Practice Exam Solution
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Course
136
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Mathematics
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Jan 9, 2024
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University of Toronto – Faculty of Arts and Science
MAT136 - Calculus 2 - Summer 2022
Practice Exam - 2022
Time allotted: -.
Aids permitted: Calculator without integral (
R
) symbol
non-programmable and non-graphical
.
Total marks: -
MULTIPLE CHOICE PART.
There is only one correct answer for each question.
(10 marks)
ANSWER THESE QUESTIONS ON
PAGE 3
OF THE TEST.
1 (2 marks)
Water is pumped out of a tank at a rate of 1
−
e
−
0
.
1
t
liters per minute, where
t
is in minutes.
Assuming the tank had 100 liters at time
t
= 0, how much water left in the tank after 30 minutes?
(A)
≈
99
.
5 liters
(B)
≈
89
.
5 liters
(C)
≈
79
.
5 liters
(D)
≈
69
.
5 liters
2 (2 marks)
Which of the following is an antiderivative of
1
x
ln(
x
)
?
(A)
ln(
x
)
x
+ 2
(B)
ln(2 ln(
x
))
(C)
x
ln
2
(
x
)
(D)
x
ln(2
x
)
3 (2 marks)
What is the value of the following integral
R
∞
1
x
ln(
x
)
dx
(A)
∞
(B)
−
1
4
(C)
The integral is not defined
(D)
0
Student Number:
4 (2 marks)
Solve:
y
′
=
1
sin(
y
)
, with the initial condition
y
(3) =
π
(A)
y
= cos
−
1
(
x
+
π
)
(B)
y
= tan(
x/π
)
(C)
x
=
−
cos(
y
) + 2
(D)
x
=
−
cos(
y
−
2)
5 (2 marks)
If
y
(
x
) =
∑
∞
n
=0
a
n
x
n
, then
∑
∞
n
=0
((
n
+ 1)
a
n
+1
−
a
n
)
x
n
is:
(A)
y
′
−
y
(B)
xy
−
y
(C)
xy
′
−
y
(D)
None of the above
6 (2 marks)
Let the region
R
in the plane be bounded by
y
=
√
x
,
x
= 1,
x
= 3, and
y
=
x
2
. Suppose that
we build a solid
S
whose base is
R
and whose cross sections perpendicular to the
x
-axis are semicircles. If
we slice the 2-dimensional region into slices of width ∆
x
, what sum approximates the total volume of
S
?
(A)
∑
n
i
=1
π
8
(
x
2
−
√
x
)
2
∆
x
(B)
∑
3
i
=1
π
2
(
x
2
−
√
x
)
2
∆
x
(C)
∑
n
i
=1
π
(
x
2
−
√
x
)
2
∆
x
(D)
∑
3
i
=1
π
(
x
2
−
√
x
)
2
∆
x
7 (2 marks)
A cell contains a chemical (solute) dissolved in it at a concentration
c
(
t
), and the concentration
of the same substance outside the cell is a constant
k
. Fick’s Law states that the solute moves across the cell
wall at a rate proportional to the difference between
c
(
t
) and
k
, towards the region of lower concentration.
Which of the following is a possible differential equation that models this?
(A)
c
′
(
t
) =
c
−
k
(B)
c
′
(
t
) =
c
+
k
(C)
c
′
(
t
) =
α
(
k
−
c
) for a positive constant
α
(D)
c
′
(
t
) =
α
(
c
−
k
) for a positive constant
α
Student Number:
8 (2 marks)
For which of the following differential equations and initial conditions will Euler’s method give
the exact value of the solution?
(A)
y
′
=
xy
, where
y
(0) = 1, and using steps of size
e
(B)
y
′
= sin(
x
), where
y
(0) = 1, and using steps of size 2
π
(C)
y
′
=
y
, where
y
(0) = 1 and using steps of size
e
(D)
y
′
=
y
2
, where
y
(0) = 1 and using steps of size 2
π
9 (2 marks)
A differential equation has the following slope field:
If the initial condition is
y
(0) =
−
2, which of the following is
false
?
(A)
y
′
(
−
2)
>
0
(B)
y
′
(2)
>
0
(C)
y
′′
(0)
<
0
(D)
y
(2)
<
0
10 (2 marks)
A function,
f
(
x
) is defined by the power series
∑
∞
n
=1
n
3
(2
x
)
n
. What is the largest open interval
where this function is defined?
(A)
−
1
< x <
0
(B)
−
1
2
< x <
1
2
(C)
−
1
< x <
1
(D)
0
< x <
1
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Student Number:
LONG ANSWER PART
–
Unsupported answers to Questions
11
–
14
will not receive full credit.
11
The sine integral function is defined as
Si
(
x
) :=
Z
x
0
sin(
t
)
t
dt.
The function inside this integral does not have an elementary antiderivative. In this problem we will use
Taylor polynomials to express this function in a series form.
11.1 (6 marks)
First, find the Taylor series for the function
sin(
x
)
x
.
(Hint: you may want to use the series expansion of
sin
(
x
)
on the formula sheet)
Student Number:
11.2 (6 marks)
Use your answer from
11.1
to find the Taylor series of the
Si
(
x
) function.
11.3 (8 marks)
Find the radius of convergence and the interval of convergence of the series from
11.2
?
Student Number:
12
Trees transform carbon dioxide (CO2) and water into glucose and oxygen using sunlight. (10 marks) This
process is called photosynthesis.
According to a paper1, during 1 hour, the leafs of the plant plantago
asiatica produces glucose and oxygen at the rate
r
(
T
) = 0
.
36
(
11 +
19
20
(
T
−
10)
−
1
40
(
T
−
10)
2
)
measured in
mol/cm
2
, where T is the temperature of the leaf in degrees Celsius.
Note that the rate of
photosynthesis is measured per
cm
2
of leaf. Now let us consider a leaf with the shape of the figure on the
right that is partially in shade so the temperature (measured in degrees Celsius) varies in different parts of
the leaf and is given by:
T
(
x
) = 10 + 2 sin(
x
2
10
)
The units of x and y are in cm.
What is the total amount of glucose and oxygen produced by the leaf of this shape during 1 hour?
Your final answer should be in the form of an integral formula. Do not calculate the integral.
In your answer, follow the step-by-step slicing procedure (in the formula sheet).
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Student Number:
Student Number:
13
Every year, the population of squirrels at UofT
(20 marks)
increases at a constant rate of 400 squirrels/year.
At the same time, throughout the year half of the
squirrels are leaving the UofT campus to live at York University. Let
P
(
t
) be the population of the squirrels
(in hundreds) at UofT at any time
t
⩾
0 (in years), then the above scenario can be described by the following
differential equation
dP
dt
= 4
−
P
2
.
The slope field of the differential equation is given by the following:
13.1 (4 mark)
Find
ALL
the equilibrium solutions of the differential equation.
For each equilibrium
solution, explain, based on the slope field, if it is
stable
or
unstable
.
Student Number:
13.2 (8 marks)
Draw the solution,
P
(
t
), of the differential equation with
P
(0) = 1 based on the slope field
given below. Draw your answer on top of the slope field.
What is lim
t
→∞
P
(
t
)? Explain.
13.3 (8 marks)
Use the Separation of Variables to find the population of squirrel at any time
t
⩾
0 with
P
(0) = 1.
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Student Number:
14
(Based on Problem 21 of Section 11.8 in the textbook.)
(20 marks)
Humans vs Zombies is a game in which one player starts as a zombie and turns human players into zombies
by tagging them.
Zombies have to ”eat” on a regular basis by tagging human players, or they ”die” of
starvation and are out of the game. If we let
H
(
t
) represent the size of the human population at any time
t
⩾
0 and
Z
(
t
) represent the size of the zombie population at any time
t
⩾
0, then, under certain conditions,
this model can be described by
dH
dt
=
−
HZ
dZ
dt
=
−
Z
+
HZ.
14.1 (4 marks)
Use the Chain Rule to get the expression for
dZ
dH
. Show your steps.
14.2 (8 marks)
Solve the differential equation
dZ
dH
=
...
obtained in
14.1
.
Student Number:
14.3 (8 marks)
Assume the relation
Z
= ln(
H
)
−
H
+ 5 holds for all
t
⩾
0. Use this relation to obtain the
differential equation for
dH
dt
in term of
H
only. Then, use Euler’s method to approximate the solution
of the equation at
t
= 0
.
1 starting with the initial condition
H
(0) = 5.