Thomas Calculus 13th [Solutions]

1064 Chapter 14 Partial Derivatives
(b) The direction toward (4, 6) is determined by v
3 4
3 (4 1) i (6 2) j 3i 4j u i j
5 5
f u
14
5 .
44. (a) True (b) False (c) True (d) True
45. f 2xi j 2zk
f j 2 k,
(0, 1, 1)
f j,
(0,0,0)
f j 2k
(0, 1,1)
46. f 2yj
2zk
f
(2,2,0)
4 j,
f
(2, 2,0)
4 j,
f
(2,0,2)
4 k ,
f
(2,0, 2)
4k
47.
48.
f 2xi j 5k f 4i j 5k Tangent Plane: 4( x 2) ( y 1) 5( z 1) 0
(2, 1,1)
4x y 5z 4; Normal Line: x 2 4 t, y 1 t, z 1 5t
f 2xi 2yj k f 2i 2j k Tangent Plane: 2( x 1) 2( y 1) ( z 2) 0
(1,1,2)
2x 2y z 6 0; Normal Line: x 1 2 t, y 1 2 t, z 2 t, z 2 t
49. z 2x z
x 2 2
x y x (0,1,0)
2y
z 2 0
0
and z 2 y
y 2 2
x y
z
y
(0,1,0)
2;
thus the tangent plane is 2( y 1) ( z 0) 0 or
2
50. z 2 2
2x x y z
1 and
x
x 1,1,
1 2
1 ( x 1) 1 ( y 1) z 1 0 or x y 2z
3 0
2 2 2
2
z
y
2 2 2
2
y x y z
1
y
1,1,
2
; thus the tangent plane is
1
2
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