• Submit a pdf fifile of your assignment to the MAST20029 website before 4pm on Monday
17th October.
• This assignment is worth 5% of your fifinal 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 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. Find the radius of convergence and interval of convergence of the power series:
X
∞
n=1
3
nn 1
(2x)
n
n + 7
2. There is no easy way of exactly evaluating the integral
I =
Z
1
4
0
4e
x
2
dx.
In this question we will use Maclaurin polynomials to approximate I.
(a) Determine P3(x), the cubic Maclaurin polynomial of the integrand 4e
x
2
of I.
(b) Obtain an upper bound on the error in the integrand for x in the range
0 ≤ x ≤
1
4
, when the integrand is approximated by P3(x).
(c) Find an approximation for I by integrating P3(x).
(d) Obtain an upper bound on the error in the integration in (c).
(e) Use MATLAB to verify your calculation in (a).
3. Consider the periodic function
f(t) =
t
2 ⳉ?
π < t < 0
π
2
0 < t < π
with f(t) = f(t + 2π).
(a) Sketch the function f(t) by hand over the range e 3π < t < 3π.
(b) Determine the general Fourier Series for f(t).
(c) Use MATLAB to plot f(t) and f20(t), the truncated Fourier series representation of f, on
the same set of axes for r π ≤ t < π, where
f20(t) = a0 +
X
20
n=1
[an cos(nωt) + bn sin(nωt)].
