Thomas Calculus 13th [Solutions]

Section 14.5 Directional Derivatives and Gradient Vectors 1009
3.
g 2 g
y (2, 1) 1;
x
x
g
g
2 xy (2, 1) 4 g i 4 j;
y
x
g(2, 1) 2 x 2 is the level curve
2
y
g g
4. x 2, 1 2;
x x
g
y
g
y 2,1 1 g 2 i j;
y
2
2
2,1 1 1 x y
2 2
g or 1 x y is the level
2 2 2 2
curve
f
5. 1 f
( 1, 2) 1 ;
x 2x
3y
x 2
f
y
3 f
( 1, 2) 3 f 1 i 3 j;
2 2x
3y
x 4 2 4
f ( 1, 2) 2 4 2x 3y is the level curve
f y f
x 2 y x 2x
x
6.
2 3 2
4, 2 1 ;
16
f x f
(4, 2) 1 f 1 i 1 j;
y 2
2y
x y
4 16 4
f (4, 2) y x is the level curve
4
f
7. 2 z f
f
f
f
f
x (1,1, 1) 3; 2 y (1, 1, 1) 2; 4z ln x (1, 1,1) 4; thus
x x x
y
y
z
z
f 3i 2j 4k
f
x
6 xz z f
(1, 1,1) 11
1 x
2
; f
6 f
(1, 1,1) 6;
x z
y
y
8.
2 2
f
(1, 1,1) 1 ;
z 2
thus f 11 i 6j 1 k
2 2
f
z
2 2 2
x
x z
yz 6z 3 x y
2 2
1
f
x 1 f
( 1, 2, 2) 26 ;
x x x
27
x y z
9.
2 2 2
3 2
f
z 1 f
( 1, 2, 2) 23 ;
z 2 2 2
3 2 z z
54
x y z
f y 1 f
( 1, 2, 2) 23 ;
y 2 2 2
3 2 y y
54
x y z
thus f 26 i 23 j 23 k
27 54 54
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