assignment 13 math 2280 dylan zwick spring 2013math 2280-assignment 13 dylan zwick spring 2013...
TRANSCRIPT
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 TrigonometricSeries
9.1.1 - Sketch the graph of the function f defined for all t by the givenformula, and determine whether it is periodic. If so, find its smallestperiod.
_/5 /L
f(t) = sin3t.
? i- yeYiJ
2
9.1.8 - Sketch the graph of the function f defined for all t by the givenformula, and determine whether it is periodic. If so, find its smallestperiod.
f(t) sinhirt.
r , -
3
9.1.11 - The value of a period 2ir function f(t) in one full period is givenbelow. Sketch several periods of its graph and find its Fourier series.
f(t) =1, —71 <t < 7i.
(T
5< / I 11 ( 11I c(11ytLcyL
i2)f
2
Thc c7(’/
4}1 i)j
hi k
± T
4
- +
kvi
coj11jv1 fcfici
(cc $ (v1)
-17
h
0C)
‘f1
(co cc(’1))
(73
iIir ( ) j
I1. /
9.1.13 - The value of a period 2ir function f(t) in one full period is givenbelow. Sketch several periods of its graph and find its Fourier series.
—<toO<t<ir
7-+ i()+
01
17
51\
cox(n)d
jt1
)
— (Co5fl) C05(QJ)
iie ñc’/ rc!lfM
rQf
co(iq) 5i (fr?)
U C)
I 1v) -
___
0 0
(0
5
9.1.21 - The value of a period 2ir function f(t) in one full period is givenbelow. Sketch several periods of its graph and find its Fourier series.
f(t)t2, —<t<
Zli JT1
o
__ _
( Li c(i))
/
I (I) )c( f C
-
ZTI
3
_L (-ri
,1
JiGI’
) e Fc’ef ci1eI
) -(z + () —
c
}
1
6
Section 9.2 - General Fourier Series and Convergence
9.2.1 - The values of a periodic function f(t) in one full period are givenbelow; at each discontinuity the value of f(t) is that given by theaverage value condition. Sketch the graph of f and find its Fourierseries.
10
e q i; A j C)
5I
$1 I RC ) (- (f()3
(
(1)+
+5)1 (i;) ++
f(t)—{—2 —3<t<O
— 2 O<t<3
—c
3—t--,
:F()ni be ,
Cl ‘-
3
—7
ccfI’4e 4f41j
odc
ri—
___
(-I)) 11 T[ /1 a 6/
/1 -t
7
9.2.9 - The values of a periodic function f(t) in one full period are givenbelow; at each discontinuity the value of f(t) is that given by theaverage value condition. Sketch the graph of f and find its Fourierseries.
f(t) = —1<t<’
3
V € So 1) Ji ‘ I 15arQ 0,
tto
vl
1V7 C’1 c5 (n
5e Qv1
IC c;fr)d(
I ei,
L / i(’i)
_______
-
-:-— CC5(Rc)
ii’ rrL
i
8
9.2.15 -
(a) - Suppose that f is a function of period 27r with f(t) = t2 foro <t < 2n. Show that
f(t)= 42
+4cosnt
—
sinnt
and sketch the graph of f, indicating the value at each discontinuity.
(b) - Deduce the series summations
i 2
in= 6
n=’
and
I in+1 21)
12n=1
from the Fourier series in part (a).
‘if L31 d
/cii ()
( cI’ j
‘L7
3
(
I;i (c5(lf’d7t
1t
jI -
ts-
—
, I‘C
flccq)I’
Il
tc c( i
9
Hi1
aJ
0çjçJ
—
1’
-dcL)‘-
o>—
L-
ii
S
—S+
c—i11
fl
9.2.17 -
(a) - Supose that f is a funciton of period 2 with f(t) = t for 0 <t <
2. Show that
f(t) = 1— 2 sin nirt
and sketch the graph of f, indicating the value at each discontinuity.
(b) - Substitute an appropriate value of t to deduce Leibniz’s series
111
1—+—+.
-IT
4
+ s (Tht)
flT
3-L C)
Z_ Lj 5
CJ) CC(r
a4
17
each
5()- 6J()
fl-L
iI (i)
______
17 T L
11
ci)Crci)0CCCci)0
5’
0
‘4-
;:4-
-5F’
c-s
I’
cC’4
L
Q)
H
9.2.20 - Derive the Fourier series given below, and graph the period 2irfunction to which the series converges.
Je he
‘l U C
)oj(
_______
— I / c 3 y— J LI IL1T —-
2 2cosnt 3t —6irt+2ir— 12
n=1
(0 <t < 2r)
/j7 )04
11
-ie?4) I/L
p’ 14y,a1_,-
C)
7La
1- ((f)) (C)
i1t
13
Section 9.3 - Fourier Sine and Cosine Series
9.3.1 - For the given function f(t) defined on the given interval find theFourier cosine and sine Series of f and sketch the graphs of the twoextensions of f to which these two series converge.
€, I 3
it:
(cuie ii,’e5
O<t<Tr.
C jtcIc
f(t)=1,
(c
z
C Y4 pi//iii ii1)
=1
14
/c --
()t1T+()(.IJ
VC
fo
‘1fiI(j)t!i
kILI)0)
)-r(1ii
4TpaaunoiJT‘TE6uijqoijJOJ11100110JAI
10f(t)= 1
10
1
JL()
1 () -
i 6 1)-
9.3.5 - For the given function f(t) defined on the given interval find theFourier cosine and sine series of f and sketch the graphs of the twoextensions of f to which these two series converge.
(05 C
0<t<11<t<22<t<3
5-eyi e ;
CG5(nc)
‘LIf 3
C(v,
‘17)
f)1
- a / Ii-- (T
o
—
AlT
€)
A (k
4:; 6/c+c
— Z’.j3 (c5(I
3(Y)
S a 2
16 L-5 -I I
flrN
cL)
S
±
N
I.
)
I
9.3.8 - For the given function f(t) defined on the given interval find theFourier cosine and sine series of f and sketch the graphs of the twoextensions of f to which these two series converge.
f(t) = t — t2,
5ft€5
qQ D 5 -
L)
-
1
f ( -(°5 - - e v
I
F 1 ((p)) - ± / (ci()
I
18
1’:,
(D 0
,
+0
•I I -
j
*
Q
rJ
C
‘I
‘S
N(.
—.“
_::D2
-k-s
‘-
_:D
Q
—.-
r
SI
—T
h
-1
,-
-j
3
0 I.—
’
‘rj
C -S
.
CD CD -S
.
CD (ID (ID 0 -S
.
0
)
More room for Problem 9.3.13, if you need it.
21
ii
(21j
1;
7()fo
__
c7G
JfE;fr)
1h-I-/±
(o7
/117/
,JLIt
U12
T=u,OL
—
_______
JL’(T_1LI+\i-OC
T=u,06=
JL J7
suomurnsai.juqo04
UUT7‘JL>>JL—+US0
IT_iIT_iiJ iL\LI
UT==j-0VE6
‘I
r
-r
—
N
—N
r-:
:o
ON
C CD 0 0 C 0 0 CD ‘D i’J 0 CD CD
I’3
+
r -I (I
-j
