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

924 chapter 14 Partial Derivatives
5 –8 Determine the signs of the partial derivatives for the
function f whose graph is shown.
x
1
z
2
y
10. A contour map is given for a function f. Use it to estimate
f xs2, 1d and f ys2, 1d.
_4
_2
4
2
y
3
0
1
6 8
10
12
14
16
3 18
x
5. (a) f xs1, 2d (b) f ys1, 2d
6. (a) f xs21, 2d
7. (a) f xxs21, 2d
8. (a) f xys1, 2d
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(b) f ys21, 2d
(b) f yys21, 2d
(b) f xys21, 2d
9. The following surfaces, labeled a, b, and c, are graphs of a
function f and its partial derivatives f x and f y. Identify each
surface and give reasons for your choices.
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11. If f sx, yd − 16 2 4x 2 2 y 2 , find f xs1, 2d and f ys1, 2d and
interpret these numbers as slopes. Illustrate with either handdrawn
sketches or computer plots.
12. If f sx, yd − s4 2 x 2 2 4y 2 , find f xs1, 0d and f ys1, 0d and
inter pret these numbers as slopes. Illustrate with either handdrawn
sketches or computer plots.
; 13–14 Find f x and f y and graph f , f x, and f y with domains and
viewpoints that enable you to see the relationships between them.
y
13. f sx, yd − x 2 y 3 14. f sx, yd −
1 1 x 2 y 2
8
4
z 0
_4
a
_8
_3 _2 _1 0
y
1 2 3
2
0 _2
x
15–40 Find the first partial derivatives of the function.
15. f sx, yd − x 4 1 5xy 3 16. f sx, yd − x 2 y 2 3y 4
17. f sx, td − t 2 e 2x 18. f sx, td − s3x 1 4t
19. z − lnsx 1 t 2 d
21. f sx, yd − x y
ax 1 by
23. f sx, yd −
cx 1 dy
20. z − x sinsxyd
x
22. f sx, yd −
sx 1 yd 2
24. w − ev
u 1 v 2
4
z 0
_4
b
_3 _2 _1 0 1 2
y
3
2
0 _2
x
25. tsu, vd − su 2 v 2 v 3 d 5 26. usr, d − sinsr cos d
27. Rsp, qd − tan 21 spq 2 d
29. Fsx, yd − y x
cosse t d dt
y
28. f sx, yd − x y
30. Fs, d − y st 3 1 1 dt
31. f sx, y, zd − x 3 yz 2 1 2yz 32. f sx, y, zd − xy 2 e 2xz
33. w − lnsx 1 2y 1 3zd 34. w − y tansx 1 2zd
8
4
35. p − st 4 1 u 2 cos v 36. u − x yyz
37. hsx, y, z, td − x 2 y cosszytd 38. sx, y, z, td −
39. u − sx
1 2 1 x 2 2 1 ∙ ∙ ∙ 1 x n
2
x 1 y
2
z 1 t 2
z 0
_4
_8
c
_3 _2 _1 0
y
1 2 3
2
0 _2
x
40. u − sinsx 1 1 2x 2 1 ∙ ∙ ∙ 1 nx nd
41–44 Find the indicated partial derivative.
41. Rss, td − te syt ; R ts0, 1d
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