Thomas Calculus 13th [Solutions]

986 Chapter 14 Partial Derivatives
29.
P
lim
2
ze y cos 2x 2(0)
3e
cos 2 (3)(1)(1) 3
( , 0, 3)
30.
2 2 2 2 2 2
| lim ln x y z ln 2 ( 3) 6 ln 49 ln 7
P (2, 3, 6)
31. (a) All ( x, y )
(b) All ( x, y ) except (0, 0)
32. (a) All ( x, y ) so that x y (b) All ( x, y)
33. (a) All ( x, y ) except where x 0 or y 0 (b) All ( x, y)
34. (a) All ( x, y ) so that
(b) All ( x, y ) so that
2
x 3x 2 0 ( x 2)( x 1) 0 x 2 and x 1
y
2
x
35. (a) All ( x, y, z )
(b) All ( x, y, z ) except the interior of the cylinder
2 2
x y
1
36. (a) All ( x, y, z ) so that xyz 0
(b) All ( x, y, z)
37. (a) All ( x, y, z ) with z 0
(b) All ( x, y, z ) with
2 2
x z
1
38. (a) All ( x, y, z ) except ( x , 0, 0)
(b) All ( x, y, z ) except (0, y , 0) or ( x, 0, 0)
39. (a) All ( x, y, z ) such that
2 2
z x y 1 (b) All ( x, y, z ) such that
2 2
z x y
40. (a) All ( x, y, z ) such that
2 2 2
x y z 4 (b) All ( x, y, z ) such that
2 2 2
x y z
25
2 2 2
x y z 9 except when
41.
lim x lim x lim x lim x lim 1 1 ;
2 2 2 2
( x, y) (0, 0) x y x 0 x x x 0 2| x| x 0 2x
x 0 2 2
along y x
x 0
lim x lim x lim x lim 1 1
2 2
( x, y) (0, 0) x y x 0 2| x| x 0 2( x) x 0 2 2
along y x
x 0
42.
along y 0
4 4
lim x lim x 1;
4 2 4 2
( x, y) (0, 0) x y x 0 x 0
along y
x
2
4 4 4
lim x lim x lim x
( x, y) (0, 0) x y x 0 x x x 0 2x
4 2
4 2
2 4
1
2
43.
4 2
2
4 2 x kx
4 2 4 2
x y
lim lim lim x k x 1 k
( x, y) (0, 0) x y x 0 x kx x 0 x k x 1 k
2
along y kx
4 2
4 2
2 4 2 4 2
different limits for different values of k
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