James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A126 Appendix I Answers to Odd-Numbered Exercises
51. (a) f 9sxd, t9syd (b) f 9sx 1 yd, f 9sx 1 yd
53. f xx − 12x 2 y 2 12xy 2 , f xy − 4x 3 2 12x 2 y − f yx, f yy − 24x 3
8y
55. z xx −
s2x 1 3yd , 3 zxy − 6y 2 4x
s2x 1 3yd − zyx,
3
z yy − 2
12x
s2x 1 3yd 3
57. v ss − 2 cosss 2 2 t 2 d 2 4s 2 sinss 2 2 t 2 d,
v st − 4st sinss 2 2 t 2 d − v ts,
v tt − 22 cosss 2 2 t 2 d 2 4t 2 sinss 2 2 t 2 d
63. 24xy 2 2 6y, 24x 2 y 2 6x 65. s2x 2 y 2 z 5 1 6xyz 3 1 2zde xyz 2
67. 3 4 vsu 1 v 2 d 25y2 69. 4ysy 1 2zd 3 , 0 71. 6yz 2
73. <12.2, <16.8, <23.25 83. R 2 2
yR 1
−T
87.
−P − V 2 nb
nR
, −P
−V − 2n 2 a
2 nRT
V 3 sV 2 nbd 2
91. (a) 0.0545; for a person 70 inches tall who weighs
160 pounds, an increase in weight causes the surface area to
increase at a rate of about 0.0545 square feet per pound.
(b) 0.213; for a person 70 inches tall who weighs 160 pounds,
an increase in height (with no change in weight) causes the surface
area to increase at a rate of about 0.213 square feet per inch of
height.
93. −Py−v − 3Av 2 2 Bsmtyxd2 is the rate of change of the
v 2
power needed during flapping mode with respect to the bird’s velocity
when the mass and fraction of flapping time remain constant;
−Py−x − 2 2Bm2 t 2
is the rate at which the power changes
x 3 v
when only the fraction of time spent in flapping mode varies;
−Py−m − 2Bmt2 is the rate of change of the power when only the
x 2 v
mass varies.
97. No 99. x − 1 1 t, y − 2, z − 2 2 2t 103. 22
105. (a)
z
0.2
0
_0.2
_1
y
0
1
1
(b) f xsx, yd − x 4 y 1 4x 2 y 3 2 y 5
, f ysx, yd − x 5 2 4x 3 y 2 2 xy 4
sx 2 1 y 2 d 2 sx 2 1 y 2 d 2
(c) 0, 0 (e) No, since f xy and f yx are not continuous.
0
x
Exercises 14.4 • page 934
1. z − 4x 2 y 2 6 3. z − x 2 y 1 1 5. x 1 y 1 z − 0
7.
400
z 200
0
10
x 0 _10
5
0
y
_5
_1
9.
z
3
2
1
1
y
0
1 1.5
0.5
x
11. 6x 1 4y 2 23 13. 2x 1 y 2 1
15. 2x 1 2y 1 2 4 19. 6.3
21. 3 7 x 1 2 7 y 1 6 7 z; 6.9914 23. 4T 1 H 2 329; 129°F
25. dz − 22e 22x cos 2t dx 2 2e 22x sin 2t dt
27. dm − 5p 4 q 3 dp 1 3p 5 q 2 dq
29. dR − 2 cos d 1 2 cos d 2 2 sin d
31. Dz − 0.9225, dz − 0.9 33. 5.4 cm 2 35. 16 cm 3
37. <20.0165mt; decrease 39. 1 17 < 0.059 V
41. (a) 0.8264m 2 34.56h 1 38.02 (b) 18.801
43. « 1 − Dx, « 2 − Dy
Exercises 14.5 • page 943
1. 2tsy 3 2 2xy 1 3xy 2 2 x 2 d
1
3.
2st cos x cos y 1 1 sin x sin y
2
t
5. e yyz f2t 2 sxyzd 2 s2xyyz 2 dg
7. −zy−s − 5sx 2 yd 4 s2st 2 t 2 d, −zy−t − 5sx 2 yd 4 ss 2 2 2std
−z 3 sin t 2 2t sin s
9. − , −z 3s cos t 1 2 cos s
−
−s 3x 1 2y −t 3x 1 2y
11.
−z
−s − e rSt cos 2
−z
−t − e rSs cos 2
13. 42 15. 7, 2
−u
17.
−r − −u
−x
− −u
−u
−t
19.
−x
−T
−x − −T
−p
−p
s
ss 2 1 t 2 sin D,
t
D
ss 2 1 t sin 2
−x
−r 1 −u
−y
−x
−t 1 −u
−y
−y
−t
−p
−x 1 −T
−q
−q
−y
−r , −u
−s − −u
−x
−q
−x 1 −T
−r
−r
−r
−x ,
−T
−y − −T
−p −y 1 −T
−q −y 1 −T
−r −y ,
−T
−z − −T −p
−p −z 1 −T −q
−q −z 1 −T −r
−r −z
21. 1582, 3164, 2700 23. 2, 22
5
25. 144 , 2 5 96 , 5 2x 1 y sin x
144 27.
cos x 2 2y
1 1 x 4 y 2 1 y 2 1 x 4 y 4 2 2xy
29.
x 2 2 2xy 2 2x 5 y 3
−x
−s 1 −u
−y
31. 2 x
3z , 2 2y
yz
33.
3z e z 2 xy , xz
e z 2 xy
35. 2°Cys 37. < 20.33 mys per minute
39. (a) 6 m 3 ys (b) 10 m 2 ys (c) 0 mys
41. < 20.27 Lys 43. 21y(12s3) radys
−y
−s ,
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