biological control systems - time response analysis - s.mathankumar-vmkvec

IVM U/VMKVEC/ETECTRONIC SCIENCES/BME{EVEN SEM(EOURTH SE$I\'20L5-201 6 I CBCS-Rz0 L2- FULL TIM E]

1. what is the pupose of Laplace transform is used in control iystem?2. Define step signal.

3. List the steps involved in obtaining the transfer function.4. Name the properties of transfer functions.5. Define time response.

6. Define differential equations.7. Whatis the purpose of standard test signals?8. Define ramp signal.9. Whatis an impulse signal?10. State various time domain specifications.1.1. Define critical damping.12. what is meant by type number of the system? Give its significance.L3. Define damping ratio.14. Define Damping.15. \{hat is the difference between type and order of a system ?16. Detine resPonse. Mention the test signals used to determine time response.17. r//hatis transient and steady state response?18. Define-rise time and settling time.19. How the system is classified on the value of damping?20. Define steady state error.

", .

$igncr :

V II{ AYAKA MI S SIO}.IS UI{ IV ER S/TY

Vinayaka Mission/s Kirupananda Variy* Engineering College

SALEM - 636 3OS

Programme B.E. BIOMEDICAL ENGINEERINGlSeme

ster |

4

Subject Code & Name

L0 - 03 - ZAL6 l Oate rf B

3471240I. - BIOLOGICAL CONTROL SVSTUNAS

Mr.Mathankumar.SName of the Tutor

Assignment. Irdo.

Date of AnnouncementI

$hort Anrwer / Two (or) Three liner Fxpgcxnstlon

$k{rk ,,{*rr#m

l.lJ.

4.

lvMU/VMKVEC/ELECTRONTC SCTENCES/BME/eVerU SEM/FOURTH SEM/20L5-20L6 I CBCS-R2012- F U LL Tt M EI

L' Derive the response of first order system for unit step/ :r.i \ rr rur.*,:.rl$ e-

2. Obtain ttre response of unity feedback system whose open loop transfer,function isG(r) = a/s(s+s) and when the input is unit step.

Develop an exPression for unit step response of undamped second order system/ o,restla-,n.r.c A,

Wrile- the expression for time domain specifications of rise time and peak time with

.Derive the expression fgltime response of a second. order system subjected. to a unit

impulse input for (<L and (>1,, where ( is a damping ratio.

6. A unity feedback system has an open loop transfer function, G(s)=10/s(s+2). Find the risetime, percentage overshoo! peak time and settling time for a step inpui of iZ units.

7. Develop an expression for unit step response of underd.amped. second order system.

8. The unity feedback system is characterized by Eu:r open loop transfer f,unctionG(s)=Kis(s+10). Determine the gain K so that the system will have a damping ratio of0.5 for this value of K. Determine settling time, peak overshoot and time. at peakovershoot for a unit step lnput

9. Derive the expression for critically damped of a second order system subjected. to a unitstep input f.or fL, where ( is a damping ratio.

10. A impulse input is applied to a uniry feedback system with G(s)=12/s(s+T). Find theresponse of the system.

5.

Descriptive Type Questionr (Erroy Type)

$"M{ffi, {rw#*/4;trlture sf the Coordiln6tor

l"

?aRr *nA'

hths* -AL

trcnfuet

fi-L f*f* f, 4^"* k-*"ff* "LL .LLLPA;''

+ TPLa- orxff+* S *X "con:br'e'L ffi, Y r*\

W ,@.tL ,]e- ,*a'a;:<-ed- bx 4s{^/"^? i-*a1p*I

s)- eWfV'

Y1

w?

fu ;oL'*re-

+,fud.-

+

{

fup- # ,ob e\-

+ x;t € -MJ,3 *

+

fu-n4-P- t

ryco..tion "*{

ie L*rv bu Ya& or\

yL >o

+^* t aD

jA -Cg,UI-d*jk

g

,jJa

WWY1A.

Datu"b ry W:-

+w

+-I+ A =tr.frLQ-rt"

d"_.a6+Ld- \ tlct) .

irtr/et"v QA

r RC+) = A

o R-(+) = D

J* JA

ObLatrnr-*? &<L *vo-ru*fit:t:

**--t

?. Lfr,U "ffi1L

aJ- WnA flo"*'* ?

+ 'To Lqfl{-a-c-e- *-ran-Y'n e1*c.J;o"^ i

#4-L

Dbbo-in

,lr.

+To.fo .ICA2* ;'y^* va'de"Ur:* '

4, AIarne-

+--fhx-

i' ,fr,\r- T, '-'av tr:*-^ou ^-'l'a'a-ur'*

pt, ;a ,,,-nt ry @"ud-

; ; .-$ ' P(s)'

@ol-e-*TP>r,o,fub**Y,-rM-*1.fr12,..oul+,d.L.)^

+ *ffLa. J{,oot& 3 'Y*a- dzrwwi"o*v otet'L "fr42-

"-l^

aN 'd\' fr\R- otped-* $ #^a- -v\^-rtYu)4o'b3-t -alL-e--

T:"ruYT,l ::trbaA;'*,kv-rn: *+ 'frlQ- )a*o#::n 5 'H^t- *t^pedz q '#"4-

i

I

I

I,&i5,

utva,r,sy f**-U3t' r

-_.|- r,

.D* J*rwP-

+ *Il'lP-

t

Ittl

Mo

*t*rva '

*i,*o

II

I--,IIta

i

t!.

Iii

+-.!.++ffiri*,

r*.f Y yc^u'bt, .,,e4- ToL

+;r* s.I-8 eCt) c

+ T"b

L,&+ ,rti{'.Q--[

frvo*

t*t * "'o-vrA

e

JALo.*&c-e-

srry tr^"'ckbof"-ct^rc-t

rye^NoL

\/*a-L

t\lh^oL*"o

,,IA

+ Wt"Lt'a, *ffi fr\-?d .UfberYlA )

*The- bors-t"c, '-b{\DddA

L#n,a'

J- ,ffi-fu\'Q*

i-,b -aA L/*

.*-r"fo rtA.L '

+ Henc€-, fr'\-a-c,L}{-al,u :H't-gt^e-

/

r{.LA-e,ct '/o.'4

tt* *e;x

*c;-vLeJf

L/aJ{.-Q- -"C-o-f'1"!-t ,AvL

8, W uory (rt'lt

J' A 'q4 -oa'No'')L*f A *L*T r- -,-tr

Jir",e-x}1

Y.g>tn '^ak

Jk .,-'-i

;O}'C'LArr- lAa-nn L

/**aL @ v*Y

f"ft"\' va''Y\ jtt'tia'L

#1 C*) 5 At )

#t(+) = O 7 1*-- ba o

?-4ft=r) JV,\an iV "i ",cq',[lLcL .rrvuit J1a-rY fmcU%n "

Wr,,vo-f ,,f^ vatw W eift'"d-1

+ AW qvun *Y:a7rf**ur^"tr'r^-ts.,avcriLrlrlr- xe,..-

tt*r-d x"l"'orr L 'atr^stq*Lon ; 'rc-o'J'l.c-lJ-

-r""rutae, /rfln-qt ?

' - .,nd n ,.a- -rr"^ifr :"fry'Tcl-e,,,l ry*- ul

f' JA,$'\RJV rf.'\ ca' df J^ovt'^t

Y -^"rt"-ir,"- )\ :' a

yr>1a *r,ar'u!A -a'b 'A)/L Jj*r- "-ff

'4k -L =o

+ Ab b = o *t\'L *-i?, u,t'u& bersmr'r'

=; 3t &r *,Y\D@ hd

t./AYe- !

$Ir Ce.fl*f,@L ryt #v+ cl-it*-lc^al

ffie-*-tru-tde

-oL@

-,S&er3@ c

+ Ueuf"o.,u

-cLffi trvi d@' fr'\'L

CD- WuU,^,b

,#"\*-

dD ,fr.Lg* a-nyru+fe-oLI

.Pt-Q.-e,-CY1-O--rlt

W-il "*f.-.g&dcLcutfx-t

Wp"44, Jk -.nr-P*aYtV

C>+t-vz ilz Uffrycnst-ue r

+ -Tt,'P* W*

'uru'-rrubelL

,q J*r #rc\-rt&teH f^e-'kt-onT

* 'TktR-

*r"Le- tr+;;e,* -{.{=t:**a*

gsat*l ,

nn*$q>t AIvvr*l

D

*iA ?. lelr1,* \

-'rurr-nrLb*t S f**

*ee{if ,

-,&b-c-fc{gi -frae-

13" b&)",LD

:$ -il4p-fT

Md"o""Yf*fl *1"ffi

,*D

,e$4,4j" sffi?

db,1s4*Jrr.-lrLG*Nlov

Ev.A? uff"* J,L'.-et CJ4r- cte*tden% ;Lo ffi

%fr,^e- We*fi) )

.D,{cflu-**^6

iJ co.-tfu-id

16,

o* trern 1

vYttrrrLbep,,

dgsL€t"!fi.fue- .-,eotc>t-o-el .-![ fruL-

w

+ Tffr u^sf, '.oL truwe-q

e*. #rLp- W'r"-rJ;]c-o.jue4

?*vex**ru #^b

iglErln cl.*._c_qjbg-3 JJ €*

#'\P'

W'i

".o}3;,r-cr.^tcA -=t

tb,

l*1 '

Mer^f*&n fr,\L *e;.v W "-L"t*€d*

*fo^-{'e- iA fr'\R* -o*q*^}

.rl^eS{-eL l'" *f* -o'L o'& 1-ttne-U-tsn 5

+ :l:t J* </-l-"..r*wL \ LCt) ,

-tuQ*v 31 P*-"^n

,Osv& fufdal

Wl4-'* J3 *raruxie-n-b 'a.v1A' -c'va-,a" s<*'t'1ov't,.*u ?U

!!€tr

i *TstM .\,b*b- '3Le)>.D,on'"9L:i

, v e--t vdFv- v- I

D.a#^{-

*{, de.b.rytrtt-ix-e-

+ Tita *;""e, fr sU4a-

-?ArrrrJ-- .

* *TP1a- *6rrurn..-G}^? ,A-l-\,eA

rual* ;J-' '/Co"*tust

)@ok-tn-\

lg"

+ *Tk1-o- Me,tL ryon;&€- 'tAJtJd- *r-*1)e.nA.e* "h

L*6!- @ 'fr'\9-

/,r?otfu -fu 'e'YLDUn*o>t '

hber-.rJa "ubr:-'tg-;{=+- Th-A-7

f*rYl*

crve,-:-

,zyea.-o\ ,rort!- "*T* ; #/e- '><eafon'LR-

,,..^-,\ -ttrrrrL, t t / IApPX-D f*a'2L=aA *S-Od '

DeL;r4L JqJd.LD

-,LL"r-e- -OJ/Ld- .\R;pfj';nq ,*1r,1-g* i

0

+ I,* .,ri Ju^s rfr-''4- 'xaif-bte-a' 1-ea'- fr/s*

y*t* Jb -rr;-rr- o - tDoT' tr*' f"-- .tr-" 2'

:f,6 ,r: MttuL -aJ-' Le1 '

Selrbt.fuX

-ll-Y.U*

lq HoLr:

=S -I .rt ;A{

fr4t* JLt-nru- 'h,zVj;Ae.ol -f**- '#'&*

.^xlSA<* ,JL fu 'LD4o*nw-\rU)51-NQ- '*o

bo-nA.-L tgqo-nce- bo-t'LoL o*"u'-

+ Neat-r'{.o-tt Lf . c.s4) 6"/ ' "{'eL

c,

-IJt Lr-t d.p,,1otta6>L --a-\ tS

*tr* tL Jlu;ltLad- L-ovl- va/'P- vo.'b'*a- %

da.mnl.n4* 1

+L;^*"? '-s1"L 'fuLa' vo'l'u,- t ]1]f^t

t tu

ry*; r^ "*,'o-x*:yea- .int' ^t'*^r^::" =o)

@,te. r : U,a-cl-a^,tyecl *f"^, t"t**f""t t

Ca,tR- 2) U't"d"ox d'rr"'fe *tr* (o t oLa-'^'T'i/tt 'rtoib ' )

La*e-!,. e)1L+;cd\ drrr",fl ff* C"f*t^pr"O xotrb =D

La;e-a,: o ve-st -d'crlr fun' "fY* td\'c'r'Ln;.:i n *tc.Luz >r) '

l"

?ne-r *nB"

S ex-flve" fi'\b

&rrLt e? /ryonw .-{gtr

"tt*",ia ZW

$t*Jrr*f

Ytr+tu@ va*- *d-s4e"ct tr rprdax Y

& $"f'"d ff l

c cs)

+ Tne-

@

eJ^S,3e€L

"&-{+ffi

J"*p +..*-^y- ,U''^"rtb^ % Ye,

.,l.A

€

;39.Cs), =

ECs)

6r Cs)

l* HCs) & Cs)

H e:-Lg- t

& fs)

H CS)

I

-t;

a+i

TS

=, tu & rs)

,+Hfs) Grrs)

Ccs) /rr .t/rst+ /r*

l/ I/7s-l-s + I

R-CS) l+Ct)*,/rs T!+-,w

c cs)

Rf S)

C cs)

TS

E

+t

ers)Ts +t

E

? Fgry .t-t*ri* '\'L T vC*) =

- l/*Tst I

I

s (tsr,)

r+1 Rfs) = *J*,S

ccs) =

.sr(s + t$

ccs) =

+81 W ?

Yr

\Sclo,HalI

r .-.4-

ff".-B

s+'lrg.

Ccsl =

flb Y eL

sfs + ,/Y)

A*fiBl-

-J-r ; Ars++)+Bs

sfs+.p scs *)J-?= A(s++) + tssT

?rrf *S so,

.,,1 s A .* t ,,

TT

A= |

-r =lrt

Ir'/'T

tOt)

,r" P* A -+1 s

L = a(s**)+BsT

! = As+ L + Bs.TT

| = (n+a)s+ +TT

E%--eJ=tn? Co- lfrlote-"E *+ f-s.r4,rJ6o""f,4

ff

A=_l/T_=ltfr

O t- P +B PL'l'b A=t

B = i-'P

B= el

A q B Vo''t-ur-ea

r-Bl€-

S+lT

_L

s*t*LT

"rrYh-(.*-s' a- =

wn oSur-b+H

c cs)

Cfs) s

j=-A.S

*Ls

*atgt_ *-

S+a-

i "- ;+b_ry_.*r

_t ?s+i r-rJ

flcr-$Cu*? J:+-vQ91";Lf,*

; ' I ccsrj

-tyttr ce-

= Et I*-

i#.rr**&f-C -FLn'"l-

r r'i)\"?

d,lrbl* &

/i

/ti

eC+) =

CC+) =

LIi) rJ-rw-o

,NeLUlu

ryoy1,-.\.L .s-

{^"r* L"eftdr}L

tr-

;u*rLv

+ T]n-e-

LAr4 c>lr:ll

<,l4>l-eA. l^e{? J*rcttlo ..l>6 f*

ccs) =& CS)

Lcs) l+ H Cs) & fs)

l-le>a ,

&cs) =I

:rs

=tt-r &) g, cs)

t-rHCs)&Cs)

c c3), .=

Rcs) l}-El+ t/r*

,_B_

TS +1,/-!x

5L13rl

Ccs) -Rcs)

t+Cr)* tlrc

,1

_l ecs)-13 tl

FC*)*k+l

? troLlr -f* RCs) = 't 3/P

ECS)

-rfs*+)

.= tlrS+ rl*r

cc+) = yr e bl,

c tt)

= -l13+t

*,

gJ,L* M fie_ ry ,;A 'bt-",,t"v n c

tsg # ,t{q>er$ t+H(s)&Cs)

J-^ .\ec-,

*' 4ls1.srr)

l+ ,4 _ CI)SCs+6")

= 4lu1r/q

c cs)

eCs)

gc)Rcs)

s6s+a-) +4

@L

-'f

St+ 5s ++

C CS) - 4 RCs)€l

C-@'

CCs) = +(t/s)

(s+ r) (s++),

eCs) = ,,4sG+t)6++)

+r3*( #t "os.;uial-d"tacHsn

. A(s+r)CS+4)+-Bs(s+4)+Cs(S+D

4 __,._ E

s(s-u)Cs*4)

&I

s6stD6*+) sGn>.6*)

4=

4=

( as+ n) C's+4)

n&++ns+Rs

{fr+B+ c)*t+

+ Bf + 4B.s :i- cs2+ cs .

2-+ +fr +BS'+ 4 Bs +cs"arcs

s r5n ++B+c)+ + A '

cLo>trnA

D{ t.S'T

.,keptrnx

1=

Em ,nDM

s+fr+

.?Az.L..,Le4rbEgpc-*-;*? "& '€ "fi$-t.,"

o = Efr +trz+c

ttB +c -54

PLr-bA=l )

..Y O.

Eqy@,AfruX

4B+c s €6

53" *escma

fr+B+c

fZ+c =- fr

O=

?r-*f fr: I t

B+C = -, I

Cclrfu@-h@'l

+ +B+c =*-5

-\ B + c = c-t7 L-) ra t+)

,.!38i = -+

y@a

B = **/g@ -"@

Pr*f

@q

ccs)

@i,n4laB=

-lbh-

a

B+C 5 sl

*1/a *c s e I

c = hr, t 1/*

*= -3+ 4 s ,/g

b +/e I --rs+r)

'l+!!-

rFr6

?8ft

Iiig{.

I t".'I

I

i!

lTB

I*tu*II

*a.r,reox"L

.+u*11*{PL

.3m

-AC Ct) s tD 4 ea + z+b '

g\'r'',3\-r'

-.*&. t -4berc)= I--+-e +*s-a

+ 'T'v'E-

c*ion 5CCS) = Ldr-,*

a _-_--.-4

RCs) <si'e-qLt:ns+t*)nz

? i, ve-oL3e- ryc^e- r*rra;n +yrrn )

tl)

vL;-xLOlca;ntfeL 1q ; o

-11 ,. - . .41:,.*r

t1

c cg).

R Cs).*t

Szq- t^Lnl

; ,..,ltLV *te1 , YCs) = I 'h RCe) = -L* l^t V&r1. TIP7

Tnr: orv\n- ** E 'e.r>*v-o;n )l)

c cs) = :911 RCs)

S11- tr-r *

ccs) ry^'Sl+ue n'

Ccs) € ,wr\z- =

sf B\-*r,')

l, .Jn',t

-' t

A(f+L"l"Usc*{fi *C s-4/"#,

lLSrr' = A Lsa u)'f) +- Bs

l,=a- a

alcf o-nar-"oo 2

)*t a"fttr^? f '-,t't-f {ao-ot<-on

-L+aB

Sa+trl6z

?ulS:o )

lpnL

Lun2

A

= A [o+urn2-)-r BCo)

= A'wnL

: i.; aL =\(,

trgp -

Vuk 492: * (-L1nL tg*{) I =j u}61 '

t/-rn 2= A (-c,u /*n') + ts(jt,r)")

[rn 2* bCjru',)

B€ /u-rnLo

L) *)6

t

_lde a.J

t,tlr',+ B=*jt-..r6 eLE

eCS) € wnn s-A+s Ee+tr)nz---@

CCS) =.S _zLA+wn

s-Ta*c o

7 . v-,JJufia

,l:t;lQ9lA4 @ -*o.rtrfrCI\rrn )

vt*) CLt)

L

I

o:EYrfi

ulr ryorv\e- 5 ova,1 **s*rud'

,^*i"t..\

The"+ A..'ru.,L*+d- A rd*4R/- fufi*

F t*nL,-z a c.u)^B +r.JnLs + L€r ilJn{

tv

.,a v ut't a.c.tt.,yz*' %*e-xL

#r,e clLl;r-r.rv';M

-":RPL -h J*w '

.dannrv;W be-

: - \wo L L^,6 ffr

f"t "fl-n '-co':n'- ba UYL'U

l f*'cI*^oo

o

Yeh*a %ccs)RGS)

grrta @

f*c*j"cn i^6J\-a'

t4 t1"'L?

-f,,* $tPo-

,4 Sz

sr ? 82

Lek

.e;3 *T"rury

jfhe-{

jr"

'o Sa- = E t,n F t'Lln ,{-fi

35 = qLu6 + L$n

M

ctepr-rn-

ces) =

Sa.- ,,&ruJ\- S t2_

f* 2Et ins+t*1'",'

v e-Lqt^,p Jr *pJFl

Fsl *o*-d- S6 = aSc-

UJnL

RCs) G+s") (s"l-sb)-e

CCS) = t.t:nt R-Cs)

(s+sa) G+sb)

Ccs) =uJr,*

sCs +sa) G-*sb)

?B*i ? *fir-ra.cLt^on

CCS) ? tfrnL

,tt

otr

-J- B

S*se-

n

C

s+sb

e

B

7Ls

rrrn* = A[o+S,a-) {DtB)+Bd5) [o*oCb))+ cco) ( o+o-)

= A(o+sa-) [o+sb)

(ilnL = ASa-Sb

+ Bto) + ccD)

s[s+s*) (s{r-.sb)

tiln* E za (s+sa) Cs+u) *e[=) C** D*"O (""")

scs +s */s*sb) a Cs-hs { Cs-rs)

t...nl = A(s+sa-) Cs+b)-r ecs) cst sE r-c cs) c*+a;

3r;s= o,

fr

= tUnu

( quu6)- .*rn W X 1*n t t,n W )f,,

= gx)n-

qu*no- *f &\l

wnz

EW-#,:**n'

= -WfiL = I

uJn-

?*s= -*Sc-)

ce* = e ( u*;{s*1 Gso- +s a) + e (-so) Gs*+sa')

+ c G s*f3o.-+84*)t

fb}sa, Gsq-+s b)

g 5= u.Jnu

Gsq- *sa-{€h)

x= (x)6

,NnZ =

;tt-.,n{fl,

Gs*) eydffir

&.so-rffi,

B= -.:ry-o ., -f -

&q-l 8a-

F dWn

?urS=

u.Jn

-sb.2 = AGsb+s*) f"/.g+ e&sb) e./*r)*c$sr)

c-sb+s u)

wu'F c 1fsb)f gb+S*)

C s fuJr,2

CC

(-sq (-sb+ sa)

wnL

Gsb) CE/- rdln W + E/"- t^ro rfftr)

(' - +' tJ., n-

\-e

Gb) (zt*rn w')C = t^lr'rf _1 _

EbAUfiIfl,

e = !{osttsnW, ab

23

Sub A, E) c vo-larx ;n eay'aVb" O

ecs)= * *^, L* | * tr'jn I '1

P r/qr-, ** S+sq' AtF ab S-rs bI

)-rn@ ota1:onr:- *t -dr** otoryrein Lc+) iA

( ,r- (>, = a : sa, S+sq, arF ab s-rsb'* ilgo_,

W r; rl1:ont?- *t f;'** oto'Yrein LC+) i^trtr)6 l,

r'| l.o,j = r'f* C, tk-.,'4,a;Cc+)= I - hin -t ;t*t-* .trrn + C

sbb

<r t_

a Gr-' uo e 'l;E ' db

, * u1 *r

lcc+>: r- t^lo f--s-u fl\ I

I I'E.r L s* su]l

.rr+)il Ctt)t'-'ttI I g=r.orl__I i -/--Tt -

I

I

III

Irr.ol.,! + &l7otL!2-, t

+.

i;

W'S+j.b

*t 8";^

ryil,a- e-,vlxt:,.gio" 1,* 't;""- ''d-erwa'm

ft3"w ar,n *;'* ' "7*,./,e-*be

'*;*'L 't'eiwL Mba'btn- '^d-otivoleion '

W%o\rffib)

.-+,T

'iBa d-o*-l-nclrrvte-

aJUJv&- *

+ Tf4-a,. Med"

-Tln-a- ,*'eJ1a:.luQ-"*

n

a -!- ts a. r^ ^ a ."-&P ru'f YLX-J^) Jv*t-r^\ t furvtz

- cJ, uc.s u ct-cwPufa Ut-'c'

*e

W& ;t*Yb

nt

*gp+tvt-l

.:pf,4"q, -#,** ttv) "*+ :.* 'b "t;r"* Wa& P

&5 eLlach. d'"-rr''P-taa'/' {*- 8\

ZYN **"u ct-:-

*? "*f* -efi 'g,etord-

Js'}r-ol-!-'-L

,Y CgL

,rJ*rfu

J^*q L,Jnb

L[+) = l- g €-^ f *.lu +s) o

"dt b= .b.rr ) LL+)= (C+rt) =t.

*qL$p*rrr-g+o

7h-e- *e>t-wt-

uYhxn # =

=5 S.-[^ce ) D/

S*b (uroLLT + @) :a

i

T, 2T,,*3tr . _ L F

rnot ?G+sib€- ryyq ) Ld"L b1 1bflr

L5

c ctv ) = I - ;q*":t /,,&, ,Lrroltr +p)t^{ r-q-

I

-gtcrntxI

JOt--\-, /\r3-, [.^r"ttrr +e) = I - t

'ITJ"*Etr>6tve

-dL L= *v1Ll

€;vlfarutn-b e ,Lc;n;--v>L bQ- O

oEC

o,r9-,;\

€4 (tr:Jtr * o) :B

e

oo

o,?

tJt-\

euag*

tr:

€ Tf'- C

Tr* ptnd,

?Urt- v&n**;"'*cd-*?

r-q qh { r -G,o d-\-ts

I

**ff?ek

ry ";*oflet**

1* q"

rfurlp F -l ,l_

l+ eJxrt- ,t

g = +c.rr'fi?+

q,

tdrnw

*TF* bq-frr f,r-f,2 , s .,-t Jtf""a- ;^ Aec. I

t'}ntrf

Xtuo;1l-sl bL @t^Yt g&

t€Lv Lr.inry )

*o.nP= A"U-e-

Adiace.rn".b ^;Ae

G^ci-1a-ortton rudL Ecryw%o*o Arj* tt'h^e- Ct-)

9 vgh,L *o*', t

,gtM,

WZ bnfrt:.) dd4-c'- t

(Y o*.r-}.-,

/ I-- P-ryh { t-E

q/"rye) UU-;L * G

q

Ci__ € [-t)dl rY

So , ,b',^ s Tf-t,G.JQ-t w&f-f

L^: g}L

?eox 'T,irng:*

,-+ -f * '^-?fr.;,.e- -furm*

Vg-lg-Q-OVc-fu./t1-,rJob

t,p :-

d-e LdL

,J+

_*_dt

* $c,:nbee,

r', 6!,r -ELuo Y I+ b e- /,Fq-

C+p) o*L C C+)

&oi

o ,e.ivel..-*3{

G quJn b

CCa) = | * e, .d)d xb, {tr:ct-b -h e} '

r*,l !*E-

c c+)

LC+) =o (- Qu^)si" (u':ctb + 0 )

c^p", (uuolr + u) ur{

* -To

t* tl4'- ^ {*" P

IJ;(.JYLAq-

ur*Lu- W uto t +e-ry'c't

LC+) I * o14 = bP

rr Q t^:n U

g ( $t*n) sints 6,^-rn b

C,^r'Uttq) € g t

FV r-€.2

I

Ces(**u +d

Iq"* twnv

I€ e , Iq*, xl",(**Y +o)-cosC-"rtllvl-'

\ r_q"

?ul tr.t = Lun .lt-q'

+B)*g

{ryE ^fnC*"rb

+e) ?G*- CG,& (*a-t -jl|jinrr .eorratru- *n -^^f "bLe'4X'tL\+1 ,J 'f,q-i'l*" Y,

in $ rl.:n b

wr\ g

{flq*

' t' Je". Il{ptf-ruoLrA ''--!Lc

Ce&&= = 3 * q+{ d"PotensuA 4idr- I

t-un ;q.*nbt,-s,-

I

,,,

.,,

6

Lun e, Qwnb

"$-^[tr:oLLt e) - q

f Cps e /';^ ( u-rd-E. +e)L

*1

!- S,:o g C€6 t lalt a e)lJ

h co& ( trrcLb -{

r-{,_q*

'i- Qwn g:t{."} n e-

7\i r- \u

-q wn?taJpl g '

fi?* €'wov

wnEre{ L_1_

t= LP ) *t'dY

g )^ :?w^b Pt-rp g

#\ar

\) t-E-

* e,' cg-rruncb

(vooLb +s) @,e d- (-s4 (t^:olu + g1 -rL 1f.uL*^

fo

L*

1,o'-1L /;t (ur.-:"t-t)l

C *r* +g>,4

dt C+) ;o

L ^i, [*+ upil

le eqP-o"L 'i']" otj

=A

? 6wnbpgfr*D.

ctex_rn- /r-?,- (*"L €p) -=o

o)"Trrffi)+iF'

an\-L

9=

M

or--o,*}:.ce t X-e>to '-rt-s

, wd**P = TfI

+p = T/orL

JIq"

,xwv 1xs*a,Lb.[-e-'

.^s{c-CUt-

c>L_

&Y

= txJn

o+r

cC&) =

= - '..[f

r,un ,lT-n, " ?

F Q utnbtrun e-

{"r p srn C*au.+9-Jft-css (**st{

-ful+rr,i^ ,-co-n-ruoE bQ- qfld Jto 7e2LD

c&E

€t

q

-',i- (urror-t +*) - ,/fq'co* (*dL -t s) =o

{ft' = s[^ (rr.rclb +e)

+ -7h;.* e.Yon-s-n:lab.-L

.'o *n 1x&. 1ructb +t) - Ltrn ',[]" t*s(t*:*U to) =o

- Qwnve C+) € I I E *^ ^,},Cu{olb

+*) - uJ*-g" ces6.',ue'9

1-: L'

CPs (t^:clb + Q)

i;)

3t

ta-n g' = *ta-r, (wdb+e)

bq-n (krcl-b

wt>lt

ruct_t

tNdY

+e) 5 ba-n ( trr+ ts)

+C rs n-ff*&'

=., nTf +gr' g

F nTf

?u-b b=+DI

to=t,

t = nTI-

L,l..loL

nTr

cAffl

.StrL t*" + 7 r1'= I

wcL

+P= Tr-- *Tr.d..-^+

uuoL rr-rn ',[E"

l"1ax-Lr.a-1.,.-y.n Aves*hoob / f "nx' ove:t"sh@t "'

--:. L* .;a *+ep'vlea '-a'/s cfr42- *'o*i S moiai'n@,n

7+vv@Diry.r-.rn

e^ra.,tge- *o 'fr/& vo'Lua-7 ld* &,.P- rn-a-}

'-W-o'y\/ Va!^c- ja qn/,a*vou.,A' foe V.]'LL.' '

I

,h "pe-o*, overt sh-oob = g C+f) - c CaD 1 ro oCC".D

&Pr-cloo*L '-&<lt-Q-'$t W?

*Tt4p* $&rulg *Wf "*TnrL'a- *f

c SrwnbC C-t, E l- e.

rFq>I tr:eLa" +*)

db

P,*

M7

b = t P2

Lt*) =

* 8 fr) e-e) t CCAD E I '-

= l_ D s t [_

{ln=

4^in C*& bp + I

,T-Y6E-

vaG '7rv+,

(rr+*)

'- E uJal ao

e dn f,*et,o + e) i

,- 1*"+pC-C+f) = t -i- ?

.IIT=

*P =Et^bl

woL= rx)n .[Q= = - 7*l

?*.,Y*aFI*P; o

= -\)g)

,lE"

\J;q,

c (+p) F r r ;qT/ G*[,. a]

w',

? (erw

.|'r\-C-Dn&*:r.-t"ue-fou11 tr \-&tLffi dtTi .-,r"*iU't-

JA"le- fl*

G"

,r'*3, G € =J-E,

0*g*n 6;r =

U.6potorr;tau-S .1*&

ryc&r)s rr;qn/JEE- W,m-

cc+p)= r+;9*irr@'1

'-.9Y

E /r+ e EE'rrk toc?

I

'/, Mp > -r tTr' 'l- * -*---

X IOO

e\E=

iv) SeL+

req.dp-ry

ccL) €

Or

i"'..** -X-t-7u[NefLr*n T:JJYLQ,.

-.}..7

I

--i'--L-J..r lA-.'

\frr-a*6

JA

d-a

-Eunbl-e-

grg*,

frrre-

-ndeLo"rA/&j eyon-o.r^*r1oL ^:tryt-r'a' /_ed'/ucsst

cJ.11c.q.,1 ry /L;v"L^/wi"L*f '-cor?onr,nt

&,ur-ve- JbD ,L,t::rL) r 'Ltl-t^ lo f e>t cnc-q bo"o'l' ^

(

* NesLrns-tt/H ) 2- */ " .^pE& 5 ,l , '.btgga.-nce- ba*d*

ery{> Y* .c-r"* t +s) :*

+ The- =

16.- *,^t-,i* '*x7 *t A%-@

*bl [war +s)

\JE92

+ 'Tnlo- JLQ-8forL'lA- -$ *Btnn-d- 'e't-oLorr *f"-" '?/"a-L

?m Um e*f o& b*f'^xn*t'.reAL t)

.- 9 u:nt

I

e) AfurdsL c'nn-,oo",enL'n'

+ A,& (( reLt +s).

+ Z* #ilA

19.&dUo.,H"qrt1-\

* Hqxtcut -t'7,1e

35

?la,- ** Jdj,'M- 4-aJrc bP- $o* 'Du'k "1

"1p* e,-f. r.'-e-'.'f,,;- 'tDrufrw* J*b /' d'oLe>to-nce-

,YtSt-*t ,

) Fo.li g-'/ ' Jr>tesr-cl,,"tec av<:<*t ba-nA* a"b € -- bs

* Q

trrnLL 2 O'O9-'

rl r _e"1

, Fe.r, J-c-aAt V*U^i- ''$ q )

--Qrrlnhs = D.og,e

+ E',rd ***n Na'fi^'.qiL *iduLrn t *u

-Qrunts F L'n(u'os)

Iek '

/qwnr6 - /4

$r^rntE F + ,

tg= *9*n

) Foq .s{-o()}LJ_ "s'ceL!}t *^p'be-rn ' 'fr'\2- *-;''e- 4D'AA'to-rr-L

0

oo-rw be- t = -l-1*n

t3 ; 1T I t"= L.f . e.rutet)

' ) Fo" 6'/' *pte-rto-nce- exeLsq ba-' -d' o-'U t= bS )

- Q urntsg = O'O5.E-

E,

? tro+a "/ iu--c'A? Va"trl,^e- '^& q ,

r $*ubsg '= 0 c Ob- .

) U*( @"? Ncr-la,tslat 'to?o,Leti";"n , *L X*jr

I *^*6

{ q trentg

E t,,6rts

; ltu LD'oC

tgF

/3

3

3

9*"*c,*,c*.-ax trero )

,fr\L *fi-lv & C-cYr'Lg-gnb

* Fo"sr Ae,rcr-rL

La-vv ba

'T- =I

q*n

bS = 3T ( {o= 6- *l ' E-n L^s"{)

Dgl&ve- #'a-

-krcn& *x-da:+

ryflq<r

n $"- .b^v\*-

**Xae,be-J-

t @

ft c,-

--,fulA€-Ye.-ru&

tuIE'gJA

I

cL-

CI.*

.rsq-8UQ=,IYL

.l o I

.xo.h."b

+ Th-a-, A^e@rvd- tre;vu '*+ M t"*P

dt),rry Yat{^ons

uI-&

ccs) tUn2^

S i 9-& UnS+ tr)I

RCs) (\

3Y

o

> 1+.,&- yd4o.2lqlbj&s+%- e4fa*bD "[E s+ "q

u)nS *-t^3n2 = o ;

uJ+lDLe-A = el-rl-r

G--1 - dy? a+-,

=TI4,a- 'rt-ootb 'a{. {'J^iu

sr tfiz s *b +.'lF+^L2a*

A- = I

b = 9-Ecun

? - &)nL\J -

=Si e-32 = -_L€,LrlP +t

7Sr rss = -9*r.t *, G{St .9z=-gbn:turn{-q*

J^y?l't^a ,

Ed = r-rJa ,lE{, Lsi)

. t^,'L y r^,n tJI1*

€r)Ss= -qq.rntj*,eL.

Ca.t*. t :- g <t ,t-undat< Aar^f e-"L

* A.puootu 3 fY .rwffiil,rrrilb o

I

.,_ RCs) : I

Ccs) = RCs) t^r n*d+ l-eturns+t'L' f

CCg) * uru=

f+2Etr;nS* t'trnL

,Affin q"*nz,

CCs) = L^ir,*

** z\ur,S *9Ln" 4Lt-r n'- E'*nt

C Cs) s Lunt

(s*E*nf+ un2 (t-q')

C;Cs) E wnL

(s+ t*.0* wd2-

Md${* a;*d- divi'd"Q-- t'L} L /

CCs)= urnz.,r^lA

ecs) E

uu&r

lrqr,ltrc&

s+EulQi:*q

b tx s[-

@)h-,r-roL2 :

c cs) = 3f d

t,,ttd-

Cr+) = f'

CCA s wnz-

;

Cct) = qlltt^)d VT!"

Cs+€l *0=+ t*,,"L2

usd,

f c cs)] = lr"1 ;- l

t t"tCL

F q.'nb8-' /L,f')

e u:A\L (s +\*n)* +r-,){

t^reLb

kAu ii :-

+ -rl'\e-

9T:^

+T1r',-o*

oJA

r.ql^lnL og v ,.3;.n t,trcLb

Ovwt ol-a-r,.pe-"[ \> t

*t oota o'lu- -'oLeal '{.,\ Jne1o*Lv Q-

-ur.iu- -r: W ryRCs) = r

.rz,,,rud- i-orucr-,,,.-- 3 ry 1*,.tl"t"

C CS) t^,)nL

Rfs) uft "qu.r6rs+.^: r|

Sl s * So,--

Sz = "-gb ,

S I E g t*)p1 'j-r (A:n al q"- t

sz : - 9*n t,-n {S-r .

CC+) = La-:n - e L$ab oe- L /YJa t''LjoLt

t*E"

Sr ?sz = qt^lntwn..[g;-r]'t

CCS) = RCs) wr,-

?hCs-r-sr)[sts*) ,

W? fa"diat {a-a-c,tr^on ,

?-,CCS) = [^)n- =-- A , + B o,LCs{€,)Cs+s, s+ s t s+8 L

ccs) = DnL r ACS+S!-)+B(s+sr)

Cs+s,) fs+s2) Cs+sr ) (s+SO .

CCs) = ta)n2 = 4 Cs+sz) + B(s+S) .

?Luts g= -Slbn, -- A(-s, *S-)+B (-{r*g,)

= t,( 1,/" + ur^ \ry'-r - #^ + *",1F-,)

runL = A (e u.rp1 Wr)A: pnt

F .rE tl 6 A=tOnarff*,ay6$r

-sF"|

P*s=+Sg-

t,nz = A (* ?L+sd) + Bc--s2 +s,)

F A(Efr,E,.rn1@l ..- gy/r"rF(-urndpr) ,

4l

wnL: A (* rcwn \4-t)

13 = L, tl) n/

&fiw

B E _ t{JD

&,q_r

Su--bs tibr-u- fr/t-a;o-L Vq-Lr-Le^ -':n eq/-th;n O

ccs)F*fu +fu

s+sl g+8.e-.

CCs)= c!4 f t- - I Ie, {f- r f-*+s ' t* *l

A farcfnq -tr,veru,-e- Jo.flqce- Tb{ rrL

t' L"c*)] = wn ,

srlF-'

1-l e I - | ?

I s+-s 1 S+s21

Cc+)= Lr:n t -srb -.-Il&\F-, Lt e I

b,

U*t:k -*fu-lxe, .

"{LU)

,t9'' ( t"r-n-et-esr

el-antp€J-)

Llr

A*oek,Y ,l4gA c\tL 3* J*fl-@rh.la,oL frrrL sr;AL

ync*ton , erc*) = to f e (s+z) , F*nd frt^-e- '27r'v'

hfrrbl ?*d*n. ove*'ushne:r_, l*h "dt""' 4-' & ^"*"?

&cs) ccs)

Sotr",ton'.-

&Iv e;.T lDa.fq- i-

Gcs) E,D

Wad.-

SC+D,

SDa-bcu'-

tiae, "k*^.r."- ) r/.Ov ) f€-cLJ,e

.-,t1-nas

t9* Ltndh e@4-d- ^*n &-u- fl *? *"f.^k %

Fo,:+-'*tu,/-,o*:-

T}&e, -L-i-r"Le- tv rd

OV€-P{-,& h"OOb ..'*

?g*3c 't na{- *p

*\- -J- I, - bwi, urJlruoLL A

F qu/,[q" x roo=Q

Tr- otu cL

^/, nnr

d *rr

.3& *,ta-n4-s- tg(

t .:OL !

-bsts

Zr

4Tfw 5 */' -Tole:to*r1 cqlSa-n

ft 2-'/ ,"To\rytance- Bo-^d'

? aW "^cJ,es+-d- *rern-ry* 'd-t-rnct{'o/'] 'c-cs) s &rs)Prs) 1r qts)

ht cs)

&ivetw **tr--b, &Ca)r tDf s(s+z)

= to/ S5s+z)C CS)

ecs)

=, $*"c -"*'&\ 1eod"a'ci&'

= to/ s c$DsG+, +jD

s (st-c)I +.,,,to

SG+r)

Ccs) IO

ecs) Siz s +roto,

,l+Ajtubnnt)

cLs) tJ-, n 2-

R-cs) LSI+ tE v)nstru n>

@

?@rv 'w-Y'*t,tJ NL =

@" O -h @ r)re- trlo

wn =- /*

2I tl;pl

q=

o-

tl-

e)-25

2 tLt6 9.x3,lgc-

til&

fN&

= trr^ rEz

= 3,162

fs

tr s;,

qvi n

O.3tb K 2.169-

g E o.gqql

)+5

Find* *y = *fT*&

wcL

e= *odi;'*l ,.-;av t- el

9

g s -l:e.nl

II

*v

,-lt,8.^n

t n LBg

T:J-UJoL

s.3lt,

r o,Qol\{"--lot o.g tb )'Pt&oL . -

F Tr * I'a33

-._ [barr (z'85 t)

= ?'l+ -- '"' 3b3.

*v = 0.63T $ec-,

ryl,o FrhA ./r ny =

-Ey/tfqg / Xloo .

-O ,Ztb-ff / K looE

ts a *3 l& x 3'lty / O ' q 8l= g, I x loo

-o.gq9/o.q8l?- x loO.

rE [* Otte, x tgO

A, 3511 x too

O. B lbL

/,Mo s 35'}-1-l '|

I

?esx Trt,a.:*t- Tr = ?'t4- i2r w& a.

I = I Sf.*,c_

1*n o'31& x3'lb2

Fcv 2'f . €!'tJ.t-.19'L, ts = 4T '- +Cl) = +ry

tro-rr. 5./- ?>r><-o,et, ts = 3r = >(-t) = 3 Al2"

Q.ea utb : -

'/. +ed< oves-a,hnst '/- nop = 36'2' / 'I

PeAX +1r.rr;*- tp = l. o 4t -\ecI

,eJ*t,ti1-? *rrvt^L +s =- L*/, s 18ec.5^*f-€g4ec,

8. .U$l,bd Fe:e;-bacK

*p^e**s"ryf*etrsnD #"a*

Jtoi& 'h^*VU vtu

*oQvMOb .u^e -lJJvt-e-,

s^it

&?ve;z Sa-|11.,'*

&r Cs) E

SCs+t o)

?zqi*tecl .}cr.*c}-;*

3@r-rT W, Ao ,frq-oe

--%,*er* 3;

t/A{ E trrn 1@t

F 4-T- ( r$o, e_y.

r- 3r C -F.,aL E */.

= ,Jf,rucL .

w)@-

TI,ri-

,.OJL

de-rYf>t*<.t

*f rynv

ld

pec-le,

Feo-le

SoLrfion'--

e*-.c, ;rt*cbee{i4 a& %arc) =3 ,- r

S CS-r- t o)

c*5

Va-l,,tlg K- D fl

-a-b @ @ob flI

$eM;nfl -b;"^L , Y-ofu c-ov o'--L *T-curue-

o',b L{r1-oJ<, ,ove-eu,hor1c_ A6a1- a- .-'*r&t 4LaP -^?t* '

troM:-ave*+,shno E y, M? € nn*/'@ ,. roo

&f-lr'.rt*- tbp) -.*r-- ,I..*r"r4-€- Cfs)

l'+CS) E I

Ccs> = Grds) = &c.s)Lts) lrF+G)&ds) ,+e,f-)

tu Hcs) Gr ffi d *f **,elion

&rs) F _j!s Cs+-r")

C{$ .F t<ls c s+{ o) K/s C sTao)Rcs) t+ K/s G-uo) s(snQ +rc/s Lyro)

C Cs) K" oR(3) ^2s + rDs+K

* T^L Vo-l-u-o- .,g-t K L,Lo*rL be, ,a_vo-L,t.o,*eol b*,T v: u(-.\ Cl;-ra{iryvw'

n

'L ,*Yo..nffua- tr^^^c*L" a,^,rt:e,u

.s.tanAo,yd- j-;o-* *t Xeeon/-- .-e%o1.4-x- br?n'\ey

c_!S2= t-un2 _ ^ i

tcs) *?"9ur,.,s+uu n2- ---@ I

I

vLDt*P

#4R-

flc-tt^on

+q

0^3n 2 F- k- .

P;;-"'"1t___

e- ELtJn = tD

g= lD, ;;.

.-.-"-,€'*-{t!

s tD

& tfl.

A, = D.5 -h L"u6 =I

C,6- = ,A

* Th-e- Varl,ut*, t^B-l K € t o$T

ltrn E \ru= too

&, dE.

. !_?_

e, Io,E)

Fy-I

{I

€i ae?w.4

'/. PecLJc

'i, n<p

OV Q*lAl7.;.Ot .'-

iF gn/ffi2-F e, I X lo0

* 0.5 xTl- I;g/ 9-[- D '5' xlDDl

E 0.1b3 X loO

Qealc fu'-W& = u-rrn ,rEt

=to

= lD

= lo,,IlTt

= lo K o,Sbba

= o' B g6 ,ha.d-l *l€e-

I "6bo

= 3.t4Etbo

ItP= cGBb3 rsec

l * o.45

txt cl.,

Se-t*td*fr ry'| *{_frL€, :*

&s=47 t $a"u **l !

_l ._.- ,

D.5 + to

w)

Frt*l.-F

€,LD nl

5

e;*oIa

'/. FEg-*q-

tr?\

S a*f-L*"-u(j

S \d€""p{** lnfre: h

**1p- '-I-*t*v'ua- bD =f

T**,t*tfl, b6 2

/" rp F lb"3*/!

D" 3 &3 Aec

D -8 3ec LL.l '

o. b -8ec L 5 */ +

Qgre<-fr*t)

@,

q. Dex.rfue- ..i}1&

tu &e;cg;'n^d-

^ef-P.qfui,'

S*, uhSo A

? lrchnn- w R(s) = t/s,

R-(s ) .b)-nL

CS+c^rOt

trJnl - tr,nl"S

u-prLL-r^-Lrcr, tr d*'*P ef

-v.-,d,#u Y ffie-ebd- f,*t,

O.-

o.* "l;.-mjlb

ry Y E=I? @ & dxI

Wa

? Th-e- ata-nda-*d- $*,* *t ,c-l'*Sed' J"Yto

*ec,nA --Wda*, W 'rtz ,

ccs) 5+rE---r-r-_

wn'gCS) 5;1+ e-q uns +tnnL

Cvi*:tcr..;u17 q=r

qcs:- u

ets)tr)n'

2.g+2 t-trns -Ful-:nL

.9CF) ,= Lj.nL

Rrs) (s*r*ni)'

Si e-rups + trr*Z = Cs{r-u'D'

o?+ 2o.b Jr- 62 = C cr:rb)-

r cs)

CCS)

CS+urO2 SCs)+r-,,-irlo

lr?.

) B1 ^fffi fW fo"d;"n ',

C Cs) .= _*^oSCs+w6o

FAr :.S

+c o,(s**r,)o S+t;r rf

Tb #AA=

,-

sx ccs) l(s=o.

;f,x wn? )'fi (s+u)

-)e-

[ . EO

€ Yl=. \C 8+wp1)z \ I =O

; tLrnl =lt J nt

To Y& E .--

B s (s+ ur6** c C-s) {

l*= -(r)6 ,

a Ce "/D"^ t,).,o

s(s"/O"S s -ulf.1 1

tL, ntls *s - t-L)n .

* -Yn,{.,wvl

/

# t^)n-.

To #ct--C = .L

B = -* tJ.)6

ots

= .L f Ldnl)E L - jl*E-t$6

= - oL [ (u* {,1'.^ k)",L 1 1ots L %*/b=llu=-r!^-

-- t-t)1..L \

S2- \S = --rr)n

= - y"i = - t*t.)n 2

+ S,rb-slil':'r,n*, A2

Ls

B) L

tun?

------

-sV o-r\rt t^g-s uJn-

*l#

E+tsn

e-l/-o*/r., O,

CS*rD-ccs)s

+EX *^t".'? itve:o+'e- ;a-p-l-o'c-e- ffiL- 1..",J =r'f*] wni I#{ -L: Itt

c-c+) = t- ,,n+;trrftbf rr

^*-Jl,r-=bzlI a .

-l' cs tct>L JL

,

cc+) = [-r,.,nba*t"t: Ctrunb

cc+) = I- e-wnL fr + Lunq

xnyub *rarlAL=

AWJr*iW

ilrr6, ffi..3ote;.}i-on..*

ja 4fPt *"L ;15 o'- wlLbd f*"""GrCs) = tv/e[s+r) . f ind- fr'12- e@A'Pn+'P- $

C CS) s acs)R"cs) l+Fl cs) G4cs)

n- l2_

s(s+Y)

t+ t2- Cr)S Cs+r)

= r2-/s6s /t>s,(s{_r)+ tt/s{l

€ lL

S 6.s+-t)+ 19 ,

ccs) E" tLRfs) si -1 s + te-

CCs) E 12 'Rfs)2_

S +T s+l2-

*r1x-*.

gCq) = l2 '.,-

Rfs5 (s ++)(s+3)

CCs) = le- R.CS)

G+O(St,

CCs) r , lLCs+a) Cs+$

+81 ffi f& ffin't2- =- A ,F B,t_

4l. r---,+ t -..-r,

Cs++) fs+3) s+4 s1=3

51

-1$

t2 4G+t + BGs+4)

(s+4)rft+a) 6S*s)

l2 s A Cs{-})+ rZ fs++)

Q*b =S= B

re- s A?,il+{>+BC-B++)

tz = B€rr)

?ur-,tS=_+,

t2 = A (*++s)+BC*/*f0

12 = A (* t)

A = -12

C(s) s L- l9- + 12*

S++ S+g

* B"d -rertl'r,n Jn reor-t+- -fl*"*- -fro-ruf m

*+v -3b,CCt) = r,E l> g + tg- s