MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
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?
Solution:
(a) See figure 1.
(b)
eiπk = (−1)k
Date: December 14, 2001.
1
0.5
–6
–4
–2
0
2
x
4
6
–0.5
–1
Figure 1. The sine and cosine functions
1
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)?
Solution:
Z
2
(f (x), g(x)) =
f (x)g(x) dx
0
Z
2
(3ix2 )5ix dx
=
0
Z
2
=
(3ix2 )(−5ix) dx
0
Z
= 15
0
4
= 15
= 60
2
4
2
x3 dx
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
3
1
0.5
–6
–4
–2
2
x
4
6
–0.5
–1
Figure 2. A square wave
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. (Use the fact that π 2 < 10.)
Solution:
(a) See figure 2.
(b) The am are zero, because f (x) is odd.
(
4
m odd
bm = πm
0
m even
(c) kf k2 = 2π
(d) Energy in bm is
T 2
b =
2 m
(
16
πm2
0
m odd
m even
m Energy in bm
1 16/π ∼ 16/3.2 = 5
3 16/9π ∼ 5/9 = 0.555. . .
5 16/25π ∼ 5/25 ∼ 0.2
The fraction of the energy in b1 , b2 , b3 is
16
π
+0+
2π
16
9π
8
8
+ 2
π2
9π
80
8.88. . .
=
∼
9π 2
9.86. . .
> 88.8%
=
4
MATH 3150: PDE FOR ENGINEERS
4.
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.
Solution:
(a)
Z
cos x dx = sin x + c
(b)
d
x cos x = cos x − x sin x
dx
(c)
Z
x sin x dx = sin x − x cos x
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
5
3
2
1
–10
–5
0
5
x
10
–1
–2
–3
Figure 3. Sawtooth function
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 , a1 , a2 , b1 , b2 . Hint: π 2 < 10.
Solution:
(a) See figure 3.
(b) This f (x) is odd, so all am vanish.
Z
1 π
bm =
x sin mx dx
π −π
2
= (−1)m+1
m
(c)
Z π
2π 3
kf k2 =
f (x)2 dx =
3
−π
(d) Energy in a0 , . . . , b2 is
T 2
T a20 +
a + b21 + a22 + b22 = 0 + π (0 + 4 + 0 + 1) = 5π
2 1
5π
15
7.5
=
>
= 75%
3
2
2π /3
2π
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 4. Three even periodic functions
3
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 5. 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 5 on the page before
with the graphs of the even periodic functions f (x) in figure 4 on page 6.
Solution:
(1) = (c)
(2) = (a)
(3) = (b)
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).
Solution:
f (x) − g(x) = 0