School of Mathematics and Statistics
MAST20029 Engineering Mathematics, Semester 1 2022
Assignment 3
Submit a single pdf file of your assignment on the MAST20029 website before 4pm on
Monday 23rd May.
• This assignment is worth 5% of your final MAST20029 mark.
• Assignments must be neatly handwritten, but this includes digitally handwritten documents using an
ipad or a tablet and stylus, which have then been saved as a pdf.
• Full working must be shown in your analytical solutions.
• For the MATLAB questions, include a printout of all MATLAB code and outputs. This must be
printed from within MATLAB, or must be a screen shot showing your work and the MATLAB Command
window heading. You must include your name and student number in a comment in your code otherwise
the code and output will not be marked.
• For the Taylortool question, include a printout of all MATLAB code and output. The output must
show the graphs, Taylor polynomial, function and parameter ranges. This must be printed from within
MATLAB, or be a screen shot showing your work and the MATLAB Command window heading. You
must include your name and student number in a comment in your code otherwise the code and output
will not be marked.
1. In your solution, you must state if you use any standard limits, continuity, l’Hoˆpital’s rule or any
convergence tests for series.
Consider the series ∞∑
n=1
(n + p)n
2pn(n + p)!
where p ∈ N and p > 0.
Determine the values of p for which the series converges.
2. In your solution, you must write your answers in exact form and not as decimal approximations.
Consider the function
f(x) = e
x2
2 , x ∈ R.
(a) Determine the fourth order Maclaurin polynomial P4(x) for f .
(b) Using P4(x), approximate e
1
8 .
(c) Using Taylor’s theorem, find a rational upper bound for the error in the approximation in
part (b).
(d) Using P4(x), approximate the definite integral∫ 1
0
e
x2
2 dx.
(e) Using the MATLAB applet Taylortool:
i. Sketch the tenth order Maclaurin polynomial for f in the interval −3 < x < 3.
ii. Find the lowest degree of the Maclaurin polynomial such that no difference between the
Maclaurin polynomial and f(x) is visible on Taylortool for x ∈ (−3, 3). Include a sketch
of this polynomial.
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3. Consider a signal represented by the function
f(t) =
{
et, −1 < t ≤ 0
e−t, 0 < t < 1
where f(t) = f(t + 2).
(a) Sketch f(t) for −2 ≤ t ≤ 2.
(b) Obtain a general Fourier series representation for f .
(c) Use MATLAB to plot both f(t) and the partial sum of the Fourier series f5(t) on the same
set of axes for −1 ≤ t ≤ 1, where
f5(t) = a0 +
5∑
n=1
(an cos(nωt) + bn sin(nωt)).
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