2210-90 Final Exam
Fall 2013
KE Y
Name
Show all work and include appropriate explanations when necessary. Correct answers
Instructions.
unaccompanied by work may not receive full credit. Please try to do all all work in the space provided and
circle your final answer.
1. (l9pts) Consider the vectors u
(a) (3pts) Find u
3
—
(
(b) (2pts) Find
=
2i
—
j
—
2k.
v
/
A
A
I + j + 3k and v
=
—
-J -2)
HuH
ii
(c) (3pts) Find u v
3
(i)()
()H) (;)(
+
1 (x) for some
(d) (3pts) Find the angle 9 between the vectors u and v. Write your answer as 9 = cos
number x.
it Il u I! tr
u =
-E3ji( t6çe
96
ii
3
K
114f-3
(e) (4pts) Find v x u
2-
—1
(-)
-2-
+(z-)
(+
3
(f) (4pts) Find an equation for the the plane that containfFöinF
,
1, 5) and has normal vector
U.
q
)
i
ç
1
o+ I
+
3Lcj
C
2. (Gpts) Find an equation for the tangent plane to the graph of the function
the point (1,—i, —2).
3-z
3ic+Zj
f
j
f-)
(
f(x, y)
-
fçJ?(-1.)
-
jh
(x
(ii)
)f)
—
2 + 2xy at
p
3. (lopts) Match the following functions with the plot of its level curves by writing the letter in the blank
provided.
+y
f(x,y)=x
2
C
f(x,y)=z+y+2
i’O
•I:, f(x,y)=x
—y
2
E f(x,y)=x
2
A
f(x,y)=x—y
4. (8pts) Consider the curve
r(t)
(a) (4pts) Compute the unit tangent
/
T
-2s
(
r’ti)I
-
-
=
2 cos ti + 5tj
vector
T(t)
si’
-
—
2 sin tk
r’(t)
=
IIr’(t)II
?-.s
t
t1 +tPL
Tt)
(-2$it
=
-
(b) (4pts) Compute the arc length of the curve between t
L
=
—
=
0 and t
3.
2 in the first
5. (lOpts) Use the method of Lagrange multipliers to find the point on the parabola y = z
quadrant which is closest to the point (0, 1). Do this by finding the point (x, y) in the first quadrant
which minimizes the function f(x, y) = x
2 + (y 1)2, the distance squared from the point (0, 1), subject
2
= 0.
to the constraint g(x, y) = y
—
V-1ix)
j
2
&2x 1>.
=i
)
9
Vf
/
a
2=—2)X
_)(l
f A_)J11ALc,)
==
£I1z_j
y
2
-)
-
Lr
(+ n ce 4cf
2
1
-çi
D
D
je_
1SILA)
2->
Sc’ I
(4iMc-
-jw
cVR..1Y)4)
?oII
(o d i. /.J .1’
7w-)
.=.
1
1 jj
2
(J.-)
+
a
i:
x
6. (24pts) Evaluate the following double or triple integrals:
(a) (6pts)
[f
3xy dA, where R is the rectangle 0 <
J JR
1, 0 < y < 2.
<
j
)o
0
3
=
2.
2..
t’1.
( (1
Jo
=
±
2- -o
[1 i + x2 + y2 dA, where D is the disk in the xy-plane of radius 2:
e._ r°”
fr
c0
(b) (6pts)
2
4.
2
+y
J JD
tirVYZLS
‘2—
2ir
2
Lf
çrr
lIc
-
[[/ 1 dv, where B is pictured solid, which is bounded by the planes x
J
3,1Sj
—r(-
-2-rgLr
(c) (6pts)
LLL,LU..
‘
=
0, y
=
0, z
J JB
0,x=1,yr=2,andx+y+z=5.
i i. ,c-x-j
cU
2.
-y-/j0 M
cO’(c
(IO2x-)
(- L
(d) (6pts)
f/f
@2
2 +p
) dv, where C is the cylinder x
2
2
2+z
+y
1, 0 < z < 2.
J J JC
U.c
C
C)
0
2-1T
JLI
=
JO
colD
‘4\9
3f
irçz+—j
/23
Cc’)
0
7. (lOpts) Match the following vector fields with the graphs below by writing the letter in the blank
provided.
A
12
I / //// 13
‘.‘
F(x,y)=yi—xj
4
////
I
/2’2’.’--
C)
E
C.
A
V
.‘ .
\ ‘ \ 4
1 1 /
F(x,y)=—xi+yj
/‘
I /
/,,
F(r,y)=1i—xj
/,----.-.‘
‘544
I,
44
44
1/1/
1/1/
4’
F(x,y)=—xi—yj
—.—,—,.
F(x,y)=li+2j
,,
i/1/
F
____/M/fr/
,1//,I//1
/
•
••_•____
\
E
p
4
44
//// /
4’4f/// 4444/44
t 4
I t I
4///4/ 4/jW//
44 44’4// 4444
4/4 4//// 4// 4/4
4,4/44/4//4444/4,4;4
_.:*\I
4
4 4// 4// 4////////
•/,
4
4
// /
8. (l6pts) Consider the vector field F(x, y)
=
IlI/,,.-r
5 4 4 4 / / ///,
4 44
/ / / / / ///
4%’\ /
1 / ////
(
/
///A
(2xy + 1)i + (x
2 + 2y)j.
(a) (4pts) Compute curlF
J
+
cxIF
(lx-zx
0
(b) (2pts) Is F a conservative vector field? Circle one:
(c) (6pts) Find a function
f(x, y) such that Vf
=
NO
F.
4
f(x
y
z
c (x).
+
-
(()
(d) (4pts) Use the Fundamental Theorem of Line Integrals to compute
parameterized by
x(t) t cost
y(t) sin t + 2
=
fF
0
=
for 0 < t < ir/2.
(o3)
ç
Fr
-
4
=
9
.
dr, where C is the curve
I
I
9. (l6pts) Let F(x. y) = yi + x
j and let C denote the square in the plane with vertices (0,0), (1,0), (1, 1)
2
and (0, 1) traversed counterclockwise.
(a) (8pts) Compute the line integral
(oh”)
c r-a
3
C
J’. F
dr using Green’s Theorem.
“
r
co’
-
C’
(I
c’
(2k— L)PtMLj
i2-
I
(X-1
oj
Jo
c2i
ojO.
(so)
(b) (8pts) Compute the line integral
1
t
ç
t4)Ifti’
)
Jo
Fr
Cl
C;
‘
-1-j
so
I
<I
s0 (%
F.,Lr
0.
0
0
.&))
1
,ç
:
2
C
3
C
<t°>
0 F. dr directly by parameterizing the four sides of the square.
f
‘I
I
(?+ I LoU
“b
C
)cL+),yIt)> <)) t>,
cLt
L)’>
1
Xtt),’
0
<D J—)
10. (8pts) Consider the vector field F(x, y,
Theorem and to compute
z)
=
(z +
ff
p)i
3
(x
F
JJ[3
7
(,Ue.
sf L.e’-tcoJ
11o
Jo
(‘j)
otu
F
+
:L
(z
+y
)j
+
(z
+ 1)k. Use the Divergence
F n
.
2 +y
where S is the surface of the unit sphere x
2 +
rector
c3/
—L,tt —j,
0
3
C
O.
=
1, and n is the outward pointing unit normal
+(i)
=
2..
3 3÷ 3
(3)( 3jt3 1
g’.(?,ta
0
J
5
b ()(‘k)
11. (8pts) Let F(x. y, z)
=
y cos xi + (3y +
sin z)j
k.
2
+ xyz
(a) (4pts) Compute divF
)LJ
4 (yex)
F
(3+)
+
(b) (4pts) Compute curiF
,‘
+ *1(x)
‘I
1’
I
I?j
jtoSX
12. (8pts) Find tile surface area of tile piece of the paraboloid z
2 +y
x
2 4.
ç
xt
XL1jX
j
=
V-Iixty)
A
-
5
(Air-
(
Jo
2 + y
x
2 that lies above the disk
pc1A’t
2-ir
L4-
,-
2
(rr9
o
.Z
11
/..L
1r
Il
‘-7
ii
13. (7pts) Use Stokes’s Theorem to evaluatefc F dr, where F denotes the vector field
.
i-y)k
3
i
2
F=x
+(y+z)j+(z
and C is the boundary of the part of the paraboloid z
oriented counterclockwise when viewed from above.
7
=
1
—
x2
—
2 which lies above the xy-plane.
p
F.tr
S
(i1 t (°)
1
x
c--
a-vlFo)
0
6