FINAL EXAM
Math 1320 5C( 004
December 20, 2013
Name:
by writing my name I swear by
he honor nate
Read all of the following information before starting the exam:
• Show all work, clearly and in order, if you want to get full credit. I may take off points if
I cannot see how you arrived at your answer (even if your final answer is correct).
• No calculators or books are allowed for this test.
• You may use a flowchart you have created that describes series convergence tests. You
cannot use one you have not created or has more information tliaii series convergence tests.
• No other notes are allowed for this test.
• Please keep your written answers brief; be clear and to the point. You may lose points fbr
incorrect or irrelevant statements.
• Please turn off celiphones, turn hats backwards, ta.ke out headphones and sit every other
seat, if possible.
• You have 120 minutes to work on this exam.
• Good hick!
Page (value)
1 (20_points)
2 (15 points)
3 (15 points)
4 (15 points)
5_(15_points)
6 (15 points)
7 (15_points)
Total (110 points)
Score
1. (20 pomt,) Short Auswer. For
y( )1 H
answer to
a.
(4
JIIs)
r(s’eiVe
Does
True or
False you iiiiist write ‘True’ or ‘False’ and justify
(redi t.
((niVerge?
Why or why not?
s y cvC
L
3
I
W
V(-’L2
b. (4 pts)
-5
3
True or False. The vectors
aiid
7
(-30
-zq
.
is
6
—8
2
-90
are orthogonal.
0
[j
Consider the following model for population growth
c. (4 pts)
the name given to tins equation? What are its equilibrium solutions?
=
rP(M
€ftVL-iIW’
Q ?(M-
)
o,
d. (4 pts)
7r1 converge to if ri
What does the geometric series
<
1?
n=O
i-Y
e. (4 pts)
Describe the level sets of f(x, y, z)
(x-3)4(yia)
=
(x
4
(
—
3)2
.
2
+ (y + 2)2 + z
—
P) What
2.
(/5 po’uits)
a. (4 ‘pts)
A (‘Oiflpliter (Ohitrolled router is used to cut desigiis into table tops. The
design being (lit today (onsists of a parametric curve giVehl by x(t) = et + e_t, y(t)
5 2t fbi’
() < t < 3 where x auid ij are in units of (In. Set Ill), but do not evaluate, mi nitegral
represeiitnig
the length (>1 tlHX (hiVe.
—
)
Jt
,
L
.‘-t
.
p,
1-
U
-z
‘•‘
“•
t:
£3
£
b. (4 pts)
Use the method to cyluidrical shells to set up au integral representing the
volume of the solid made by rotating the region enclosed by y =
y 8, mmd x = 0 about the
x—axis. Do not evaluate this integral.
I3)
C. (7 pts)
Consider a spring with rest length 10 cm. If 40 N is required to stretch the
spring from 10 cm to 15 cm, how much work is required to stretch it from 15 cm to 18 cm?
LIQ
(.us)
ica
-
‘2
.0
0’i
—
‘4
j
1)
)(
9
3\(. 00
•1
o’t
U)9 9
qu
00%
(.o
i
•.
-
ax
oz)
u
II.hc
Lo(
•
l)
3. (15 pin1.s)
a. (A pt’)
Solvt’ tlu’ dill(’r(’iItHli (‘qlIa{iou
4
wii.li
i
ntial
COI1(litiOll
y(l)
=
2.
J
C
“
9,x1c
\
N
C
carbou-14 is a radioactive isotope with half life of 5730 years. If e model the
b. (7 pts)
= ky what is k?
amount of carbon—14 in au object at time t with tl1e differeuitial cquatioIi
2
.1
ç.:9
)‘5
3A’.
C:i13O
—r
Ii
I
I
--
‘V
‘I
V
V
—s
-:
•-
:rD
-I
-t:)
—
‘1
-4
‘I,
x
C
-d
C
C
-
0
,—‘
0
—
f____’
L—,
F—’
0
I’
5.
(15 points)
Wha,t is the angle between the vectors
a. (5 pts)
[i
[ ] [ ]
uici
5
/-
\\\icuW
CCL
Ca5
-fl-I
(2)
3
the phmn 5x+2y+z
b. (5 pts) Consider the line x = 1 +t, y = 2—t, z = 4—3t and
in one point?
plane
ct
the
Does the line 1i(’ in the plane, lie in a parallel plaii or interse
S
I
2(2) t(H3
f
ti —2-!j
Lj?
-
)
[[
L
2
(c-z-
0
it-
01
c.
(5
pts)
Taiisforni the point (p
=
2,8
Cartesian coordniates.
2sw/3
1
Zcos
Sw\ILg
z( ) -r
‘k
2-
Z,
-z
=
ir/4,
=
ir/3) in spherical coordinates to
v--I
j1cj-
6.
(15 points)
a. (6 pts)
?()
Find lie mat tmigcnt vector of it)
/J, 4t,
t± -F t
2 + t2 at the ponit (1, 1/2, 1).
i>
‘(t
K
K’ ‘)
b. (9 pts)
A boat is traveling across a river with uct velocity (t)
the boat starts at the origin and the opposite bank is located at y
ha.s the boat moved when it reaches the opposite bank?
=
=
__(o•)
v’(t
J
L
ç():
90
c
0
-
(
1
k
o
L5
-
3
U
aos# 1uw’
[
]
II
40, how far downstream
7’.
(15 points)
a. (6 pts)
,2
2 with z pointing
4 —i- xy y
x
Consider a uiouutai i modeled ly z = 50e
y northeast,
east and y po itmg north. If you are at pout (2,0,34) and heading on a path directl
what is the rate of change in elevation?
—
—
1;
v
U
Lt)
1
(7,
3
-zJ
L
2-
j
i
(-3l4L)
-
30
‘3
“[2-
I
Consider finding the local maxiiiia/uiiniina aiid saddle points of
b. (7 pts)
x What are the critical points you need to check? (you do not need to
Y
y2)e
+
.
2+ 2
f (x, y) = (x
find any local max/miii just critical points)
—
r
JI
)(i
,
4
(zx
(
)clhjZ
2xe
?
\
()(2Y)e
/
2)
Lj
1
4
2
X
)
(I
j i-,34-j;2
0
0
Xo
,
‘-r
Ci\L(,L i
(v, C))
1_
I c
I
‘1.-
When finding extrema of an objective function f with constraint g
the method of Lagrange multipliers, we solve the system of equations
c. (2 pts)
Vf =7g
g=0
Circle the Lagrange multiplier.
=
0 using