Name
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Student I,D.
Math 225 0-4
Exam #1
October 4, 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 yourwork and justify 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
10
2
25
3
20
4
15
5
20
6
10
100
TOTAL
jj Concepts:
j What is a linear differential equation?
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(2 points)
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jj What is an autonomous differential equation?
(2 points)
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icL What is an equilibrium solution to an autonomous differential equation?
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lcD.
(2 points)
I
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For a square matrix A,,
what condition on the reduced row echelon form of A is equivalent to
A having an inverse rnatrixA?
b\
(2 points)
1’
LS
cL I
i). For a general rectangular matrix A,,,
3i
I
what is the condition on the reduced row echelon form of A
that guarantees that each matrix equation A
.wilI be consistent (i.e. have at least one solutions s.), for
every possible choice of the vector?
(2 points)
r
(A)
<
c o
(iJj
t)
2i.
Consider a boat which starts at rest at time
t
0 sec, is accelerated in a straight path byanengintiat
N.
provides aconsti-t 1500 Nofforcead which is also subject to drag forces
veIoci. The boaing the pilot) has mas0k
Use your modeling!abiIi to show that the b::: veloci1 (in meters per second) satisfies the initial
value problem
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3
(5 points)
CL
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2k).
Solve the initial value problem in
2.
Use the algorithm for separable differential equations.
(15 points)
v/=
S
-.Cv
vCoD =)
-
m2. (v-2S)
zb
5
C
S3
0,
—
v_\z -2#
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ç). How far does the boat travel in the first ten seconds?
(5 points)
1)
0
sç
-2
e
Consider the following input-output model: A brine tank initially containgallo of brine with
an injtial concentration is 0.1 lb of saltioTeachgaIlon of water. At time t 0 waer begins to flowii and’
(u,t’6f the tank
the water volume constant. The salt
-
concentration of the incoming water varies with time, with concentration 2 e
005 pounds per gallon, and
the brine solution in the tank remains well mixed so that the concentration of brine leaving the tank is
always the average concentration in the tank. (There is space below for you to sketch the configuration, as
it may help you with the rest of the problem.)
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3a) Use the description above to show the salt amountx(1) pounds at time 1 minutes solves the differential
equation
x’ (1)
=
160 e°
05
—
.05 x(t).
(6 points)
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t’
2
3b) Solve the initial value problem for the salt amountx(1) in this problem.
(14 points)
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5) Consider the homogeneous linear system
23
.
0
x
1 2 -1
y
0
0
—
z
0
20 6
Compute the determinant of the matrix in this system. (Be careful to check your work!)
(4 points)
(2
5j What does the value of the determinant in Satell you about the possibilities, for the number of
solutions to this homogeneous system? Explain.
(4 points)
i
c) c
0
tZ L4Y
C) , J
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4j CA
4P
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Find the solution set to the linear system a ove, by c
augmented matrix and backsolving.
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co 1
-c
reduce row echelon
rm of the
(12 points)
2
0
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