FinalpartialsolMAT1332_2019
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Course
1332
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
12
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Universit´e d’Ottawa
University of Ottawa
1
Facult´e des sciences
Math´ematiques et statistique
Faculty of Science
Mathematics and Statistics
613–562–5864
613–562–5776
www.uOttawa.ca
STEM 336
Ottawa
ON K1N 6N5
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Marks
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MAT1332 B: Instructor Monica Nevins
Final Examination
April 23, 2019
Duration:
3 hours
Family name:
First name:
Student number :
Seat number:
•
You have 3 hours to complete this exam. Do not detach any pages.
•
This is a closed book exam. Except for Faculty-approved calculators (models:
Texas Instruments TI-30* and TI-34*, Casio FX-260* and Casio FX-300*),
no notes, cell phones, smartwatches or related devices of any kind are per-
mitted. All such devices, including cell phones,
must be stored in your
bag under your desk for the duration of the exam
.
•
Read each question carefully — you will save yourself time and grief later
on.
•
The questions are long answer, with number of points as indicated.
You
must show your work and your work must be legible and well-
justified, to earn full credit.
•
Where it is possible to check your work, do so.
•
Good luck!
Very important:
Cellular phones, unauthorized electronic devices or course notes are not al-
lowed during this exam.
Phones and devices must be turned off and put
away in your bag.
Do not keep them in your possession, such as in your
pockets. If caught with such a device or document, the following may occur:
academic fraud allegations will be filed which may result in you obtaining a
0 (zero) for the exam.
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Universit´e d’Ottawa
University of Ottawa
2
1. (2 points) (a)
First find the following indefinite integral
Z
e
cos(
t
)
sin(
t
)
dt
=
-
e
cos(
t
)
+
c
(b)
Now evaluate the definite integral below and circle the correct answer
Z
π
0
e
cos(
t
)
sin(
t
)
dt.
A. 0
B.
e
-
e
-
1
correct
C.
e
-
1
-
e
D.
e
π
-
1
E.
π
(
e
π
-
1)
F. 1
2. (2 points)
(a)
If
z
= 1 + 3
i
and
v
=
-
2 +
i
then
z
v
=
A. 4
-
5
i
B. 1
-
7
i
C.
4
3
D.
1
10
+
7
10
i
E.
-
1
2
+ 3
i
F.
1
5
-
7
5
i
correct
(b)
The polar form of
w
= 3
√
2
-
√
6
i
is
A. 2
√
6
e
-
iπ/
6
correct
B. 2
√
3
e
-
iπ/
3
C.
-
√
24
e
π/
6
D.
√
12
e
iπ/
3
E. 2
√
6
e
-
iπ/
4
F. 2
√
3
e
-
iπ/
6
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3
3. (2 points)
Consider the following integral:
Z
1
0
ln(
x
)
dx.
(a)
Why is this integral improper? Choose the best answer.
A. Because lim
t
→∞
ln(
x
) =
∞
.
B. Because ln(
x
) is not the derivative of an
elementary function.
C. Because the domain of ln(
x
) is not the en-
tire real line.
D. Because ln(
x
) has a vertical asymptote at
x
= 0.
correct
E. Because ln(
x
) is zero at
x
= 1.
F. Because ln(
x
) has a horizontal asymptote.
(b)
Find the value of this improper integral, if it converges.
A. It diverges.
B. 0
C. 1
D. 300
E.
-
4
F.
-
1
correct
4. (2 points)
The rate of change of a wolf population
P
over time
t
is given by the differential
equation
dP
dt
=
P
ln(
P
)
500
Use Euler’s method with a step size of
t
= 0
.
5 to estimate
P
(2) if
P
(0) = 1000. Round your
final answer to the nearest wolf but round intermediate steps to two decimal places.
y*ln(y)/500
0 1,000
0.5 1,006.90775527898214
1 1,013.8701592000317
1.5 1,020.88769205263853
2 1,027.96083873335377
A. 978
B. 4234
C. 546
D. 1002
E. 1028
correct
F. 1245
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Universit´e d’Ottawa
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4
5. (2 points)
Consider
P
=
4
6
1
1
Q
=
2
2
3
4
1
-
1
2
0
.
(a)
What is the size of
P
-
1
Q
?
A. impossible to say
B. 2
×
2
C. 2
×
4
correct
D. 4
×
2
E. 8
F. 2
×
6
(b)
What is the (2
,
1)-entry of
P
-
1
Q
?
A.
-
2
B.
-
1
correct
C. 0
D. 0
.
5
E. 2
F. 4
6. (2 points)
Consider the function
f
(
x, y
) = (
y
-
3)
e
xy
-
2
.
(a)
Find the partial derivatives of
f
:
∂f
∂x
(
x, y
) =
(
y
-
3)
ye
xy
-
2
∂f
∂y
(
x, y
) =
e
xy
-
2
(1 +
x
(
y
-
3))
(b)
Give the equation of the tangent plane to the graph of
f
at the point (
x, y, z
) = (2
,
1
,
-
2):
f
(2
,
1) =
-
2,
f
x
(2
,
1) =
-
2,
f
y
(2
,
1) =
-
3; so
z
=
-
2
-
2(
x
-
2)
-
3(
y
-
1) = 5
-
2
x
-
3
y
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5
7. (2 points)
A population
x
of rabbits on a farm grows according to a model
dx
dt
=
x
(100
-
x
)
-
hx
where
h >
0 is the harvesting rate and
t
is measured in years.
(a)
What are the equilibria of this dynamical system?
A. 0
, h
-
99
B. 0
,
99
-
h
C. 0
,
99
D. 0
, h
-
100
E. 0
,
100
-
h
correct
F. 0
,
100
(b)
Under what conditions is the nonzero equilibrium stable? Choose the best answer.
A.
h
= 1
B.
h <
100
correct
C.
h >
100
D.
h
6
= 100
E.
h >
0
F.
h
= 100
8. (2 points)
On the graph below, sketch and shade the domain and sketch two level curves
of the function
z
=
p
x
-
y
2
+ 2
.
Label each level curve with the height of the corresponding part of the graph of
f
.
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6
9. (5 points)
Evaluate
Z
3
x
3
-
2
x
2
+ 28
x
-
10
x
2
+ 9
dx.
3
x
-
2
x
2
+ 9
)
3
x
3
-
2
x
2
+ 28
x
-
10
-
3
x
3
-
27
x
-
2
x
2
+
x
-
10
2
x
2
+ 18
x
+ 8
.
Since
x
2
+ 9 = 9
(
x
3
)
2
+ 1
we make the substitution
u
=
x
3
,
du
=
1
3
dx
or
x
= 3
u
dx
= 3
du
so that
Z
3
x
3
-
2
x
2
+ 28
x
-
10
x
2
+ 9
dx
=
Z
(3
x
-
2) +
x
+ 8
9
(
x
3
)
2
+ 1
dx
=
3
2
x
2
-
2
x
+
1
9
Z
x
+ 8
(
x
3
)
2
+ 1
dx
=
3
2
x
2
-
2
x
+
1
9
Z
3
u
+ 8
u
2
+ 1
3
du
=
3
2
x
2
-
2
x
+
Z
u
u
2
+ 1
du
+
8
3
Z
1
u
2
+ 1
du
=
3
2
x
2
-
2
x
+
Z
1
v
1
2
dv
+
8
3
arctan(
u
) +
c
=
3
2
x
2
-
2
x
+
1
2
ln
|
u
2
+ 1
|
+
8
3
arctan(
u
) +
c
=
3
2
x
2
-
2
x
+
1
2
ln(
1
9
x
2
+ 1) +
8
3
arctan(
1
3
x
) +
c.
10. (5 points)
Consider the following system of linear equations in the variables
x
,
y
and
z
,
with parameters
a
and
b
:
2
x
+ 5
y
+
az
=
b
x
+ 3
y
+ 2
az
= 4
b
3
x
+ 8
y
+ 6
z
= 0
Write down the augmented matrix of this linear system, then use row reduction to determine
all values of
a
and
b
for which this system has
(a)
a unique solution,
(b)
no solution or
(c)
infinitely many solutions. Justify your answers.
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7
11. (5+1+2+2=10 points, over two pages)
The autonomous differential equation
dP
dt
= 2
P
2
-
70
P
+ 500
.
models the dynamics of a town population, with immigration and emigration.
(a) (5 points)
Rewrite this differential equation to show that it is separable, then solve it.
Your answer must express
P
as a function of
t
.
(b) (1 point)
Suppose initially
P
(0) = 15. What is the particular solution in this case?
P
(
t
) =
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8
11, continued.
Recall that
dP
dt
= 2
P
2
-
70
P
+ 500
.
(c) (2 points)
Identify all equilibria of the differential equation and draw the phase line
diagram corresponding to this system. Label each equilibrium as stable or unstable on your
diagram. Show your work.
(d) (2 points)
Consider again an initial population of
P
(0) = 15.
Give your long-term
predictions for this population, with explicit reference to each of your answer in (b) and your
phase line diagram in (c). Explain and justify your answer succinctly and clearly.
Universit´e d’Ottawa
University of Ottawa
9
12. (2+2+2=6 points)
Consider the following system of linear differential equations:
dx
dt
=
-
5
x
+ 20
y
dy
dt
=
-
2
x
+ 7
y.
(a) (2 points)
Identify the matrix
A
and the vector
~x
such that this system can be written
as
~x
0
=
A~x
. Then show that 1
-
2
i
is an eigenvalue of
A
.
(b) (2 points)
Find an eigenvector of
A
corresponding to
λ
= 1
-
2
i
.
(c) (2 points)
Determine the general solution to the system of differential equations.
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10
13. (2+4+2+4+3=15 points, over two pages)
A disease propagates through an isolated
community.
The population consists of
x
susceptible individuals and
y
infected individuals.
There is a constant immigration into the community, and at any time, a certain percentage of
susceptible individuals who are not infected leave the community. Infected individuals some-
times die, and sometimes live but remain infected. A system of nonlinear differential equations
describing the dynamics of this disease is given by
dx
dt
= 20
-
5
xy
-
5
x
dy
dt
= 5
xy
-
10
y.
(a) (2 points)
Calculate the nullclines and equilibria of this system of differential equations.
Simplify.
x-nullcline:
y
= (20
-
5
x
)
/
(5
x
) = 4
/x
-
1; y-nullcline :
y
= 0 or 5
x
= 10, that is,
x
= 2.
Equilibria: (4
,
0) and (2
,
1).
(b) (4 points)
Sketch the nullclines in the phase plane, below.
Indicate the vector field
direction along each nullcline as well as in each of the regions of the first quadrant defined by
the nullclines. Circle the equilibria. Label the axes.
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11
13, continued.
Recopy your equilibria from the previous page here:
(c) (2 points)
Compute the Jacobian of the vector-valued function
~
F
(
x, y
) =
20
-
5
xy
-
5
x
5
xy
-
10
y
.
(d) (4 points)
For each of the equilibria in the box above, find the eigenvalues of the corre-
sponding Jacobian matrix, and use these to classify the equilibria according to their stability.
Show your work and explain how you decided the stability.
(e) (3 points)
For each of the following two initial conditions, sketch a corresponding solution
on your phase plane diagram in (b), respecting the vector field there. Using this, determine
the populations in the long term:
(i) If initially
x
(0) = 1,
y
(0) = 1, then for
t
very large we have
x
(
t
)
≈
,
y
(
t
)
≈
(i) If initially
x
(0) = 1,
y
(0) = 0, then for
t
very large we have
x
(
t
)
≈
,
y
(
t
)
≈
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12
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