Math 237
Assignment 5 Solutions
1.
Let
f
(
x, y
) = ln(
x
2
+
y
2
).
a) Find the directional derivative of
f
at (

1
,
2) in the direction of the vector
~v
= (3
,

4).
Solution: We ﬁrst ﬁnd the unit vector ˆ
v
in the direction of
~v
. We have
ˆ
v
=
~v
k
~v
k
=
(3
,

4)
√
3
2
+ 4
2
=
±
3
5
,

4
5
²
.
The partial derivatives of
f
are
f
x
=
2
x
x
2
+
y
2
,
f
y
=
2
y
x
2
+
y
2
,
which are both continuous at (

1
,
2). So, we get
D
(3
,

4)
f
(

1
,
2) =
∇
f
(

1
,
2)
·
ˆ
v
=
±

2
5
,
4
5
²
·
±
3
5
,

4
5
²
=

22
25
b) Find the direction in which
f
is increasing the fastest at (1
,
1). What is the magnitude
of this rate of change?
Solution: We have
∇
f
=
³
2
x
x
2
+
y
2
,
2
y
x
2
+
y
2
´
so
∇
f
(1
,
1) = (1
,
1)
.
Therefore,
f
is increasing
the fastest in the direction of
∇
f
(1
,
1) = (1
,
1) and the magnitude of this rate of change is
k∇
f
(1
,
1)
k
=
k
(1
,
1)
k
=
√
2.
c) Find the equation of the tangent line at (1
,
1) to the level curve
f
(
x, y
) = ln 2.
Solution: Since