# 辅导案例-MAST10006

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Student
Number
SAMPLE EXAM, Semester 1 2020
School of Mathematics and Statistics
MAST10006 Calculus 2
Instructions to Students
• This sample exam is intended to give an indication of the length and format of the written
MAST10006 Calculus 2 exam in 2020 semester 1.
• The questions on the real exam will be different to this sample exam.
• This sample exam may not be indicative of the difficulty or topics covered in the real exam.
• You are recommended to try to complete this exam within 2 hours, and under exam conditions
for practice.
• Answers to this sample exam are available on Canvas. Full solutions are not available.
• The questions in this sample exam are drawn from the 2020 summer semester MAST10006 exam.
• There are 8 questions with marks as shown. The total number of marks available is 60.
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 1 (10 marks)
In this question you must state if you use any standard limits, limit laws, continuity, l’Hoˆpital’s
rule or the sandwich theorem.
Let f : R→ R be given by
f(x) =

1
x − sinxx2 , x < 0
kx, 0 ≤ x ≤ 1
cosec
(
pix
2
)
, x > 1
where k ∈ R is a constant.
(a) Find lim
x→0
f(x), or explain why it does not exist.
Page 2 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 1 (continued)
(b) For which value(s) of k is f continuous at x = 1? Show all your working.
Page 3 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 2 (10 marks)
In this question you must state if you use any standard limits, limit laws, continuity, l’Hoˆpital’s
rule, the sandwich theorem or series convergence tests.
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
Page 4 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 2 (continued)
(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
Page 5 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 3 (5 marks)
Evaluate
d47
dx47
(
e−x sinx
)
.
Page 6 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 4 (5 marks)
Evaluate the integral ∫
2x4 − x3 + 4x2 − x− 2
x3 + 2x
dx
Page 7 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 5 (5 marks)
Find the general solution y(x) of
dy
dx
= x
(
ex
2 − 2y
)
Page 8 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 6 (8 marks)
Find the solution of the differential equation
y′′ + 2y′ + y = 25 sin(2x)
subject to the boundary conditions y(0) = −4, y(pi) = pi − 4.
Page 9 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 7 (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 interept with its
value.
(c) Sketch the surface S in R3. Label each axis intercept with its value.
Page 10 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 8 (10 marks)
Let f : R2 → R, f(x, y) = 3x2 − 2x3 − 3y2 + 6xy.
(a) Find the directional derivative of f at (0, 1) in the direction from (0, 1) towards (1, 0).
(b) Find the equation of the tangent plane to the surface z = f(x, y) at the point where
(x, y) = (0, 1).
Page 11 of 12 pages
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MAST10006 Calculus 2 SAMPLE EXAM Semester 1, 2020
Question 8 (continued)
(c) Find all stationary points of f , and classify each point as a local maximum, local  