x + 8 + 2 x 6 x + 20
= =
= 3x2 − x − 4 x( x + 2) x( x + 2)
2(3 x + 10)
18. (2 x − 3)2 = (2 x − 3)(2 x − 3) =
x( x + 2)
= 4 x2 − 6 x − 6 x + 9
2 y
= 4 x 2 − 12 x + 9 28. +
6 y − 2 9 y2 −1
(3x – 9)(2 x + 1) = 6 x 2 + 3 x –18 x – 9 2 y
19. = +
2
= 6 x –15 x – 9 2(3 y − 1) (3 y + 1)(3 y − 1)
2(3 y + 1) 2y
= +
20. (4 x − 11)(3x − 7) = 12 x 2 − 28 x − 33 x + 77 2(3 y + 1)(3 y − 1) 2(3 y + 1)(3 y − 1)
= 12 x 2 − 61x + 77 6y + 2 + 2y 8y + 2
= =
2(3 y + 1)(3 y − 1) 2(3 y + 1)(3 y − 1)
21. (3t 2 − t + 1) 2 = (3t 2 − t + 1)(3t 2 − t + 1) 2(4 y + 1) 4y +1
= =
4 3 2 3
= 9t − 3t + 3t − 3t + t − t + 3t − t + 1 2 2 2(3 y + 1)(3 y − 1) (3 y + 1)(3 y − 1)
= 9t 4 − 6t 3 + 7t 2 − 2t + 1 0
29. a. 0⋅0 = 0 b. is undefined.
0
22. (2t + 3)3 = (2t + 3)(2t + 3)(2t + 3)
= (4t 2 + 12t + 9)(2t + 3) c.
0
=0 d.
3
is undefined.
17 0
= 8t 3 + 12t 2 + 24t 2 + 36t + 18t + 27
= 8t 3 + 36t 2 + 54t + 27 e. 05 = 0 f. 170 = 1
x 2 – 4 ( x – 2)( x + 2) 0
23. = = x+2, x ≠ 2 30. If = a , then 0 = 0 ⋅ a , but this is meaningless
x–2 x–2 0
because a could be any real number. No
x 2 − x − 6 ( x − 3)( x + 2) 0
24. = = x+2, x ≠3 single value satisfies = a .
x−3 ( x − 3) 0
t 2 – 4t – 21 (t + 3)(t – 7) 31. .083
25. = = t – 7 , t ≠ −3 12 1.000
t +3 t +3
96
2
2x − 2x 2 x(1 − x) 40
26. =
3 2 2
x − 2x + x x( x − 2 x + 1) 36
−2 x( x − 1) 4
=
x( x − 1)( x − 1)
2
=−
x −1
2 Section 0.1 Instructor’s Resource Manual
, 32. .285714 35. 3.6
7 2.000000 3 11.0
14 9
60 20
56 18
40 2
35
36. .846153
50
13 11.000000
49
10 4
10
60
7
52
30
80
28
78
2
20
33. .142857 13
21 3.000000 70
21 65
90 50
84 39
60 11
42
37. x = 0.123123123...
180 1000 x = 123.123123...
168 x = 0.123123...
120 999 x = 123
105 123 41
150 x= =
999 333
147
3 38. x = 0.217171717 …
1000 x = 217.171717...
34. .294117... 10 x = 2.171717...
17 5.000000... → 0.2941176470588235 990 x = 215
34 215 43
x= =
160 990 198
153
39. x = 2.56565656...
70 100 x = 256.565656...
68 x = 2.565656...
20 99 x = 254
17 254
30 x=
99
17
130 40. x = 3.929292…
119 100 x = 392.929292...
11 x = 3.929292...
99 x = 389
389
x=
99
Instructor’s Resource Manual Section 0.1 3
, 41. x = 0.199999...
( )
4
52. 2− 3 ≈ 0.0102051443
100 x = 19.99999...
10 x = 1.99999...
53. 4 1.123 – 3 1.09 ≈ 0.00028307388
90 x = 18
18 1
x= = 54. ( 3.1415 )−1/ 2 ≈ 0.5641979034
90 5
42. x = 0.399999… 55. 8.9π2 + 1 – 3π ≈ 0.000691744752
100 x = 39.99999...
10 x = 3.99999... 56. 4 (6π 2 − 2)π ≈ 3.661591807
90 x = 36
36 2 57. Let a and b be real numbers with a < b . Let n
x= = be a natural number that satisfies
90 5
1 / n < b − a . Let S = {k : k n > b} . Since
43. Those rational numbers that can be expressed a nonempty set of integers that is bounded
by a terminating decimal followed by zeros. below contains a least element, there is a
k 0 ∈ S such that k 0 / n > b but
p ⎛1⎞ 1
44. = p ⎜ ⎟ , so we only need to look at . If (k 0 − 1) / n ≤ b . Then
q ⎝q⎠ q
k0 − 1 k0 1 1
q = 2n ⋅ 5m , then = − >b− > a
n m n n n n
1 ⎛1⎞ ⎛1⎞ k 0 −1 k 0 −1
= ⎜ ⎟ ⋅ ⎜ ⎟ = (0.5)n (0.2)m . The product Thus, a < n ≤ b . If n < b , then choose
q ⎝ 2⎠ ⎝5⎠
k 0 −1 k0 − 2
of any number of terminating decimals is also a r= n . Otherwise, choose r = n .
n m
terminating decimal, so (0.5) and (0.2) , 1
Note that a < b − <r.
1 n
and hence their product, , is a terminating
q Given a < b , choose r so that a < r1 < b . Then
p choose r2 , r3 so that a < r2 < r1 < r3 < b , and so
decimal. Thus has a terminating decimal
q on.
expansion.
58. Answers will vary. Possible answer: ≈ 120 in 3
45. Answers will vary. Possible answer: 0.000001,
1 ft
≈ 0.0000010819... 59. r = 4000 mi × 5280 = 21,120, 000 ft
π 12 mi
equator = 2π r = 2π (21,120, 000)
46. Smallest positive integer: 1; There is no ≈ 132, 700,874 ft
smallest positive rational or irrational number.
60. Answers will vary. Possible answer:
47. Answers will vary. Possible answer: beats min hr day
3.14159101001... 70 × 60 × 24 × 365 × 20 yr
min hr day year
48. There is no real number between 0.9999… = 735,840, 000 beats
(repeating 9's) and 1. 0.9999… and 1 represent 2
⎛ 16 ⎞
the same real number. 61. V = πr 2 h = π ⎜ ⋅12 ⎟ (270 ⋅12)
⎝ 2 ⎠
49. Irrational ≈ 93,807, 453.98 in.3
volume of one board foot (in inches):
50. Answers will vary. Possible answers: 1× 12 × 12 = 144 in.3
−π and π , − 2 and 2 number of board feet:
93,807, 453.98
≈ 651, 441 board ft
51. ( 3 + 1)3 ≈ 20.39230485 144
4 Section 0.1 Instructor’s Resource Manual
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