Math 209 - Assignment 2 Answer Key - Course Hero Page 3
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1
2
3
4
52
2
2
5. Find the directional derivative of the function f (x, y, z) =
x
+ y
+ z
at the point
(1, 2, 2) in the direction of vector v =
6, 6, 3 .
Solution:
We first compute the gradient vector at (1, 2, 2).
x
y
z
f (x, y, z) =
,
,
2
2
2
2
2
2
2
2
2
x
+ y
+ z
x
+ y
+ z
x
+ y
+ z
1
2
2
f (1, 2, 2) =
,
,
.
3
3
3
Note that v is not unit vector, but since v = 9, the unit vector in the direction of v is
v
2
2
1
u =
=
,
,
.
v
3
3
3
Therefore
2
D f (1, 2, 2) =
f (1, 2, 2) u =
.
3
2
2
2
x
3y
9z
6. The temperature at a point (x, y, z) on the surface of a metal is T (x, y, z) = 200e
where T is measured in degree Celsius and x, y, z in meters.
(a) In which direction does the temperature increase fastest at the point P (2, 1, 2)?
(b) What is the maximum rate of change at P (2, 1, 2)?
Solution:
We first compute the gradient vector:
2
2
2
x
3y
9z
T (x, y, z) = T
, T
, T
=
e
400x, 1200y, 3600z
x
y
z
43
T (2, 1, 2) =
400e
2, 3, 18 .
The temperature increases in the direction of the gradient vector
43
T (2, 1, 2) =
400e
2, 3, 18 .
The maximum rate of change is
25
43
400e
2, 3, 18 = 400e
337 .
2
2
2
7. Find the points on the ellipsoid x
+ 2y
+ 3z
= 1 where the tangent plane is parallel to
the plane 3x
2y + 3z = 1.
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