Student
Number
Summer Semester Assessment, 2020
School of Mathematics and Statistics
MAST10006 Calculus 2
Writing time: 3 hours
Reading time: 15 minutes
This is NOT an open book exam
This paper consists of 6 pages (including this page)
Authorised Materials
• Mobile phones, smart watches and internet or communication devices are forbidden.
• No written or printed materials may be brought into the examination.
• No calculators of any kind may be brought into the examination.
Instructions to Students
• You must NOT remove this question paper at the conclusion of the examination.
• All questions may be attempted.
• Marks may be awarded for
– Correct use of appropriate mathematical techniques
– Accuracy and validity of any calculations or algebraic manipulations
– Clear justification or explanation of techniques and rules used
– Clear communication of mathematical ideas through diagrams
– Use of correct mathematical notation and terminology
• Write your answers in the script book provided. Additional script books are available
from the invigilators if required. Any answers you write on this exam paper will not be
assessed.
• Start each question on a new page of the script book. Clearly label each page with the
number of the question that you are attempting.
• There is a separate 1 page formula sheet provided, which you may use in the examination.
• There are 10 questions with marks as shown. The total number of marks available is 104.
Instructions to Invigilators
• Students must NOT remove this question paper at the conclusion of the examination.
• Initially students are to receive the exam paper, the 1 page formula sheet, and two 10
page script books.
This paper may be held in the Baillieu Library
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MAST10006 Summer Semester, 2020
Question 1 (10 marks)
In this question you must state if you use standard limits, continuity, l’Hoˆpital’s rule, the
sandwich theorem or any tests for convergence of series; you do not need to state if you use any
limit laws.
Let f : R→ R be given by
f(x) =

1
x − sinxx2 , x < 0
kx, 0 ≤ x ≤ 1
cosec
(
πx
2
)
log x, x > 1
where k ∈ R is a constant.
(a) Find lim
x→0
f(x), or explain why it does not exist.
(b) For which value(s) of k is f continuous at x = 1? Show all your working.
Question 2 (14 marks)
In this question you must state if you use standard limits, continuity, l’Hoˆpital’s rule, the
sandwich theorem or any tests for convergence of series; you do not need to state if you use any
limit laws.
Let
an =
4n
n3 + r2n
,
where r ∈ R is a constant.
(a) If r = 2 then does the sequence {an} converge or diverge? Justify your answer.
(b) If r = 2 then does the series
∞∑
n=1
an converge or diverge? Justify your answer.
(c) If r = 3 then does the sequence {an} converge or diverge? Justify your answer.
(d) If r = 3 then does the series
∞∑
n=1
an converge or diverge? Justify your answer.
(e) For which value(s) of r does the sequence {an} converge?
(f) For which value(s) of r does the series
∞∑
n=1
an converge?
Page 2 of 6 pages
MAST10006 Summer Semester, 2020
Question 3 (6 marks)
Evaluate
d47
dx47
(
e−x sinx
)
.
Question 4 (12 marks)
Evaluate the following integrals:
(a)
∫
x5
√
1− x2 dx
(b)
∫
2x4 − x3 + 4x2 − x− 2
x3 + 2x
dx
Question 5 (12 marks)
(a) Find the general solution y(x) of
dy
dx
= x
(
ex
2 − 2y
)
.
(b) Make the substitution z = x+ y and reduce
dy
dx
= (x+ y)2 log
(
x2 + 1
)− 1 (1)
to the differential equation
dz
dx
= z2 log(x2 + 1).
Then find the general solution y(x) of equation (1).
Page 3 of 6 pages
MAST10006 Summer Semester, 2020
Question 6 (10 marks)
Suppose that the population p = p(t) of a new type of coronavirus in a patient’s body satisfies
dp
dt
=
p
100
(
1− p
1600
)
− h, (t ≥ 0) (2)
where h ≥ 0 is a constant depending on the drug used in the medical treatment to kill the virus.
(a) Determine the value(s) of h for which there is exactly one equilibrium solution of equa-
tion (2).
(b) For each value of h that you find in part (a) determine the equilibrium solution and its
stability.
(c) Determine the value(s) of h for which p(t) strictly decreases with t irrespective of the
initial population.
(d) Given h = 0 and p(0) = 1314, does p(t) have any inflection point for t ≥ 0; if so, find the
population when inflection first occurs. Explain why.
(e) Given h = 0 and p(0) = 520, does p(t) have any inflection point for t ≥ 0; if so, find the
population when inflection first occurs. Explain why.
Page 4 of 6 pages
MAST10006 Summer Semester, 2020
Question 7 (12 marks)
(a) Find the solution of the differential equation
y′′ + 2y′ + y = 25 sin(2x)
subject to the boundary conditions y(0) = −4, y(π) = π − 4.
(b) Find the general solution of the differential equation
y′′ + y′ − 2y = sinhx.
Question 8 (6 marks)
Consider the differential equation
d2y
dx2
− 8mdy
dx
+ 25m2y = 0 (3)
where m ∈ R is a constant.
(a) Determine the value(s) of m for which y = e−4x sin(3x) is a solution of equation (3).
(b) For each value of m that you find in part (a) find the general solution of equation (3).
(c) Determine the value(s) of m for which lim
x→∞ y(x) = 0 for every solution y = y(x) of
equation (3).
Page 5 of 6 pages
MAST10006 Summer Semester, 2020
Question 9 (7 marks)
Let S be a surface in R3 given by z = cosh
√
x2 + y2 for (x, y) ∈ R2.
(a) Find an expression for the level curve of this surface when z = c. For what value(s) of c
does the level curve exist?
(b) Sketch the cross section of the surface in the yz plane. Label each axis intercept with its
value.
(c) Sketch the surface S in R3. Label each axis intercept with its value.
Question 10 (15 marks)
Let f : R2 → R, f(x, y) = 3x2 − 2x3 − 3y2 + 6xy.
(a) Find the gradient of f .
(b) Find the directional derivative of f at (0, 1) in the direction from (0, 1) towards (1, 0).
(c) Find the equation of the tangent plane to the surface z = f(x, y) at the point where
(x, y) = (0, 1).
(d) Find the second order partial derivatives fxx, fyy, fxy and fyx of f .
(e) Find all stationary points of f , and classify each point as a local maximum, local minimum
or saddle point.
(f) Evaluate
∫ 1
−2
∫ 2
0
f(x, y)dxdy.
End of Exam—Total Available Marks = 104
Page 6 of 6 pages
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