MTH240 Final Exam W23
Toronto Metropolitan University
Apr 21, 2023
Time Allowed: 120 Minutes
Family Name: Last Name
First Name: Name
TMU Email: test@bitbolide.com
Signature: —————————————
Instructor
01
05
11
17
23
Section (Circle One)
02 03 04 29
06 07 08 9 10
12 13 14 15 16
18 19 20 21 22
24 25 26 27 28
Instructions:
1. Calculators, notes, and other aids are not allowed in the Final Exam.
2. Answer all questions in this booklet.
3. Some questions have extra pages for their solutions. Each extra page must only be used for the related question
if needed. If you need extra room, use additional pages, clearly indicating where your answer continues.
ANYTHING WRITTEN ON THE BACK OF ANY PAGE WILL NOT BE MARKED.
4. In every question, show your work, presented clearly and in the correct order. Unjustified answers will be
given little or no credit.
5. Cross out all irrelevant or incorrect work, as marks may be deducted for work, which is misleading, irrelevant,
or incorrect.
6. Make sure your test paper is complete; there are 6 Long Answer Questions in this Final Exam Paper.
7. The exam invigilators will collect the scripts of handwritten solutions from all students. Please hand in your
solutions at the end of the test as instructed. Failure constitutes a breach of academic integrity.
MTH240 Final Exam W23
test@bitbolide.com
1. [9 Marks] (a) Use the method of separation of variables to find an explicit solution of the differential
equation
dy
cos(x)
=
.
dx
sin(5y)
(b) Solve the initial value problem
x
dy
− y = 3x ln(x), y(1) = 2.
dx
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Q1 extra page ...
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MTH240 Final Exam W23
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2. [9 Marks] Determine if the following series converge or diverge. If the series converges, find the sum.
(a) 4 +
(b)
4 4
4
4
4
+ +
+
+
....
3 9 27 81 243
2n + 5(−1)n
.
n=1
3n
P∞
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MTH240 Final Exam W23
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3. [8 Marks] (a) Find the open interval of convergence of the series
P∞
n=1
(−1)n+1 (x + 3)n
.
n5n
(b) At what value of x is the interval of convergence centered?
(c) What is the radius of convergence of the series?
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Q3 extra page ...
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MTH240 Final Exam W23
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4. [6 Marks] The Maclaurin series expansion for cos(x) is
cos(x) = 1 −
x2
x4
x6
x8
x10
+
−
+
−
+ ....
2!
4!
6!
8!
10!
(a) Write down the first five nonzero terms of the Maclaurin series for cos(t3 ).
(b) Using part (a), find the first five nonzero terms of the Maclaurin series for
Z x
f (x) =
cos(t3 )dt.
0
(c) Using part (b) with the first three nonzero terms, estimate
Z 1
cos(t3 )dt.
0
Leave your answer as a fraction.
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5. [18 Marks] (a) Evaluate the limit
lim(x,y)−→(2,1)
x2 + xy − 6y 2
.
x2 + 4y 2
(b) Find the limit
lim(x,y)−→(0,0)
xy 2
x2 + y 3
by approaching (0, 0) along
(i) the x-axis
(ii) the y-axis
(iii) the y = x
(iv) the y = 3x
(v) the y = 2x2 .
What, if anything, can you conclude about the existence of the limit at (0, 0)?
(c) Determine whether the function g(x, y) is continuous at (0, 0)
2
2
3x + y
if (x, y) 6= (0, 0),
2
2
g(x, y) =
x +y
0
if (x, y) = (0, 0).
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6. [10 Marks] Find all critical points of
f (x, y) = x4 − 2x2 + y 2 − 28
and classify them using the second derivative test.
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