2210-90 Exam 2
Fall 2012
Name______
Instructions. Show all work and include appropriate explanations when necessary. Please try to do all
all work in the space provided. Page 6 is blank in case you need extra paper. Please circle your final answr
1. (l9pts) For this problem, consider the function
f(x, y)
(a) (3pts) Find
=
5
xe
+
—
1y
3
f(x,y)
flx-3
(b) (3pts) Find f(x,y)
if
xe —3,<
)
1
LX
(c) (3pts) Find Vf(1,0), that is, find the gradient off at (1,0).
Vf) (-f’
° (‘°)>
1
(d) (3pts) Find the maximum rate of change of
o
1
Il Vi
at (1,0).
.
(4, 4).
(e) (3pts) Let u =
the direction of u.
TLL-c
f
Find Df(1, 0), that is, find the directional derivative of
f
at (1,0) in
-
U,b)
<3)
(f) (4pts) Find the equation of the tangent plane to the graph of
2—2.
3(x-i)2j
1
f
at (1,0).
2. (2Opts) Evaluate the following limits, derivatives, or integrals. Be sure to show your work. Remember, a limit might not exist, in which case, write ‘DNE’ as your answer.
urn
(a) (4pts)
(xy)—*(OO)
2y
x +3
(Hint: use polar coordinates)
x2 ± y
2
x
)
0
va
()(o,v)
(b) (4pts)
—) 0
—2y
2
x
urn
(x,y)—*(O,O) X
2 ± j
2
x-1c
Q)
Sb
(c) (4pts) Find
wh z
=
ysin(xy).
cc
s (x y)
&5
y))
2
(d) (4pts)
-
J
(oc).
ji[-
/
/ /
2-..
2-2..
IOrX
2-
/ /
JO
(
2
0
=ç
2 dx dy
xy
Jo Jo
2
(e) (4pts)
1
6
3
(11
1
(x + y
) dx dy
2
JO
Io
3’
V
/iL
2
3. (8pts) A bullet is manufactured with an intended mass of .1 kg, but errors in the manufacturing process
result in bullets which can be too light or too heavy by as much as .001 kg. One such bullet is fired
from a gun and observed to have a velocity of 1000 m/s, give or take about 20 m/s. Use differentials
to estimate the possible error in the calculation of the kinetic energy of the bullet K = mv
. Note:
2
/s
J=rkg.m
.
Energy has units of joules (J); 2
i#vI’j
LO0i
toDD -/c
v
vvV
(.)CI
D0c)
C2-c)
=
4. (l3pts) Consider the function
f(x, y)
(a) (5pts) Find the 2 critical points of
o
—
p
f.
—
2x.
2-
?<•-I>
-
50
y
2
x
—
C:_I
2xj
x--L
(b) (4pts) Find the discriminant, D
—
.
2
(f)
fcc2)
LY
L o)
<0.
-
2-
(c) (4pts) Use the discriminant to determine whether each of the critical points found in part (a) is
a local minimum, a local maximum, or a saddle point.
cc
<
‘h4_
C kL CL
3
ks
0
-d
S
0
(I] ‘1
0
II
cq
0
12
0
0
-
x
Ii
0
0
II
d
I’
0
0
Ii
x
-
0
C
)l)
ci)
C,]
II
-d
+
01
H
c’ld
N
Cl
Cl
+
N
H
-e
H
0
0
0
C.)
S
N
C
ci.)
-e
cr:jcr:,
-d
J)
01
0
ci
1iK
‘-4
-4-
U
+
-
Z
,
1
r
4-
?,JY
)(
ç-%4
0
r
7. (lOpts) Find
ff(x+2Y) dA,
where S is the region bounded by x
region S.
=
0, y
=
x2,
and p
=
sketch of the
1. Note:
5
S
3_)(4
(xt
5
)
(ç+zx
2.
5/
3
0
I
‘2.
t÷2)
J
(
0
=( I’
z+’1
0
2-’
8. (lOpts)
J
z
Using polar coordinates, compute
lID
s
!
—
2.o
2-0
‘l-t)
2-”
e_X dA,
2 +p
where D is the disk z
2 <9.
yM
cc
—‘1
z_
WI
I
S.e-f
—‘I
0
cA- -2roLr
5
3
F-,
I
I
p
4