Name____________
Student I.D.
____________________
Math 2250-4
Exam #1
February 14, 2013
Please show all work for full credit. This exam is closed book and closed note. You may use a
scientific calculator, but not a “graphing calculator” i.e. not one which is capable of integration, taking
derivatives, or matrix algebra. In order to receive full or partial credit on any problem, you must
show all of your work and justit~’ your conclusions. There are 100 points possible. The point values
for each problem are indicated in the right-hand margin. Good Luck!
Score
POSSIBLE
1
30
2
30
3
10
4
30
TOTAL
100
fl
There has been a Valentines Day accident! A truck
an accident and spille~ns of hi h
fructose corn syrup into Scum Pond! Scum Pond• spring-fed and maintains a constant volume 200
acre feet because water flows in a stream out of the pon an directly to Heart Lake below it. (An acr
ot
is the volume of water in one acre, if the water is 1 foot deep.) This stream is the only water flowing into
Heart Lake, and it flows at a rate~?~~≥~Øeetper day. Water also flows out of Heart Lake at 20 acrefeet per day, and so Heart Lake’s volui e of 400 a~~~jj et stays constant. Before the accident there was no
high fructose corn syrup in either of Scum on or in Heart Lake. As usual, assume that the lakes are
well-mixed so that the concentrations of corn syrup are uniform in each (but changing in time as the total
amounts in each lake change). You may wish to use the space below to draw a diagram in order to
organize the information above and your subsequent work.
-
‘I’~2ov
ç t71o
iit)
xDt
so
j~ Let x(r) be the tons of high fructose corn syrup in Scum Pond t days after the spill. Use your
modeling ability to derive the differential equation and initial value problem for x(t). Solve it (or recognize
the form), to explain why
x(t)
=
50 e01
(8 points)
La~.vc.4vn~ (K~1-
x’L&)t
C
V
.it~ 4C1
7-fl
-.gt
Ce
-~~)
t:p~~ ~x1tNSO
R0~
So
,C-çoe
ft)
Lety(r) be the tons of high fructose corn syrup in Heart Lake (days after the accident. Use your
modeling ability and your knowledge thatxQ) = 50 e01 ‘to derive the initial value problem fory(fl:
(44 y’ (1) = 5y(O)=O.
e0~’
.O5y(t).
—
~t~L~tv7(c_r0Co
t20X
(6 points)
_zo4-
‘ti
.t~2.0
tUv
—it
,
2-O
~-~‘5e
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e05~
HRS
~tC4tk~.t
j.ç) Solve the in~ttia1 value problem in lb.
(12 points)
--it
Sc
e
•ot
-oct
,
it ~
—.tt
5€
~Se
.0çt
.oct
jse
—.05
€
C-
_.lt
-~
.oS I;
e
-.ocf
—loDe
-~
~iave
j~) Find the maximum concentration of high fructose corn syrup that occurs in Heart Lake, from this
accident.
thit)t ~
C)
:i~~
(4 points)
-‘It
-1*-i
z
‘~tft;~oe
t05tt214t
-24-i
-~
t
~ &ostt2_
•c,St ~_L2-
SQ kr-A,( Lm~t.
~Z5
fr’S
_i;_~i-~fw\
~L~L’_~i
~).
Consider the following differential equation for a population P(t).
+
F’(t) =-2P2
UP—
10.
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We considered a differential equation of this na4 in our distsions of applications. Describe what
sort of population and situation could be modeled in tqis case.
(2 points)
Lo~4~t +
2k).
c.c~vda$r~k
Find the equilibrium solutions.
(4 points)
?‘!flt .-≥...(?t—6P+6)
-a
ni~ I);
Zcl
Construct the phase diagram for this differential equation, and use it to determine the stability of the
equilibrium solutions
I’ MCO
(6 points)
~1
S
°
2≤J).
Suppose P(t) is the solution to this differential equation, with P(O)
diagram predict for the value of,lim P(t)? Explain.
suit
=
3. What does your phase
It P0<.S +Ltpkw SJAyv~
(2 points)
1~
~).
Although it wouldn’t make sense in a population model, one can consider the IVP with P(O)
1.
What are you able to conclude from the phase diagram, regarding the ultimate behavior of P(t), as I
increases from 0?
sh~ct
?‘lk)<o
‘~‘°~
e(utcftit<o
c~Ay~a_~pV’~
(2points)
P(t—
iZC4a543+
o~c
ov2
21)
SolvethelVP
F’ (t) =-2 F2 + 12 F
F(0) = 3.
Hint: The partial fractions decomposition
I
1
(1
(x—a)(x---b)
—
—
10
b—a L~x—b
—
1
x—a
might be of use.
—i
—1
C?- s)Cy— C)
(12 points)
a”
I
‘
‘
,
I
L~LFsl_Lt.~w_Ii__tti_tl
OF’
F-5_ ~
-t
—t
---I
-
(j’(o~
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2~
W~
_st _~4~1
e
te
F
~tt3
v)
-
—
64€
I
Veril~’ that your soltition to
21 has the limit you predicted from Zd•
54-0
s
—Ct’
—
-
—
—
-
S
(2points)
31
Consider the initial value problem for a function y(x) ~c<
y (x)x +y
~)
stziu 4t4*
y(O)=l.
Use Euler’s method to estimatey(0.2) using step size h = 0.1.
(10 points)
K0t 0
~a.M.:
.1
1’
t.v+ •~ &(Di”I
4-k
r14 ~j (cfl-i~’)
.2_.
‘1)
~_j
.i
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L-t 4 .12.2..
~.
1.12.2-
4). Consider the matrix A defined by
201
A
:=
-3 1
-1
613
4~). Compute the determinant of A.
(4 points)
4k). What does the value of IAI above tell you about whether or notA ~ exists, and about what the reduced
row echelon form ofA must be? Explain your answer by quoting the relevant facts relating determinants,
reduced row echelon forms, and inverses, for square matrices.
&.~-I,nm~r: (Alt .t_ajl —1
V3
.....0~j
3 1~
(4points)
IAItO
yrefLA’~I
≤~nct
wcki’as.r
a~o% K’
szic.’s4~.
4ç). Find A’. You may use either of the algorithms we learned in class. It maybe wise to check your
answer.
zo
—3
1
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(60
63
010
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R,4-ft~-1121
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So’ j1o 101
08 Ii
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4.~). UseA~ to solve the system
2x+z= 1
-3 x +y z = 2
6x+y + 3z 3.
—
(5 points)
@,
p-’A’~~Fr’L
f~j
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r,fl
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1—39
101
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a(-3fl7t1 /
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Solve the matrix equation below for the unknown vector [x, y, z].
[2011
[xyz]~ -3 1
6
1
-1
23].
3
(5 points)
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