M
ath 105, Section 208 & 209 –
M
idterm 2
S
olutions
1. Consider the following function:
f
(
x, y
) =
x
3
+
y
4

3
x

32
y
+ 10
(a) (5 points) Find
all
the critical points of
f
.
(b) (5 points) For any one of the critical points you found in part (a), determine whether
it is a local maximum, local minimum or a saddle point.
You do not need to check this
for every critical point, one is enough.
2.
(a) (3 points) Draw the graphs of the two curves
x
=
y
2
and
3
y

x

2 = 0
in the (
x, y
)plane and shade the region that is bounded by these curves.
(b) (5 points) Find the area of the shaded region in part (a).
3. You are standing right above the point (1
,
1) on a surface whose height is given by the
equation
z
=
2
3
x
2
y
+
5
3
xy
2
.
(a) (4 points) Determine
z
x
and
z
y
at (1
,
1).
(b) (4 points) If you wish to move in a direction that gains height at the fastest possible
rate, in which direction should you start moving from (1
,
1)?
All directions should be
expressed as unit vectors.
(c) (4 points) Starting from your initial position (1
,
1), you intend to follow a path on the
surface lying directly above the parabola
x
(
t
) =
t,
y
(
t
) =
t
2
.
Find the initial rate of change in height with respect to
t
. Are you climbing up or down
at this point?
4.
(a) (6 points) Suppose you open an investment account with $10,000 that earns interest
at the rate of 10% per year compounded continuously, but you withdraw money con
tinuously at the rate of $2000 each year. Set up a differential equation that the value
of your investment
y
(
t
) at time
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 Fall '10
 MalabikaPramanik
 Calculus, Critical Point, fxx, Zy

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