Math222
Assignment 6
due Wednesday Nov. 7, 2007
1. The function
z
=
Z
(
x, y
) is defined implicitly by the equation
x
2
+
y
2
+ 4
z
2
+
z
4
= 64
(a) Find
∂z
∂x
,
∂z
∂y
,
∂
2
z
∂x
2
, and then evaluate these at the point
P
=
(4
,
4
,
2).
(b) Find the equation of the tangent plane to the surface defined above
at the point
P
.
(c) Find
∇
Z
at
P
and find the direction of maximum increase of the
function
Z
.
2. Find the critical points indicating (with your justification) which are local
maxima, local minima, and which are saddle points of the function.
z
=
xy e

(
x
2
+4
y
2
)
/
2
3. Let
z
be defined implicitly by
z
3
+ 4
z
= 2
x
2
y
+ 12.
Taking
x
and
y
as independent variables, find all the first and second partials of
z
and
evaluate these at (
x, y
) = (1
,
2) given that
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 Fall '07
 Harris
 Critical Point, minimum values, minimum surface area

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