Name
Student ID.
.Soz~cenoNs
_______________________
Math 2250-0 10
Exam #1
February 14, 2014
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 justil~’ your conclusions. There are 100 points possible. The point values
for each problem are indicated in the right-hand margin. Good Luck!
‘4-
10
2
25
3
25
4
10
5
10
6
20
TOTAL
100
fl Concepts
~) Define what it means for a first order differential equation for a fUnction y(x) to be separable.
‘CC
C-a ‘t
b~-
(2 points)
f6t71(3)
jj~) Define what it means for a first order differential equation for a function x(t) to be linear.
bEE cm... k ~ci,4-$ep.. t~
4 y,~
x’(fl ‘i- Pethe
~.
(2 points)
Q(4)
j.ç) If the square matrix A nxn reduces to the identity matrix what can you say about the possible solution
vectors & to the linear system
Ax b?
Be as precise as possible.
(3 points)
Wck~QI It
So
A
•(~, Ate.
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ave (Ar$fk(. 1k
mi4 +Lt
.5ac4? c4ca Arcg&bctg~ I
I
S• (i’t%..’S
j4) Let A, B be square matrices. Which one of the following two equalities is always true? Explain, using
matrix algebra properties.
±d-B~2 =A2 + 2AB+~
B)2~2+~
(3 points)
(fln)e • (f’fl)B
bt,~ d4+$bib.c
~ A2flA ÷B3i-3
S~kct
AF)tM hei+L-tl.t124i..
ç~~+g~t:fi nn+r
kS~’4 k 4rw~t4~1k.
Consider the following linear drag initial value problem for the velocity function v(t) of a falling object,
initially dropped from a great height. Here we are using the English system of units, for which the
acceleration of gravity is (approximated by) 32
ft.
sec2
v’ (t) = 32— 0.5 v
v(0)0.
Z~) In the differential equation above, have we chosen the up or the down direction to be the positive
coordinate direction? Explain.
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ska.n...4 U +3z
,
(3 points)
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2~ What is ihe erminal velocity in this differential equation? Explain with a phase diagram.
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(5 points)
~
~
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lv
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~ Solve the initial value problem above, for v(t). Use the method for linear differential equations.
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$1
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A
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vco)tO~C:_~~1 =~[v&)t’~y_~ttfe—.c& I
241 How far does the dropped object fall in the first 4 seeo’iid~? There is no need to find the decimal value;
an answer in terms of exponentials suffices. But, do include the correct units.
4-.vcc)4s
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(Spoints)
to
so
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t
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ft
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Consider the following differential equation for a population P(t).
F’(r)=-2F2 + 8F— 6.
I
~) We considered differential equations of this naturJ in our dis~wssions of applications. Describe what
sort of population model and situation could lead to th1s differentiakçation.
(2 points)
logs .kt
1-
‘mit1-mk+ tti-t
3j~ Construct a phase diagram for this differential equation, and indicate the stability of the equilibrium
solutions.
—2
(6points)
t
-≥
(r’—qr÷;)
(P-3KY-t)
~t~”1
sr€4h3
P~ i 3
no
S
3
~
3ç). Use your favorite technique to find the partial fractions deMmposition for
1
(F—3)(P—1)
(7points)
+
(14) (P-i)
t
7—3
2
Q7ZiT~r~n
-
if
__
‘...jj’l
3
1:A(2)
r-’Ij
c)
__
b.~’
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az
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_—
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j:j
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P—i
jçj) Solve the IVP
P’(t)=-2P2+8F—6
P(0) =2.
.-z(P.4)lP—’)
Your work from k should be helpful, and your work from 3h. should let you check whether your solution
seems reasonable.
(10 points)
tar’
Qr3 ) Li—’)
r
.5
~!_)aF=
I.
i—
J
I
——
P~-3
P-I
P—i
t
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C-
c-cit
tee
cx-’
-14
€ttb,
r~z
~.
C’t ft
C~1t
0~
C.
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(p_1)(_j’tt) t
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P.-3
(
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Ptt~ 3~L3
I—
—
p~tftrf~9t
4). Consider the initial value problem for a function x(t):
x’(t)31—2x2
4~). Sketch the (approximate) graph of the solution to this initial value problem onto the slope field below.
(5 points)
3t—2x’
-.
.
.
1.2
~
1
~
\~ \
-
Arrow of slope -1-1.0,71
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+- —
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x
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0.4
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—02
Done.:
0
02
04
06
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08
.
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4b) Use a single step of Euler’s method, with step size h 0.2, to estimate x(O.2) for the initial value
problem above. Add thecorrespOnding approximate solution pointto the slope field above. Does this
point appear to be above or below the actual solution graph?
-
.
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.
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(5points)
Consider a brine tank which initially contains 100 gallons of water, with concentration 0.2 unds of
salt per gallon. The
apacity of 500 ggjJnn&----Attiniet~& hours water begins t flow into
tank at a rat o 50 gallons
our, with ~ñ~&ntration 0.5 uouii~1~5salt per gallon. At the same time,
the well-mixed brine solution begins t~ov1öiit of the tank atarate of 30 gallons~f]i~, until the tank
fills up (and the system is shut down).
~)
At what time will the tank become full?
V’tt) t So-Jo c20
VLoW Ira
~ V(t)tluv+zo1,
fi~twLss...Vtt1t Sovttuve-zot
(2 points)
4,~Qo kea~.ç.
~ Find the initial value problem satisfied by the salt amount x (t). You do not need to find the solution
function x( t).
(8 points)
x’Lt)
t~C1
—
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(~1).
3
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ant
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it,.
Consider the matrix A defined by
1
A:=~-1
2
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fik).
Compute the determinant of A.
tj. 4opfl’i.r~ L~
t (~‘,)
0-21
1 3 I.
1
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—o _z(_37z —744:--
(3 points)
What does the value of Al above tell you about whether or notA’ exists? Why?
≤~bCL
ixi~a.,:A-’i,~-~s+s
(2points)
~ç). Find A1. You may use either of the algorithms we learned in class.
o
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~j Check your answer above by veri~’ing that a certain matrix product is the identity matrix.
discover that your answer is incorrect)
(3 points)
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