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Solution manual for chen

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Solutions Manual for

Linear system Theory and Design
Third edition

Chi-Tsong

h,**aat'

Chen

C T Ur€+u

ECE

suN y et te{

gn-{

,t/z'T 0//r4, N Y t t
ffr - W


e4y6L
2, t kr)

.Ltrtza.y

(6)

g..1



e4

{ n* ,*Vtut< / = / _ lo
. "-, _i.f
cfu ,Vr.rrZV
/6c;t7 6 (0.ls|.(fu* xfu
(u.i)

2' Z
d4!

/-,r-/

e"td

4Q4
2.3

j-*rt2^?6

rtaa ./-Uza,

t<

4;

4+tl

2,7


{.r

= H qc''

H u c -@ ,e
J

a .a )=

E s= PnH4

:

c >eC

tbtaxz-

a4(

alo nal

WF
,ud du

,4-€1f

- du.n

a6f,:a


- 4(4,.7o)

{t-f
/-.t

/*

";."2-r/
;6-et ;
Tfa2 = Zta)

7(a) = T0) - z(p)
, Tfuq

au

trtn/ae . ,Jtt (2),

a< /tatc

til)

r H(-nrt,-ntc))
= - ft (nU, h Irl

J G&,
(| )

t7


t'-

*=*

ceu -&V't

ufu-r-t naa}m

u = n'V

e,tu't

flit'

Tr= mXo
,.iuoL

=
t4 (t/.Xo ) = * H( tr, rc) H(6

n7'/,t"
u, i't,)

Caz+rAa

H (ae,otre) " H (* r, ft ro) - h H(iu. i t,)
= L H (u,To) = o1H (0,2cc)

r Z(a)


( r) at". G )

= -r7 H(a,2t)

u,'p&^en

H {u, xe) = 2"ll ( I Y")

t*a,

,u
*

(u,ka)

?x-rl"tc rn xzT"zZn* +

fu

ttx.
,ro) * o. da r;?t4;/
(J)

t'l)

)4er L4(


H tn4, x ?c)'=n Hk,4)

t>q.

att /fu >tznit-o

2.€ ^f'1ua.pe"2?L

4

? tl'f

zfs.n

-rtu,t *%

PnHPn u

(PotHu)G)= (Hpau)G)

/*

Ut (t-t)

n {4 ,t Vtutr*

H ?n4,-rr7o)

fu /1q / HP,(tl


7lr4u

du *f?ut

o*4

Ht4o2--o.

=11(nu, hte)

(Pu/la)rct=o fu

(p p, u)ttt

nqt

(fsr,t/

(u,/c/, tL tt))z

*

a = #(h4r(-40),nrc-(-h

H Pn il c-*.
ro)
.4n1^1

B.ernu.az.


= 4{

11 h u, 27te) = H (e, tr) * // ( u, zo) r'.'

zth.

=
44

r,t)tx,t
m)

{aan

= nH

d r-n.o,
Thrq.

'te)
. 4 . i t -h
tl
il. (t-i)

,-{- *aAAi;fr,
*d

Arr,-,

.


ua
%

_.
-= q

7-/'*4 a4/-At4,f

z-,

4/Ll

H (tl, to ) : H ( u r 0, .r* r o) .= H (u, xo) r H (o. a)

/rrnzaa- , *t+a/

ta-y),g
(
{

.-

( tc-t
-t) )

4t

)z{


-d< A.-rZr

raaxor

o,

baru

,.!+ a

2'4

AIH

;./e.r/

ua,aar'

n'r/t( a)
q u(c-i)
q t [v)
urJ
tct

,t4 d

,4

u, = 4 tr. {i{^e.a


$

14 /*uc+.

8
A'tc)=g, tfu4* (() * (3)

f,trr ,
4
&tz ,

Merr/*rtto-_

4 (h).

Pci'

-//t"t (z)

cr,f ( z€) - z'r4) = 4'(o)

z

T/s< 4dZ;tr

4

tl (,f Q, sl 7o)


)(/") =
h

ry

,v<5.rr.q(/

ry€oo
= a H /a,Ac)
2+u'*-/
d, ,


2.8 /g

z-ttt-"U

t= (r-X/)/z

at4

2,ic ,4

rAtz

7=t-2.

= = (x-g)/2,


(t't*2r-

z{(t,t)
=r-t(#
__
)x

A

ry)

t ( :

A;t-og

2, t t 4'{

Ax

aa

da'

=o

c* t-2.

4


4z -
2.7 (;) t<
o.
sii) p < t :

*;

.fu;

a

tJ (t)
v

2i2
: :::J;;']
i,M=[:;:,':
f;i'i,';:,',

o)

=o
f b)

6at= :,';:)'
[::,':,::; ]
f:,'

I

q(r)

2 , 1 3{ t o = u ( t - t ) ,
t'/t
{ ( t 1=
(i,,)

4-t

t < t 1 z

{(ttr)

. . 1

{tt1'J-It=tt
_ 2x

t I

f-

(z't)

it(t)

:

t1'


I

FI

,;

Bzca<-

( * 1 - ), '

= 1- !-.-t)'-(t-t)z
2

t(+)

0

-/-c*<

= L
4f

{(d

=
J fll4 J r

{t')utt-c)clz
,ry


4= -.2 .

4

f ft,2)- f (l-2.

t-zibt?'4t

f k) tc ztu a*p*./"t

r
{(4 =J

Ax

a4p

*>c

T/tzi,

=hs(#.4)-t(F4)
dr?ct^al/

-/)

= e-r"

lr*l'{-'tirtrl


=^l;f(+-y,+,#)qe +)
Ax- 0
TAq lh'")

5'-,
(s+3s(s

_--__

T.ry-th 4z4l6re4-r

AX

dx + o

tr-,

/;rr'./
(.r-rtirst

lS irct:

fr., - tkt
-u6)

} Y

4*.444.,*7

e+*a)totts


d(r)=

ry)-try{t

O / X + A r a {

= /rr4'-

,4,24/)

F."

C4x4dr-r-

,Q{y'aq

=A=c.447

Q=o.f :

.J

t yG)

: I i

"

0


tu

{ltt

=/

/*

| ,t,

=:

f*

o:e

= o,t

Bze'ua-. {(t)=a
t(t)=o,f

t 3 t < 2
> ( t

< J

at

( | r {k))

h
/*

dlz++

osttsi<2

= a,7f
{ (c) = o.J' 7 t; o,5)
!tt1 : o.r(tra,7t)
"a,t?f

- z -

tl&r.

< /.

t/t)=o,t(fld+7G))

/rt+r)

)

4(t-t

= | +
{(i)


/k)

-L
| 1t,f) = a -- o.4z!

/ (z>

c f(t)

t>o

r(o=t fo

/*
h

z t t< J
Jst<+


F",LhVet/^.
Q: /

/ t e 1= s

tu

I
=o
= /


-n
O
- - v . !

<'

aSt
h

z3 t
h

J * t<4

{u) = o
= o,r

ztlz.n .oa&;e

e*t,:t

t*/,azr

/ a t< 2

ft


/tc;r)

Ji e H u*a-//

= | - f (*)

{(t;,)

9zc rtau<.

{t4'

./"8*

/
..

=^, lr4

Al..r- *r=4, .r.- i,

/ < r 1 2

=e , t ( / - e , 5 ' ) = 0 , 2 f

Fz+ tt *th Bz

Til;(e2-0,)-n,ltup,


7l4z -

/

o < t <

h

*

tta7 gr'^rl'+u f,'n,/z
4

= o, € ( r - f cD
/*

t nr!

t

tl =9"
z * t< 3

f,

-ft&

i
,r**= 4


a

'o.5(r

-p,zr) =o.,]7s

f-,

x., = t,

< t<4

-3

-t
7z
'

ia4,
I

'
- x r - - - 5x .
-?

f

4 ,q


--l

,
24:

/+

64rf d*f2!

x.=i

o
o

.14 9, to.

zyzahz
e

a

o
d
0 -{il,

tOrtg
t)e/cn-

t l
o)


I
I
l',/n,t,J

c-fre,<.[*r-,

a /cicttt?e/.

2-/6 oi=J:-{

a&;LeEh-

,Dt/trr<- Lt=g

/**zt

c,a xs ),
6T'

; 4 Z-rt/u'ta4t>
,h4 e"'&, + o. z4t+t

rni

y'a^uy,t+ 4/.

t1(zd g - ,g a^a.g

i


Tld

I
L

n/z

,t/eot&n's

( t' -'z' ) 4

ol ( o I
o
t o
i , l ' l r o y n ) / a , l to 4 { / n , 1 ,a l z , l o l ,
e

R/t

2.ts (4' 4v/,+

cctzJ au (zJ- zr)

ffi

- zcl

i,


Rc=t

hz?

'oti z' t/;

A,

=-3Tt-rox2+lr.l
R/z

z, *

=i,e-l^,

(t,

= (l,t,l.).fr-t,d,

(2)

*t = 4, trti,

*"= 0, .t , 6

( 2 r ' z z

= ,xlI i'

) t . =* +


ft2.-

* .

)ur='o

*' '= x\
|
4
| ;, =
dt* x,
rL

:V

7/w n a zat4ia*

slzZ. etr"&L

I i , = - P "- ! +

ed'dx,

- J -

7ho +64

ty.,v{'*-r.


,zoty'a,rtz-

.

r (/, rlr)

.*crrt

-./Ct

{, u
t/*

7lutt


nft;f

du {yt-/,t-t<

(, c44
tT*"f
'1 s'-

Arw/rr4*

d

(t) a-o.t


faA

r ea /;//

,

= (/,. /r.)
{rirsl -r(,.trf,r, t
(2,'f'l-4,
frt)=
itl
6s + *,2,

(S)

- a,)irtt

.

A

z

;,

1;tJ-5=_;
-<,

s
uG)


t

fu

ry

xJtTt

--(t .

{,=4

a

-;

7fut;.

^

't

=
-

)-t

+


fr,'t
ll { , = x t
i,
/t(il

!/Y)=

v)

-

.\

;
T

I

, - l
|

4,R,)

-

i

(,i, . r.J -t/,

4t


C'ri. = 4
4*7.

{
1\

/

/.^

-/t-*cfi-t*
/

/ /r*'

i r i l = i r r r i ,Lr r t = -

4 5 d u

(ArP,,s'*t)(Arfrs

tu A.*oP

E? ). r ) o&Vzasa/o c,+,
n,

/

rr)


,r G /, /1,

fu".Eo-, l*,
&

Alt

az.z,l
at:6+
tLc
un

- Rr*.t

=
fr"l=&
Ar?rs + |
irs)
6a.q

Ssl
= L,frt;

F'*,

t-


Tfu

.\

Ttst

8+ |
g^sSrl

& 'q P,(rt-a.)
rJ e rlet<

A,P,s+r
rl"*v

=

rk -zstGy ctiau

I

'f"i'

f.srt

2.2 0

u

T


.
t J

Rrr-

tl(t)

i

')

* ( s * r)

z^B_

:

o
t*< 4a.e.


z4 aA

u

'f/r'ttt

A,clx,=(q-{,).tt


&-

1'

,t4//<

Xt

(*r4

t clx

r 4t t+ . )*zrtz .A+;J
rl

"fe;
4il

{=to

c kz -4)'16

r-ttj€

= Zt={

|I
,L;J'


f/ee6dL

ayzdr.*'(

t Q

*r',
!

Ct el

G/.o/-',t

=-x,

Y3 + i^ = u

*, ' ll - -ol o - t
lx
lo | -tJ
t

",f.

,f^au1

, t''

til
tJ

. t ,

*,t,)

i,4rrr*r--f

i,a)=

z4r

l

r-{l-+1

7l-

2.18

di

2.i1 rr ri - I

- {zbs

^o't (ls+

r

lrrul


ao o t+nJ

t6t = 8@ frrt)

I sfts

tt

itsl

{ftl/

i G) > l, tts.t - ,1, i r, t

ns'ntc,=(
ffi
fot= i2 = 6 t i r t z

__-[
tzu trua-f +r*fo6
.; fua.'/*-2t
2

t 4r

7/rr4t

x,1 | a -,1c,1
I I
l[u,l

*'l'l

' o,
IL',,J

',) li 2l

' (tr ar) l, , 't,

R,I Iu, 7
Lut J


!'' z tto/ ,ry

t; cr,rr7'"tt, t-nrL^-

,/ttn ,a a

e t = o.

/'*'*-,-,4

{.U

'/c's
/"t ['"
a,d'
i R ' !i,t
J=l-Jl-l


.

4+,ztJ

.

.

tl

lt):

,I

fr

s e c e z are= i t , * b i ) ; t

/ts) - -

= Ctt i,' , J i ' ! t I

-

u,lsl

LYtlI

t


to { E.t,irl-*
,^-e4.pecf

? Lt s

C2. S

haaz/

s'-1@'1r ,
( e,-

i(r) :

ilTT\

,

9,

*)(,"t

t , '[ i

7aJ",41^-

t

f ut=f,r', i) rst - f,o tir' t


*

,4f

I z-ll , = | f I r | = l,

7/tt

g rrlla

4te,,

t/u. /.^Aot

{ 4e du >tamzafm
u
Az

J'l

t

7tz**

4.

Ip' ='tru - fr,(et,)/,-1r(/,g- !) /,
nrX'= Ar(trc'{)
0r//^1 x,= 0. zrtd,

t
'
t
,l rlrt)/ r

/^ I l^"2/ .
i"'i
{ =zt d
t 0
o 6,tr/t
0

/^,

a

tl:r
,^
ul r r l :

a\

ttst =
-

tl (s)

-- t,TJ2

= t


u 'rll,

a*al zrr,t4d t:/En/

0

I

t,= ,,/tt x,ll= , f z l
J:+ I ,. ,|

'r=xr-(fi*r,) =
=
t,
[:J ft t,

14

t r = u r / t t ' t , l -l -

tr,1

il'
l.,lf
o

,144 Ant

8?'d


t , ei

3'4

l,L /,'
- 5

1
.+.t/t

+, a4/

. ze', /*-

rct^+lryA

oJz

/o se, [n, (1,(,t,(,lrt)'ir$r]ti

tl

,,r-

It

o -1,/r, o
o


-- 3..7+

ll z,llo - 3

s

2,)i ut
h*

t.

t =g

ttt,i(r=JW

i, r't i

-/r44

frr.lrt

=[i]
: lL,:,]

1 t1c r l l r =2 r a f

=Li,r, i,,stI' lIt i':'
' t a J1

/*,*.


'otJ/o

*,

ut 'ts^e

\(#)"(1,t)i

erkl-A).

o,'i,")

du 6 td',?,] a;
[-s r.jJ'

lltpzrf

iJ*{.a-ddvW4

r/.J'fy

)

ti
: t'lt

(n,s**)(t-*)


cr$/r+cr+{
fu

=-lt

f
I t

f,rFr.rfr,6,t,t

t_

J

f ; ? i,

. uq Aaue

Sccttt4t

c2

lG)

i o t ,trr(t)
U'-

wlfh

"=fi il[i;J=[:J


t'^r(Rr/tt)s

=

X

olg

'Up.Ut-Un{e7tA,

ne

R' , r 0
,r!-rt
L
e,t

T'

J

Rrc L, S

I

Jf- u,= o, a\tl*
y'cr,
.


Cia4/I*

az /.t c
7^

t,

A/L(
u4(

nt/u'
?.

on

dttt-

ruc{a<

n)r1 , AA'a
2r,

[1

4

h=n.

* q r , t + t + c < - t.4

qla,

ah

=i,

/-.&t
a4

-

{


A u na4//T/a4
A A'=

- Ir,

A A''

7/u ahan

.r"^/

.

Oo ar, orctr/t&.

t abu


f,

4

.. /*

f.

i
fAL,Ltt6 o1z

A''- L1', ?"1

.

f.4^o/Azrr,,l,J, tx lltta
,
!.t o
I
QA-

'-f:l

az.(

. Thtt4 A-t= A'

1.7


o - c t ' e 4 t z0 ' + ' / 6 2
0d'io'eri
I
l'o,nlst o,f7t2 o,tryo
l
aseat)
lo,+fdz 44lO

Ftorrt (J.r/ )

.,

1

Ux tl''-

is

luU,{, (A,)=J-2=

f'(A,)= z

f

P( 4 t ) = g , e i f r

'(Ar)
= 3 - J= 0

((AtS= j


kt)= 4''3 = |

".r//'4

p*+.a.tu6.t

fu

rt t3
fi

( -+t,st, + 2dt)t*

ort*

n]

= o

J,;(, t J{"'

*

= / o d z ? 4 a ,- r l = o

rsPvrou
trltf l; n*trq*[iJ
q . = t '( + - t t , ) = ! ( * - j l ' #
R*yrryan( *"c([iJ

t? 7/yl4c41tp-/*z,+t^. *= II # #
Iiltilftl;
n*ye/boa
trl L:llil;uilra,.H
+

€x(t)

- i2):

itC, - 1

e,'- 4/tt

=a

# ]'

3i0 ,=[-:;"
I

J.7

t,.Itl
n au/efrh,.,
L IJ

ht',/&E
q


q O.

t4 t4r4flz
a

At .e^g.Atlir'

Szotttta-

-t
1
f , -t
J lt = z * f ll - . 1J

,[;

l J

l-t

2

{ , ( r

3'6'p

=

= 6 q'+


r

Ttuo & p./.hz

dt

t r -t//

L+ #

+ t)'4'-

2-h*+rr

tatt&a.f

t ) '
J,i/

| ; I

I r)

it e uLrzrr,, r/u,uLcf
?ac.

4 +lonnzd, * uTufrl
u, Pd(ra-t.e,
-t oJ'
Tt4td d, Sa^d

4 t-t .r
*a ^/a,4*-x.

. L c*

,iB
: tzc + 2 = o.
A

ft

('+ |

B = l i Y t l ' '(-' t - * ) ' * + a ' +

:,1='

art zV+a-Aat 4,u n+t ea.At

lL -, . 1
l

gtz*ae.dz

.lt

,4

*


4

ttr Ab
Ot l*att.t*t-Xt<,la,t

l, ia

"/

dil

,tx h oro.h,d

a'-'L)
et

/-U

4"o(t)

(2)

2 -h?rt t ,

4r4 du tae&(aal

='B

PC, - i6tt;t(


S4rr*.trr'.41,c2-

= 64t 1 z*<, - I

n,t,

K

a ( r ) = e l , t - f( s E - A ) = ( " - . ) J
=J{-/,sl-leJ'-2e,s

-1arr/ -rat*1

ts -, )
tg


Tiu

-i;-

A4 -

r s ^ c - A- t e A '

- 7 AJ

F * A = t


4 6 =[ u A b t t ' bt ) b ] l 1 , l
l a l

( ) t - A ) t = l: :l :r = o
Lo e -i l

Lol

j

CAL-A) f4i ts*

A L4 ' LA '[ 1l : 1 I

.t1l
A
D

l-,,I
L 7J

7-h4/,0
at azf.rzzs"{"ira

?4tlZq

t*d4edt*f

$ a ,,;/L


o

c

t
o

I o
Lc

,., c/A

LoJ

oo'oolrr=o,tr=lll
(,,rr-O
)trlt

.{

44a/A/4-Gfr-h

L;//, -t:apu/ 6

&a

Fn )-- z

I


o - 6
, l )|
r
dL/

hrllr(

l^oJ,

o o ,'o.l

[=lr
7/rq

'-+'

ct+t Ee

Tlaa iAou

t , = [ i lt , = l ' , 1

fr {t,Ab,4'[.4t1] -,
i d

2 ,

t a.,,4,**f44:

A


{ f . aT, A'6.At[ I

G,=li;
i)

.A r-2f lt ,o oo 1
r
\

'

A,=l!
il:,1

L o c 2 )

:

i;-4

8,

q sI
t d

l,
l o

1l


I

L oo , ) .

-t
oi
t, ; eI |

l o -+ -s |

l-z
A;(^)

er(tr-A,\

= )

(A:.r)IAtr>A

I L

-t

/-*

(Ar-s Z\ -- 44

J


(A+t)(A+tt

a

eeryat irr

-t

,?

J;t.r +)

r z

)

),

,I

I

o
d -t-gi

0

o

n , = [ ' : o ,?;ay&.

,1
.v@
t,

{

a4q(

Tlru at Jaula'tt /*"
/1c"4

A'A'*

. tile. c4tf"'^A

2^e+

e
a
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Ja
a^r
2

rrttzsf

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J'''la4*


|

+ r l
zo,tl

- L r - 2 o ) L o -ts -zo)
I o

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AL

4t44 ^.'4

(t*-^r)'=foo
i :r\l:

i ) () + , - J )

I

,*( t-t
-zs zi I

3 ,

-u)0l r-)
A+u) = ,rtf (t z - 4*) = I
lS
=A(A'-+rvr+aa) =)J

) -- o, o, o

:l:::l
A,tr,'=[1
I:I; l:,11"


e*

v=(o

. /
t o j

( r t o - tz ) J v =
o 4

"ru. L

(1o-t t1

O'/otz-

Li =(,t+4 z)-v =

:i,':.^:i
t'
^'
)

''flJ* 1,t
q-,t*,
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; D * r!'
:'::': ::i"
'
,
t
JI
= lT il,-lj )
arr[t't,-)'l(/,,tttA+) UL-4);tLtLlL)

i
[[ J

t

t,= (A+_) E) v =

r/uq ut /,t ,t

t:l

J./4

^"o, orr
1r-'
A
*(,t) =atr 0Z - A) =azr |


_
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=14+4tl
arlf l

o

iil

-, A o *l _;
- , l,

L,

;)

o

"i')

t,l;-Aj)

rliet ^f(

-:
L:

ol


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d , )

[e

ltl

Lo

ow ctu.6nc{

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a'tfe:tzaz.ttaaf

i

du t*afrh

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;

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4%


. f/t4

t%

rrr.d

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t7u

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-

aiaLtz*Zz,f

J, i6

O.ui+r wA./..af+

t
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[LLLL]\l
a
I

o

At Ai ) = O


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fiAu Acxv

,*7fu;

af = - o,rt|- - 4),lj - or ); - o(4
h,{

dr;

c-7*a6e;,

faru

&

Atiil=[ifJ=^,fi;
1
fiuo
3. ti

[^i i
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I

t


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lr'
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r

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ah e.i41"ta-/'tz'

n


i

uaf,**

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ovz4

^,'J

0
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-8-

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) T" , l f ' / , I
s 1
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o ^r

L o o
a*dt T>a-

I


o )

(l-,tZ)

olr )

, orr zl;

A

eryac-a/u-t

t/

rtaf

.

li ^i ai'i,fr't'r ^t, 1
^; *li,-,; ti-ti ti-ri ,tiI
^, l=
A'-A,,lr-A

A', til

/12e_

itt

t+nt'ta,a. d42C h4t

J,/7

H a4 t-rfiL+ec6"

du

l

14 = O, ^A l on

eAt,

li'Jt;;I hJ
(,1! tj

l

=
-A1) t't,-l' )('l''-/")(A'-'tt)
,!lrrt't'

'.',
rfi i: i I i.=[o,I
L0 -Lt o I

x

A,-)r


^^'^,"'J
=,T1r*t
t,-l) {A''l^111'-/)
at
[^'''u,u'

[*r)

\, = o =

n'
;^'

r+q *.*fr 2

)E


&n<{

J. 7au4 du {q,{,rt,

*4
4

icn/a*,

+ar;


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U - tr ) '

*
h*
at ,r*f*t,

J. it
A,(a) '=(A-f,,)J

=[2000^'{'l

\(^)

A r - ( A=
) (A_At)a
(A -A,)J

Y r { ' 11 =

(a-,ttSt = o

Arta) = ( A -A,)1

CL,*Q,

,r=fo

o t J'


,Lz" r/c.(

(t -,tr)rzr= s (A -.t i)'v

f?n{t

11-Az )
= (A-At)3(l-)r) = a,(A)

Aol,ur'*
Li=(a-)z)'u=

, ,,r:
| ;

i a

t

17('t)=

( rl-A,)'

Aotl) =

I,t-A,)+

- - ( , 1- A , )


f*t,t7

)

\ = ( 4 - , toz =
)f)r\

L o )

(rol t'= (lor *f,A +...*fr-,An',
)*

b, =tr = [1.l
l ; )
Thqa

= (f,

t F,A + - -. 'fr-, ^''' ) ,
=f,tt)-r

are +(e.r<

^'['t,
j :l

a=[^'{2 i,:';]
z*4'a'a

e+rtV,:Jr..-V a.'tAg- =i

"{ruv catVu&.

,

Aa= ^l^IX^'t'
jrz"1
[:
l , ; ^ l L e: i J
= [ ^t r'
| "
Lo

4'T
o

i^"t
I o
l o

&

t<.

g-,

u*t-

Jtr

1o


Aal

A

a

ut.

tar{a-,

1"e n

,

cn- A

n

v

/L =

= ef

O a*t7,

o<;tt*n/ttz

dat


'

A,. ,42,.- J , rAz.*
"tr+ {

I o , o .n ! ,

t,

el tzcLA'

4 a J,*-cltzrt
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4,h.6
aua,<;fi./
o, Az4-, u tuu- cE41..1r.L
@
,fu

'r.l

&

cv-zlu- 4

a*( ra,&uL.ap.
taoa

-,/-


A Q= o / 6 o - ,

aa,u*z

( r , + o ) , A 2f = o

e-,Ag=i

,16 /Al

, 4/ a laz -'t/ rct(r^a
fu
324{,rzazz aQ.L el * o
,t . T/'14 z.f- "tl /;,+4
aldlL
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h*4-c,zzo,7*n&2r
ft

/

l a ' T 7 tt t T r l
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I
A l
c

fluat are4&14Q
ag=af


f/u4

zta'ftp(aze

AT I
, A)

n i = [ ^ : r ' : r ::: w
:1
L d o ,JD o ;l
:

J,2o A = et

>tqal

t ,rJZ ^rr2f,

a ,u jarzr-Lrz

Ttua fut

f*42t*

n ,?'t e+

ie',x t/ A

4a444

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.

; 2 n a -

7hut


f / | o I
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l::
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J,2 /

'Lr

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ii))

=l n l' ;

.J4
qftal.a&Ud

t

o4


+(€)t-e')

/ t

e'E

a.+4 C

Fo , F,) n frA,

1ir')

3
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F6.=c

. 1= z

Ar=1 :

I = !3,-/]t

):J

:>

f-itl = /''( t) =2 16t =ili , t,r,


A"t

71ua

Bc

- - l

t )

1

e

[ o

€ - t
e t

n

pn/-&*

e't

d

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t'^c.

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:l
e,''i=lt
J:i'.]l' l l
i il[1,'
d o llt

o t)Lp
-4

'f

t


0

J

I

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I
q o
f,r t oll o I o
A r = l '2 0 ' l l o a t l f " 2 r , l ' '
ll,
I
t
r
o
r
r
o
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J
l
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o
o
Io
J
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)

g.

o
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t_l

lo

l. o r l

(;)
a4 ur*/*td

.v"
t"il

,o 1

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a,-2r --

I

r ' 2 2t , = [ : i ' : l

- 2,f e "


4 ( e ) € -e " ) 5td'

lc

e'-t *e'r-e'rtl
e A ,= [ o *

l "

-r, r eJt

Io
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t'+r.ftLa

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l r'

cl-lcLr-

. (/ | q
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ctztt

, ax

slrt =Pot-r,[;:

r I
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t*7 carl

fu.a

c
tc
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=r/ / lo:-1
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eJ"=,,)rr3Frrll,

/at = 4e)'

I i , o l ,,,r L , r ]
=_,"rf;
: ,,)* l: ,.i )
l e c
l e c


io, i, , ii.
et" = i" ., tlt, , *i,

8.=o,f(eit-2et'*et)

t , 3 , A . , 5 r 'A - ' = - / c i A r / 6 2

=,i, i

) t
(ut ' e

ni=

E*r,l

'1.= /€ z .,,j,= | -i2"= -tg r

eJ''

t l t A ) = 1 1 "* , 1 , t | + f r r t ^

A=(

ftr)''fi(r)

o

e


a/.r( ^*/

"t,l

,f(ett-

J
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l,

-

I
|

t

qr+),, t'

C

ti2.6;E

rdc

-)tt

t-Lct J


lo
,ild, f tct

8, = |

ln'tr)'fit)

f,' t

J tt rtL

l"(c),{?o,0r=r'h
- lc -

I
I


e

o"'

s,

1 lo,

)r
4 t ]t
z c f t 1I {t'

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t 6( tg

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a
c

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l

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lt

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r-

o

l a

o

t-rJ

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G

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0

o

ts -tS,j

0

I

-t luu

ts(s)

. ,, Y
(s)=ffi

3,26
t*<

{atS,

A k ) Z = ( R c s r - , + . . .n R r , - , ) ( r z _ n )

tL'


a'"4(,

=7(A)t@
4 e4t

I

(s-r/s-2) o

0

'n B) - t-l

nA

. r 1 3 , 4r , . . * / i r n A ' ' t
f U) =Fo

,t prr*t)z/o-t-.

,
|
- l
6(s) |
I

L

( t l l = , x c ? 4 , 4 + " ' + d n- , 4 o - '


ca.cy'at*LfU)fU)

( s - r ) ( s- z )

l ' r

tLx tL. lLv /cr,<

Lt-c't"oa-c A cannttra

, )_,

(s-E'At)=l a

':
l

,;{L

o

-t I S'l

, ,:rr)

2:., .l,f l t

J ],

l


J 2j

l"
L:'t?

=fr. If , c A*+ Ant
P, 4
,c

s'n I

=

/

+ 4t8''l

c "' + un 7

+ Rn', (-A)

Ror'r(tt'RoA)sn't+"'

"AtA

EtrA?

rua'htuo q
"t4/*L;r/

of r 7.dda dr +74/'/b4'

ttu

&1, ?-l%
uile

e.ya6et-a .
l; , .i =/, 2,,"', ,L ,, /v

obf

:

du erlztrUodrar
(/

,t A . D/.,)'lrt = ^:. 4 =,,2,"'.
E

'L' ='Z A;
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t =l
.

' f t r] = a A a
-,
/*

-',


.-,
' lJecautz

aigtc

i,

o: uf3zdatr

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/+<

4z,t

a m..t<(xt/*

O

.uo|

h U) = h (aA a-t) = fi, (a''01 )
:fi 6)= 2); "It

cltt

c =I6 e.-'
pfo,1t t


u La Iador,

5a^t/4.
"/a4r

/*rr,

B=A,C=a(A,€)s-,
J,ld

A:s) = (s -r )' ( s -:> .
Ea,
tuy fuA
2/ts; = (-r-t)(s-z)

a(A4)'La.

/"tz z.<<4 t&
;Qazri7

n r l t ( , - L A - t- ' " '

l/etfu"5

r ( - , n , ' 4 ' t O ' 'o

retleto
h

4 = /, 2, - -., ,L .


eri/

Thu n,+ /tar<


{e=
=

t /-tro, (, Ll*t -t... , 4a_t
A, J

c o e t { t L - A )n=r r I t

s-r- 2' ^/l = u , r * ) ) r 2

lt

/ k, GQI , (, h (Ai-') t,. ?,*-,riu)l

'=sLsLSt

= (St,*jXf

2

rr._j)

t,

tz.Tzutz/rrz c/ B
J
,J

=

+ & [ A 4 , , ( ,A t l - t* ' - ' , + 6 - , 4 ' ]

=

6''* t, A14 *.- + (A-,r)J
i h I A 14

e?*+vztuta a/ cfu {V"Vurrn,
t

=/h({Ro',;.
Tlrl ca7'4-4-ita

Nc .7t-ta uf4a/a

A. fu rrt./o'

,&al"M{

*/Zox

C

a -...<-trru ea4z04;-


O = r 4 R r - ,T 4 n Z '

du

*in-r4

e4.&4
{

fu d',4't

ats,etz+.

tlai^ta,

q

=Q.

+ Lnr

(r)-itS,

3 3 2

u4adrtil&)-

^,)


I

:-7

lP,

r/-te,ydq/4

l,

4

€t

lt

At
A,,

lo'

i ^,Pi;

t : lt / t l t^ , r I,l
L P,,J

et

(ti - A
)


[,,

A;QtP.

''

,A

n

t P|

J , 3| /.t

au4^

l

f , t ,l l f r , i ,
l o ; ,
L

[ -l'
l.

Lhr

I


*t'@,/

B

t s - r l
= t (sr2)n I
l, / st, )
: -/ - -l

I
tt?/'+ttY

W*of

d4

z7u
P,1
P>l

; 4';N*

"f*fu
aitft

4a'f

4


/

7tt"
-ht

l

qou,

+

T/cttc 6a

= )

)tt,

-\rr>r
=)

)n.

= -rt/,

- H,

-/2-

%


= 3

Dtr+Ht='J

cf4f4t-,

a,L. u,enau&*f.

fAa/tu,

.[:"'1,.8]
:ll::J

/4

, = . t S l -= c
r l
;
l
;
l
n
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]
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)
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J
J
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J
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T4r^

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(7frrr', fn'1

ia

I"f"

u=[.n'l

A

7/r. /fo.f*""u

= e . ( s f - A f ' pP

L J,
i=t

t/


A .tiltzneL

)z

=

.q)= j

4r=O.

:

-

L/r'41

',Jc

(z)

--r'*zrfr=(s*t)z

' J1l ^' ^, ,. Il ' *

I r, 1

-cri+7s

-1iltrn/,=3


=

fh,=Q

dtr(si-A) - o,

3 , 2 ,l

Aq =

Qtf

/*

tt)

''2ht-)itl>+3tl2

Tt{44 d

a.4

/4 )k?r1-{-/af./a*

httrSn,=J

7.t//a

J,z8


Tlu /Vapaatn-

tfu

Rn_, ,r"-6

4nr4rAh-l*

?xe"f.zr-

2*J'. z_j

-r.;4
tu da4L€eL
iAl,*
Cz= fi -Jf

h

furi b, = 3
'k,t -Dtt=-3

tuhrrh,

=J

C/


ftuqi!*. y',- *7 ',

, [!,,r,1 i 4

a,,(,.!
*::fftr;o.=:

IJJ

f;i ;r

[t

o

'&ee1 3i l = > - q : _ r

L:;

t,36

J,3+

NP"

V oei la e az,;,/,L/'"'f'

qP,

['r o t l
2 -2 |

1, H,
'2 f
Lz-ro l H
J
[..

"
g A - z )01-,t) - +

qH,

,/

f

"tt

6)t,t

tt, , *teZzd

I -t
L.L

c4u.p4/

"{
':AL-zrA

6 |

l

= ()-r)(^-u)

$

-J/

lW
{ffi
J, l 'f
tl'A

= A z

A at4

= ,n, 4tr (' rirt - B A )
6

az

= 4d

-i8)

( rZr'BA),

At = X, Ua
A mrrL eke(

r-rVd;u
(A) = n, ,*/L
h >n ,

n > lrt -. ,t/r"

A'A

/4 t xn

eu/

)'4

Ax

U^Lr-QiA

aU.-

/< du

*o,

taeca
- zJ-

VJ-/4a

{


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aat itL.E-'-

lrgA'
$

n,"

e'(tt+')''dr

7a44 e'(4tt')-'{

. e/ ,14,n , fu+

A-tq(, cal

"c/Ud.4

.rr4( fa'nIh'tr

a.d n-to)7o'/aa' , .24oAAAV

= +,7ct/
= r.7Q/6

dt"-

kxrl,


tvd.

; nof a 6,€.4rtp; . 8t2ru.-

,
"f Hr.

,/<.r ).

s,&f (Ap)

7/r.u.,,(Azr4ft or not *F

+ 6+

J.l A a egrnr&<,L.

= /2r Q' 4.rP

(NP) = atr N.cl4P

"nX*dtut*ft

z ( A - z z . l o + Z ) ( r l - z . e ?s 3
)
f,tlf t /*u y-,e44

= .e.l e


t:

4t^( (sZn

o/ /ti/1,,

L6u J

/{ll,

t(mtn)/z

-GA

n =rn, anC

8l

-?,)t

:1

cb{fl=

eZn

gn &r (sIm-,a g)

H:H, = tl d


(r.s?j.

o

['+
L - cB

- i )

S.',yat*" ya/ua 6 H,' JT ! , Ae, rur
,
.<-v u44,

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4,ro /6 a'4 ,.7d/6 .

'40

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: A )- - 7 A rr 65
=( ,l

4-t!

erir/,uFZuz"

sh'BA


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, U

du Hz t;,.

, h

"lr /*hzo ,tubel* fu

/oa1r&k W,
n*+ pih&
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f a,lc a, 4t arl
e z4 t
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I
lrqt q, 4 t Q t q s e t ll ' L| a s l

c*at*F(t

a'4

uoArq au

4 t 4 ' 4 rQ s
lo,o,

I
o'0, Q z d >

q!

"a

J,t4

*,7arl

k{

z J

44

Patl?et"

FGa"1rtLb
ji*,
heL P&14
atrxr-trtr!-r4

aeu't4't:, -: l' -, . u
;' I
t,

lo o


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it ; q

r)

;

6dA.dtta

r '.

"-cly'u'ct'
l,< '5,-


4?a
dI .t; r

4
i l-

i=lo

) = - t tJ

=At

P oJ'


I = 'r 'J :

't1

AU)=.tlf f A

,'

*t.
I r AJ = n

{rr

iq,t).= id, r
i?,,1

,l=,1',

en't=

A r J

e'./€ =,,Jo 7'il i

T-

gr=

-( t


+

t

EI

4(0=t
&t,

?zrt;tA=[n,,

ett

:l

/'t

"J

t: :lz,-[,,J,
L z
ftdt41

t

' L 2

t l
s


----

t+

A-^1

2 5 *2 [ ' : '

.fs

0 rs)-

i,rt=ir"tirrl'= dFl
,/trd"./

z

t/4r7
rq

4,J

: l[,,J

I

dtft) . f e'ta.n

"


u,.(t-z)dz

''
r - o' . u ' a ?- €-' - t t' *+tl\r=' t l - )f {et 'u ^* r
'-''{ft)=',f Ie
J
L

/-,a

l ' [ li

s'azsrz

'

0

'*ot

flaa w rb//€

,*',

"[:

t'i

C=o


/"<-.

lcr=re

-

u- ft-')a (+-.) ctz : uJ

,ti"'"-c)4*(t-z) - J2-,'-')tun
(t-z)n. J
I" o

3 J z
/,V

tla

{ r t s =r z t 7 J . A k - " { : l , t . a .
t
=
,r )r' r*-,r-.r, t+a/.c
f'{
0
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,/

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a

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41

,e6/4

* "'
, brcsl 4t '=Ialtlal' * 1a13

4f,a/,4

(tt + a ir;t> )

= r.2 g, 4t41 ( tt

t

aa4 Au q*afrh;

4r../&
€. t /

fk

b6//-d.

t,7

=r:'l31
ITf]f:l=


balr^p fa4 J.f;t

."4r-

-/. o, 4 )

i zat afryry
ry AlzfA&:orp

/, tt)

= g-,t4";'
fr,1
J''[-;J
"-',o

n*/;ra14

aa*z;ZJ

dz ry&
ait(

(t -/ z) v =
+
sr I

Al zfea&/

) =o '


o o' , 7 ,
['J

=o

L o o al

Jr 4'4 *{
- 2t _

a4 /Ltze/

ad4z.*az,f


*fwcf?r.

Tto" )6 {etdarr

^ f -t:
A s l D _l ,o_oto .I
L ei o a lI

.f, tS

u

f_*


j,J=
7:l[;,,
l::,::lir TJL::
[ ) i)
i t. t):

hI.- p.) f nr, -- |
-a,,
|
( t.l1
f hl+ + t,,f h,r = o
( l,zS t H.r2 - Drt
- Lhr.l - hr"=

JrAAdawfu+da,t
64l ^wffir:ft
L' h6r na6*z(

?:P
5.r2

4a//4.

fan,

naE,.r aea otttuzt*,

fk

d.Jt fu


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&{afa*da

ad

a/al

f

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b4d

4

4^6/}lr '

n

teulaa

aa i

fnrt,

P,t{.


A&t,


u

.'t.t r H4 =

| , -t J1',,,,
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o,,)*l
n,.r::]Li,,
ab P,,jlt* az7:
;

(z,zS ,
( l, z) '

-g,S Dtzhrr-

Dtr.ilrtr.-)ttrr=

'o,f ilrzl

f r,r'f

tl=1,
f


r

/ tA-ln

/*^+J

frtt- Htr=O

t

-l

4>

Ptrr= I

t

hrro t,5

*

,rrn

= 4,d,o,ao,f ), f,-^, ^1r-^,
I
l,'1,',t;-"i lr'-'lil

/.7f


= (, 4r ej (/,-41)0t-At)ur

I

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= 4,4r 43t l'-A,)Uj'A,77

ru,r,a'r r.7 f 7 o
clLP,'t/",

:,,lrl

L)'' 'l: A; )

.751 t.f-lr/>O

M u pe;rt,w

't?q,tot J
L^,-n,

a t g = e t 4 , o*,r [ ) ,

rrlt= -l

qt

[:,i,'':,:
(J-et


";A=i:J"'a'a''
f

ell -,i,aaua/'zt"
/ e it'4
Af ; o&-a* t(al
'rral
/r/+.1 tt ti pai/tlF*rs
ffa6t
At n'L/4)r.L
, tltz enpuft
o' Q> clr
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f
I
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Y'4t
|
0
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{
l
l
o
n , , n ,r l ( r , , n
f

( t, l)


I

- Ft'li--' - N

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f i = l _ : - -r
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?*a/

2-/,2'*'ai4d

naTri'te

Jst'

La,e.,

a/;/r&, hlrl% ry ry,?6'14r.

6,t4

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Lao AtJ

natytaZQ

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au

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f/*

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"z-^( prud atd
2a/ -6-a'u /,


tlcV 3441 Y'aZrza O,l. t

ffu n"A^i

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d,i J

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-,'
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e'he
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fu,4

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d'/"r.&o
f, : J i d p,o^fri€

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I

dt/t'"t4

%

il

f z
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o

r4*^ o, c+ pc )" (Ar,/t)
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t ul

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1
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= gr /

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cr-tz 2

ru"rr-tl

rlxa
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hd

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Secorraz-

S,to

!6,u, * , ,'nj *)
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fu

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J
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tt< tc/4

cVO,

- (e't- ,)J

"t")

I o

= s-Lt [-e'" - eco + zi

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P

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tuuz
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J

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lr?tt*L& o1t 2
/--

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to d

xra/ild.4.i

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, d i .
:b
'ea/a

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ot*4 pt>o, dt44 el(erfir4ftc*

"tl

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= -N


-N

(A'*./I')

"/ G+,kr)

(on- eto) < |

= i'

I,IXrc-zslrz

fa4

I

t

tt(

f

W

A,-c( zrt-0.4Y)'=

tL,e /z4

B =d"ft;e'd. , l*u--u.f eilt t e ( o


,*fu

,l-f 'a n*r H;h.€ o{4;";n
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P*'-?"/
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to(t1o,

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t,

fr,=t (i:, !1, t i r'J)=
[-i,;",']
O a/4

+o

e&/4,).Oxzz,

,tzc4 t7"., h. g-x"34

t >, C

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to


tu.{nre6*L

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zltt -tc(

-- e

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eT

tediZt"

IA;(A)| < F

N {t,.5

t'ttt* x =

ax zafl ulleft

d/..

c r - . dl J > o .

I^'(n
/ ) I"

o


r. Mt

=f)N

ftu-ArrulA

J

p eoi-Cue

%rra
,ahra

ue = rli ( A ) +/

'ti (A +7 r)

,4{

3lg ,

/v*Lc-7itu(

tz'4{

.f, 1 8

J, /?,


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de

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%

4,

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.
,

t

e4rxazit*.

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eal
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=e


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x

L1 1.a. .
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rt'- t))

h


-e
(t

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e-.'u"

(a

4<*-.:t

e-",

6'"tSv
U


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t4

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t"t',,e

t + @- k

d n*t

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of,rLl4 .,')t&
rynpfeiaea
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F =€

-t',

a

or*, Ftxl=

=0,i


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e

?

= e-'E^* ,

t

'%
Tb

at

fu

1,
f icrr"S

-;tt

z, or,,aEil
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%-f

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e-z

a' BtAoafaaa,

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zo

n{q /y-V-,r-*ryrh
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Ttu/*

p'/rtl=ett

A {A44d

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afaaZofiza

t to-

4Af

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4

dEJL.

y

ti $t4botll+o ctat+@

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ft
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d

ructt'tazV

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a,-2 e/'.h,

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o, 7Au4
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7/u f,ara/.ft^a.A,:tu

;

c V to. */

o1

d ^"Trh-q

n

;zv

i = 0,2

ro>,6t 4

u

B (z): lr /xe


t- tt'6)

L o , r .( u - r ,

f = i

f t t , z ) = c r x lp 6 z )
rqz-u?'
o1

.f
a qa) =xb)x-(bo)

i

e7r-a^fr.U

ef)r"ofu

x(o)= x(t)=
r_
t; L
[:,-' ,.-u ,)J
x(d=Lf t?JI
x(o)=
Lf ?
, l|


=
x-t(t)
,*, "-4*)i l
L".o

tr =fPrctAo+vptlP-'ft)
=frr"-t' - 2t e-"-f t' = o
"
fAou Al

'

Io,t (e-t"-t) t J

(i,7o), & fu44

%

- x(t) '+ x(u = e'+ z,{a)
*rct)= - €' t' r,(.r) = - e-1"=,1o)
*
\rr)= o,r. fe-t*- , J z4o.t+ t20)

X r U = [" - ' . -

nyq
4fi/4..

p lt) = e- tt


tt

a/! toq/

f"

to

zilt

hto)o

i,rfl:

-o

rh/'o

o I

i-t

x = L - u - t 'o ) r

-tz

=J

J l{k,Dlc/z
t)Co


.i23 ,

+ t>- az

8 rtrz) = e't' 4 tt, t; ,

?w,

, T/ura 44 1r7',',ZA^

aEU<.


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