Calculus III 2210-90 Final Exam
Summer 2014
gy
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. (l3pts) Consider the vectors u
=
i
—
j + 3k,
v
5i + 2j
—
k.
(a) (3pts) Find u + v
3)
(b) (3pts) Find u v
‘3?
(‘)(c)
2.
:)(-‘)
-(-i)()
-3
(c) (3pts) Find the angle between u and v. Give your answer inradians.
‘17
0
o-r
tcs’(9))
(d) (4pts) Find u x v
)
fri
j
I
3
c
2-
2. (lOpts) Consider the function f(x,y)
/
iJ
I(i-)
=
(a) (6pts) Find the equation of the tangent plane to z
x
()
fr
=
f(x,y) at the point (2,1,1).
(2I)
-L
—
-,
L--
icr’(
vr4ze-
.
1(y—i)
(b) (4pts) Find a unit vector which is perpendicular to the tangent plane to z
(5,2,1)
A
(c)
(tc+i)
f(x, y)
at the pOnt
t
fx
3
<y(),
j(),1>
c)
-
1
a
LL%A,L
Ezz/--
.1-
v.ecft4”
-L
t?
i3
3. (l2pts) Match the equation with the graph of the surface it describes by writing the appropriate capital
letter (A-F) in the provided blank. Each answer will be used exactly once.
(b)
/2—
B.
Al
_y_z
2
x
O
(a)
—
x
+
2
=
—1
y
z
(c)
V
(d)
C-
2 = 1
y +z
2
—y+z
x
=
2
O
(e)
A
2 +2
4v
4y +
(f)1
2
x
+
2 =
z
2 =
z
V
4
F
2
:1,
:1:
>“
:;
4. (lOpts) Use Lagrange multipliers to find the extreme values (both maximum and minimum) of the
function f(z, y) =
x on the circle (x 1)2 + y
2 = 1.
—
—
4(j)2_x
l)
>
2
.
(x-i)y-(
0
—i
CP
,==,
V
L
3
y)
c2.x-2,’2j)
if
j
2
\jf
/
: i.
I
if
?lJt44t
cD
—Ix-Z
3-)
foI1(3
we-
±112..
i
4
tt
(o,o), ()
0))
(i.
2)
i.L
ü-J)L
1
fio,o
)
f(i
o
t
)
ç(.L-JrTr
)
(J
2.L
5. (26ps) Evaluate the following double or triple integrals.
(a) (5pts)
i/R(x2
+ y) dA, where R is the rectangle 0
x
1, 1 <y
<
3.
3
1
f
3
(
3
I
J!Jo
z3
L
2-Ia
3
f
4
l
/1
‘I
-
/
2
z.
‘
“3
2-7
(b) (7Pts)// xy dA, where T is the region in the first quadrant between the curves y=x
2 and
ff
çjx
/__
i.
JJ
çj
+
±2
r
2
A,tY
(.t
i
<-2-
(c) (7pts)
x’j
I .
1
_xI,
dA, where D is the unit disk x
2+y
2
x5)
J-
1.
%(Zv
2-’rr
0
jL_
(d) (7pts)
f/fç.e-rVt\
2 dV, where S is the sphere x
z
2 +y
2
2 +z
2.iTT
L J0
.7-
[jr 2
4.
,-ø’rL
2-
2(
3
toS
1
3Z’
(L
6. (lopts) Circle T if the statement is true and F if the statement is false in general. Let F(x,y) denote
a vector field and C an oriented curve.
The value of
L0
J’. F . dr is
the same regardless of how you parameterize the curve C.
T
There is always a function f(x, y) such that F(x. y)
T
If divF
If curl F
T
If
0. then
=
F . dr
f.
I
F• dr
=
=
Vf(x.
).
0.
0. then F is conservative.
=
0, then C must be a circle.
3
3/
7. (l8pIs) Consider the vector field F(i. y)
=
(eY
± 8x)i + x&’j.
(a) (4pts) Compute div F
tv
4 (xe)
F
=
(b) (4pts) Compute curl F
A
‘1•
,J
011
xe’
/
o
(c) (2pts) Is F a conservative vector field? Circle one:
(d) (8pts) If C is the curve parameterized by r(t)
Theorem of Line Integrals to compute
=
cj
NO
(cost, sin t) for 0
t
r, use the Fundamental
(
C.
3j
19
+
)()
fiii
Fr
(io)
C-
-1
)
(‘io)
-
(i)
8. (6pts) Let C be the curve and F tile vector field given by the picture below. Indicate whether each
statement is true (T) or false (F) by writing in the blank provided.
J”
f
F
0
.dr<0
P
There is a function
f
such that F
=
Vf.
/
4
cc
9. (l6pts) Let F(x, y) = (x y)i + (x
2+y
)j and let C denote the boundary of the triangle with vertices
2
at (0,0), (1,0), and (1, 1), traversed counterclockwise. See picture below.
—
(a) (8pts) Compute the line integral
S
f. F . dr using Green’s Theorem.
A
S
)(
(0,0)
y
0
‘
(1,0)
(4 )
(b) (8pts) Compute the line integral
Plane Divergence Theorem).
v F
f. F n ds using Green’s Theorem (this version is also called the
(-)
((cx
ff(iy)
(lfZy)iLX
1
fx
4
.2/Xq
3
L
0
Si
10. (8pts) Use the Divergence Theorem to compute the flux of’ the vector field F(x, y, z)
k through the boundary of the cylinder determined by x
z
2
x
2 +y
2 < 1,0 < z < 1.
(+y
v F
ScF.s
S
=
xy
z
2
i
—
j +
3
x
)__
J9V
IC-
I S
cc-czS
çL
-
2
)(çr)
0
(ç
f 2. 1 / r
1iL- tq
2
0
V
2-y
11. (llpts) Consider the surface G in 3-space determined by the parametric equations
r(u,v)
where 0 <u
=
(u+v)i+ (u—v)j+ (2u+7v)k,
1,0< v <1.
(a) (6pts) Compute hr
. x rjl.
1
rLL-
A
rv i -J 7fr.
)J
f
2
r
-i/
(b) (5pts) Compute the surface integral /f/(s + y) ciS.
Recall: For surfaces defined parametrically, dS
ctS
4
J
=
1. x
hr
rhI
du dv
ffv)÷(u-v))
12. (lopts) Let S denote the surface determined by x
2 +y
2 + (z 1)2 = 2 where z > 0 (S is the part of
the sphere of radius
centered at (0,0, 1) with positive z component ...see picture below). Use Stokes
Theorem to evaluate
—
/f(curl F) n ciS,
.
where n is the outward pointing unit normal and F denotes the vector field
to
F
=
—yi + xj + zk.
LALj D
rL-t)
f S
‘-b-t
-
cter)
Ikt7)
o>.r’t{
12
çIF4s
tlJ
t2ir
fFr
f(ciw *t&).(-f1
6