1
. (a) Determine whether
f
(
x, y, z
) =
xy
1+
y
2
z
is increasing or decreasing
at (1
,
1
,
0) toward the point (3
,

1
,
1)
.
At what rate? What is the maximal
increase rate and in what direction it occurs?
(b) Write the equation of the tangent plane to the surface
S
:
xy
1+
y
2
z
= 1
at (1
,
1
,
0).
Answer.
(a) A vector pointing from (1
,
1
,
0) toward (3
,

1
,
1) is
~a
=
h
2
,

2
,
1
i
. The rate of change of
f
at (1
,
1
,
0) in
~a
direction is the number
D
~u
f
(1
,
1
,
0) =
∇
f
(1
,
1
,
0)
·
~u
, where
~u
=
~a/

~a

=
h
2
,

2
,
1
i

~a

=
2
3
,

2
3
,
1
3
.
So we find
f
x
=
y
1 +
y
2
z
, f
y
=
x
(1 +
y
2
z
)

2
xy
2
z
(1 +
y
2
z
)
2
, f
z
=

xy
(1 +
y
2
z
)
2
y
2
,
∇
f
(1
,
1
,
0)
=
h
1
,
1
,

1
i
and
D
~u
f
(1
,
1
,
0) =

1
3
<
0
.
The function
f
is decreasing at (1
,
1
,
0) in the direction of
~u
at the rate
1
3
.
The maximal increase rate is
∇
f
(0
,
1
,
1)

=
√
3 and it occurs in the direction
of
∇
f
(0
,
1
,
1) =
h
1
,
1
,

1
i
.
(b) The tangent plane passes through (1
,
1
,
0) and is perpendicular to
∇
f
(1
,
1
,
0) =
h
1
,
1
,

1
i
:
(1)(
x

1) +
y

1 + (

1)(
z
) = 0 or
x
+
y

z

2 = 0
.
2.
(a) Determine the set of points at which the function is continuous:
f
(
x, y
) =
(
x
2
y
2
x
2
+
y
4
,
(
x, y
)
6
= (0
,
0)
,
0
,
(
x, y
) = (0
,
0)
.
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 '07
 KAMIENNY
 Math, Calculus, Derivative, cylinder x2, answer., adirection

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