Thomas Calculus 13th [Solutions]

Chapter 14 Practice Exercises 1061
15. Let
2 y 2
y kx , k 1. Then lim
lim kx k
( x, y) (0, 0) x y 2
( x, kx ) (0, 0) x kx 1 k
2
y x
different values of k the limit does not exist.
2 2 2 2
which gives different limits for
2 2 2 2 2
x y x ( kx) 16. Let y kx, k 0. Then lim
lim
1 k
( x, y) (0, 0)
xy
( x, kx) (0, 0)
x( kx)
k
xy 0
different values of k the limit does not exist.
which gives different limits for
17. Let y kx . Then
2 2 2 2 2 2
x y
lim
x k x 1 k
( x, y) (0, 0) x y x k x 1 k
2 2 2 2 2 2
which gives different limits for different values of
k the limit does not exist so f (0, 0) cannot be defined in a way that makes f continuous at the origin.
18. Along the x-axis,
y 0 and
not continuous at (0, 0).
sin ( x y) sin 1, 0
lim lim x x
,
( x, y) (0, 0) x y x 0 x 1, x 0
so the limit fails to exist
f is
g
g
19. cos sin , r sin r cos
r
20.
y
2x
x
2
x
2 2
y
2 2 2 2 2 2 2
1
x
f 1
y x y
,
x 2 x y x y x y x y
1
2
x
x
2 2
y
2 2 2 2 2 2 2
1
x
f 1 y y x y
y 2 x y x y x y x y
f 1 f 1 f 1
21. , ,
R 2 2 2
1 R R
1 2 R R
2 3 R3
22. hx ( x, y, z) 2 cos(2 x y 3 z), hy ( x, y, z) cos (2 x y 3 z), hz
( x, y, z) 3cos(2 x y 3 z)
23. P RT , P nT , P nR , P nRT
n V R V T V V 2
V
24. f ( , , , ) 1 T , ( , , , ) 1 T
1 1 1 1 1
r r T w f r T w , f ( , , , )
2 2 T r T w
2r
w 2r
w 2r w 2 T 4r T w
1 T 1 1 3 2
, f ( , , , ) T 1 T
4 w r T w w
r T w 2r 2 4r w w
25.
2 2 2 2
g 1 g
, 1 x g g
0, 2x
g g
,
1
x y y 2 2 2 3 y x x y 2
y x y y y
x
x
26. gx ( x, y) e y cos x, g y ( x, y) sin x gxx ( x, y) e y sin x, g yy ( x, y) 0,
gxy
( x, y) gxy
( x, y) cos x
27.
2 2 2 2 2
f 2 2 f f 2 2 f f f
1 y 15 x x , x 30 x x , 0, 1
x 2 2 2 2 2
x 1 y x ( x 1) y y x x y
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