MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
Name:
Work out everything as far as you can before making decimal approximations.
1.
f (x) = sin x
0≤x≤π
Extend to an even periodic function with period 2π (note that it also has period π,
but we will treat it as having period 2π).
(a) Draw the graph of this function for −π ≤ x ≤ π.
(b) Calculate the amplitudes a0 , a1 , b1 , a2 , b2 of its real Fourier series.
(c) Calculate the energy of f (x).
(d) Using the fact that
π 2 ∼ 10,
how many terms of the Fourier series do you need to capture 97% of the
energy?
Hint: you may want to rewrite your sin and cos functions in terms of eix type
functions, in order to calculate the integrals. Or, if you prefer, try integration by
parts.
Date: July 16, 2001.
1
2
2.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
If g(x) is a function with period T , show that
Z
g(x) dx
has period T precisely when the average value of g(x) is zero. Hint: write out a
complex Fourier series for g(x) and integrate it.
MATH 3150: PDE FOR ENGINEERS
3.
MIDTERM TEST #1
What is the smallest period of
1
h(x) = e 2 cos x
3
4
4.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
Write f (x) = (1 − x)2 as the sum of an even function with an odd function.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
5
5. Among the functions f (x) in figure 1 on the following page, which one of them
has am amplitudes decaying fastest as function of m? Which has the slowest decay
in m?
6
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
0.8
2
0.6
1.5
0.4
1
0.2
0.5
–3
–3
–2
–1
0
1
x
2
–2
–1
1
x
2
3
–0.2
(a)
(b)
6
4
2
–3
–2
–1
0
1
x
2
3
(c)
Figure 1. Three even periodic functions
3
MATH 3150: PDE FOR ENGINEERS
6.
MIDTERM TEST #1
7
You may assume that
Z 1
2
(1 − x2 ) cos(αx) dx = 3 (sin α − α cos α)
α
0
for α any non-zero constant. Let f (x) = 1 − x2 for 0 ≤ x ≤ 1 and let f (x) be even
and periodic with period 2.
(a) Calculate the real amplitudes for f (x).
(b) Calculate the energy of f (x).
(c) Find the energy of f (x) stored in the amplitudes am and bm in terms of m.