MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
Name:
Work out everything as far as you can before making decimal approximations.
1. Calculate the integrals:
(a)
Z
eax cos x dx
where a is a constant.
(b)
Z
eax cos bx dx
where a, b are constants, with b 6= 0.
(c) Check your answer to this last question by differentiating the result.
Date: July 16, 2001.
1
2
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
2.
(a) Use the result from the previous question to find the real Fourier amplitudes
of the function
f (x) = ce−c|x|
(b)
(c)
(d)
(e)
− T /2 ≤ x ≤ T /2
with period T , where c is a positive constant.
Find the energy in each amplitude.
How much energy is in the function f (x)?
Taking the limit as c → ∞, what happens to the energy in am for m c?
Draw the function f (x) with c = 1 and T = 1. Include at least 2 full
periods in your graph.
MATH 3150: PDE FOR ENGINEERS
3.
Calculate the value of
e4001πi/2
MIDTERM TEST #1
3
4
4.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
Suppose that f (x) has period 2π, and average value 0.
(a) Show that the energy in df /dx satisfies
2
df ≥ kf k2 .
dx (Hint: you will need to write out f (x) and df /dx as complex Fourier series,
and use Parseval’s theorem to write the energy of f (x) and of df /dx as
infinite sums of energy from each amplitude.) [This is Wirtinger’s inequality; Wirtinger was your instructor’s thesis adviser’s thesis adviser’s thesis
adviser’s thesis adviser’s thesis adviser.]
(b) Which functions f (x) satisfy
2
df = kf k2 ?
dx MATH 3150: PDE FOR ENGINEERS
5.
MIDTERM TEST #1
5
You may assume that
Z 1
2
(1 − x2 ) cos(αx) dx = 3 (sin α − α cos α)
α
0
Let f (x) = 1 − x2 for −1 ≤ x ≤ 1 and let f (x) be periodic with period 2.
(a) Draw the graph of f (x) over 2 periods.
(b) Calculate the real amplitudes for f (x).
(c) Calculate the energy of f (x).
(d) Find the energy of f (x) stored in the amplitudes am and bm in terms of m.
(e) Show that more than 90% of the energy is contained in the terms a0 , a1 , b1 , a2 , b2 .
Use the approximation π 4 ∼ 97.