Millersville University
Department of Mathematics
MATH 467 Partial Differential Equations, Homework 2
February 12, 2008
Please work the following problems for homework and turn them in at class time on Thursday,
February 14, 2008. Each problem is worth 10 points unless marked otherwise.
1. Consider the function:
f(x) = (2(x+ 1) if −1≤x≤0,
xif 0 < x ≤1.
(a) Sketch the graph of the Fourier Series representation of f(x) on the interval [−4,4].
(b) Sketch the graph of the Fourier Sine Series representation of f(x) on the interval
[−4,4].
(c) Sketch the graph of the Fourier Cosine Series representation of f(x) on the interval
[−4,4].
2. Using trigonometric identities (e.g. product to sum formulas and half-angle identities)
find the Fourier Series of the following functions without computing any integrals.
(a) f(x) = cos2(πx) sin2(πx) for −1≤x≤1.
(b) g(x) = sin(x) [sin(x) + cos(x)]2for −π≤x≤π.
3. Consider the function
f(x) =
1 + xif −1≤x≤0,
0 if 1 <|x| ≤ 2,
1−xif 0 ≤x≤1.
Find the Fourier Series for this function.
4. Find the Fourier Series for the derivative of the function given in Ex. 3.