SOLUTION TO 18.02 PRACTICE MIDTERM #2B
BJORN POONEN
1) A particle is moving in the
xy
plane so that relative to the origin it is rotating coun
terclockwise at a rate of 2 radians per second while its distance to the origin is increasing at
a rate of 10 meters per second. At a time when the particle is at (

4
,
3), what is
dy
dt
?
Solution: At the given time
r
=
p
(

4)
2
+ 3
2
= 5 and (cos
θ,
sin
θ
) = (

4
/
5
,
3
/
5). We
have
y
=
r
sin
θ
, so the chain rule yields
dy
dt
=
r
cos
θ
dθ
dt
+
dr
dt
sin
θ
= 5(

4
/
5)2 + 10(3
/
5)
=

8 + 6
=

2 m
/
s
.
2) Let
f
(
x, y, z
) = 2
xy
2
+
z
3
.
(a) (10 pts.) Find the unit vector
u
in
R
3
that
minimizes
the directional derivative
D
u
f
at the point (0
,
√
2
,

1).
(b) (5 pts.) Find the value of
D
u
f
at (0
,
√
2
,

1) in that direction
u
.
Solution:
(a) We have
∇
f
=
h
2
y
2
,
4
xy,
3
z
2
i
,
which is
h
4
,
0
,
3
i
at (
x, y, z
) = (0
,
√
2
,

1). The length of
∇
f
there is 5, so the unit vector in
the direction of
∇
f
is
h
4
/
5
,
0
,
3
/
5