2210-90 Exam 2
Name
Summer 2013
K Y
Instructions.
Show all work and include appropriate explanations when necesary. Correct answers
unaccompanied by work may not receive full credit. Please try to do all your work in the space provided
and circle your final answers.
1. (2opts) For this problem, consider the function
cosy—3xy.
2
f(x,y)=x
(a) (3pts) Find the gradient of
3
f,
Vf(x, y).
Vffty =<Zcos-j.,
(b) (3pts) Find the maximum rate of change of
f
i
2
—
—
4
3x>
i
at the point (1,0).
Vf(igo)
oH)
1
lIVfU
J13
—
(c) (3pts) Find unit vector which points in the direction in which the maximum rate of change occurs
at (1,0).
Vf(to)
1
(d) (3pts) Let u = (,
the direction of u.
).
-
Find Df(1, 0), that is, find the directional derivative of
C)io- 7f(io)
f
at (1,0)
iii
(z-3>.< ,
(e) (4pts) Find the equation of the tangent plane to the graph of
f
at (1,0).
I
(f) (4pts) Use part (e) above to estimate
f(, —h).
Zn
2. (l3pts) Consider the function
f(x y)
(a) (5pts) Find the three critical points of
2 +x
y
2
+ 2y
=
—
1.
f.
C+Xj) j>
Vfr(xcy)
OZxCJij)
W.4L?0
1)e’
X-°
9
=
1
Q_L
O-ftx2
(b) (4pts) Find the cliscriminant, D
=
—
.
2
(f)
ZI--y
k)c
4
L.
-
(c) (4pts) Use the cliscriminant to determine whether each of the critical points found in part (a) is
a local minimum, a local maximum, or a saddle point.
ioc
7(o,o)
0
(
=z
-r’)
&
0
-
L
3. (l2pts) Use the method of Lagrange multipliers to find the maximum and minimum values of the
function f(x, y) x
2 +x
2
y+y
2 +y
2 on the circle x
2 = 2.
j
2
-
xrj
I
)X: zA
VfXV
<X12j) 2)(42j(>
liAi1
—
xj
>
O,i) ,,-(—i--))
ji
k
x
1
-F(-1 i) -ç()
1
9
4. (28pts) Find the following double integrals:
pl
(a) (6pts)
2
p
I I
Jo ii
(3
—
) dy dx
3y
x
2
Cx!,
I
J(x3— 2,)L
—
(b) (6pts)
[f
(2
+y
) dA, where R is the rectangle R
2
=
JJR
{(x,
)i
—
1
1,
x
‘Lo j(W+i
fJ
+pzJ fi L
Ij(c) (Spts)
[1
2.fc
dA, where T is the region in the xy-plane bounded by the
uT
2x.
y=-x
+
2
—?;
cçxotA
1
x2+2x
(ho) =
/
(d) (8pts)
(,Ut_
2 +y
2 +y
2
) dA, where D the disk x
2
I’D cos (x
r
tc-
cctv..a.4ts
9
1
cc f scsi1T
3
w
t’S
and the parabola
5. (l6pts) Evaluate the following triple integrals by using cylindrical or spherical coordinates:
(a) (8pts)
(,Lk
fff
2 dv, where S is sphere x
2 +y
2 <2.
2 +z
2 +z
+y
5p(v-vi ovyC
21J-(S4
(b) (8pts)
fff
-ii
()C) &4ir
2
2z dv, where B is the piece of cylinder x
Lke
<
1 with 0 <z
I e S.t’ca1
Jc2Sv J
<
1 +x
2 +y
.
2
cos
LL
I
2’irj
=
rrr)oLr
t
6. (llpts) Consider the map from the uv-plane to the xy-plane given by
x(u,v)=2u+v
y(u,v)=u+3v
This map takes the unit square S in the uv-plane to a parallelogram R in the xy-plane. See
below.
(a) (4pts) Compute the Jacobian of this map. Recall that the Jacobian is the determinant of the
matrix
/ 8x
6x
J(u,v)=j
8u
8v
T(uv)
(b) (7pts) Compute JIR(’ + y) dA by using the change of variables formula
fIR
V
(1,1)
ff
f(x(u,v),y(u,v))J(u,v) du dv
S
5
LM
6
pC ti
S
(0,0)
=
f r
3v)J
.Jo Jo 2LL ,) -(
y
(0,1)
dx dy
(:1,0)
U
(0,0)
=(v2v
f—