Thomas Calculus 13th [Solutions]

1022 Chapter 14 Partial Derivatives
2
56. (a) fx
( x, y) 2 x( y 1) f x (1, 0) 2 and f y ( x, y) x f y (1, 0) 1 df 2 dx 1 dy
df is more sensitive to changes in x
(b) df 0 2 dx dy 0 2 dx 1 0 dx 1
dy
dy 2
57. (a)
r 2 x 2 y 2
2 r dr 2 x dx 2 y dy dr x y
| 3 4
(3, 4) ( 0.01) ( 0.01)
r
dx r
dy dr 5 5
0.07 0.014
5 0.014 dr
r
100 5
100 0.28%;
y
1
x
2
x
y
2
y
2
x
x
d dx dy
1 1
y
0.04
dx x dy d | 4
3 0.03
2 2 2 2 (3, 4) ( 0.01) ( 0.01)
maximum change in d
y x y x
25 25 25 25
occurs when dx and dy have opposite signs ( dx 0.01 and dy 0.01 or vice versa)
0.07 1
d
0.0028; tan 4 0.927255218 d 100 0.0028 100 0.30%
25 3 0.927255218
(b) the radius r is more sensitive to changes in y, and the angle is more sensitive to changes in x
58. (a)
(b)
2 2
V r h dV 2 rh dr r dh at r 1 and h 5 we have dV 10 dr dh the
volume is about 10 times more sensitive to a change in r
2
dV 0 0 2 rh dr r dh 2h dr r dh 10 dr dh dr 1 dh ; choose
10
dh 1.5 dr 0.15 h 6.5 in. and r 0.85 in. is one solution for V dV 0
a b
59. f ( a, b, c, d) ad bc fa d, fb c, fc b, fd
a df d da c db b dc a dd;
c d
since | a | is much greater than | b|, | c |, and | d |, the function f is most sensitive to a change in d.
y y y y
60. ux e , uy xe sin z, uz
y cos z du e dx ( xe sin z) dy ( y cos z)
dz
du| 3 dx 7 dy 0 dz 3 dx 7 dy magnitude of the maximum possible error
2, ln 3,
3(0.2) 7(0.6) 4.8
2
61.
1/2 1/2
1/2
Q 1 2KM 2M , 1 2KM 2K
K
Q ,
2 h h M
and Q 1 2KM
2KM
2 h h
h 2 h 2
h
1/2 1/2 1/2
dQ 1 2KM 2M dK 1 2KM 2K dM 1 2KM 2KM
dh
2 h h 2 h h 2 h 2
h
1/2
1 2KM 2M dK 2K dM 2KM
dh
2 h h h 2
h
(2)(2)(20)
1/2
(2)(20) (2)(2) (2)(2)(20)
dQ| 1
(2, 20, 0.0.05) =
dK dM dh
2 0.05 0.05 0.05 2
(0.05)
(0.0125)(800 dK 80 dM 32,000 dh)
Q is most sensitive to changes in h
62. A 1 absin C A 1 sin , 1 sin , 1 cos
2 a b C A
2 b a C A
2 c ab C
2
dA 1 bsin C da 1 a sin C db 1 ab cos C dC ; dC |2°| |0.0349| radians, da |0.5| ft, db |0.5| ft;
2 2 2
at a 150 ft, b 200 ft, and C 60°, we that the change is approximately
dA
1 1 1
2
(200)(sin 60°)|0.5| (150)(sin 60°)|0.5| (200)(150)(cos 60°) |0.0349| 338 ft
2 2 2
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