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rye nA uR , '/e t^r.*o_ ,t AR€4PTA 7U SPAT/() SI PT.ENUL @ c/" ab/.4- VLanelz tel 3*-kt-E+-4:o, [p") X+h**-2 -o ^) Ea' lk ao4;a e-.u-oAi.;L e--a- vLel-1-('r< o& d^r-fhe4 0l 4!- rn-(*I<#- a- eu-LatL d-ott /eo.vq. 6) fa' /k )a^z < o^-aA.''(a <.ye-*-t< ./< ohlltei D . .c) 7a' /k Socacaztct f^r":4_ e_aa-e_ h.*_ /*l d^"a/Fz (0) ?' 46 ff.no&_c,-a^_ f 7x_a+,,Q (P) x -27-+)t -4 :o d") /a" /k aehana*ctt-e- c-ct-r-afr6,_ f4yL,/-^^ A ca^4 h..* N* p"4 M.(,r, -5)^i; 4' ..,A /44/*^ ot;,ulalv F 'f^,,^-(t (P, ) +t' ( P.) e ) ha k ea&czL,*( u^f L:,-"(- d-tzd.u^ lr rhet ff",}"il,_ (p, )4 (P"1 f ) ./a' .1< ca-a-c^'!<-2c d-c,rt-ac4/o d4 e"- /Y- 4f.) ac444 .*oll4t ff.tro na-Q f f.ru,_! C p). X ) -fa-e /)' &" dru-Vtzt O) &XW ^) ttto-u-ael4 il, 4 t^ /^ (*/- d.ou€ for* 4.,*; il,=(3,-4)t); fif (lr4,|, Aa'z4z (J ) = ( Po) n ( P, ) a-1-e (/€d",.,-A -L n&z afr.,v*'q71-a^ F4xilz l: ln i{:^2r-zj+4t &,End- fat-n,a d.a #*:;) {,f.t,.,.^- ^-Y ;' # -i.l;' :' n' nu- x,^3 44
Transcript
Page 1: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

rye nA uR , '/e

t^r.*o_ ,t

AR€4PTA 7U SPAT/ ( ) S I P T . E N U L

@ c/" ab/.4- VLanelz

t e l 3 * - k t - E + - 4 : o ,

[ p " ) X + h * * - 2 - o

^) Ea' lk ao4;a e-.u-oAi.;L e--a- vLel-1-('r< o& d^r-fhe4

0l 4!- rn-(*I<#- a- eu-LatL d-ott /eo.vq.

6) fa' /k )a^z < o^-aA.''(a <.ye-*-t< ./< ohlltei D .

.c) 7a' /k Socacaztct f^r":4_ e_aa-e_ h.*_

/*ld^"a/Fz (0) ?' 46 ff.no&_c,-a^_ f 7x_a+,,Q

( P ) x - 2 7 - + ) t - 4 : od") /a" /k aehana*ctt-e- c-ct-r-afr6,_

f4yL,/-^^ A ca^4

h..* N* p"4 M.(,r, -5) ̂ i; 4' ..,A /44/*^-

ot;,ulalv F 'f^,,^-(t

(P, ) +t' ( P.)

e ) ha k ea&czL,*( u^f L:,-"(- d-tzd.u^ lr rhet

ff",}"il,_ (p, )4 (P"1

f ) ./a' .1< ca-a-c^'!<-2c d-c,rt-ac4/o d4 e"- /Y- 4f.)

ac444 . *o l l 4 t f f . t r o

na -Q

f f.ru,_! C p).X ) -fa-e /)'

&" dru-Vtzt O)

&XW ^) ttto-u-ael4 il, 4 t^ /^ (*/-d .ou€ f o r * 4 . , * ; i l , = (3 , - 4 ) t ) ; f i f ( l r 4 , | ,

Aa'z4z (J ) = ( Po) n ( P, ) a-1-e (/€d",.,-A -L n&za f r . , v * ' q 7 1 - a ^ F 4 x i l z

l : l n i { : ^ 2 r - z j + 4 t&,End- fat-n,a d.a #*:;) {,f.t,.,.^-

^-Y ;' # -i.l;'

:' n' nu- x,^3 44

Page 2: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

TEM A NR,1Ot'a?2.'la-a- 2-' a

'

* = ot d"^ Lc,. , '-& ,lf<o^t!*t (f, ) 4"' ( P, ) d.? "Uc44/u-

t 7 x - 1 + { : - o t x ^ ! -( d - ) ) ' 7| " ' u 2 - o ) 1 +r ̂ * t - t t = t

9^-- t//Luza1! ,, tu ltuu-* c.8 a*/et (O) a,nz bctv4ou^Y4

( 1, *, o). tu^ch)'-(! caupLLtc(- a-& oU/Vtea(l lJrLd-| 4 t T ) " /

tD)^1 &d 4

x t I

4) tc^,-a*a&-

L%"l* a^-

n o-t{^-0 ) I

t

1 4n .4-

ex-f-Ucll4 nx-

,Lln>*'-'L 4'V^>bc faJ-And- f"''4

al dvr& q e-.( ^1h-r""-,4

v 4 . r {' \ = - t f - 4

9 : - + F + - - tO L ' 4

Aa rbe/\tr /v,aL- cq-a zu4nh'.& L//-L'*A- 4<

ptr rf b,a d-rr- .v*h-*r-L fi r,--ot d/'^ -(-cttql-;-a

7+t., yz-k @ ) 4^' , ,'r-) aal c,4L /-,a\"- fu + l- Lq-

QrV at r.L^lA urfr .

.c) ?l.v/^-(- cac u,tzt "/-t ,'t'* /-*,* a-44 4-

( P ^ ) P , + ) P r = c : )

(P " ) (>+Dx + - (A -n )T *0+ r ) { +4^ 2 , , \=o

l"lt &{^- dr4- (f^; a.ul Fo,* ca42 .><

C.tarl-^* f ( n x- 2h r b* - 4 : o . tF.---f.^-

a. gz ".rt<

, Yt

A/v'!-u-a- Md^,f q c4 1z-oryu-,,a/tA l<

(e^t ih' L (0 /6 fr fupa,.,t--*_ ,//vvr,^_a

f r ^ = ( z r r , 2 - t ) A + 4 ) , t : - ( r , - r , 3 ) r

l i l f r : o , \ + b * z ^ t L + 3 A - + 3?p-,.",.'L c3.4=& 'tr F-, : -x- 5J- - ?f +-g--o {o,n-

' P - 4 : X - r | h + 3 * - 3 : o

Page 3: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

taTu-q_ I

d") A/t--'-u4^l** i, , fr^ d-/A yaaaa-el-on- @, )5 Cn)

/1 ,^4.t c.< UL-il, (A ) , J4 cu-u-o*

t^,lr.- f^*

a4- .h^^- 0 ,, A4o ( l, -r, -3) ,'/t->1;

(&) (n-E' , fr,, fr,) :- e :-) \"-r ) '+5 I+i

o = l 3 - 4 4

l t t 1: ) - 2 [ x - _ 4 ) - r C * - 5 ) o 4 ( * + 3 ) : o 4

X _ _ , \ + 2 | - * f 2 F . - e : o 1

TeM* NR-ao

C-"n CP

- z k - z - o

: a (?.t, Pt) :),--'

Nt Nz 3

//N-.,11 //N.11 fifrF ) ,( C Art . , (P)) : Ax. +- t l V- * C?o * D

a + r r o - g - 4 _ 3= + -

. ( =

tq) x *3:

e) -1/ r+=*,^ V r=- j (trr,il+

F+ 6+ :)

9"-eL M, fu afe; t,lrli/L\/t-\-n b-.' 4f4 r,.a|,'| ( /) , a k

l/et24co-t [^

1

n4,'t:la-

^ ' ' ' l t t

/ t2*1u_Q_<-a

a- teu4Ln. ir_,/q)74^?- -4:'il l(3) ,//t ./ l- l,

2:;i;+;+i- I

*r-ae.fa- D/ /k a{-2; /^ fi,',trrkfto..fto h/ 14'

tP)1so^,r,a&c/+0,a'-v fx /Xo,",^-A- (?) 0i A< w'tt/*'& 'fu'o^Ln,^t. (s)' h""* 1* M,t - y'u^d''t ,d" r"'fr*62

/L An4Fr 0I .r'.^ /4^","t ( P) ry +, h^^!'!* ar t/' 4' Al

," (n'-;^,'i, il =' ilt ̂ *'i+3*- #' = 0 ) ('l l:;;:-rtf-Ei=,

rt"+BTC'

Page 4: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

u A , . 4 o

/tun4!

7a" /4 af& <-u-a-!.h Vl^'v*&,'i c-a4-L h'eB- f*u

cfr-A B' , z[-l'wz-h..u! f--fu2.,; B(0, ', -4)

y*-fu A(4,0,2)4 ryrt n"f ' / 4q dq!4

(a,r { ';:L:;:;:., (o,t :"+ *:.

Efu fr* = t(fir+ ,Eu,) =>

frn, = 2 Q-* ̂ 78 4,^c t)

/gu*.'L- cA.'bf- !.* P* :'ffi /h''.b

tq ,eu il^" A^ F-, u',^.b l)" )rn /-^tq4

-(r*," b, I tan G^ ^E w ito/\..i- ch^t*ut *( /^rrrZc.'

r A \P L I

O l(.^l

ffi.rtc u< !-^a_,

Ua

T T Z2 -4 -4, i ! - 4z b '

f-&tv Cft-c.'tt-c'c,s't-z' -e< a-^e-fA+ .

: ( 4 , 4 0 , ^ F ) , 6 ^ : ( 4 , - 4 ,

, Fu_\; l \P,fur.'2)

3*-t4.^ 7><auru-&'i: c;'{''{=-t 'a-'A*

(P ) ( t r - Xo , Gn , t r - ) :o 'Qar * \

( ? ) x - + 1 o 5 t - 5

4

4

4 t-4

-f- r2

= o +

z\ (x-+) - r t(y +s) -4gG-r)=,o =) z) x -ntk -4qk-444=

,fua^+, (p) 2bx_- t5-y 492 - 444 =o-

Page 5: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

T€ ML /VR, 10

6ayun i

9d le -totie- -zeu-a$a 47aa-n^t€,.i @/ .""n- hnv-

n^^Wc,'-!- 4 n zt^/2.-C-) 48, rr.,-4 4 ( t, g, 2) 1'-5) 1), /& Vana6/- --{r..- oUzar*\

( D ) y = - L = F ^ GT 3 v

^' .aG V<^7x,watct'Lo--- /e f^h,'! (&- ) X +7 -4 =o.

f**4( - , ,

da oa r<c-i-- TbLav^l-a @ /'r'A

6 N 4 ,

&.fu ,{,"$er.L Mo ^-(- t<7m'evffi'; +6 a44

rc@ ya-lz .-et-L zku)zu"rr.a ,-4orao*t^A.r-bL

,/t *:/tar1-a afu- <-xft. vr*#A+-, 4-o Q' M.(O, o, - A

Fzotu,-L G',-be G) tu-* P /v2& 4 t<-7La-&-(- .cu- d,-.1ra (D , d-oo- tl+, ttn& F=1;,1,1

,Lfr 4/<fun'!- fu6bn- "-!- d/''/{'u: @) a' fr,1fLvLx^u4n

4tt yaaTt/,&- (A) , d^eu' fr t il, , ux-Lt- N. = (t ,4, o)

"4q- ywr*<aL< -l- ./-(^o<ul @). &"rAf t trc.&n^L

*.',t

N' , 4r1A/Lt/re-0-* /XA'vt/'.Ln

2$ A*at- 5 P ) t /-D aQrcror;

,.aL hrvdlrrh*L trc-den<-aA & X il,, =l: ! t\=' / t - , , l a 4 o l

nl (/,,1^ *t*.n c^Jent*e,,\ c-.1 J x n". f";r- fr=Q,u;)

q' ( P) (E -ii'}rt : o +2 ( x - o ) - 2 ( 1 - 4 ) * ( f + t ) = o e

( ? ) P x - t t - k * t : o ) A a ' ^

C P ) 4 x - 4 y - 2 * + 3 : o

figoo,ya*r<. JL h,otea- 4 c'1^-a io, d, in)= "t^a/'L- a^ qrc"tvrv) d

Page 6: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

G)/a'

TCM,+ Al /?,4Qrfa'tt-a G

* 7|orrlavt-ofuk h'r*eh) u-{^i p" ff-ai

f-iS d4 -V-tatu'! (P ) 3 x+ S-

- 5* + 2 = o.

it,Qlrd , M. (t,s,t)

/k

3)N t ( ^ 2 , 5 ,

&wM"( - \ ,+ , ^ )

N#^).^, /a4 M o ,7^.^.arl dz ir,rtu<%r.(N) rctrn"L,cd"^; f" f aT,'t- ( P) ry c-cutL

tlrunc,E! ^1q. l)"*r''^*t otl,r&fu/L .-!- A4q

I #'aL 6at-{ftt

*uu ./'i t'",-'' /- \ ,

(/V 4/'A

--,

N , da cr' ecuat)Le, Q-Ytatiq /krt

/ . , \ X + 2 , = Y - 5 . - 4( N ) T - 1 v7n-narw -U .,u-o-L;eo ,^.---,--*'q "/4

-&n(nl)7"'Ur<

t /Laf^'a'"'f"h- <qa-I< :

r , , I x j - z + 3 t( A / / 1 q = o + tI 0 -

k: b -5 t , *a [R

/,,,'Wt fr2J^ (N).-u,.- (P) f^h A' ofru [1'-

f^"fra- ,ftum,H,,.- A14 f< pba.u! ( P) 344u'a 4o'

ne#t^ra*}t;- ,Att">nru',/ fur''<"t A< a qtq$*4- (,*to*-h c1

a4" "U,qh CM f.' 1e*< oEa /v(^y',^4,-. J2 f*+ 2 : ) x o : - 1 t u t ' T , * - : t l . ? u u & l ' 4 2 )To >

E ;) g t do T

Ar*-fu*,1 /'*, Nt foto-t oa favl'aA

(P) 2rfr 4

)-, cet M o +)t '>";fac-.-A /t-r' lY4l\42, &o' i"=ry

at uvu"Le- da al'-u 4.1,r Ez = 2 E' - Fn^j ( ̂ L ' u! '2)- F2'5'4

Page 7: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

/7a-y',-t.ra- yv 4

pnrt !* r,'rz

4. /; /k "-fU

<-cz-afi4 ,t4avu41-; c-tu-(- h.q f*)r"-

axa- 0* f' /a,tr^yaat2i /4r-L

"Uv*'-L

2x +/ ^*t6-7"0

uotol-n*4 oU- 6 O")

%-*e2. 1/',^- No* co-1a co''frut c4x4 o* 4"1t

-(cu-o4^ X f A7 = o, u,'wAe- ] /-& />aaovn-t"'l 'ut-Q -

tr bw,fu/vLL *v1ol'to /c-a-- "nfru

d''"d"tt- &tt"-h fu-' v , r - ! a 2 x + [ - / _ [ E - + = o 4 ^ x + 2 7 : - o ' t o ' F u 6 0 ' .

A r r 2 :) x+ 3!. = o i' 3x^1=0.- , / 3

\ _ a3

2 /a' /k ^f,- e,-o&q h<^->-,,,^-('; e-'a''-'ta htq-

/f4,,>w at-,t^ryn- d,4- ivrtnrl<+<- & ./o%x<b @ *5]+!:o

+" ( Pr) x- * + 4 : o +' ctu1L /-rw''ea2- 4n- '"-'yh' &

9 ,c /4 7 r<arwu @) x - 17 -8 t +42 ̂ o .4 /

/-a. rofi. 8-(o ?,^'Lt- c6''t=-f y'^* N A<nt-

/z<.oz^z f^f; fh.%L-A ( Pr) +/-tta6{/'l- c4/!- aht' ca-

(P*), dtet- fu a,u <,t af,a Po +) Q- O

+ 4 ^ -;, ha*vrno-(q l<

rUav'tal< X^ fa *g!

/Yu1^4- qad'fra c'<

/ )*k,. !E A- -*

/Jat4r- (\+q)x * 5y + C'l- A) *

fua d- t,^ -- (^ i, , 5, 4- A)

@) .e# f r r - (4 , -4 , -_8 ) y ' .

+(d, frt) - €,

/'L ,-t'tot-e-, - r

tKa-'ltfnt"Y\A ,t -

) . + 2 o ? + 4 f - 4 2 = p

Page 8: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

3 /"" 4< ̂ fu

c ) l{

ren * NR .4eM"^* t

eu-ralti,h. 1')9 -ree&,"L Yda.eTy-fe-<-

x - 4 L + 2 T - 5 - oo

3 x + Y - k + 2 ^ oo

( P ) 9 x + 3 t + X - b : - o 'U

& 4aLarut-L/ /

,QA.l+urltt.

@. &-!- da a-(- cl-rtttq.- fo* 4n (o' )''fr

fttatu'-L- 7'tr'-'*-^'ut ^-l- aruVXe^ (D) T ff"M (9

%r'a'u-t, ,X4av*,/ ftrcc-b't /aa d'L

-fuita"At-!- .U < u'' cq-- /'?(^ r',-^( d*h-

/-14r'(4 ca,l-Q- .U-/; n-e4c

f (O I

4 /ci ,A" m^fi.q doca* uront*oon 4t /"r^"rl,n "A .U-ryte

/k ttnfu-t<Sa^7^- /Ja/.4. 71't4. g, co-t "3

+-^t ..zO'rTutp&.

Aa' /R ooie <atafr'a- f,^nr.€*r'

do-byvznot 42 ntQ.

l x = 2 i . - ) x ^ - A + + ba-) (0, ' ) { l=-N+bt 4 ' [Dz)12= s:31- 7 ' ( u ; - = t

, l e l R L ? - = 3 + A , a e R

- ( K : 2 + t-E) ()r) I 'a

: 4- 3t 4'L; = -4+t ' tetR

I X = , l + A(D')1

1: stln , ,re rR

Uw. a-) (o,) z-; (Dz) ,t-rt co,ract.tu,-'b. 'il-r"'-(,

Arh/Ly,trro-t oU a.cot dl,a p *E (P) J x +4J- ?/ -2r0

&) Mo (2, ?,-() e (D, ) , $.t : (/,-3,4) "e

v<c-tv7v,/ o-t &rchu

Nt(n, o, 5 ) e (0, ), F, : ( 4, - 4, -) - (1ee&ruu2 &.'atY'

&oa,.pz ( W, , & ,6r-i + f n*p''-ta-" cq- (J-'1 'u"

( 0r) /t ut zt**t' ry €p L"

Page 9: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

5 y'o'

0,)

(D')

R6.tTrqwr.t -

ATaf;r' \ex,b*

andt ^44 ( 2,- - l

4 U L = ( 4 , - ( t

finvn'-!*^*

/, +"'rnt0 t

Te/v+ Nf?.40

Ei^?q.,1< ca-(- cr'* y dt-fa/ufa dtrh- d^t/fr'A

e =4

x-4

3 - t- - 5

tul= - L

- 4

* + t4

*-5- 3

JAr4tL -/-,,.,4t *t^e*€ ?.P 'L

c* /br"od//nu/ o,.uri- (frffi, , D,d/+g4, -4) e(0,) , Nr(, o,e)dD') ,&=(,,4,/)-3) bttzh-'vda /R caJ-^"Uqfu' aa

A : l(ffit", J,, F.)lJ t l

l lT, x 'uz I l

d . _ | ,64

6. %:' 4z .ca,l-a.-/-V

to ,s l x+2t " t -4.= o

\ - / I x - z t + * * 4 : a

* ry fi; L'f l'c,vffiL "'|^"/fre/ r \ l x - t - * - 4 = o

r l D z ) \ ^ - t + z t + 4 = o

a; rUtuYul . a - ry3

Page 10: Gheorghe Asachi Technical University of Iașimec.tuiasi.ro/ro/images/tema10.pdf · 2014. 3. 13. · Created Date: 4/19/2013 12:48:18 PM

q, # 41 eh;

/dv,nff--e M. (t, 2, a )

A) bx- zy +-a = o

T€M+ NB '/O

FVuq 4o

ap<.taf"'a-

4' enQ

/,h' ( Pz)

VAa rl.u&-; ,.c(- he- Tttlr<-

t^/.-oL c,-r-tr-a't

f" /4" ^tlo

2 x + * - 3 = - o

R.hryuM ( P ) 9 x + - 2 7 - 4 * - 4 : . o ,

4 O . /a, a4 ca,tc z.o^afio /"hrr4,,ne)c,4.1e h.er<, Tnia

/v*nrt"/L Ylo (4, 2,4 ) , llz (n , 3, 4 ) h' -e/tu ff-&cu!a/\

N l l t a v * ' ( ( q ) € x t u l " F : o

@ / - s ( P ) 8 x ^ 5 J - * + 3 - o

4/. /a' ^ Tacc<az>ca- V*'atnefi; ^+ f ry l'"X't /4""4(Pr) t^ :

f t +3t -4 :Q , , ( Pz)^n?- . * f=o, (Pt )x i l1 -67+1%

7+a' af,$;'.'

4) u* , ?un"t 47Yt<'<av,2.

2) X"ft*Ctffil|'.t* ^' h"4' d",//rt a"6" &i 3 )

lJ,^..(".!L ( fr^ru-ea;la-* u'1.L '&L

f^"''^o-Lc" ) '

/ 2 - 4 3 \ : I t ^ ' t \

W n c A = ( ; ; : : , ) " A = ( A \ i )

' moo,hi cza r /*-4f oCV ruo-h^ cr._ a*fr>-at A- tttqU>vcr/:-<4

,{trt4toA .-€)r, zrtl-ofi;4n Na)*/-rr f. rrurbqu- C-tnoLt;Q:' r) 4 = 3 (<=,d* A + o) i "l An = AA - 2 ; 3)' ho= z, Li = 3

@ry. t) 1 + r ) z) ) ,= + / F = i ; r) )=+,/rs

,/ 2 /o' a< ft4.o-rc{ /u/n- 7u nr.t f axa Oy A"ht

' ,X-^ dr.hu-te 7"-l- 42 lft ' '^!- )x-47 + 2T =o 4^44

' t * t m o Z A N 4 "

( 4 , o , 3 ) '

' Tn\- , - ,^ , -^ t t=o, t r ,=Yoo =: Ma= o ; [42(o,

Tt" )


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