Math 2280 Assignment 13
-
Dylan Zwick
Spring 2013
Section 9.1 1, 8, 11, 13, 21
-
Section 9.2 1, 9, 15, 17, 20
-
Section 9.3 1, 5, 8, 13, 20
-
1
Section 9.1 Periodic Functions and Trigonometric
Series
-
9.1.1
Sketch the graph of the function f defined for all t by the given
formula, and determine whether it is periodic. If so, find its smallest
period.
-
f(t)
_/5
?
=
sin3t.
/L
i-
yeYiJ
2
9.1.8
Sketch the graph of the function f defined for all t by the given
formula, and determine whether it is periodic. If so, find its smallest
period.
-
f(t)
r,
sinhirt.
-
3
9.1.11
The value of a period 2ir function f(t) in one full period is given
below. Sketch several periods of its graph and find its Fourier series.
-
f(t) =1,
(T
—71
I
c7(’/
j
c(11ytLcyL
f
i2)
Thc
hi k
11 4}1 i)
11 (
7i.
fcfici
coj11jv1
5< / I
<t <
(cc
$ (v1)
-17
2
±
)
0
C
T
h
‘f1
(co
cc(’1))
kvi
4
(73
r
-
iIi
+
I1. /
(
)
j
The value of a period 2ir function f(t) in one full period is given
below. Sketch several periods of its graph and find its Fourier series.
9.1.13
-
0
1
—<to
O<t<ir
17
51\
cox(n)d
co(iq)
5i
fU
rQ
(fr?)
C)
jt1
)
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1v)
0
(Co5fl)
—
0
C05(QJ)
(0
iie
7-
ñc’/
+
-
rc!lfM
i()+
5
The value of a period 2ir function f(t) in one full period is given
below. Sketch several periods of its graph and find its Fourier series.
9.1.21
-
,
2
f(t)t
—<t<
Zli
1
JT
-
ZTI
o
3
JiG
(
Li
c(i))
/
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(-ri
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f
e
1
ci
I
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Fc’ef
c
)
-
}
C
(z +
()
—
1
6
Section 9.2
gence
9.2.1
-
General Fourier Series and Conver
The values of a periodic function f(t) in one full period are given
below; at each discontinuity the value of f(t) is that given by the
average value condition. Sketch the graph of f and find its Fourier
series.
-
—2 —3<t<O
f(t)—{
—
2
O<t<3
—c
3
—t--,
:F()
ni be
10
e
,
ccfI’4e
A j C)
q i;
3
$1
I
RC
3
—
I
—7
‘-
odc
ri
5
Cl
4f41j
)
((1)
(
7
+
(f(
)
+5)1
(-I))
(i;)
[
++
a
/1
11 T
/1
-t
6/
9.2.9
The values of a periodic function f(t) in one full period are given
below; at each discontinuity the value of f(t) is that given by the
average value condition. Sketch the graph of f and find its Fourier
series.
-
f(t)
—1<t<’
=
3
V€
arQ
1)
So
0,
Ji
‘
I
15
tto
5
e Qv1
1
vl
1V7 C’
IC
c5 (n
L /
c;fr)d(
i(’i)
-
i
-:-—
ii’
CC5(Rc)
rrL
8
I ei,
9.2.15
-
(a)
-
Suppose that f is a function of period 27r with f(t)
o <t < 2n. Show that
=
f(t)
cosnt
42
+4
=
2 for
t
sinnt
—
and sketch the graph of f, indicating the value at each disconti
nuity.
(b)
-
Deduce the series summations
2
i
6
in
=
n=’
and
in+1
I
1)
2
12
n=1
from the Fourier series in part (a).
I’
c
(
j
L31
‘if
(
‘L7
d
/cii
3
()
ts
flccq)
1t
I’
I;i
(lf’d
5
(c
7t
jI
Il
-
tc
9
1
Hi
c( i
-
—
,
I
‘C
cL)
-d
>
o
1’
‘-
—
—L-
ii
c—i
—S
+
11
0
çjçJ
S
fl
aJ
9.2.17
-
(a)
-
Supose that f is a funciton of period 2 with f(t)
2. Show that
f(t)
=
1
—
2
=
t for 0 <t
<
sin nirt
and sketch the graph of f, indicating the value at each disconti
nuity.
(b)
-
Substitute an appropriate value of t to deduce Leibniz’s series
111
1—+—+.
-IT
4
Z_
Lj
5
a4
r
CJ)
L
17
3-
C)
+
5()
s
(Tht)
-
flT
iI
(i)
17
CC(
TL
11
6J()
fl-L
each
ci)
C
r
ci)
0
C
C
C
0
ci)
c-s
‘4;:
4-
-5
F’
I’
0
L
5’
Q)
H
c
C’4
9.2.20
Derive the Fourier series given below, and graph the period 2ir
function to which the series converges.
-
3t 2 —6irt+2ir 2
cosnt
12
—
n=1
Je
he
/j7
C
‘l U
)o
j(
(0 <t
<
2r)
)04
11
-ie?4)
/L
—
—
I
J
/ LI
c
3
IL1T
y
—-
p’
7
La
1- ((f))
I
(C)
t
1
i
13
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y,a
14
C)
Section 9.3 Fourier Sine and Cosine Series
-
For the given function f(t) defined on the given interval find the
Fourier cosine and sine Series of f and sketch the graphs of the two
extensions of f to which these two series converge.
9.3.1
-
O<t<Tr.
f(t)=1,
(c
€,
I
it
3
:
z
4
CY
(cuie
pi//iii
ii1)
ii,’e5
=
C
1
jtcIc
14
/
-
-
c
()t1T
+()
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.I
J
V
C
fo
‘1fi
k
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0)
1
(j)t!i
ii
4T paau noi JT ‘TE6 uijqoij
JOJ 111001 10JAI
9.3.5
For the given function f(t) defined on the given interval find the
Fourier cosine and sine series of f and sketch the graphs of the two
extensions of f to which these two series converge.
-
10 0<t<1
f(t)= 1 1<t<2
10 2<t<3
5-eyi e
C
(05
;
1
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If
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C(v,
‘17)
a
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(nc)
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3
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6
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1)
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A
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I
9.3.8
For the given function f(t) defined on the given interval find the
Fourier cosine and sine series of f and sketch the graphs of the two
extensions of f to which these two series converge.
-
f(t)
=
t
—
,
2
t
5ft€5
5
qQ D
f
F
1
( (p))
1
L)
-
-
(
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ev
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More room for Problem 9.3.13, if you need it.
21
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