Math22lO Final Exam
Summer, 2012
Dylan Zwick
Date
Name
7
Instructions: Please show all of your work as partial credit will be given where
appropriate, and there may be no credit given for problems where there is no
work shown. All answers should be completely simplified, unless otherwise
stated.
1. (15 points) For position vector given by r(t)(t—2t+5t)i+(r+t+1)j
the velocity and acceleration vectors and the speed at t=1
()
(i t
-
I÷(&
±
i)
ft5) (?+/);
v(t) (6points)=
zj
a(t) (6 points)
(I)
I
(6
(- i±)
iJv(i)I)
speed at
=
t=1
(/
—
+
1
(/ti)
5
1
-;j
(3 points)
=
-
?
,
find
i_)l
=
-/
I-?IIt
‘(‘Z)-
.L’!
(suiOd E) (J0PaA !Ufl Lj)
q
=
oiuo
n
(p)
f
o uo!pa[ojd
S
l +b(
1-?
.9-?
d
2
()
(51U!Od g)
q
o:uo
40 UOLpDIOJd
(+q)v
T=ih- ?
()
-‘(-Z)
/
(SulOd E)
,
—
I
b)
(q)
(+q)v
—n
=
(-
/
)
‘O
2h7Z
-?
-‘(2)-92h)
)
(su!0d
()
_
bu!MoI I°1 aqi 40 qe
PULl
PU2
(T—’T’o=3
(E’T_’,=D
ç’E’T)=q
11
(S1U!OdcT)
3. (20 points) For the points A(1, 3, 2), B(0, 3, 0) and C(2, 4, 3)
(a) (10 points) Find a normal vector to, and equation for, the plane through
points A, B and C.
1,
-
2
/\
1’
fr
CJ
A
-Zy
Normal vector
Equation of plane:
2
?/
=
1/ /
fr\/
(b) (10 points) Write a set of parametric equations for the line through point B
and perpendicular to the plane in part (a).
-/;-
Line:
3
Y
()
2
y
(1
&±
(I)
4. (20 points) Find the directional derivative of
p—(1,O,1)
x at
2
2 z+z
7 y-l--y
f(x, y, z)=x
a—i+2j+2k
in the direction of
Zxy
69/
ZX
)/
t
1
(o,i)
/
/\
11/(
i2 f
(i,c;)
5
1
/
277
I
I
3,
/7
Answer:
4
3
5. (20 points) Find all critical points of the function 2
-6x
f(x,y)=xy
—
3y
Determine if each critical point is a local minimum, a local max, or neither. If there
are none of the given type of point, just write “None”.
Pd!
fl1’V
(6(11
(
I
c1I,+icc
I)
I
c Ct
5
?Lc{
/
(
c/q
f
4 c // /2o i4/.
1
tc/ay
-ix
_f
C’
y C, 2 C
y
7
yc.:
( 01/ ( 6), (-()
C
7
1
7L
Y4
(“i
ci) ::
19(1,6)
L
7
Y
7L?6
/LIL/
-
)
<()
rn
Jc dd/e
Critical Points:
Local Max:
Local Mm:
C
I
1 (,
(O o)
1
iV, i ‘‘
(p.6),
D
-
)
I ci (
/
--p—C
y
1
r
1/
/e/7i!c
1-?
E7x
‘O)
dI1
/ 0
(
Zxy-
\/
Cl
i’/i
C(
H.-)
J-’ ‘V
)/
(i
-
6. (20 points) Calculate the integral
1’
j e
Note
—
—x
dx
You must provide a correct calculation. Don’t just state the answer.
I
1
)
)
4()
j
e
yJ
—
ei
x
1
I
L
1
Answer:___________________
6
7. (20 points) Calculate the surface area of the part of the elliptic paraboloid
—y above the xy-plane.
z—16—x
2
L7
fj.
7
±
&)
Ic
-
Q(
f
--
J
&A
dqd8
61 )6/
2(
6
)
I
Surface Area:
7
6
(629
8. (30 points) Calculate the following integrals:
yr
2
siny
e’cosydxdy (10 points)
ff
a)
-
—S
1,Y
5 11
f
E
dy
i17
L
e
Y_i)orydy
e
0
5
._
)/
(
--
(0
r’V?
)
(i-
Answer:
iT
2zy
fffsin(x+y+z)dxdydz (lOpoints)
b)
000
Id
—
J:/L
(y)
(yt )
(5 i’fl
it
Wi
(&
)O
(
‘1
(c(?y-)
-
ja
t
(z )
() .( 1)J
111_(
-
-
H4
U)
-
2
(cc(7)
2
—
z
Answer
8
j
3
I
c)
1S6xedydx (lOpoints)
0
/ jcj”l
f/i
i
)
e7
Jcf4
)CLJ
I
/7
?(
C
ci,y
-/
Answer:
9. (20 points) Given
following.
4
F(x,y,z)=5xy
z
k
zi--2yxj+y
calculate the
(a) divF(5 points)
?y
-
divF=
9
I 5/i_
L
2
‘a
1- yyLj yk
(ky/)
(b) curl F (5 points)
(y)
5y
d)
I
(-
5Y?)
YXL
(c)
V(VF)
/
(5 pnts)
7 ?YzfZy)
V(5
7
(
-
/
L
I5
/
1
7
Jo
(d)
V•(VxF) (5pohit
=
o
c/
1/
V(VxF)
10
‘
W
K
/
2
/i2i2
10. (30 points)
a) (20 points) Determine if the field F=(2xy+z
)i+x’j+(2xz+rrcosrrz)k is
2
conservative. If it is conservative, find a function f such that \
(f=F
7
-1
/iifr
yfr T(cTT?
0
4
t
(Z ?-Z)f
()
t
5ey [,‘v’
ZytT
:
Yi-
((y/
‘J
(
C
(1
514
(
/
t C5(Tj
1
(h?)
)
(
yZL /‘/(m?)+
Conservative? (Circle One)
f
(ifitexists)=Zyt
%ny
11
False
1
//1/)
/2Ci/,(/7
e
(
/
j/
(
(This is problem 10 continued, so
rz)k
x
rrcosixy+z
i+
+(2xz+
F=(2
)
2
j
b) (10 points) Calculate the line integral
.1 Fdr
where C is the line segment
connecting the points (0,0,0) and (1,2,3).
J
F-J/
( ci)
it
Answer:______________
12
).
11. (20 points) Calculate the line integral xydx+(x+y)dy
triangle with vertices (0,0), (2,0), and (0,1).
Hint
—
where C is the
Use Green’s theorem.
x
•
( )
C
—I--fl
(t-y)/
/
Y f
&
-
X
/L
Z-5t
6
13
-
(c,
/
12.
(20 points) Calculate the surface integral
g=x+y and G is the part of
Note-
ç
j
dx
z=\f4-x with 0
f fg(x,y,z)cis
x
3 and 0
where
y
1.
._ix
,S11I
a—x
(—)
a
-/,
JL
dy
L
y
3
X
-
\Iq +
Answer:
14
•+
Jo 9
Extra Credit:
(10 points) What did you think of the class, and what could be done to improve it?
/ccFi
Vy/Qi
t/e /
c c
q ii
y
C
c / / yc
15
°
]/ /
/I
cocJ
(
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