2210-90 Final Exam
Spring 2014
)\.
Name
/
(._
Instructions.
Show all work and include appropriate explanations when necessary. Correct answers
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. (l5pts) Consider the vectors u
(a) (3pts) Find u + v
=
—
=
2i
—j
+ k, v
=
—i
+j
+ 2k, and w =4i + 2j + k
w
(2-)(-+21 ‘(j+)
(b) (3pts) Find Jw
l13IH
J
(c) (3pts) Find u v
(z-j
)Ijt)
()(-‘)+ (-i)b)
()()
(d) (2pts) Which two vectors are orthogonal? Circle one:
B.u,w
A. u,v
(e) (4pts) Find u x w
-t
Li
i(-/-)-)
-‘
(q)
=
2. (8pts) Find an equation for the plane which contains the points (0,2,3), (1,—i, 3) and (1, —2,1).
P
-<I-/,3)
-L
1
l
I
—i,I,3)
p.’
0
J
_L
3
0
2X
-I
J
—2
*%V
4(c (O2i)
-)
1
fL
3. (l5pts) Consider the function
f(x, y)
= x2
(x
+p
.
2
f(x, y).
(a) (5pts) Find Vf(x, y), the gradient of
h- (2?c)
(x2tj/(
I,,
(b) (5pts) Find the equation of the tangent plane to the graph of
-;-r
-(34
z
=
f(x, y) at the point (3, 4, 5).
3
(-1
+ x
( )(-)
s_
(c) (5pts) Use part (b) above to estimate the value of /(2.9)2 + (4.2)2.
Sc.e-
2.’?Z3
V
f(i,j z
()L
-
,
()
+
ç
(-)
+
4. (l2pts) Use Lagrange multipliers to find the extreme values (both maximum and minimum) of the
function f(a’,y) = 2 + p
2 on the circle (x 1)2 + p
2 = 1.
—
(j ()_I
2(x-i)
ct 2x
/1
1
?i
_)
A
0
i
---
ca
=
2x
?ot
—
C
O
c
1
’)
c*..4_
(2o)
2
9
Z7
5. (26pts) Evaluate the following double or triple integrals:
(a) (5pts)
If
zy dA, where R is the rectangle 0 <x
ff
(x + 2y) dA, where T is the region bounded by the curve y
<
2, 2
4.
y
J JR
(b) (Tpts)
J
I
x—x
Jofo (
(
(c) (Tpts)
x
—
2 and the x-axis.
x
.X ?(
0
ii
fr2)
/
#xi
3
-3x
2
11 x dA, where S is the piece of the disk x
2 +y
2
r’
(x
-
+ (x-x)c
3
3
1 in th first quadrt.
ifs
Ic-
=
JT
ct)
tO-iDi
17
)r
A
-
•I1,
(d) (7pts)
If /
z1x2
J J JO
f
(
cs&
2 dV, where C is the cylinder
+y
I
Ia
2
2 <1, 0
+y
ttj
ç
z...
2-’j
Q-
I
r
3i
2
-
ç
2c
rZ
c
ç
3
3
3
-‘do
< z
ios&
<2.
-
6. (lopts) Match the following vector fields with the level curves to which they are perpendicular by
writing the letter in the blank provide
B\c
F(x,y)=xi+yj
A
L’
E
F(x,y)=—i+j
F(x,y)=i+j
F(x,y)=xi—yj
F(x,y)=xi
E
D
7. (l8pts) Consider the vector field F(x, y)
=
2 + p)i + yj.
(x
(a) (4pts) Compute divF
F
(x
2-c-l.
(b) (4pts) Compute curiF
‘I’
(7
4.
/y
(c) (2pts) Is F a conservative vector field? Circle one:
YES
(d) (6pts) If Ci is the curve parameterized by r(t)
(t, sin (7rt)) for
—
$t
1, compute
/
rj)
H1i-
ç
s
::1)
1
Jc
fr))
7rcs(1rL-)>
1’[(s;ir)f
)
1
J
()
st()u1
D[
I
ir
(os (TJ-1)
to
I
1
Jc
fr)
ci (1r)
(e) (2pts) If C
2 is another curve connecting (0,0) to (1, 0), would you expect
/
JC
2
F dr? Circle one:
.
YES
4
“
F dr tual
.
8. (l6pts) Let F(x, y)
=
)j and let C denote the unit circle traversed counterclockwise.
3
)i+(y —x
3
(3x+y
(a) (8pts) Compute the line integral
fF
0
.
dr using Green’s Theorem.
(3)
F
(3j))
F’tr
0
I
(b) (8pts) Compute the line integral
Plane Divergence Theorem).
JF
.
(-
cs
9. (8pts) Consider the vector field F(x, y, z)
Theorem to compute
=
-
(2xz + y)i + (2
—
3y)j + (3z
=
1, and n is the outward pointing unit normal
FV
5
2 + y)k. Use the Divergence
z
.
F
Th1
—
F n
2x + 2
2
where S is the surface of the ellipse 2
y +z
vector.
S
3++.
A(s
ff
v
—
n ds using Green’s Theorem (this version is also called the
(x 3t
i’V
tcTfrvS
10. (6pts) Let C be the curve and vector field given by the picture below. Indicate whether each statement
is true (T) or false (F) by writing in the blank provided.
_.‘\‘\\ ‘S
fcFd3
F is conservative.
F
.
,
I
1
1,
.
,
I
1’
I I
/
/
11. (l6pts) Let S denote the surface determined by
z
where x
2 +y
2 <1 (S is the graph of
9
=
—
f(x, y)
y
2
over the unit disk).
—
(a) (8pts) Find the surface area of S.
)
-
(s
f
-
c
0
cc LctS’.
cc
-
r
tf
(b) (Spts) Use Stokes Theorem to evaluate
Lv= rdT
ff
F
—
9
=
1F
Lt
(curl F) n dS,
where F denotes the vector field
(x 2 +y
1)i+(z
2
C
F
+y
)
2
k.
Hint: What does F look like on the boundary of S (the graph of z
2 = 1)?
+y
=
—
over the circle
2
o-
xi
1
c)
S
So
s
fF
6
f Dc
0