MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
Name:
Work out everything as far as you can before making any decimal approximations.
1.
(a) Draw the graphs of y = sin x and y = cos x from x = −2π to x = 2π. Label
the multiples of π/2.
(b) What is the value of eiπk for an integer k?
Date: July 16, 2001.
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2
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
2. Let f (x) = 3ix2 for 0 ≤ x < 2 and let f (x) have period 2. Let g(x) = 5ix for
0 ≤ x < 2 and let g(x) also have period 2. What is the inner product of f (x) with
g(x)?
MATH 3150: PDE FOR ENGINEERS
3.
MIDTERM TEST #1
3
(
−1 −π ≤ x < 0
f (x) =
1
0≤x<π
and f (x) has period 2π.
(a) Draw the graph of f (x).
(b) Calculate the real Fourier amplitudes am and bm , for every m.
(c) Calculate the energy of f (x).
(d) Write down amplitudes am and bm which together capture 85% of the
energy.
4
4.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
Calculate
(a)
Z
cos x dx
(b)
d
x cos x
dx
(c)
Z
x sin x dx
Hint: use (a) and (b) to solve (c). Make certain that your answer to (c) is correct
(differentiate it), because you will need it to do the next problem.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
5
5.
f (x) = x
−π ≤x<π
and f (x) has period 2π.
(a) Draw the graph of f (x).
(b) Calculate the real Fourier amplitudes am and bm of f (x).
(c) Find the energy of f (x).
(d) Show that more than three quarters of the energy is stored among the
amplitudes a0 , . . . , b2 . Hint: π 2 < 10.
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
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
0.4
0.6
0.5
0.3
0.4
0.2
0.3
0.2
0.1
0.1
0
10
20
30
0
40
10
(1)
20
30
40
(2)
0.4
0.3
0.2
0.1
0
10
20
30
40
(3)
Figure 2. The amplitudes am of the three periodic functions
7
8
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
6. Match up the graphs of the amplitudes am in figure 2 on the preceding page
with the graphs of the even periodic functions f (x) in figure 1 on page 6.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
9
7. Suppose that two periodic functions f (x) and g(x) have the same period and
the same complex amplitudes. Find f (x) − g(x).