9
vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord pcltaro4 DDllli z)fi e-l lx e esrtc nrluad lenuelu PtlqcPN ultPlPc PllqcPN orlad INTNISUSNII IS IIuYIU)USJ .If IiYf N(S'INUIISINIIAI

Matematica - Clasa 12 M2 - Manual - Petre Nachila, Catalin ... - Clasa 12 M2... · vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord

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Page 1: Matematica - Clasa 12 M2 - Manual - Petre Nachila, Catalin ... - Clasa 12 M2... · vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord

vruo,*B

oluuolsaJord egrgcgguc oluolpc13o7ouqa7 oanrl

IIJnleu e1e eiurrlS :erezqercedg

IEar luordpcltaro4 DDllli

z)fie-l lx e esrtc nrluad lenuelu

PtlqcPN ultPlPcPllqcPN orlad

INTNISUSNII IS IIuYIU)USJ .If IiYf N(S'INUIISINIIAI

Page 2: Matematica - Clasa 12 M2 - Manual - Petre Nachila, Catalin ... - Clasa 12 M2... · vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord

t97,

gtl """"""' Ilicury reun ele^rtrlultd.z.Ll9tI """"""' p^rlrrurJd ep eeun4ou eI cnpuoc or€c euolqord'I.AI

e^pltulrd .AI InJorldsf,Yf ["rYr{[IYW YZnYNY gO gJ,NfiI^rf,Tg

ttl """"""' """"""':""" """"""""""""""''""' eJ€nIeAe ep eqoJd

I€I """"""' ':"""""""" ruud Jpunu d'oZu! {uercgeoc nc erueorrrlod 'tI'mLZI """"""' elrqrlcnperr erusorrrlod 'tl.ilIET,l """"""' 'eureourlod ep eleq .A.ilIZII """"""""" €e4u1 {uercgeoc nc oru?outlo4 .ryuuoriur

}iuelcgeoc nc erueourlod .II.mtII """"""" "' rTeer riuercgeoc nc eru€ouriod'gl'ilIZ0I """"""' """" (etetn my eprieler) 1ulcpppr rS riuercgeoc e4ug nieleg '6'111

Z0I """"""' rorredns per8 sp ecrrqe3le rdence Jorm ?eJBAIozoU 'g'III96 """""""' """""";"""" lnoz?B rnl €rueJoeJ :oyeureourlod epurcpppg 'Z'JII

26 """""""' """" JolerueourTod €el"lurqrzrlr(I '9'IIIgg """""""' "'reruoH rnl €uleqcs .S.JII

?8 """""""' rolau€ouqod eeni.lpdrul 'y'111

08 """""""' " pl€ruourlod ericung 'ruourlod mun eereole1'E'III9L """""""' """ uom1od rrum Inp€JC 'urouqod rnun e gcrrqeSle euuod 'Z'ilI2L """""""' riceldruoc liuercgeoc nc rolerueoullod ueunfptr41 '1'111

apulntuo, droo un-rlug $ualogaoc nJ oruuoullod ap e1au1 .111 p1o11du3

0, """""""' eJenI€Ae ep oqord

99 """""""' " pmd;o3 'y'11

V9 """""""' eleq uI 1ncl€c op p8eS'g'11I9 """""""' eleul ep erusrJrol { 'ocr4€ru op eleq 'z'[l,9 """""""' """'eleul 'I'il

,s :::l:":T:.1jT:f;8t """""""' rmdru3 op eusrJroruozr rS erusgrontr'ZI'I

9€ """""""' """"" """"""'ecrrl€ru ep undrug .91.1

St """""""' .........'dru3 un-4ul Inclec op gn8eg .6.1

62 """""""' "' undrug 'g'1

92 """""""' "'zrouotrAl '/'I82 """""""' "' Iquzl4eluls lueurolg '9'I07, """""""' ............ru1neu

lueulelg .S.I

tI """""""' ....'....e181l^IlemruoJ .e1€I^rl€rcosv.t.J

II...'..'....... . .................pllqe1s eu€d .r.Ig """"""""' 1l oFporu rrnlseJ es os3l3 .z.I

t """".'"""' '..""""'.guls1ur e{rzoduroc ep e8el .1.1

r-rndnrg .1 p1o11du3

YU g g3'rY -q tI -rtI N.qL{ sn'd

l

'rolgJserJ lcr.rts rs i aA.r : \J t_-

It0<.rf IUnJ'0=r nES _

pirqe^rJep rS pnurluoc ets; : <

'0=r uJgllq€-\Irel:-

eceJBoOC[ '0 = r ut tS e:.-.--

'epqBAtJep rS anunuoc u::--:

!(

<(t''t+"/)'-'=(+ tr9 ; \l

?Lt

,P{-,l*'l# i='

: (0'l )r,, -

lI

t

Page 3: Matematica - Clasa 12 M2 - Manual - Petre Nachila, Catalin ... - Clasa 12 M2... · vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord

v9z

0002 ilselld 'St €loIeJBd BJrUIpg ' ,DcllDtualDLtt oztlttuf'- A ncseJpu€xelv ''l ncseiq3 .

,96 I It6erncng'qct8o8epe4

rS gcrlceplq uJnilpg ',,Pc'llttwayttu pzllDuo ri ptqa81y" - 'C -ropuolg ''51 ntcuoq .sg6I rlserncng

'gcrpedolctcug 13 pcgliulu$ e;nypE ' ,1tt.t?a1ut ri 1ofiuan{tp pc1o3" - 'qg 1qcie45 .

t86 I pseJncng 'gcrSo8upe4 rS gcqceplC eJnllpg ' ,,Pcllowalrttrt ttzr1rtuV" - '1,q ielncSoa .086I

rlSarncng 'pcrSo8epeg rS pcuceprg eJnltpg ',,PclryttualoVy pzllDuy"' 'Jzt{ ncseloclf{ o

/66 t FSerncng 'uorfuocg

BJnllpg ' o,"toLtadns 1n1ugwpipru3 uJ Darailtupv n.tptad atualqo)4" -'J EcIolS ' d PllrlcgN .t66I llselld 'ist elel8rEd

ernllpg ',,a.tailwpD n4uad ayqrsod a1oarqn;" - '(I IazugJE[ ''tr 1 nu€oqceg 'd Ellt{cPN .

Ig6I pserncng'pcrSo8epeg rS gc4ceprg BrnllpA '.,12,rqafl7' 'E eBIocrN 'uol 'C['I .

IILIYUSOITflI{

" argerSotlqtg"""""""" ilryos

1€oJnuluceq ep Inpuetuexa uertlp8e;d ru1ued euolqoJd....'.....,...,.........:.'.......;.'...

.........1..:....... ............1:...."....... r̂Jsnle^e op eqoJd"""'adelor op fuoc Inun lmunlo1'Z'IA

eueld eie;erdns Ieun eFV 'I'IAalpgep la1uria1ul o1u gierlldy'1n p1o11du3

Arc """"""' "' gzelurs ep otuelqord'g.ng02 """"""' ellulJop epr8elul n4ued gIIqsIreA op eer€qluqcs 'Sn

102 """"""' """' ollulJep roleler8elut e Wpd urrd eerer8elul 'tns6I """"""' r"r"":"'irrrri"' "' ellulJep telerSelut e1e tigteudord'€A

Z6l """"""' enulluoo Ilicury Ieun ? ellulJop rsler8alur ?eJIuIJe( 'Zh

68I """"""' """" PllulJep glur8elul ep eeunliou "I

cnpuoc er€c etuelqord 'InEllugop upr8aluy 1 p1o1gdu3

oJenle^e ep eqoJd

i9zWZ0tz9ZZ

9ZZ

IZZLIZ

981

7,81

6Ll9LI89Iz9l6SI9SIZ9I8rtivI

....................... ........................................:.:i'*'o'o3ir1rolrrfcun; uere,3e1u1 '7111

""' aleuorier rolttiouny eare-r8e1u1 '1111

eldruts eleuotier roitricun; role,trltuurd eor€ulruJeleq'0I 71l"""" e1druts sleuotier tticunl ut elerrotie-r:oltticun; uereunduocseg '6,1I

..'..';;;;;;l;*ffif ::r}#.iil ri,rpd urrd es.rer8elul '911""""' anqltilud ltrupe nu oJEc 11icun-I

'a..u1ttut-td llrupe elso tticung '7'61

"""" ellllurlrd lturpe arec tticunJ nc rrierad6 'g',11

Page 4: Matematica - Clasa 12 M2 - Manual - Petre Nachila, Catalin ... - Clasa 12 M2... · vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord

'(,t'x){ 'nydurrsr€ru'nes ({,f '";)ynr gzeetouasrs(leurp.rogtseecern) r{ncxmlpmdruocetSeunu es/(eriucrtdu) ericuq urrd (.{'r) (rrqcared) rnlnldnc rolpztmdseroc Iqueurelg' W <- At x W: I efiecrlde o (yg ad Woulq aiotado nas IN ad pntqaBp afiotado nes)

W ad (puta1ur) afitzodwoc ap aBal elSerunu eS .W pprleu ueurilnur erg .e$1ugaq

'gg a,4cedse; '79 ayeteurnu uelnd BI oJecrpu ap erferedo uud pundsetoc

at (Z'S) rS (S'Z) rolqce.red rer'g 1rugurnu er€unpu ul.rd..epundseroco'et (S.S),(S.g)ropqcered'nlduexa eq 'pluriuese lereue8 ut olse (..11uerura1" nes),,ruo1ce;" gJeprsuoc es eJec

uJ SeuIpJO '.,1o1 elsed oilulJep g lod nu" ngs (,1o1 elsed elruryep" U lod rS o.eJBurq" rS cserunu

as trfe.redo €euetuosv'(pqrsod o6e pugc rcurqu) IeJnlpu Jprrrnu Belle4 IB un eleJrueu eretunu

ep (q'r) eleuopJo qcered reun pz€roose e{eredo ecrro'elereprsuoc eldurexe re4 elec uI' 0 = g,0 - D ff\nzece{decxa

nc qD lrup{unu ,.llulJep" ruele el€Jn1su JoleJerunu e aralnd el rrJpcrprJ Inzec uJ ..,llulJep

elsonu" p'zeclsece ug'ecrueunds't',{ uJEnlceJe eluodesnu q-, rcunlB,q>o gtep,gcu?AJasqoPS'q-D=plelouelsetSroleiuere;rpelseeJ€cplBJru"uJgunuunercos€rueind

el q <, opJqBu eJerunu €nop JoJgcrJo' q +o lelou alse rs Jol uluns e ss eJBc r lsJntpuJprunu un urcos? uelnd el q'D eleJnl?u eJerunu pnop JoJgJIJg .p eeurilnul uI srelnd e'

arucrpu rS areppcs 'aleunpe ep elniuredo rS elerqeu roleJeumu setulilnru urnce ruFeprsuoJ' (e1ce4sqe rarqaSle cryceds ldeg) eleuyu:etap a ;

elcerqo n3 pzeenlceJe 3s ec elBuruuelapeu rolderedo BeJeJeprsuoc 3l snpuoc nu ('qe'Folc: ,-

'eruuourlod 'ecr4eur) rou runriou;oun eeJecnpo4ur rs ec4uureleur ropriqou esreuo{ceg:a4'(exeldruoc eJetunu 'eluuotier ereurnu) ppeoued eaoe uJ alezcerdeu eJeurnu lo'x

lS lndecul u1 necgdu es eJec 'lncyuc ep eyereue8 rln8er;oleruud eerefunue srured e (ruqa8-lerirrede pclpg) auelqo-rd ur ur^Jelu er€c Jolalueruele B eJalrl nc BeJeloN 'lncluc ap rlni;-Jolsocs eerecrJrlsnfl€cJecuJ nB eJec uullrd luns (1ueJorq 'pllcng) rcerS llusrcu?ru3lel\

'erelncrged llr]{Zec u1 eleiunua 'recrqo ep ,nele rp8sr alsecy .Belrop Iu rS lgug lnpe-6 :c

eptiunce'eleurn1ol tS epue'a1mu8rm1 'arrqrzod aleuorfer eyaleurnu'a1ern1eu aleJer,unu pu. ,- j:InclBc ep rln8er ap lelduroc ruelsls un nee16oun3 rueruolrquq rS rualdrta rruercrleulet?l\

;: : :

'vIAIDIS ilrnlrpl ;eped orcr51 'VWDIS Iu.-

'9007'7r

IB ulpro ulrd gtuqoJdBalsa Is e^pBlIIEc llrPnlB.\a15 grglor.ro3 la$ucnpg rn1

ltl

Puralq elilzoduloc ap e8el .1.1

rundnucI TNTOIIdVf,

Page 5: Matematica - Clasa 12 M2 - Manual - Petre Nachila, Catalin ... - Clasa 12 M2... · vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord

:rrieJedo eleJuolqurjn rue^v. 9l= rZ e$o Wed elrugepecuqs8p roprieredo lruprunu Icunlv .elueuala

t = zZ erc W x 7y .a1jn1og

' {t'O} = 1,ty ad elwrlep acrrqe8le ep{eredo elpol eucs as pS

' (f ) t ad elrugap arirzoduoc

ap t3a1 luns /u y=(;fy)s,,{oX=(.1.,x)!'(r)a *(r)a"(r)a:B'! avficunJ rouqv '{r- xlx}=(y)r;"a*v erg (p

elrrurilnur urp Bun else

'({to'D)l Fuue.uele gue es /euuoloc nr z rerurl

ericesralur q :edrzodruoc ep rr8el ..e1qel.. pcpq es goup

plnrsounc g eleod 79 ed pruryap / arirzoduoc epee?al '

t"r, ...,zo,rzj

= W gtlurJ eeturilnu erg

uD

;

!D

i

ZD

tD

(f) t ed erirzoduoc ep e8al oe$e q

" a = (q' 3) !' (r) t * (r) o " (r) t: .,f rcunly . {r * y : B} = (r) t au @

i(e plcunduy ep elrunilnru urp uun olse W opun,tx =({,x)!, j,t[ <_ Wx W:./ (q

: (a)"w'(a)"w,J ou, o,z,Nl

W.epun , ,( +x=(,t,x)! oIAl<_

Wxht: !e.rfpoduroc ap;ia-J ^ffi,

'c1e ' r{yx'l T x,tgx,,( @x ,{ nx,t v r ' [ *.x ',( o r :rriulou alle rS csesoloJ Ieur as (n,x) ! 1nsnduroc ru1ue4 .(gnrtecrld4ynur odutou)

( trr nes; '{'r

nes (q'rlllPe erielou) tr + r .,rsoloJ rnlnsnduroc eeJelou ru1ued .(pcueurnu)eerrilnuug nes (pcuarunu) eoruunpe nc rr8oleue eJeun laurzeJd erirzoduroc ap ue8el eceq

?

Page 6: Matematica - Clasa 12 M2 - Manual - Petre Nachila, Catalin ... - Clasa 12 M2... · vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord

s

' !t[ ) { o.r rcep li [S'g-] r (t + @ - A@ - x))

pc etqil rcuntv'[t' t-] = 0 * Ob - r) lcop 16 [t' t-] ) ? -,('l - r rrunl?'19' E-) = ['xgceo 'V+(t-f)(V-x)={ orierr3s aleod reu es E?Bp eeBel gc urg^resqo 'a!ryog

'A{ ),t'*A' OZ+ IV - xy - tx = r( o x

epun '[9 '€*] : 1tg ed efirzoduoc ap etel alse ..o" Be8el po ezeJlsuouep es pS '€

7,1 9

99

NZ

tt

ZZ

II

ve7C

?I

I€

7, I

II

7, I

7, 1

zlII

7, 1

II

7,t

9

v

IZ

I

8fztvvztz7,ztIIII

acr.lqeSlu rop{erado lruEiij- - ]

'tt'ol= lt *.

(v)t;+-(r)ax(r)6' a ., ;

erse ry " A =(q,B) {,(y) !, * i r

i(e lnlcund e1 ap aprurilnu u:

::ltulllnlu urp eun elsa ,i1' al

-.r ',{ o r :rriulou e11e rS csasolo:

-r nus),{.r nes (putppe arlao,' ;-ri1nuu1 nss (pclreunu) Bereun;

8

v

z

I

:1uns e{tzodruoo ap olqet ?nop ele3 ' \U'e\\'e 'Il=7b'{g't'Z'I\= !6r ruany

'zlo l5 86' ru1uad e{tzoduroc ep tr8e

errcs es qS'(,{tS r rnl IE umuoc Jozrlrp ereru ruru 1ec) (,{'r) = r( ox e{rzoduoc al

alSeur;ep es '6, e4 'Z <u IBJueu rnlrupunu re rleJnleu JollJozrlrp eeunilnu ue 41,

N9ZENETZnEz€ntzr

b

c

Z

I

ep atey ruele nu lrep ' W t g =(g'y)t uc

'ler1qpl" yleluezetdat else rl3e1 ,.€lqB.L"

ep?]sp I^i {-alilzodruoc ep eEel elsg '

{V't'Z't} = 11

alEAIOzeJ

i,,leiuel€^lqce" eyqel

also oru3' (+-) eriecqdury

'(") ericunlsrp '(")ericunluoc ruelSeouncer

rrierado atsaJB eJluI

T

rmImt

8tzl

,D ..... {o..... zo iD i .

ZI 9NEZI

nEzl

Page 7: Matematica - Clasa 12 M2 - Manual - Petre Nachila, Catalin ... - Clasa 12 M2... · vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord

' SI v (6 v g) 16 (Siv O)v 9'SI v 9 ezepclec as gg (q

'8^(ZInOt) 16 (SnZt) ^OI'ZIn 0I ezelnclecesgS(u

'c'ru'ru'tu'c Aqcedser ''c'p'ru'ut'c ' q v onc .,.r1cadser ' q Aonc ..,*lou.*t=|u,?#ii::;' *

rtelrunsnuee,irsdur,,ur.,"?;::fjj,i"1trru;ffi1'ffi 1;lir?'1f :;*

adzodurocepa'al)ecuq"r,,3,i,1,B.f."H[IIi];f-:1,'ffi1T;,1'$1'''

(ooo)'etuetlrele lert erz ht '{zY 'Y "O) : I^I nep IS

tO : ,V'l O 0 0 l= ztr urelY 'a1iryog

[, o oJ

I (o oo) ll ' l-p=r'l t 0 0l=vl,yf =7g uouritnualeelue.,aleelgf,'s'lt [oto] l]

'l*t* { - x - {x7 ePuneP'0 # l* n' x - xZ

eclpe' o.(1-rx?-,)z *v,", {f}- - > t' x urp tprepe-4u, {f}- u3 d (r ecuo

ru1ued t?)-- >t-x-&z+I ?culprsuourep?s emqar 'w )(AY'@)vpcrulued

'({-r-tu1)y=

(,t-x-,tx7+1 0(d-r-,c*zF)(t s{-l.) f ,0,-l)=l o o o l=l o o o ll o o o l=Qr)Y'(x'1r

[1r-*-rr7; st-*-,*z*r) [n-, s { J["-,0 *)(21

tpnepu-r1u1 tii -H 3f 'recyronrlued w =(AV.$)v PcerlsuouepuIoA

( x o x-r)

{f}-u=''l 0 0 0 l=(r)7ue1o,ri'a1in1os\r-t o x )

'q ed efirzodutoc ep e8el else Jolecl4"ru ear{imuul pc eze4suouep es PS

t- ,( * o x-l)lr {f}-, = "ll 0 0 o ll= w axuteur ep Eeu,irnu au 'r

l. ,'-'o;l

Page 8: Matematica - Clasa 12 M2 - Manual - Petre Nachila, Catalin ... - Clasa 12 M2... · vruo,*B oluuolsaJord egrgcgguc oluol pc13o7ouqa7 oanrl IIJnleu e1e eiurrlS :erezqercedg IEar luord

i

{r=,re -zri'o ,n*1(i ir)}=^ ,, ,{r=n '",1:

o ,)l, oll=w (,o,))

:

'c'ru'ru'ru'c Aqcedsar''o'p'u: : :

:.:t3e1 luns nu ee:ri:pdrul rS e;:::-:

-le{rzodtuoc ap e3a1) pcuqaS': ;:

'eluetrrele IaI ere h' ".1

' ;

-ecrp€ ' o.(t- ..)(i - , r. r

ruruad {f}--={-r- i;-

'.Ip^epe-rlur {f} f, 3 .. :

t fu= i o x'12 > tt I t + ttzl : t + zT, : w {p

: gg+,(g-rS- tx - { " x'[9'l] : ltt G:71+ ng-xt- tk - t "r'(-'€l : W @

: 71+,tg- xE - tx = to r'(oo'e) : Ilt G:allJnz?s {np

uJ ptrBcrput eerurilnur ed erirzodruoc ep eEel elSe ,, o " e{uredo pc aze4suotuep es ES

'{E'z'r 'o'l- 'z-tl'{Z'I'O'l-'Z-l: tf,{

"g-tilnur ed.,uluJ" rS ,.xeur" royEel elaleqetr al.lcs es 95 (r

;e{rzodwoc ep rtel }uns ..urru" 16 ..xetu" 'J 'U 'O 'Z '1.{ e11u4ilnur u1p eJec ad

,{,-. ,t(';' ::;,)}=w (q'{,-. ,t('Io,' ;,,;,)} =w (s

:plecrput eaurrilnru ed pu:e1ut e$yzoduroc ep a8el else .0." gc eze4suouep es PS '8

'e$yzoduroc ap eEel ag gs/lgcu1 IaJF? Z 3 p eullruetep es PS

.tmmc elt?lelec uI 'lr - ,l- ,J _ ({ , x\ [ (,>{'Z--r'lr-xlJ .r

u!:d glrur;ap W <- ht x yg : ! eficurg elg '{, 't '2

.'l '0} : W eeunflnur erg '1

.E=n!^.r ={ox.(I .*-) :we z- ix

{x +t :d=r{ox '(l'l-):W@

W*, *= (s'r)urur'

fifi* o= @

i' .1'1 = :

({-*-t Z+t 0 (,..- r -I

=l 0 0

[1,r-x-rr7; o i-r-,

:urene 'X; ) q'o ecyto n-r1ued 'gc eze4suourep es pS

LZ < u 'Nl 3 a '(u)q ed edlzodtuoc ep r8ey,,v" I8 .,A" lm<

tOt)O'(gt)O'(8)O ad e{tzodtuoc ep r8e1.,Voo IS ,,A" lm,<

'@do'(gt)o'(9)6r eutuueleP as 9

'c'ru'Iu'(u'c e ,rr.qcadser ''c'p'tu'urc e ,,eJ€nl" ep ecFqetye epriursdo ,ovo' rS . 7 "

'{o l* | .Nl = x) :(o)o eeru{ptu eU Is z < ,'Nl f D ;.: t

' D =DV D'D - OV l'0 =D AD '1=::tuele '*f{ 3, ecuo rulued'pc eza4suoruep ;! . :

'.;zg ed e{rzodri.ro:

I

l{f }-, =

t

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8

'(u poiu) z : i. rcop rS z - r I u p qpzu'Z - q + D IS (q a o)u : (z - O + (/ - r) * z - xutn)'qlt:z i'ou-.,i-r1gcu1 :re;lseZ=q.optstxa e(aporu) z: to(u poru)f : r(c

<=Z >p-'u.(p-):x-[eup:t- xnxZ3pplsrxo <= l--xlu e(a pour) ,=t"1;l 'i(n poru) n : :re Z < u'1,\ 3 aA,0 lu,0 : x - x (e .afio"r1suowaq

p)ep u olnpotu I nc luan;3uoc olsa r ec ueundg .7'(r.t pour) ,t' : r ruploN '[ - x lu

3 ri'r rS Z < u'Nl 3 & eld .a1i1ugaq

:luns nV ruluad sns rsru ec elrurJep

rr.r{lnruugrirFpunpeolelqeJ'Q.D=q.o 'q*o=q+t7 :leJ}sBelrurJep,trleclldqlmursn4rpe

e1e1ou rrie;edo enop .rrrporfi1') p"1J;[ , o1"po* rrnlser op rolaselc eeu{ynur e6'n) = y: U = A: Z: = 8- Icep lue^€ 9 olnpou unlser

ep Joiosels Inze) vI 'l€Jnl€u Jpr.unu luelueze"tda; cnu teur IaJ tupnl luuluezarder ec ,recrqo

eC ' l,J resBlJ re duuluezardeJ luns 9 t '01 ', 'Z-,g- alotatunu ,9 olnpou rJnJSeJ op Jolasulc

lnz€3 uJ 'nldurexe acl'tasDp 1o Tuo\uaza.tdat elsaunu os psBlc glrrunu€ o-4ulp r€runu ocrJo"qa'{Z>*ft+ uru}=1: rc'{Z= wlruu}=

9 = oJ rcopruelv {Z=*ll +wu}= l = t)IS I-a>7>0 apun'y nes 13 urgtou el es€iJ olseov.a olnpou rJnlsoJ ep

esel3 u lcsxe Elsrxa gc gllnzeJ'I--u ' "' 'E'Z'|.6 alereurnu urp Inun elsa a el Ber1ul rprunu

rluPcrJo ruri.rpdul InlseJ eseJEoa(7'u oupolu Lmlsar ap t)sol) elserunu es gsels ep IeJlsE

O 'lse.r rSulace nep u q etri-rgdurl e;ec r8e;1u1 eloJetunu aleol epuudnc ps€lc oJecarJ :Jol€urJn

Inpour ur ,,os€1c" ut r8e4u1 JoleJeunu eeurilnru rurj;pdur1 .Z<u IeJn1€u Irupunu erd

u olnpotu IJrysaJ ap asBIJ .z.I

'(u poru) z : r <= / ) z'ri'r'(a poru) z : !{'(uporu),{ : x :p^tlLZUE_q else (c:(a pou) r :,r€ V =,t'x'(u

pour) I : r :gcularurs elss (q

iV =

xp'(a pou) n : r :glrxege; else (e

:elriglaudold er€ a olnpotu piuen.rSuoc ap erieleg .prualoel