Thomas Calculus 13th [Solutions]

Section 14.6 Tangent Planes and Differentials 1015
5. (a)
xz 2
xz
f sin x 2 xy ze i x z j xe y k f (0, 1, 2) 2i 2j k
Tangent plane: 2( x 0) 2( y 1) 1( z 2) 0 2x 2y z 4 0;
(b) Normal line: x 2 t, y 1 2 t, z 2 t
6. (a) f (2 x y) i ( x 2 y) j k f (1,1, 1) i 3j k
Tangent plane: 1( x 1) 3( y 1) 1( z 1) 0 x 3y z 1;
(b) Normal line: x 1 t, y 1 3 t, z 1 t
7. (a) f i j k for all points f (0, 1, 0) i j k
Tangent plane: 1( x 0) 1( y 1) 1( z 0) 0 x y z 1 0;
(b) Normal line: x t, y 1 t,
z t
8. (a) f (2x 2y 1) i (2y 2x 3) j k f (2, 3, 18) 9i 7j k
Tangent plane: 9( x 2) 7( y 3) 1( z 18) 0 9x 7 y z 21;
(b) Normal line: x 2 9 t, y 3 7 t, z 18 t
z f 2 2
( x , y ) ln x y f ( x , y ) 2x
and f 2 y
y ( x , y ) f (1, 0) 2
x y
2 2 x and f y (1, 0) 0
x y
9. x
2 2
from Eq. (4) the tangent plane at (1, 0, 0) is 2( x 1) z 0 or 2x
z 2 0
10.
2 2 2 2
x y x y
z f ( x, y) e fx
( x, y) 2xe and
f y (0, 0) 0 from Eq. (4) the tangent plane at (0, 0,1) is z 1 0 or z 1
2 2
x y
f y ( x, y) 2 ye f x (0, 0) 0 and
11.
z f ( x, y) y x f ( x, y) ( y x ) and
1
2
x y 2z
1 0
x
1
2
f y (1, 2) from Eq. (4) the tangent plane at (1, 2,1) is
1 1
1 2
1 1 2
1
2 x 2
f ( x, y) ( y x) f (1, 2) and
y
( x 1) ( y 2) ( z 1) 0
2 2
12.
2 2
z f ( x, y) 4 x y f ( x, y) 8x and f ( x, y) 2 y f (1, 1) 8 and f (1, 1) 2 from Eq. (4)
x
the tangent plane at (1,1, 5) is 8( x 1) 2( y 1) ( z 5) 0 or 8x 2y z 5 0
y
x
y
i j k
13. f i 2yj 2 k f (1,1, 1) i 2j 2k and g i for all points; v f g v 1 2 2 2j 2k
1 0 0
Tangent line: x 1, y 1 2 t, z 1 2t
14. f yzi xzj xyk f (1,1,1) i j k ; g 2xi 4yj 6 zk g(1,1, 1) 2i 4j 6 k;
i j k
v f g 1 1 1 2i 4j 2k Tangent line: x 1 2 t, y 1 4 t, z 1 2t
2 4 6
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