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MATH7501 Problem Set
Questions: 45 marks in total. Weight: 15%.
Important points about submission of this assignment:
• No coversheet is needed. However, by submitting your assignment on Blackboard (using
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I hereby state that the work contained in this assignment has not previously been submitted
for assessment, either in whole or in part, by either myself or any other student at either
The University of Queensland or at any other tertiary institution except where explicitly
acknowledged. To the best of my knowledge and belief, the assignment contains no material
that has been previously published or written by another person except where due reference
is made. I make this Statement in full knowledge of an understanding that, should it be
found to be false, I will be subject to disciplinary action under Student Integrity and
Misconduct Policy 3.60.04 of the University of Queensland.
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be FFFF LLLL SN-ProblemSet3.pdf where FFFF is your first name, LLLL, is your last
name, and SN is your student number.
• Do not submit code files - instead format your code into the PDF file. Both handwritten
notes and typed notes are acceptable. A combination of handwritten and typed notes is
acceptable as long as it is formatted nicely and continuously in a single PDF file. All
graphs, plots, source code, and other figures must be clearly labeled. All questions/items
must appear in order.
• You may use mathematica as an aid for the computations, however make sure to do the
hand calculations where suitable as well. Ensure that your working is set out clearly and
neatly, that you state any assumptions and define notation. If rounding a numerical answer
(at the end of a question), three significant figures is enough.
1 of 3
MATH7501 Problem Set 3 Semester 1, 2022 Due 30/05/2022 at 14:00
1. Let M > 0 be a finite constant, and let f : (a, b)→ R be differentiable on its domain, such
that |f ′ (x)| ≤M for each x ∈ (a, b).
(a) Show that for all x ∈ (a, b) and y ∈ (a, b), we have |f (x)− f (y)| ≤M |x− y|.
[3 marks]
(b) Give an example of a function f : (a, b) → R (for a < b) such that f is not differen-
tiable everywhere on (a, b), but there exists a finiteM > 0, such that |f (x)− f (y)| ≤
M |x− y|, for all x ∈ (a, b) and y ∈ (a, b). [2 marks]
2. For a natural number n, let f : R→ R be defined by:
f (x) =
{
xn sin (1/x) , for x ̸= 0,
0, for x = 0.
(a) Show that ∣∣∣∣f (0 + h)− f (0)h
∣∣∣∣ ≤ hn−1.
[2 marks]
(b) Show that, for n ≥ 1, f is continuous at 0. [2 marks]
(c) Show that, for n ≥ 2, f is differentiable at 0. [2 marks]
3. Let f1 : R → R, . . . , fn : R → R be a system of n functions, such that fj > 0 and fj is
differentiable for each j ∈ {1, . . . , n}, and let
F (x) = f1 (x)× f2 (x)× · · · × fn (x) .
Prove that
F ′ (x)
F (x)
=
n∑
j=1
f ′j (x)
fj (x)
.
[5 marks]
4. Let f : [−1, 1]→ R be defined by f (x) = x3 − x.
(a) Find the global maximum and minimum values of f . Justify your answer, rigorously.
[4 marks]
(b) Consider instead that f was defined on the domain R instead of [−1, 1] (that is,
f : R→ R). Would the global maximum and minimum values be the same as in part
(a)? Why, or why not? [2 marks]
(c) What is the largest interval domain [a, b] (that is, suppose that the domain of f is [a, b]
instead of [−1, 1]) for which the global maximum and minimum values of f remains
the same as the answer in part (a)? [2 marks]
5. Use the fact that ex =
∑∞
k=0
(
xk/k!
)
to derive the Maclaurin series of sinhx = 1
2
(ex − e−x)
and coshx = 1
2
(ex + e−x). [5 marks]
6. Let f (x, y) =
√
x3 + y3, for x ≥ 0 and y ≥ 0.
(a) Produce a linear approximation for f (x, y) at (a, b) = (1, 2). [4 marks]
2 of 3
MATH7501 Problem Set 3 Semester 1, 2022 Due 30/05/2022 at 14:00
(b) Use the linear approximation from part (a) to compute
√
1.023 + 1.973. [2 marks]
7. Compute the following integrals (with full justification):
(a)
∫
x−1/2 (x+ 1) dx. [2 marks]
(b)
∫
sin4 (x) dx. [2 marks]
(c)
∫ x ln(x+√1+x2)√
1+x2
dx. [2 marks]
8. Use the Fundamental Theorem of Calculus to prove that yp − xp ≤ p (y − x), for p > 1,
and 0 ≤ x < y ≤ 1. [4 marks]
