1. For
f
(
x; y
)=3
x
3
y
3
9
x
+12
y
+3, which of the following statements is true?
1) (1
;
1) is a saddle point 2) (1
;
2) is a local maximum
3) (
1
;
2) is a local maximum 4) (1
;
2) is a local minimum
2. For
L
= lim
(
x;y
)
!
(0
;
0)
5
xy
x
2
+
y
2
, which of the following is true?
1)
L
=
1
2)
L
=0 3)
L
does not exist 4)
L
=5
=
2
3. Let
w
=
f
(
x; y
) =
xy
3
e
x
2
where
x
=
u
2
+3
uv
and
y
= 2
e
v
+
u
2
. The partial
derivative
@w
@u
evaluated at (
u; v
)=(1
;
0) is
1) 108 + 4
e
2) 4
e
3) 108 +
e
4) 108
4
e
4. Let the directional derivative of a function
f
(
x; y
) at a point
P
in the direction of
1
p
5
~
i
+
2
p
5
~
j
be
16
p
5
and the partial derivative
@f
@x
evaluated at
P
be 6. Then, the direc-
tional derivativeinthedirection of
~
i
~
j
is
1) 1
=
p
2 2) 1 3)
1
p
2
~
i
1
p
2
~
j
4)
p
40
5. The Maclauring series for
f
(
x
)=
3
1
2
x
has the open interval of convergence
1) (-1,1) 2) (
1
2
;
1
2
) 3) (-2,2) 4) (2,4)
6. For a collection
f
a
n
g
of positive numbers which of the following combinations cannot
occur
1)
1
P
n
=1
a
n
diverges and
1
P
n
=1
(
1)
n
a
n
converges
2)
1
P
n
=1
a
n
converges and
1
P
n
=1
(
1)
n
a
n
diverges
3) lim
n
!1
(
1)
n
a
n
does not exist and lim
n
!1
a
n
exists
4) lim
n
!1
(
1)
n
a
n
exists and lim
n
!1
a
n
exists
7. For positivenumbers
p; q;
and
r
, consider the three series
1
X
n
=1
1
n
p
;
1
X
n
=1
1
q
n
;
and
1
X
n
=1
(
1)
n
n
r
:
