2210-90 Final Exam
Fall 2012
/
Name
Instructions. Show all work and include appropriate explanations when necessary.
all work in the space provided. Please circle your final answer.
Please try to do all
1. (l5pts) Let u=2i—3j+kandv=31+5j—k.
(a) (3pts) Find u
—
v
(b) (4pts) Find uH
ii
iE
1
T
(c) (4pts) Find u v
(d) (4pts) Find u x v
2. (lOpts) Find the equation for the plane which contains the points (1,0, 1), (3, —1, —1), and (2,5,0).
—
Z_IS, b’)
—
<iOiI>
-LXV
—1
a
I
C
<1lc)—I).V
-2—
CL
1
1l
1
3. (l5pts) Consider the function
f(x, y)
(a) (5pts) Find Vf, the gradient of
f.
Vfxj<J
;>
(b) (5pts) Find the equation of the tangent plane to the graph of
.9.(3,L”) 31
f at the point (3,4).
(.
(c) (5pts) Use part (h) above to estimate the value of (3 +
-,
3’
--
1
;4--;-=
4. (l2pts) Use Lagrange multipliers to find the extreme values (both maximum and minimum) of the
2 = 1. Hint: Solve for A. then use the constraint equation.
function f(x, y) = ig on the ellipse x2 + 4y
<JiX>c2xrj>
@
X/ry
4
2.
2c
2.
-
>
2
x
T
YJ-)
(‘/‘/)
2
cY”)3
5. Evaluate the following double integrals:
(a) (Spts)
ff
R
2 +2
(x
i) dA, where R is the rectangle 0
x
1, 2
y
3.
=
(b) (5pts)
ff
x dA, where R is the region bounded by the x-axis, the y-axis, and the line y
=
=
3
-
( Hz
jj
t
0
2 +y
2 +y
(c) (5pts) “sin (x
2
) dA, where R is the region x
2
4
ç
2—
,trcL
1
(‘ iwrr
0
42rLr
J
1T(-CASL1-
3
3
1
—
x.
6. Evaluate the following triple integrals:
(a) (5pts)
fff
+ y) dV, where R is the rectangle 0
.i:
x
1,0
1, and 1
y
z
x
3
a
-
9
9
dl / where R is the cylinder x + y
.
(b) (5pts)
.
,
+
2
x
fuR
2
1, 0
<
z
2.
_r
.ct CAj L.v1L
t
(,
,1
1..
( I I..
jooJO
(c) (5pts)
fff
U-ce
5o
r
rklrøl&
-
2-ni
2
2 +y
z dV, where R is the hemisphere x
2 +z
çt4L:m
3.
0.
2—
uvrtui-.j4ve
e°k ea4
1, z
2r
k 44)
oUA,
cocrcd4’
2_ir(i
I
0
4
7. (8pts) Match the equation of the vector field with its plot by writing the letter in the blank provided.
F(x,y)=yi+xj
F(x,y)
A
F(x, y)
C.
F(z.y)
A
t
r=i+sinyj
(x
—
1ii+
=
B
3
‘.
‘
‘.
‘
2)i + (r + 1)j
j
S
‘ *
H
\
•\
..
——,/
— , / f
-
3
‘‘1
—5
—.-.----‘‘1t
‘1
ii,——
‘
-.---—-—-‘1t
.—-.--.--r-- ‘
I
..— .-.
D
3
5
,c
\\\\
-.
-.-.
.
.
—3
.
‘
41’’
---i
/s”
— — / \
— —. / •\
.-——- /“-,--.-
.-
“
.—
—.
3
•
—5
—.
.,-
(2r
(a) (6pts) Find a function
f(x, y)
—
such that Vf
2 t2x
FJD
= Lj.
y)i + (4y
2
2x)j.
F.
+c.Cx)
(b) (6pts) Let C denote the curve with the pararneterization
for 0
t
=
(1 + 2t)i + (et sin (2irt)
1. Use part (a) above to find
fF.dr.
r(o
L’S
çFr
(L)e)
(3,—’)
e
1
ç(;,i)-(i
5
—.
—.
—. —.
—..
--*
—.
/ / / / /i
(1&Xlx
r(t)
-..
7/7 / /
—. —
—5
—
‘*
,
-
—3
8. (l2pts) Consider the vector field F(x, y)
—.
.
\N NNN
.
,
“-
—5
—3
C
‘
—
)j
2
t
5
9. (lopts) Consider the vector field F(x, y)
xzi + xyzj
=
—
.
y
k
2
(a) (4pts) Find divF
otiv
(b) (4pts) Find curiF
cirIF
(_z-
/?
_:
(c) (2pts) Is F a conservative vector field? Circle one:
10. (l8pts) Let F(x, y)
parameterization
=
YES
NO
(—2y + x)i + (2x)j, and let C denote the counter-clockwise unit circle with
(cost)i+ (sint)j
r(t)
for 0 < t < 2i-.
fF
r’1-) -$(S”
4J
(a) (6pts) Compute directly the line integral
ç
.
dr.
Lit
(2.(A$
$((-2iv
o
C
-5 ft
t ÷
Theorem to compute
(b)
(.
f
7
‘I)’ (—c
t-
÷c.ectj)
-
s
Jo
fF
.
dr.
—S
(c) (6pts) Use the Divergence Theorem (plane version) to compute
cE
26
fF
0
.
n ds.
.1
11. (Gpts) Indicate whether each statement is true (T) or false (F) by writing in the blank provided.
F
*—.-—---.——
—
vv.—.- — vv_._—-
fF.nds<O
F is conservative.
,/fr
.-
,
:E—N
I’
,
--1’
•
I
I
•
-
-
d
dl
‘1
—
‘
I
—j----.
----
12. (l4pts) Let S denote the surface determined by
z
where x
2 +y
2
<
)
1
=
1 (5 is the graph of f(x, y)
0
—
—
1
—
—
y,
2 over the unit circle).
y
(a) (7pts) Find the surface area of S.
= —2-
tIX
-
,1
f
2-jr
f
Dir
LO
c
5.
q’rR.tL&
jqr
S*A_
(b) (Tpts) Use Stokes Theorem to evaluate
ff(curlF) n dS,
•
where F denotes the vector field
F
=
pi
—
zj + xyk.
r1k -ch1+O
r’ Gt-
—‘-
.+
2r
S
(F)
L
0
I
*fsr
-
0
2.
Jo
7
+ CtI(t
14%
1-1 o
,tk