babeș-bolyai universitymath.ubbcluj.ro/~abuica/sem7-solved.pdf · created date: 5/27/2017 1:03:29...
TRANSCRIPT
-tJo/wui""a), +
E,X^',r,"*-*-t W,"""ru 2o/+
L"!" *.a,A
;rl { , A* fR '[L canfr'zxt*(r4 ' 1t4b +{"* lhL r}k/'"fu^ 11 -
in= i"+, 4r:+.{? -.., {u--{,{Q, -, +&''2'
ln'# tf,* t@1,4re**V,.4 #',** 1' n '/4* &k-'*'{2' d-'^( f'= 1'
o,Ie /te{+ c)er 2o b a V:*; f* -f + 4'$-A'-" !::'r=2o'
[e.t, rf,* e f*t "{ { {' *"t /^'>'* { +* ' 4 rz>'2 '
Nt ry #*r ?* A * 4*A& {' #* f* t*J' E *-4/"*'dy
,/*, ;* r* ; #"^e W{rr)= ?*
5; ,a,,,1 ^altna.tm 2* /l,e ,rd A2* ' )7*A ' h{tU)= {lt ra/.(,d v\u I ol* '
[frr,*r^:r, ,q/tv/"*/- J1,nl yr"(fte) i^/ Wr ol* i " f'-"xp'$'
Xt ty'Q*) I . A #e* 2* 't" a'wl n'tf'narfa> F*Y#ffiJt ){'e*)l >( 'fne' 2* L 't/"'"/"'//' ( 't//'/-tb-')
Zt-e'r*-* i " 't4fr c*w&m lfi,o {:fRnQ, +$=r-t ,zTi.
a) &;-a- :ff* t** -f,*-t', 4 { aund. lffi$ A#w;* *f*{,*!"V
irt lhht ffn .eawayt*,acac { *\ h tf^" iVf
Tfr*t=:z6*tff-e) , zo=?
c) ilU*ry fif" &ott''tuP aQ'a;ilq'o"m'
#u %u.?-e',",s (l *)pr. o 'oy'fi'n oC 4")
fl&,*'**,, a) T'{* *txz'{ '7+''""ato
Af"* #* 'kf*"-a'* 4.f* *ru ^a**J *oh'e luQ"
A,,L.e f,&u .*ett*'b*- "{ {f"'
f" P'&'<'
(,ea 'Lr =- ,tr' tr1 't2 = f5 '
: *{i an"J gJ :E'
*^at'e* $H = x-" b rue {''a'*
x- J, t +a: : 2d. z*:> ** a: o
Nc {**d a f*r yo*fi" ?:
J-^- ,s7,'*,e4 te at*-ty" i&'e d*tr{"fu ntre' ry* f^*{'tr\ = 4-t"2,_=l1'(-rt)i : lt*Yl
.4-'z , :!nZ
'7 'l
A , lvvfri/s'r{t w*e{*dz Md Tr*=-{7*i! = {Z )n a.4,1. ,**tx*&z ,
&) fue*.*t*t 'CIA,'<-t*-t, 'Hw- "-*"X* "x&''&wn 4 {l'; iVP 'U
y{fi,U) . o,f&r ^ti,bp. 1&*f rV* ftu*fu)un a Yfr*r={fz6\"
:f,** Z* =*{U A d- {od f*vzt "t 4 rm€ #'**
V/ a,*E)=*E ,4i='CI"&*uruff..*r*d, f (&,,,17)= E' '+fr21 o"
f,** o
rro= {f ;^ ,o,u- *th*&n f* t rr,e #'a"*' 4/-** '
fu 7 ffi.etr,,,.tfor *ls-( tv vE *;rf(fr,?)='E- '
,) 4AE ,nzel W fu l*faad 14* ffi!" I ( on/- ff-cr-
,{8, ft-t *fu W -{ f a 'a' *W t>fot-*ff,&, -t"r,z U= z
" /^ /A" /*A-,-.IE ) o"nJ (\/>, Ir ) ;
,h,to,"Ae& *A, /*r. 3=o "'at (*r, o) a,^,( (xr, o\ 'wi#t xa2=2lGi
,,{Lt*, n A*'ry 'fi* &z's"t4"s,+
4 rO -- -42 /14'< -{'h"l
-{z a'n'L il*ttr) '
.,',-.. 4.
-rnrd {r*t a .n1/14-yrn/r't}'t-/1'a'L af ( ,, 7
'vii "TG4a"z /fraf '
I . -\? tu=Z c. (a,-tz) q,t A** -&)*, Yt fr, '7) = ' c2a "'*"{
h.n **
i **"-d3-'ltcnza-a'*b
f* k*-=? € (*,ra, ,l;) ,4^,< /'o^e "k;f(&'1)="v- 'z'v'uL
"ti "fr"rf b" 4'Atc-nz+:-e"-y
fu 1b.1 € [,n, ,J n4 i-"" **Y(&'2) = G ^n^t
i 'l*';tb Jtu*o"m.^2'
+* 2Q,2 € o, t.u;) ,ve fr',a"'<'r4(*) e(-{7 ''z) & '(k
W*u (,F tt,2) ) k>,4_ $u-/'*o* t,d* )"^ +t'- /k&w\at'u)
c1Qo24 r
t* e-<:=- 4+ff. ,'[w€ &/'*' f 1&'|f,.y'= -'fL ' '+ {tz' !
W ry lee 4+,ti L xn*'<*t*a-L&*-t4fu f:* n
,A,k-a ,al'ro ,trwt *l* ryx^ect- *- {r*, = (7 &n t -& 2 lt'avIt t
^D a-aL 4 ^'lZ , llo*u ll-vtr ;
ff .i d-z cz.,nl ,f ( E , h*yZ) ='fT,
r0* {onwrr"tu*!r'r*
"g fzr 4- "
.D {- x4 = 2 e (4+JT, *" ) n'lu< ito,rc 7,r *- {f*')Ai +4" Yy.Letu,
u ffte ? ) A, ,, (.e'*"u, ,Z-/* /"^
n/LA- ue"r&u /z tr*t : 4,.\z = (- \F , a+vZ)
e G* ,- ,/r)+4* fd '€a-4
,W;
-G*"il-u*e*rl
e (- fi,,r* r; ) . f le' -€-) =
= 't/& , q *'E)= -6, flz !
{af *( * ('L 4f* 4 =i2"? *^t ({e2'h)=4}*+o
l-\d. { ' fP-' lD ' {txr -- z'- I '
4 1 a^^'/ W .1&a^x 'doll&Ut '
@ -t t o"""/ a^'/u r?** ablilbt"'
CIltwgio",w, o'ttf''ct ft$l- a'A'h @?
=2 e4^,1 ,,n*y"r*fu- 2=-+ at'<"/
t-l = t <-> {- *- '1 = o G-\,2= /J\?7
rw'rt {"oru^/ !- $*X 'V*';";h' ' 2, : ''}' 41 'l(L z
,F(E,l\ ='{T , +^ **3 2
*'{(e,2):-&t
C*rrrr;.* Z fufz c6.,,ru'do>
a) {**"a .Hu t*l ri'e""fr'
o") fr'rrt #* "2- Y"t'u*e'Y
c) tllAr,-4 t1* 'nfrqn - 4,rtv li* ,tw*'o! oafi' 2
/.to"z"'{e f&u""' l<A^o'*-*H n
{*"-h*n 4 {t*>=/ (4>
/
l-l<rr.,t J'dl, *'*J f*'hi' /a'4'/
^a) 1r(*>=((+e,ir) =/+@ +'c*)=-24-
{'t*1 =7 €) *'*2xz-7=o (-)
rtaf'e fr,,*,.- frr*t P0* frlt€ +'/"-d 'f*-'>'f' 4 + "*t'*t "0e
l-vt r,*',.4 4 1' . Tfr'j-; {Y\ x- 41 );* aro'dtX' ff,
frter( 7N2''***a( '
y'(x'-z-l) +x3* .:--* z*-l; =o (-.) t*'-x-_r)("*=)=o
dd> r-r,L a '+ ',:r;=o "' 7+=-L
Ju, +' fr*-" 4 4.*l /*'""'fr" ' T'''u 4 t&"^"- /ovtz
't1' +''/'/ P:ffi *t + "u'/ #' 'nff"n' tt"* +** &*
l- y,u'r-t;c E"e"Z: 6) '4 t ot -4 ' o '*4 '
n[4- l,a*< (+)' ( o) = o , fu tt^' 2- 'y'u e*Le 'n-h't
/. e4,'1,1 ,c,bb'a-&rl' -
<-) r,.'rt- e4f lL/) 'fr'o^a-'*-t' fr '
llLdt 'f e c ''( tz) dz'n-l {t"*tf-1@
1+ l+'h^>'{'(o?,)\ <t\
4 l {'(v,) ' {'(l) l' 4
ftkx,u- *l"t '.t'{'t
-Lfa-, W 4"'M i,rlt, ctltt o&k'afft "
'1r",t J& .
(n'(il = f '(f @)" {'(')'J"**4?r,'lrtAu''-7't"t'a-*['4-'&t4$4d-aw<{ hr) = ,,,
'**t {'q,) = 1, ' '#V'*? "t'Ir^a -rta44
e4 ih* {'{*)= {rfc:ar) /t'L'{"
Ci)' ('t,) : {'(f rz,t) +'0/) =
'TAo ce-oLeeeLn' snt 4-dr."'" efr T€*u*'- /.l *ff t-acff*^n, .P'ut"La 7t*-L# ' D
{'(?,) , {'('),) .
q.'n"/ {tu -tu{'"*;L'v-'--
WF[rn"L ,al.l
x)
nr| a,ar'n{.u, He *^'t f : to, il "''Q ,T@)= I -le'-a } "
,&* fth 1 T, Lt 2r*= o c< I*e Y^* 7
t4* {*r f**, 17"*fil dr0*4 E8zni./8-/''/*'t ta *€* ^*'*''o/ at&-
a-) 'X--* *l euk-e'e/'> /"rtu'€- te" !{"'a"be-'a "
T(r)=a I \(x)=4; -Tf-)=* "
T-t:c)=o "T'{-)'=4"-r'Q) = J- 2
Z.o
,t) b*fe -72{D ' .. .-z r *.,-3
e) R'/ 't--/ #* f/^'4 'T' a'n'''''{
*r- '*t''* '"*a'W
4"*e 7E hd, #ry lw"Yt 2
{) 'T fi*t G e*y"to't^" frtt^4- ? 0a a u'l^-t ?
T A'o') a 4+ 9Z - Yttu>l"z' -"ht /
Sr.etLx 3
Jrtu&"*^- T d1 =
(,)
r(Jr)
*ru/ 7'or""f 4 T'
r>tLui,o! Rft-z) = z nN fi*"to)* : Z ). a'ur,o#o, ('ft23*f AaA 2 {"o/ /t*"*D ,
xe(lra)
f/o) = o *{ oaur?/>t T'o= o
4-lR"?-rl=4-* = * )
4) = L t 9* 3 t'Ya7t2 'rBLt
)^*,
7r,n-,r,
'x ef u, ,1
:lu*uAdL
b-d#,.t
Tfr,,.
T=? , r(7)-r(f)=,,
/\ .4 )(4)=o;)-t'i
fi* .urt,t /'" + ' *-
1
3/t
tz
!-\
t 4., O1 o ) CIl
8 1 '
51
n 3 ^1^,
qd -)*4
i g T(zc) r -T(z) . LO, *)))s-eTua)r T(z)*(trD
= 4- l-i)=)t.2 - 4lt3l,.) l).1,q, z
ri !-\ =\z^ )
Z8
3)I
4-,G);fr-lt;t A
,1, oro)"- -
) '*'>.)c1
zrT1_.1
2
o,+)
) t ,'y ,i,?l
,,}
'* u f u,+)
.r(t,i7? .x.(t,71
1 y* (7,q
h,x
=1e-4x
a- 1(t**)
h (t- x't
ln, 1+*
4u,
mz 1 1t*,.-
3ll'1 )I
({* .r.,_\ri- L
c-) T(z) = o
T(z) = 4
T(r) > J'7
.1rtfe dl J.rrz/ eW
T(r(zr) = o /s
f ( r(rt) -- A tc)
'T(rtur)= J^ b)Z
d) fC*>= T(rr:.r) =
harr!" furw n*&-ka"'t '' O t'nnJ '{
tr*r lgMi l$' ' *
fi-. fio'o- ">t! " t a'n'l ?l#or* {^'* "U- f4'
Ttx)=o "oa T/.v)=4 ''=) :zle|
T(z) *t ('=) 7e\+ '})tb)= + '01 TQ)--? (':D 'x€r\^/-
I 1
I 4, o, ozot ""
,')- ")-1Jt),\ a1 Q-4Z2
'3
e) {ns- c) ,41<
an*L T'( +) =
€nr* d) '4/L'c
al l{* 't.'"76or'*hI
fieu,ru :.
'Tz(o) = 'T2 ii) = 'T'(.i) = 6
{cr,"a"- /-gr"t T /'
/0, i) , 1{'i) )
'Wz ,nat'ct- *0*i
,&W-rro.u -On^''€ctc/u
( / , *) a"^J G,')'
T= AaA 4
\;7Br^ln, tg
-ff*l = c)
p ,*4o,t,*,i,41d^.J elrw *fu #_
Tb(z)= 4 (-) r(t'h;)= * a> T"*>= Le-.')b''r'37u\
fi,k rua-Q,rx tA,^* Tb ;. A'rw-" 'oh zacl' 4 f& "at*ez'v*L:
(o,$), I i, il, ( *, b), (3,+\, (!-'5)' [$'])'fl/b 7\ /? ,) fio.rsa(i,e/, (gt t/
^ -J-'("?ra 'Foi"tP) .
Tr*e -( /'&n'* .o"* , '"{ cetud '
bru {"*' 'P'"f' "{ { 'T
'c .c,nJ 7 .
/Cr"A Tj ft''tu)NZ ,vt-E{l&
8 {*/ fc;"ffi '
7,rr* *t l4*'* 'ovte, I a&tetuw t
{0" {*/ P'l,,c, '"{ T)
O at^4 Z
te *y"-d *t* t** 4 f 44< 'a4'1 {a* g''
(=) 'T= ("rf=)) = <: <'-) ak>'4 o ' t ' a\ (=)
9.tsii id )7;:2
+) l.i^* T, $44 trrt J-,*rt 7^^r- /74i,.er. .a'.lL -r*+ {*u
Ne *N T, # 44-e &e&*u' i&,-t /A"7 fu*' 6L p-P'Jz
fr--rl"tt .f^ T-. u{*l"roLL2,1r*? €<a..m l. {*.-d ex*x$
lr1 *a*-* &-.-qx = z t-) -x: 2 "t^J */45q 4-hx-x &) x-**5r, T Aa- 4 w.{* u-Vr'z'-v*!" 'u'l''* ) ? .,+\ &
d,,u 73 A,-ar, 6 $-*'a '7o''>,;t', ""t"'('''c'h -ac'?t-e nna4 {*/ V+'t*'':D
D otu,eua .&*t tey {o-," @ 'rLr- ^o''A't 'O{ l/ 6"4
*{ ''- T 4w-, p-f /u';t az v-a'c '$"Lt , 'M '"&Ju '-.]ttue-t- I
qrxw {€r J/rtu-A*'^'4P-" 7/***-" /+"'t -T ?"a- rtt'*vL./
n ht" 4 on7 l**J "
fue,zo*'* q 4/tL @tt'x'c!&v ,*i' TL*/' (i:[o't1 'tiil' GL)=\x(t*x)'
,.) frrT-*t f4* .yq 4 cr - X" 2'* = o a {'-'x /^'^i ?
fr>r,t nil /4,, ?on /tffi '{ C{ '
L) tr""* *"^*? ,'*4'"'t'e'ut 4**o bfr* '{"ai' *""" :
GCx)=o; G(z)=4 ,, G(z)=L ;Gt*)=o', G'(o)=4"€{-'*-}€
G3Cr)=o ', G3(a)= L ?
c) R"7 r*x ,t'0* 7*fr* 4 G' a'n'/ G3 ' A'G'*' /'1'"<j)"-'t-?
4*e /t-rr',.,fr *fr-fl &o'u" ?
.l) G "'fr*^ a a- fr-aa z'z'!'ct ( O'" o
G {**^ a- '{ 4 gz * /**'*t t n-'k"t z
&{Ld-t"s1,L 4,A d,$^* G(o)=G(t\*O @r,kt ff" ffr4 G
; a /r*M *a{A A ioua4i,ttz,,c.*- ^/ (tr 4)
TfL"", G
,{;S F/,*"-
&(y) - o
G(x) = 4
&(x) = J-L
6a(x) = o
G'0) - ,1
c*Cr, = *lZfi4+{rt:0a1
clczl '= e3
*a) =t
c) *ffu
#^*f 14*
fi** Gco) :c t 4 a*<aw 2'* =o
L" /;, *'rr* f''# *{ G '
4tk ,^frf! lL'*t G fe*+ e*e4fu* 4 f
."*fr.A Ea-rut /t 'f8"*' "{ &: N"*2'
)z(t'x].'=7- , viaq-+T=4 /=i 4rn=
.t'ura' !-'Y/ Vn>'ffi O a'o<J 3
il* 'w4 '04e tdv'X:
.-(*o-" .{-uro' aCt&oTvJ: 0 a""'/ 4
4*. 'aul ''4ht* t-" t !g"oo f,*w ^'(atwm
t en * h, !) "nu't Q= € fl{ , ,)
rb\ e(6C2,)'* (t1 GCv>e )',+)<:: x€\o'!'tJ
(<i) G(&cz>i)*4 €> 6h)e L (=) xe2'or'a'\
c>) g(6rz't) '- { .=> Cbl e )a',, *-\ a4"r fC'
n] r,'\ t:-, ?c' 1 o,4',,'! ro,, nl
!- fro, ? ,.6/u-{ Ln-oZ
5
E
)7
a,%
q-#r4
&',,-', 4 ra'&"'au-" '
ce) 6-( 6LczD = o &> G'(')
t=) e(e?:z))=4 (:) €u)=
4 &' lert,^ "e"fr.
-t &t r' "5/v1/1'{-?-#t '
",*? 0
A4* N 4 t-? o'*Y
.ah l-Tfq d'tb'czlutoiwn
a./v l"/" 4r-I*Ztv?I a44.,( I ,
I,9?'*U*9 "