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App. 2 Answers to Odd-Numbered Problems
A27
Problem Set 10.8, page 462
1. x ⫽ 0, y ⫽ 0, z ⫽ 0, no contributions. x ⫽ a: 0f>0n ⫽ 0f>0x ⫽ ⫺2x ⫽ ⫺2a, etc.
Integrals x ⫽ a: (⫺2a)bc, y ⫽ b: (⫺2b)ac, z ⫽ c: (4c) ab. Sum 0
3. The volume integral of 8y 2 ⫹ [0, 8y] ⴢ [2x, 0] ⫽ 8y 2 is 8y 3>3 ⫽ 83. The surface
integral of f 0g>0n ⫽ f ⴢ 2x ⫽ 2f ⫽ 8y 2 over x ⫽ 1 is 8y 3>3 ⫽ 83. Others 0.
5. The volume integral of 6y 2 ⴢ 4 ⫺ 2x 2 ⴢ 12 is 0; 8(x ⫽ 1), ⫺8(y ⫽ 1), others 0.
7. F ⫽ [x, 0, 0], div F ⫽ 1, use (2*), Sec. 10.7, etc.
9. z ⫽ 0 and z ⫽ 2a 2 ⫺ x 2 ⫺ y 2 ⫽ 2a 2 ⫺ r 2, dx dy ⫽ r dr du,
a
⫺2p ⴢ 12 (a 2 ⫺ r 2)3>2 ⴢ 23 ƒ 0 ⫽ 23 pa 3
11. r ⫽ a, ␾ ⫽ 0, cos ␾ ⫽ 1, v ⫽ 13 a ⴢ (4pa 2)
Problem Set 10.9, page 468
1. S: z ⫽ y (0 ⬉ x ⬉ 1, 0 ⬉ y ⬉ 4), [0, 2z, ⫺2z] • [0, ⫺1, 1], ⫾20
3. [2eⴚz cos y, ⫺eⴚz, 0] • [0, ⫺y, 1] ⫽ yeⴚz, ⫾(2 – 2> 1e)
5. [0, 2z, 32 ] • [0, 0, 1] ⫽ 32 , ⫾32 a 2
7. [⫺ez, ⫺ex, ⫺ey] • [⫺2x, 0, 1], ⫾(e4 ⫺ 2e ⫹ 1)
9. The sides contribute a, 3a 2>2, ⫺a, 0.
11. ⫺2p; curl F ⫽ 0
13. 5k, 80p
1
15. [0, ⫺1, 2x ⫺ 2y] • [0, 0, 1], 3
17. r ⫽ [cos u, sin u, v], [⫺3v2, 0, 0] • [cos u, sin u, 0], ⫺1
19. r ⫽ [u cos v, u sin v, u], 0 ⬉ u ⬉ 1, 0 ⬉ v ⬉ p>2,
[⫺ez, 1, 0] • [⫺u cos v, ⫺u sin v, u]. Answer: 1>2
Chapter 10 Review Questions and Problems, page 469
11. r ⫽ [4 ⫺ 10t, 2 ⫹ 8t], F(r) • dr ⫽ [2(4 ⫺ 10t)2, ⫺4(2t ⫹ 8t)2] • [⫺10, 8] dt;
⫺4528>3. Or using exactness.
13. Not exact, curl F ⫽ (5 cos x)k, ⫾10
15. 0 since curl F ⫽ 0
17. By Stokes, ⫾18p
19. F ⫽ grad (y 2 ⫹ xz), 2p
21. M ⫽ 8, x ⫽ 85, y ⫽ 16
5
x ⫽ 87 ⫽ 1.14, y ⫽ 118
23. M ⫽ 63
20 ,
49 ⫽ 2.41
5
4
25. M ⫽ 4k>15, x ⫽ 16, y ⫽ 7
27. 288(a ⫹ b ⫹ c) p
29. div F ⫽ 20 ⫹ 6z 2. Answer: 21
31. 24 sinh 1 ⫽ 28.205
72p
33. Direct integration, 224
35.
3
Problem Set 11.1, page 482
1. 2p, 2p, p, p, 1, 1, 12, 12
1
1
1
1
cos 3x ⫹
cos 5x ⫹ Á ) ⫹ 2 (sin x ⫹ sin 3x ⫹ sin 5x ⫹ Á )
9
25
3
5
15. 43 p2 ⫹ 4 (cos x ⫹ 14 cos 2x ⫹ 19 cos 3x ⫹ Á ) ⫺ 4p (sin x ⫹ 12 sin 2x ⫹
1
Á)
3 sin 3x ⫹
p 4
1
1
cos 5x ⫹ Á b
17. ⫹ acos x ⫹ cos 3x ⫹
2
p
9
25
13.
4
5. There is no smallest p ⬎ 0.
p
(cos x ⫹
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App. 2 Answers to Odd-Numbered Problems
2
1
1
1
⫺ acos x ⫹ cos 3x ⫹
cos 5x ⫹ Á b ⫹ sin x ⫺ sin 2x ⫹
4
p
9
25
2
1
sin 3x ⫺ ⫹ Á
3
21. 2 (sin x ⫹ 12 sin 2x ⫹ 13 sin 3x ⫹ 14 sin 4x ⫹ 15 sin 5x ⫹ Á )
19.
p
Problem Set 11.2, page 490
1. Neither, even, odd, odd, neither
3. Even
5. Even
4
px 1
3px
1
5px
asin
9. Odd, L ⫽ 2,
⫹ sin
⫹ sin
⫹Áb
p
2
3
2
5
2
1
4
1
1
11. Even, L ⫽ 1,
⫺ 2 acos px ⫺ cos 2px ⫹ cos 3px ⫺ ⫹ Á b
p
3
4
9
13. Rectifier, L ⫽
1
,
2
1
1
1
1
⫺ 2 acos 2px ⫹ cos 6px ⫹
cos 10px ⫹ Á b ⫹
p
8
9
25
1
1
1
1
a sin 2px ⫺ sin 4px ⫹ sin 6px ⫺ sin 8px ⫹ ⫺ Á b
4
6
8
1
4
1
15. Odd, L ⫽ p,
sin 5x ⫺ ⫹ Á b
asin x ⫺ sin 3x ⫹
p
9
25
4
1
1
1
17. Even, L ⫽ 1,
⫹ 2 acos px ⫹ cos 3px ⫹
cos 5px ⫹ Á b
2
9
25
p
1
p 2
19. 38 ⫹ 12 cos 2x ⫹ 18 cos 4x
4
px 1
3px
1
5px
23. L ⫽ 4, (a) 1, (b) asin
⫹ sin
⫹ sin
⫹Áb
p
4
3
4
5
4
p 4
1
1
25. L ⫽ p, (a) ⫹ acos x ⫹ cos 3x ⫹
cos 5x ⫹ Á b ,
2
p
9
25
(b) 2 (sin x ⫹ 12 sin 2x ⫹ 13 sin 3x ⫹ 14 sin 4x ⫹ Á )
3p
2
1
1
1
27. L ⫽ p, (a)
⫹ acos x ⫺ cos 2x ⫹ cos 3x ⫹
cos 5x ⫺
8
p
2
9
25
1
1
1
1
1
cos 6x ⫹
cos 7x ⫹
cos 9x ⫺
cos 10x ⫹
cos 11x ⫹ Á b
18
49
81
50
121
2
1
1
2
1
(b) a1 ⫹ b sin x ⫹ sin 2x ⫹ a ⫺
b sin 3x ⫹ sin 4x ⫹
p
2
3
9p
4
1
2
1
a ⫹
b sin 5x ⫹ sin 6x ⫹ Á
5
25p
6
29. Rectifier, L ⫽ p,
2
4
1
1
1
(a) ⫺ a # cos x ⫹ # cos 3x ⫹ # cos 5x ⫹ Á b , (b) sin x
p p 1 3
3 5
5 7
Problem Set 11.3, page 494
3. The output becomes a pure cosine series.
5. For An this is similar to Fig. 54 in Sec. 2.8, whereas for the phase shift Bn
the sense is the same for all n.
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App. 2 Answers to Odd-Numbered Problems
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7. y ⫽ C1 cos vt ⫹ C2 sin vt ⫹ a (v) sin t, a (v) ⫽ 1>(v2 ⫺ 1) ⫽ ⫺1.33,
⫺5.26, 4.76, 0.8, 0.01. Note the change of sign.
4
1
1
11. y ⫽ C1 cos vt ⫹ C2 sin vt ⫹ a 2
sin t ⫹ 2
sin 3t ⫹
p v ⫺9
v ⫺ 49
1
sin 5t ⫹ Á b
2
v ⫺ 121
N
13. y ⫽ a (An cos nt ⫹ Bn sin nt),
An ⫽ [(1 ⫺ n 2)an ⫺ nbnc]>Dn,
n⫽1
Bn ⫽ [(1 ⫺ n 2)bn ⫹ ncan]>Dn,
15. bn ⫽ (⫺1)n⫹1 # 12 >n 3 (n odd),
Dn ⫽ (1 ⫺ n 2)2 ⫹ n 2c2
ⴥ
y ⫽ a (An cos nt ⫹ Bn sin nt),
n⫽1
An ⫽ (⫺1)n # 12nc>n 3Dn, Bn ⫽ (⫺1)n⫹1 # 12(1 ⫺ n 2)>(n 3Dn) with Dn as in
Prob. 13.
17. I ⫽ 50 ⫹ A1 cos t ⫹ B1 sin t ⫹ A3 cos 3t ⫹ B3 sin 3t ⫹ Á , An ⫽ (10 ⫺ n 2) an>Dn,
Bn ⫽ 10nan>Dn, an ⫽ ⫺400>(n 2p), Dn ⫽ (n 2 ⫺ 10)2 ⫹ 100n 2
ⴥ
19. I (t) ⫽ a (An cos nt ⫹ Bn sin nt),
An ⫽ (⫺1)n⫹1
2400 (10 ⫺ n 2)
n⫽1
Bn ⫽ (⫺1)n⫹1
24,000
nDn
,
n 2Dn
,
Dn ⫽ (10 ⫺ n 2)2 ⫹ 100n 2
Section 11.4, page 498
4
1
1
⫺ acos x ⫹ cos 3x ⫹
cos 5x ⫹ Á b , E* ⫽ 0.0748,
2
p
9
25
0.0748, 0.0119, 0.0119, 0.0037
4
1
1
5. F ⫽ asin x ⫹ sin 3x ⫹ sin 5x ⫹ Á b , E* ⫽ 1.1902, 1.1902, 0.6243, 0.6243,
p
3
5
0.4206 (0.1272 when N ⫽ 20)
1
7. F ⫽ 2 [(p2 ⫺ 6) sin x ⫺ 18 (4p2 ⫺ 6) sin 2x ⫹ 27
(9p2 ⫺ 6) sin 3x ⫺ ⫹ Á ];
E* ⫽ 674.8, 454.7, 336.4, 265.6, 219.0. Why is E* so large?
3. F ⫽
p
Section 11.5, page 503
3. Set x ⫽ ct ⫹ k.
5. x ⫽ cos u, dx ⫽ ⫺sin u du, etc.
7. lm ⫽ (mp>10)2, m ⫽ 1, 2, Á ; ym ⫽ sin (mpx>10)
9. l ⫽ [(2m ⫹ 1)p>(2L)]2, m ⫽ 0, 1, Á , ym ⫽ sin ((2m ⫹ 1) px>(2L))
11. lm ⫽ m 2, m ⫽ 1, 2, Á , ym ⫽ x sin (m ln ƒ x ƒ )
13. p ⫽ e8x, q ⫽ 0, r ⫽ e8x, lm ⫽ m 2, ym ⫽ eⴚ4x sin mx, m ⫽ 1, 2, Á
Section 11.6, page 509
1. 8 (P1(x) ⫺ P3(x) ⫹ P5(x))
8
3. 45 P0(x) ⫺ 47 P2(x) ⫺ 35
P4(x)
9. ⫺0.4775P1(x) ⫺ 0.6908P3(x) ⫹ 1.844P5(x) ⫺ 0.8236P7(x) ⫹ 0.1658P9(x) ⫹ Á ,
m 0 ⫽ 9. Rounding seems to have considerable influence in Probs. 8–13.
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App. 2 Answers to Odd-Numbered Problems
11. 0.7854P0(x) ⫺ 0.3540P2(x) ⫹ 0.0830P4(x) ⫺ Á , m 0 ⫽ 4
13. 0.1212P0(x) ⫺ 0.7955P2(x) ⫹ 0.9600P4(x) ⫺ 0.3360P6(x) ⫹ Á , m 0 ⫽ 8
15. (c) am ⫽ (2>J 21(a0,m)) (J1(a0,m)>a0,m) ⫽ 2>(a0,mJ1(a0,m))
Section 11.7, page 517
ⴥ
1. f (x) ⫽ peⴚx(x ⬎ 0) gives A ⫽
冮 e cos wv dv ⫽ 1 ⫹1 w , B ⫽ 1 ⫹ww
ⴚv
2
2
0
(see Example 3), etc.
ⴥ
2
p
1 ⫺ cos pw
3. Use (11); B ⫽
sin wv dv ⫽
p
2
w
冮
0
5. B (w) ⫽
2
7. p
2
1
冮 1 pv sin wv dv ⫽ sin w ⫺w w cos w
2
p 0 2
ⴥ
冮 sin w wcos xw dw
0
9. A (w) ⫽
ⴥ
cos wv
dv ⫽ e
冮
p
1⫹v
2
2
ⴚw
(w ⬎ 0)
0
11.
2
ⴥ
冮 cos1 p⫺ww⫹ 1 cos xw dw
2
p 0
15. For n ⫽ 1, 2, 11, 12, 31, 32, 49, 50 the value of Si (np) ⫺ p>2 equals 0.28, ⫺0.15,
0.029, ⫺0.026, 0.0103, ⫺0.0099, 0.0065, ⫺0.0064 (rounded).
2
17. p
19.
ⴥ
冮 1 ⫺ wcos w sin xw dw
0
ⴥ
p 冮
2
w ⫺ e (w cos w ⫺ sin w)
0
1 ⫹ w2
sin xw dw
Section 11.8, page 522
1. fˆc (w) ⫽ 1(2> p) (2 sin w ⫺ sin 2w)>w
3. fˆc (w) ⫽ 1(2> p) (cos 2w ⫹ 2w sin 2w ⫺ 1)>w 2
2
2 (w ⫺ 2) sin w ⫹ 2w cos w
Bp
w3
7. Yes. No
9. 12> p w>(a 2 ⫹ w 2)
2
3
11. 12> p ((2 ⫺ w ) cos w ⫹ 2w sin w ⫺ 2)>w
2 #
1
2 #
1
1
2
2
w
13. fs(eⴚx) ⫽ a⫺fc(eⴚx) ⫹
1b ⫽ a
⫹
b⫽
w
w B p w2 ⫹ 1 B p
Bp
B p w2 ⫹ 1
5. fˆc (w) ⫽
Problem Set 11.9, page 533
3. i (eⴚibw ⫺ eⴚiaw)>(w12p) if a ⬍ b; 0 otherwise
5. [e(1ⴚiw)a ⫺ eⴚ(1ⴚiw)a]>( 12p(1 ⫺ iw))
7. (eⴚiaw(1 ⫹ iaw) ⫺ 1)>(12pw 2)
9. 12> p(cos w ⫹ w sin w ⫺ 1)>w 2
2
11. i12> p (cos w ⫺ 1)>w
13. eⴚw >2 by formula 9
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App. 2 Answers to Odd-Numbered Problems
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17. No, the assumptions in Theorem 3 are not satisfied.
19. [ f1 ⫹ f2 ⫹ f3 ⫹ f4, f1 ⫺ if2 ⫺ f3 ⫹ if4, f1 ⫺ f2 ⫹ f3 ⫺ f4,
21. c
1
1
1
⫺1
dc d ⫽ c
f1
f1 ⫹ f2
f2
f1 ⫺ f2
f1 ⫹ if2 ⫺ f3 ⫺ if4]
d
Chapter 11 Review Questions and Problems, page 537
1
3px
1
5px
sin
⫹ sin
⫹Áb
3
2
5
2
1
2
1
1
13. ⫺ 2 acos px ⫹ cos 3px ⫹
cos 5px ⫹ Á b ⫹
4
9
25
p
4
11. 1 ⫹
p
asin
px
2
⫹
asin px ⫺
1
1
sin 2px ⫹ sin 3px ⫺ ⫹ Á b
2
3
15. cosh x, sinh x (⫺5 ⬍ x ⬍ 5), respectively
17. Cf. Sec. 11.1.
1
4
1
2
1
19. ⫺ 2 acos px ⫹ cos 3px ⫹ Á b ,
asin px ⫺ sin 2px ⫹ ⫺ Á b
2
9
p
2
p
p2
cos t
1 # cos 2t
1 # cos 3t
21. y ⫽ C1 cos vt ⫹ C2 sin vt ⫹ 2 ⫺ 12 a 2
⫺
⫹
2
4 v ⫺4
9 v2 ⫺ 9
v
v ⫺1
1 # cos 4t
⫺
⫹⫺Áb
16 v2 ⫺ 16
23. 0.82, 0.50, 0.36, 0.28, 0.23
25. 0.0076, 0.0076, 0.0012, 0.0012, 0.0004
1
p
27.
ⴥ
冮 (cos w ⫹ w sin w ⫺ 1) cos wxw ⫹ (sin w ⫺ w cos w) sin wx dw
1
2
p 0
29. 12> p (cos aw ⫺ cos w ⫹ aw sin aw ⫺ w sin w)>w 2
Problem Set 12.1, page 542
1. L(c1u 1 ⫹ c2u 2) ⫽ c1L(u 1) ⫹ c2L(u 2) ⫽ c1 # 0 ⫹ c2 # 0 ⫽ 0
3. c ⫽ 2
5. c ⫽ a>b
7. Any c and v
9. c ⫽ p>25
15. u ⫽ 110 ⫺ (110>ln 100) ln (x 2 ⫹ y 2)
17. u ⫽ a( y) cos 4px ⫹ b( y) sin 4px
3
19. u ⫽ c(x) eⴚy >3
21. u ⫽ eⴚ3y(a(x) cos 2y ⫹ b(x) sin 2y) ⫹ 0.1e3y
23. u ⫽ c1( y)x ⫹ c2( y)>x 2 (Euler–Cauchy)
25. u(x, y) ⫽ axy ⫹ bx ⫹ cy ⫹ k; a, b, c, k arbitrary constants
Problem Set 12.3, page 551
5. k cos 3pt sin 3px
8k
1
1
7. 3 acos pt sin px ⫹
cos 3pt sin 3px ⫹
cos 5pt sin 5px ⫹ Á b
27
125
p
9.
2 acos
0.8
p
pt sin px ⫺
1
1
cos 3pt sin 3px ⫹
cos 5pt sin 5px ⫺ ⫹ Á b
9
25