Thomas Calculus 13th [Solutions]

1012 Chapter 14 Partial Derivatives
26. f 2xi j f 2,1 2 2i j
Tangent line: 2 2 x 2 ( y 1) 0
y
2 2x
3
27. f yi xj f (2, 2) 2i 2j
Tangent line: 2( x 2) 2( y 2) 0
y x 4
28. f (2 x y) i (2 y x) j f ( 1, 2) 4i 5j
Tangent line: 4( x 1) 5( y 2) 0
4x
5y
14 0
29. f (2 x y) i ( x 2y
1) j
(a) f (1, 1) 3i 4 j f (1, 1) 5 Du
f (1, 1) 5 in the direction of u 3 i 4 j
5 5
(b) f (1, 1) 3i 4 j f (1, 1) 5 Du
f (1, 1) 5 in the direction of u 3 i 4 j
5 5
(c) Du f (1, 1) 0 in the direction of u 4 i 3 j or u 4 i 3 j
5 5 5 5
2 2 2 2
(d) Let u u1i u2j u u1 u2 1 u1 u2
1; Du f (1, 1) f (1, 1) u 3i 4j u1i u2j
3 2 3
2
25 2
3u 3
1 4u2 4 u2 u
4 1 1 u1 u
4 1 1 1 u
16 1 u
2 1 0 u 1 0 or
u 24
1 25 ; u 1 0 u 2 1 u j , or u 24 7 24 7
1 u
25 2 u i j
25 25 25
2 2 2 2
(e) Let u u1i u2j u u1 u2 1 u1 u2
1; Du f (1, 1) f (1, 1) u 3i 4j u1i u2j
2
2 25 2
3u 4 4 8
24
1 4u2 3 u1 u
3 2 1 u
3 2 1 u2 1 u
9 2 u
3 2 0 u2 0 or u2
;
25
u2 0 u1
1 u i , or u 24 7 7 24
2 u
25 2 u i j
25 25 25
30. f 2 y
i 2x
j
2 2
( x y) ( x y)
(a) f 1 , 3 3 i j f 1 , 3 10 D f 1 , 3 10
2 2 2 2 u
in the direction of u 3 i 1 j
2 2
10 10
(b) f 1 , 3 3 i j f 1 , 3 10 D f 1, 1 10
2 2 2 2
u
in the direction of u 3 i 1 j
10 10
Copyright
2014 Pearson Education, Inc.