04 Esperanza Matemática
September 30, 2022 | Author: Anonymous | Category: N/A
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Igpåtuca =
Nspnrgkzg egtneçtjig =.7
Enmjg mn ukg vgrjg`cn gcngtarjg Nk nc igpåtuca 7 nstumjgeas cg enmjg eunstrgc, qun ns cg enmjg grjteêtjig mn cas mgtas. Gharg iaksjmnrn cg sjfujnktn sjtugijùk4 sj mas eaknmgs sn cgkzgk 7< vnins y X ns ns nc kôenra mn igrgs qun rnsuctgk nk igmg cgkzgejnkta, nktakins cas vgcarns mn X punmnk punmnk snr 2, 7 y 0. Yupakfg qun cas rnsuctgmas mnc nxpnrjenkta sak4 inra igrgs, ukg igrg y mas igrgs, uk tatgc mn =, 6 y > vnins, rnspnitjvgenktn. Nc kôenra praenmja mn igrgs par cgkzgejnkta mn cgs mas eaknmgs ns, nktakins, (2)(=) + (7)(6) + (0)(>) 7< 5 7.2 + (7) + (0) 5 7.2/7< mgk iaea rnsuctgma mas igrgs, ig rgs, nc kôenra enmja mn igrgs par cgkzgejnkta snråg 7.2, l (0) 5 78/9> y l (9) 5 =/9>. Rar ca tgkta, X ) 5 (2) ¾ 5 N ( X
7 70 78 = 70 + (7) + (0) + (9) 5 5 7.6. 9> 9> 9> 9> 6
Mn nstg egknrg, sj mn uk catn mn = iaepaknktns `unkas y 9 mnlnituasas, sn sncniijakgrg gc gzgr, ukg y atrg vnz, ukg eunstrg mn tgeg÷a 9, êstg iaktnkmråg nk praenmja 7.6 iaepaknktns `unkas. Ijnrta måg uk vnkmnmar mn ukg neprnsg mn gpgrgtas eêmjias tjnkn mas ijtgs. Iaksjmnrg qun nk cg prjenrg ijtg tjnkn 62 par ijnkta mn pra`g`jcjmgmns mn inrrgr ukg vnktg, par cg iugc pamråg a`tnknr ukg iaejsjùk mn $7222. Rar atra cgma, irnn qun nk cg snfukmg ijtg sùca tjnkn =2 par ijnkta mn pra`g`jcjmgmns mn inrrgr nc trgta, mnc iugc iug c a`tnkmråg $7>22 mn iaejsjùk. ±Iuçc ns su iaejsjùk nspnrgmg iak `gsn nk mjihgs pra`g`jcjmgmns1 Yupakfg qun cas rnsuctgmas mn cgs ijtgs sak jkmnpnkmjnktns. Yacuijùk4 Nk prjenr cufgr sg`neas qun nc vnkmnmar, nk cgs mas ijtgs, punmn a`tnknr = iaejsjakns tatgcns4 $2, $7222, $7>22 y $0>22. Kninsjtgeas igciucgr sus pra`g`jcjmgmns gsaijgmgs. Enmjgktn cg jkmnpnkmnkijg a`tnkneas
Ndnepca =.04 =.04
l ($2) 5
(7 ∐ 2.6)(7 ∐ 2.=) 5 2.78, l ($0>22) 5 (2.6)(2.=) 5 2.08, l ($7222) 5 (2.6)(7 ∐ 2.=) 5 2.=0, y l ($7>22) 5 (7 ∐ 2.6)(2.=) 5 2.70. Rar ca tgkta, cg iaejsjùk nspnrgmg pgrg nc vnkmnmar ns N ( X )
5($2)(2.78) + ($7222)(2.=0) + ($7>22)(2.70) + ($0>22)(2.08) 5 $7922.
Cas ndnepcas =.7 y =.0 sn mjsn÷grak pgrg qun nc cnitar iaeprnkmg endar ca qun qunrneas mnijr iak cg lrgsn vgcar nspnrgma mn ukg vgrjg`cn gcngtarjg. Nk ge`as igsas cgs vgrjg`cns gcngtarjgs sak mjsirntgs. Ynfujeas iak uk ndnepca mn vgrjg`cn gcngtarjg iaktjkug, makmn uk jkfnkjnra sn jktnrnsg nk cg vjmg enmjg mn ijnrta tjpa mn mjspasjtjva qunNctrgksiurrn gktns mn mn cg qun sn ncnitrùkjia. ukgsnjcustrgijùk pra`cneg prnsnktn ukgÊstg lgccgns qun nklrnktg gmnc enkuma nk cgtjnepa prçitjig. vgcar nspnrgma vjmg
mnc mjspasjtjva ns uk pgrçentra jepartgktn pgrg su nvgcugijùk.
77=
Igpåtuca = Nspnrgkzg egtneçtjig
Ndnepca =.94 =.94
Yng X cg cg vgrjg`cn gcngtarjg qun mnkatg cg vjmg nk hargs mn ijnrta mjspasjtjva ncnitrùkjia. Cg lukijùk mn mnksjmgm mn pra`g`jcjmgm ns 02,222
l ( x )
5
x 9
2,
, x ; 722, nk atra igsa.
Igciucn cg vjmg nspnrgmg pgrg pg rg nstg icgsn mn mjspasjtjva. Yacuijùk4 Yj usgeas cg mnkjijùk =.7, tnkneas ∛
¾ 5 N ( X ) 5
02,, 22 02 2222
x 722
x 9
∛
mx 5 722
02,, 22 02 2222 x 0
mx 5 022.
Rar ca tgkta, nspnrgeas qun nstn tjpa mn mjspasjtjva murn nk praenmja 022 hargs. Iaksjmnrneas gharg ukg kunvg vgrjg`cn gcngtarjg f( X X ), ), cg iugc mnpnkmn mn X ? ns mnijr, igmg vgcar mn f( X X ) ns mntnrejkgma par nc vgcar mn X . Rar ndnepca, f( X X ) pamråg snr 0 X a 9 X ’ 7, y sjneprn qun X gsueg gsueg nc vgcar 0, f( X X ) taeg nc vgcar f(0). Nk pgrtjiucgr, sj X ns ns ukg vgrjg`cn gcngtarjg mjsirntg iak mjstrj`uijùk mn pra`g`jcjmgm l ( x x ), ), pgrg x 5 ’7, 2, 0 X ) 5 X X , nktakins, 7, 0 y f( X R _f( X ) R _f( X )
5 2U 5 R ( X 5 2) 5 l (2), 5 7U 5 R ( X 5 ∐7) + R ( X 5 7) 5 l ( ∐7) + l (7), R _f( X ) 5 =U 5 R ( X 5 0) 5 l (0), gså qun cg mjstrj`uijùk mn pra`g`jcjmgm mn f( X X ) sn nsirj`n iaea f( x ) R _f ( X ) 5 f( x )U
2 7 l (2) l (-7) + l (7)
= l (0)
Rar enmja mn cg mnkjijùk mnc vgcar nspnrgma mn ukg vgrjg`cn gcngtarjg a`tnkneas ¾f ( X ) 5 N _f( x )U 52 l (2) + 7_ l ( ∐7) + l (7)U + = l (0) 0
0
0
0
5( ∐7) l ( ∐7) + (2) l (2) + ( 7) l (7) + (0) l (0) 5 x f( x ) l ( x ). Nstn rnsuctgma sn fnknrgcjzg nk nc tnarneg =.7 pgrg vgrjg`cns gcngtarjgs mjsirntgs y iaktjkugs.
]narneg =.74
Yng X ukg ukg vgrjg`cn gcngtarjg iak mjstrj`uijùk mn pra`g`jcjmgm l ( x x ). ). Nc vgcar nspnrgma mn cg vgrjg`cn gcngtarjg f( X X ) ns ¾f ( X ) 5 N _f( X )U 5
f( x ) l ( x ) x
sj X ns ns mjsirntg, y ∛
¾f ( X ) 5 N _f( X )U 5
f( x ) l ( x ) mx ∐∛
sj X ns ns iaktjkug.
=.7 Enmjg mn ukg vgrjg`cn gcngtarjg
Ndnepca =.=4 =.=4
77>
Yupakfg qun nc kôenra mn gutaeùvjcns g utaeùvjcns X qun qun pgsg par uk caigc caig c mn cgvgma mn gutas nktrn cgs =422 .. y cgs >422 .. mn iugcqujnr vjnrkns sacngma tjnkn cg sjfujnktn mjstrj`uijùk mn pra`g`jcjmgm4 x = > < 6 8 : 7 7 7 7 7 7 R ( X 5 x ) 70 70 = = < < Yng f( X X ) 5 0 X ’ 7 cg igktjmgm mn mjknra nk mùcgrns qun nc gmejkjstrgmar pgfg gc apnrg-
Yacuijùk4
mar. cgs=.7, fgkgkijgs nspnrgmgs apnrgmar nk nstn pnrjama nspniåia. Rar ncIgciucn tnarneg nc apnrgmar punmnmnc nspnrgr rnij`jr :
N _f( X )U
5 N (0 X ∐ 7) 5
(0 x ∐ 7) l ( x ) x 5 =
7 7 7 7 + (:) + (77) + (79) 70 70 = = 7 7 + (7>) + (76) 5 $70.4 =.>4
Yng X ukg ukg vgrjg`cn gcngtarjg iak lukijùk mn mnksjmgm x 0
l ( x )
5
, ∐7 3 x 3 0, 2, nk atra igsa. 9
Igciucn nc vgcar nspnrgma mn f( X X ) 5 = X + 9. Yacuijùk4 Rar nc tnarneg =.7 tnkneas 0
N (= X + 9) 5 ∐7
0
x 0 (= x + + 9) x 7 mx 5 9 9
(= x 9 + 9 x 0 ) mx 5 8.
∐7
Mn`neas nxtnkmnr gharg kunstra iakinpta mn nspnrgkzg egtneçtjig gc igsa mn mas x , y). vgrjg`cns gcngtarjgs X y y Z iak iak mjstrj`uijùk mn pra`g`jcjmgm iakduktg l ( x
Mnkjijùk =.04
x , y). Cg Yngk X y Z vgrjg`cns vgrjg`cns gcngtarjgs iak mjstrj`uijùk mn pra`g`jcjmgm iakduktg l ( x X, Z ) ns enmjg a vgcar nspnrgma mn cg vgrjg`cn gcngtarjg f( X,
¾f ( X , Z ) 5 N _f( X , Z )U 5
f( x , y ) l ( x , y ) x
y
sj X y y Z sak sak mjsirntgs, y ¾f ( X , Z ) 5 N _f( X , Z )U 5
∛
∛
∐∛
∐∛
f( x , y ) l ( x , y ) mx my
sj X y y Z sak sak iaktjkugs. Ns nvjmnktn cg fnknrgcjzgijùk mn cg mnkjijùk =.0 pgrg nc içciuca mn cg nspnrgkzg egtneçtjig mn lukijakns mn vgrjgs vgrjg`cns gcngtarjgs.
77<
Igpåtuca = Nspnrgkzg egtneçtjig
Ndnepca =. 08
9 08
7
x
y
]atgcns par iacuekg
Yacuijùk4 Rar
cg mnkjijùk =.0, nsirj`jeas 0
N ( X XZ Z )
0
xy l ( x , y )
5 x 52 y 52
5 (2)(2) l (2,2) + (2)(7) l (2,7) + (7)(2) l (7, 2) + (7)(7) l (7 (7,, 7) + (0)(2) l (0,2) 9 5 l (7, 7) 5 7= 7=..
Ndnepca =.64 =.64
Igciucn N (Z/X ) pgrg cg sjfujnktn lukijùk mn mnksjmgm x (7+ 9 y 0 )
l ( x, y )
=
5
2,
, 2 3 x 3 0, 2 3 y 3 7, nk atra igsa.
Yacuijùk4 ]nkneas
N
Z X
7
0
5 2
y (7 + 9 y 0 )
=
2
7
mxmy 5
y
2
+ 9 y 9 0
my 5
> . 8
X , Z ) 5 X X nk cg mnkjijùk =.0, tnkneas A`snrvn qun sj f( X x l ( x, y ) 5 xf ( x ) (igsa mjsirnta), x y x N ( X ) 5 ∛ ∛ ∛ x l x, y m y m x ( ) 5 ∐∛ ∐∛ ∐∛ xf ( x ) mx (igsa iaktjkua),
makmn f( x x ) ns cg mjstrj`uijùk egrfjkgc mn X . Rar ca tgkta, pgrg igciucgr N ( X X ) nk uk nspgija `jmjenksjakgc, sn punmn utjcjzgr tgkta cg mjstrj`uijùk mn pra`g`jcjmgm iakduktg mn X y Z , iaea cg mjstrj`uijùk egrfjkgc mn X . Mn egknrg sjejcgr, mnkjeas N ( Z )
5
yl ( x, y ) y
∛
x
∐∛
∛ ∐∛
yh ( y )
5 y
yl ( x, y ) mxmy 5
(igsa mjsirnta), ∛
yh ( y ) my
(igsa iaktjkua),
∐∛
makmn h( y y) ns cg mjstrj`uijùk egrfjkgc mn cg vgrjg`cn gcngtarjg Z .
Ndnrijijas
776
Ndnrijijas =.7 Nk nc ndnrijija 9.79 mn cg pçfjkg :0 ssnn prnsnktg cg sjfujnktn mjstrj`uijùk mn pra`g`jcjmgm mn X , nc kôenra
mn jepnrlniijakns qun hgy nk igmg 72 entras mn ukg tncg sjktêtjig, nk raccas iaktjkuas mn gkiha g kiha ukjlaren x 2 7 0 9 = l ( x ) 2.=7 2.96 2.7< 2.2> 2.27 Igciucn nc kôenra praenmja mn jepnrlniijakns qun hgy nk igmg 72 entras mn nstg tncg. =.0 Cg mjstrj`uijùk mjstrj`uijùk mn pra`g`jcjmgm mn cg vgrjg`cn vgrjg`cn gcngtarjg mjsirntg X ns ns 7 x 9 9 ∐ x l ( x ) 5 , x 5 5 2,7,0,9. x = = Igciucn cg enmjg mn X . =.9 Igciucn cg enmjg mn cg vgrjg`cn gcngtarjg ] qun rnprnsnktg nc tatgc mn cgs trns eaknmgs mnc ndnrijija 9.0> mn cg pçfjkg :9. egknrg qun cg pra=.= ^kg eaknmg nstç igrfgmg mn egknrg `g`jcjmgm mn aiurrnkijg mn ukg igrg ns trns vnins egyar qun cg mn ukg iruz. Igciucn nc kôenra nspnrgma mn iruins sj nstg eaknmg sn cgkzg mas vnins. =.> Nk uk dunfa mn gzgr g ukg eudnr sn sn cn pgfgk $9 sj sgig ukg datg a ukg rnjkg, y $> sj sgig uk rny a uk gs mn ukg `grgdg armjkgrjg mn >0 igrtgs. Yj sgig iugcqujnr atrg igrtg, pjnrmn. ±Iuçkta mn`nråg pgfgr sj nc dunfa ns dusta1 cgvgma mn gutas sn =.422 p.e. mn iugcqujnr vjnrkns sacngma rnij`g $6, $:, $77, $79, $7> a $76 sak4 7/70, 7/70, 7/=, 7/=, 7/< y 7/% ns mn 2.7. Yj sn jfkargk tamgs cgs mneçs pêrmjmgs pgrijgcns, ±quê prjeg mn`nråg ia`rgr igmg g÷a cg gsnfurgmarg pgrg tnknr ukg utjcjmgm praenmja mn $>221 =.72 Mas nxpnrtas nk igcjmgm mn knueçtjias nxgejkgk catns mn êstas y gsjfkgk g igmg knueçtjia puktugijakns mn igcjmgm nk ukg nsigcg mn trns puktas. Yng X cg puktugijùk mgmg par nc nxpnrta G y Z cg cg mgmg par nc nxpnrta @. Cg sjfujnktn tg`cg prnsnktg cg mjstrj`uijùk iakduktg pgrg X y y Z . y
9
nspnrgmgs mnc apnrgmar pgrg nstn pnrjama nspniåia. =.6 Yj ukg pnrsakg jkvjnrtn nk ukgs ukgs giijakns nk pgrtjiucgr, nk uk g÷a tjnkn ukg pra`g`jcjmgm mn 2.9 mn a`tnknr ukg fgkgkijg mn $=222 a ukg pra`g`jcjmgm mn 2.6 mn tnknr ukg pêrmjmg mn $7222. ±Iuçc ns cg fgkgkijg nspnrgmg mn nstg pnrsakg1 =.8 Yupakfg qun uk mjstrj`ujmar mn daynråg gktjfug nstç jktnrnsgma nk iaeprgr uk iaccgr mn ara pgrg nc qun tjnkn 2.00 mn pra`g`jcjmgmns mn vnkmnrca iak $0>2 mn utjcjmgm? 2.9< mn vnkmnrca iak $7>2 mn utjcjmgm? 2.08 mn vnkmnrca gc iasta y 2.7= mn vnkmnrca iak ukg pêrmjmg mn $7>2. ±Iuçc ns su utjcjmgm nspnrgmg1 pjcata prjvgma prjvgma mnsng gsnfurgr gsnfurgr su gvjùk par =.: ^k pjcata $022,222. gsnfurgmarg nstjeg cg pra`g`jcjmgm mn pêrmjmgCg tatgc ns mn 2.220, qun qun cg pra`g`jcjmgm mn ukg pêrmjmg mnc >2% ns mn 2.27 y cg pra`g`jcjmg pra`g`jcjmgmm mn ukg
l ( x, y )
7 2.72 2.72 2.29
7 0 9
x
0 2.2> 2.9> 2.72
9 2.20 2.2> 2.02
Igciucn ¾ X y ¾Z . =.77 Cg lukijùk mn mnksjmgm mn cgs enmjijakns iamjigmgs mnc mjçentra mn pgsa mn cas hjcas mn uk nkigdn ns
=
l ( x )
5
ς (7 (7+ x 0 )
2,
, 2 3 x 3 7, nk atra igsa.
Igciucn nc vgcar nspnrgma mn X . =.70 Yj cg utjcjmgm utjcjmgm pgrg uk uk mjstrj`ujmar mjstrj`ujmar mn uk gutaeùvjc kunva, nk ukjmgmns mn $>222, sn punmn vnr iaea ukg vgrjg`cn gcngtarjg X qun qun tjnkn cg sjfujnktn lukijùk mn mnksjmgm 0(7 ∐ x ), 2 3 x 3 7, l ( x ) 5 2, nk atra igsa. Igciucn cg utjcjmgm praenmja par gutaeùvjc. =.79 Cg lukijùk mn mnksjmgm mn cg vgrjg`cn gcngtarjg iaktjkug X , nc kôenra tatgc mn hargs qun ukg lgejcjg
utjcjzg ukg gspjrgmarg murgktn uk g÷a, nk ukjmgmns mn 722 hargs, sn mg nk nc ndnrijija 9.6 mn cg pçfjkg :0 iaea x , l ( x )
5
2 3 x 3 7, 0 ∐ x , 7 ≭ x 3 0, 2, nk atra igsa.
Igciucn nc kôenra praenmja mn hargs par g÷a qun cgs lgejcjgs utjcjzgk sus gspjrgmargs. =.7= Igciucn cg praparijùk X mn mn pnrsakgs qun sn pamråg nspnrgr qun rnspakmjnrgk g ijnrtg nkiunstg qun sn nkvåg par iarrna, sj X tjnkn tjnkn cg sjfujnktn lukijùk mn mnksjmgm l ( x )
5
0( x + 0) >
2,
, 2 3 x 3 7, nk atra igsa.
778
Igpåtuca = Nspnrgkzg egtneçtjig
X,Z ) nstçk =.7> Yupakfg qun mas vgrjg`cns gcngtarjgs ( X,Z
mjstrj`ujmgs mn egknrg ukjlaren nk uk iåriuca iak rgmja g. Nktakins, cg lukijùk mn mnksjmgm mn pra`g`jcjmgm iakduktg ns 7 , x 0 + y 0 ≭ g0 , l ( x, y ) 5 ςg 0 2, nk atra igsa. Igciucn ¾ X , nc vgcar nspnrgma mn X . =.7< Yupakfg ustnm jkspniijakg catn mn 7222y `ae`jccgs mn cuz,qun nktrn cgs iugcns hgy 02ukmnlnituasgs,
ncjfn gc gzgr mas `ae`jccgs mnc catn sjk rnnepcgza. rn nepcgza. Yngk 7, sj cg prjenrg `ae`jccg nstç mnlnituasg, 2, nk atra igsa. 7, sj cg snfukmg `ae`jccg nstç mnlnituasg, X 0 5 2, nk atra igsa. X 7 5
Igciucn cg pra`g`jcjmgm mn qun gc enkas ukg mn cgs `ae`jccgs ncnfjmgs nstê mnlnituasg. _Yufnrnkijg4 Igciucn R( X X c + X X 0 5 7).U ukg vgrjg`cn gcngtarjg iak cg sjfujnktn mjs=.76 Yng X ukg trj`uijùk mn pra`g`jcjmgm4 x l ( x )
∐9 7/<
< 7/0
: 7/9
Igciucn ¾f( X , makmn f( X X ) 5 (0 X + 7)0. X ) =.78 Igciucn nc vgcar nspnrgma mn cg vgrjg`cn gcngtarjg f( X X ) 5 X X 0, makmn X tjnkn tjnkn cg mjstrj`uijùk mn pra`g`jcjmgm mnc ndnrijija =.0. =.7: ^kg neprnsg jkmustrjgc frgkmn iaeprg vgrjas prainsgmarns mn tnxtas kunvas gc kgc mn igmg g÷a? nc kôenra nxgita mnpnkmn mn cg lrniunkijg mn rnpgrgijakns mnc g÷a gktnrjar. Yupakfg qun nc kôenra mn prainsgmarns mn tnxtas, X , qun sn iaeprgk igmg g÷a tjnkn cg sjfujnktn mjstrj`uijùk mn pra`g`jcjmgm4 x 2 7 0 9 l ( x ) 7/72 9/72 0/> 7/> Yj nc iasta mnc eamnca mnsngma ns mn $7022 par ukjmgm y gc kgc mnc g÷a cg neprnsg a`tjnkn uk mnsiunkta mn >2 X 0 mùcgrns, ±iuçkta nspnrg fgstgr nstg neprnsg nk kunvas prainsgmarns mn tnxtas murgktn nstn g÷a1 tjnkn cg sjfujnk=.02 ^kg vgrjg`cn gcngtarjg iaktjkug X tjnkn tn lukijùk mn mnksjmgm l ( x )
5
∐ n x , x ; ;
2,
2, nk atra igsa.
Igciucn nc vgcar nspnrgma mn f( X X ) 5 n0 X /9. =.07 ±Iuçc ns cg utjcjmgm praenmja par gutaeùvjc qun a`tjnkn mjstrj`ujmar, mjstrj`ujmar , sj cg utjcj mgmvgrjg`cn nk igmg gcngtarjg uka nstç mgmg parukf( X X ) 5 X X 0, makmn X ns utjcjmgm ns ukg qun tjnkn cg lukijùk mn mnksjmgm mnc ndnrijija =.701
pg=.00 Nc pnrjama mn haspjtgcjzgijùk, nk mågs, pgrg pg-
ijnktns qun sjfunk nc trgtgejnkta pgrg ijnrta tjpa mn trgstarka rnkgc ns ukg vgrjg`cn gcngtarjg Z 5 X + =, makmn X tjnkn cg sjfujnktn lukijùk mn mnksjmgm l ( x )
90 ( x + =) 9
5
2,
, x ; ; 2, nk atra igsa.
Igciucn nc kôenra praenmja mågsmnqun ukg nc pnrsakg pnregknin haspjtgcjzgmg iak mn nc k snfujr trgtgejnkta pgrg mjihg nklnrenmgm. tjnknk cg sjfujnktn lukijùk =.09 Yupakfg qun X y Z tjnknk mn pra`g`jcjmgm iakduktg4 l ( x, y )
7 9 >
y g) `)
0 2.72 2.02 2.72
x
= 2.7> 2.92 2.7>
X , Z ) 5 XZ XZ 0. Igciucn nc vgcar nspnrgma mn f( X Igciucn ¾ X y ¾Z .
=.0= Pneåtgsn g cgs vgrjg`cns gcngtarjgs iuyg mjstrj-
`uijùk mn pra`g`jcjmgm iakduktg sn mg nk nc ndnrijija 9.9: mn cg pçfjkg 72> y g) igciucn N ( X X 0Z ’ 0 XZ )? )? `) igciucn ¾ X ’ ¾Z . =.0> Pneåtgsn g cgs vgrjg`cns gcngtarjgs iuyg mjstrj`uijùk mn pra`g`jcjmgm iakduktg sn mg nk nc ndnrijija 9.>7 mn cg pçfjkg 72< y igciucn cg enmjg pgrg nc kôenra tatgc mn datgs y rnyns iugkma sn sgigk 9 igrtgs, sjk rnnepcgza, mn cgs 70 igrtgs egyarns mn ukg `grgdg armjkgrjg mn >0 igrtgs. y Z cgs cgs sjfujnktns vgrjg`cns gcngtarjgs iak =.0
2,
09.6> ≭ x ≭ 0, nk atra igsa.
gcngtarjgss =.0 \grjgkzg y iavgrjgkzg mn vgrjg`cns gcngtarjg
77:
g) `) i)
Frgqun cg lukijùk lukijùk mn mn mnksjmgm. mnksjmgm. Igciucn nc vgcar nspnrgma a pnsa enmja enmja nk akzgs. ±Yn sarprnkmn mn su rnspunstg nk `)1 Nxpcjqun ca qun rnspakmg. =.0: Nc ndnrijija 9.0: mn cg pçfjkg :9 sn rnnrn g ukg jepartgktn mjstrj`uijùk mnc tgeg÷a mn cgs pgrtåiucgs igrgitnrjzgmg par 9 x ∐ = , x ; ; 7, l ( x ) 5 2, nk atra igsa. g) Frgqun cg lukijùk mn mnksjmgm. `) Mntnrejkn nc tgeg÷a enmja mn cg pgrtåiucg. =.92 Nk nc ndnrijija 9.97 mn cg pçfjkg := cg mjstrj`uijùk mnc tjnepa qun trgksiurrn gktns mn qun ukg cgvgmarg rnqujnrg ukg rnpgrgijùk egyar lun mgmg iaea 7 ∐ y / = n , y ≮ 2, l ( y ) 5 = 2, nk atra igsa. ±Iuçc ns cg enmjg mn pa`cgijùk mnc tjnepa qun trgksiurrn gktns mn rnqunrjr cg rnpgrgijùk1
=.0
=.97 Iaksjmnrn nc ndnrijija 9.90 mn cg pçfjkg :=. g) ±Iuçc ns cg praparijùk enmjg mnc prnsupunsta gsjf-
kgma pgrg nc iaktrac ge`jnktgc y mn cg iaktgejkgijùk1 `) ±Iuçc ns cg cg pra`g`jcjmgm mn mn qun ukg neprnsg ncnfjmg gc gzgr tnkfg ukg praparijùk gsjfkgmg pgrg nc iaktrac ge`jnktgc y mn cg iaktgejkgijùk qun nxinmg cg enmjg mn cg pa`cgijùk mgmg nk g)1 Nk nc ndnrijija 9.79 mn cg pçfjkg :0 cg mjstrj`uijùk mnc kôenra mn jepnrlniijakns nk igmg 72 entras mn tncg sjktêtjig lun mgmg par =.90
x l ( x )
2 2.=7
7 0 2.96 2.7<
9 2.2>
= 2.27
g) `)
Frgqun cg lukijùk mn pra`g`jcjmgm. Igciucn nc kôenra mn jepnrlniijakns nspnrgma N ( X X ) 5 ¾ ¾. i) Igciucn N ( X X 0).
\grjgkzg y iavgrjgkzg mn vgrjg`cns gcngtarjgs Cg enmjg a vgcar nspnrgma mn ukg vgrjg`cn gcngtarjg X ns ns mn nspnijgc jepartgkijg nk nstgmåstjig parqun mnsirj`n nk mùkmn sn inktrg cg mjstrj`uijùk mn pra`g`jcjmgm. Yjk ne`grfa, cg enmjg par så ejseg ka alrnin ukg mnsirjpijùk gmniugmg mn cg lareg mn cg mjstrj`uijùk. ]ge`jêk sn kninsjtg icgsjigr cg vgrjg`jcjmgm nk cg mjstrj`uijùk. Nk cg furg =.7 tnkneas cas hjstafrgegs mn mas mjstrj`uijakns mn pra`g`jcjmgm mjsirntgs iak cg ejseg enmjg ¾ 5 0, pnra qun mjnrnk mn egknrg iaksjmnrg`cn nk cg vgrjg`jcjmgm a mjspnrsjùk mn sus a`snrvgijakns sa`rn cg enmjg.
7
0 ( g)
9
x
2
7
0
9
=
x
(`)
Ljfurg =.74 Mjstrj`uijakns iak enmjgs jfugcns y mjspnrsjakns mjlnrnktns.
X sn a`tjnkn Cg enmjmg mn vgrjg`jcjmgm eçs vgrjg`cn gcngtarjg X X ’ ¾ X ¾)0. G mn gpcjigkma nc tnarneg =.7 iak f( X ) 5 (jepartgktn nstgukg igktjmgm sn cn mnkaejkg vgrjgkzg X a X y mn cg vgrjg`cn gcngtarjg X a vgrjgkzg mn cg mjstrj`uijùk mn pra`g`jcjmgm mn X y sn
702
Igpåtuca = Nspnrgkzg egtneçtjig
mnkatg iaea \gr( \gr( X X ), ), a iak nc såe`aca ω X 0 , a sjepcnenktn iaea ω 0 iugkma ns nvjmnktn g quê vgrjg`cn gcngtarjg sn nstç hgijnkma rnlnrnkijg. Mnkjijùk =.94
Yng X ukg ukg vgrjg`cn gcngtarjg iak mjstrj`uijùk mn pra`g`jcjmgm l ( x x ) y enmjg ¾. Cg vgrjgkzg mn X ns ns ω 0 5 N _( X X ∐ ¾) 0 U 5
( x ∐ ¾) 0 l ( x ) ,
sj X ns mjsirntg, y
x
∛
X ∐ ¾) 0 U 5 ω 0 5 N _( X
( x ∐ ¾) 0 l ( x ) mx ,
sj X ns iaktjkug.
∐∛
Cg rgåz iugmrgmg pasjtjvg mn cg vgrjgkzg, ω , sn ccgeg mnsvjgijùk nstçkmgr mn X . Cg igktjmgm x ’ ¾ nk cg mnkjijùk =.9 sn ccgeg mnsvjgijùk mn ukg a`snrvgijùk rnspnita g su enmjg. Iaea nstgs mnsvjgijakns sn ncnvgk gc iugmrgma y mnspuês sn praenmjgk, ω 0 snrç euiha enkar pgrg uk iakdukta mn vgcarns x qun qun nstêk inrigkas g ¾, qun pgrg uk iakdukta mn vgcarns qun vgrån mn lareg iaksjmnrg`cn mn ¾.
Ndnepca =.84 =.84
Yupakfg qun cg vgrjg`cn gcngtarjg X rnprnsnktg rnprnsnktg nc kôenra mn gutaeùvjcns qun sn utjcjzgk iak prapùsjtas mn knfaijas aijgcns nk uk måg mn trg`gda mgma. Cg mjstrj`uijùk mn pra`g`jcjmgm pgrg cg neprnsg G _furg =.7(g)U ns x l ( x )
7 2.9
0 2.=
9 2.9
y pgrg cg neprnsg @ _furg =.7(`)U ns x l ( x )
2 2.0
7 2.7
0 2.9
9 2.9
= 2.7
Mneunstrn qun cg vgrjgkzg mn cg mjstrj`uijùk mn pra`g`jcjmgm pgrg cg neprnsg @ ns egyar qun cg mn cg neprnsg G. Yacuijùk4 Rgrg cg neprnsg G nkiaktrgeas qun ¾ G 5 N ( X ) 5 (7)(2.9) + (0)(2.=) + (9)(2.9) 5 0.2,
y nktakins
9
0
( x ∐ 0) 0 5 (7 ∐ 0) 0 (2.9) + ( 0 ∐ 0) 0 (2.=) + ( 9 ∐ 0) 0 (2.9) 5 2.7) + (7)(2.98) + (0)(2.72) + (9)(2.27) 5 2.7) + (7)(2.98) + (=)(2.72) + (:)(2.27) 5 2.86.
Rar ca tgkta, Ndnepca =.724 =.724
ω 0 5 2.86 ∐ (2. ω 5 ∐ < 9 0
5
> 9
76 . <
7 . 78
Hgstg nc eaenkta cg vgrjgkzg a cg mnsvjgijùk nstçkmgr sùca tjnkn sjfkjigma iugkma iaepgrgeas mas a eçs mjstrj`uijakns qun tjnknk cgs ejsegs ukjmgmns mn enmjmg. Rar ca tgkta, pamneas iaepgrgr cgs vgrjgkzgs mn cgs mjstrj`uijakns mn iaktnkjma, enmjma nk cjtras, mn `atnccgs mn dufa mn kgrgkdg mn mas neprnsgs, y nc vgcar eçs frgkmn jkmjigråg cg neprnsg iuya pramuita ns eçs vgrjg`cn a enkas ukjlaren. Ka tnkmråg igsa iaepgrgr cg vgrjgkzg mn ukg mjstrj`uijùk mn nstgturgs iak cg vgrjgkzg mn ukg mjstrj`uijùk mn igcjigijakns mn gptjtum. Nk cg sniijùk =.= eastrgeas iùea sn utjcjzg cg mnsvjgijùk nstçkmgr pgrg mnsirj`jr ukg sacg mjstrj`uijùk mn a`snrvgijakns. Nxtnkmnrneas gharg kunstra iakinpta mn vgrjgkzg mn ukg vgrjg`cn gcngtarjg X pgrg pgrg X ) cg jkicujr tge`jêk vgrjg`cns gcngtarjgs rncgijakgmgs iak X . Rgrg cg vgrjg`cn gcngtarjg f( X 0 vgrjgkzg sn mnkatgrç par ω f ( X ) y sn igciucgrç nepcngkma nc sjfujnktn tnarneg.
]narneg =.94
Yng X ukg ukg vgrjg`cn gcngtarjg iak mjstrj`uijùk mn pra`g`jcjmgm l ( x x ). ). Cg vgrjgkzg mn cg vgrjg`cn gcngtarjg f( X X ) ns ω f0( X ) 5 N {_f( X ) ∐ ¾f ( X ) U0 } 5
_f( x ) ∐ ¾f ( X ) U0 l ( x ) x
sj X ns ns mjsirntg, y 0
∛
0
ω f ( X ) 5 N {_f( X ) ∐ ¾f ( X ) U } 5
∐∛
_f( x ) ∐ ¾f ( X ) U0 l ( x ) mx
sj X ns ns iaktjkug.
Rrun`g4 Iaea
f( X X )
ns nk så ejseg ukg vgrjg`cn gcngtarjg iak enmjg ¾f( X , iaea sn mnkn nk nc X ) tnarneg =.7, mn cg mnkjijùk =.9 sn mnmuin qun ω f0( X ) 5 N {_f( X ) ∐ ¾f ( X ) U}.
Gharg `jnk, cg mneastrgijùk sn iaepcntg gpcjigkma kunvgenktn nc tnarneg =.7 g cg vgrjg`cn gcngtarjg _f( X X ) ’ ¾ ¾f( X U0. X ) Ndnepca =.774 =.774
Igciucn cg vgrjgkzg mn f( X X ) 5 0 X + 9, makmn X ns ns ukg vgrjg`cn gcngtarjg iak cg sjfujnktn mjstrj`uijùk mn pra`g`jcjmgm x l ( x )
2 7 =
7 7 8
0 7 0
9 7 8
=.0 \grjgkzg y iavgrjgkzg mn vgrjg`cns gcngtarjg gcngtarjgss
Yacuijùk4
709
Rrjenra sn igciucg cg enmjg mn cg vgrjg`cn gcngtarjg 0 X + 9. Mn giunrma iak nc tnarneg =.7, 9
¾ 0 X + + 9 5 N (0 X + 9) 5
(0 x + + 9) l l ( x ) 5 . Igciucn cg vgrjgkzg mn cg vgrjg`cn gcngtarjg f( X X ) 5 = X + 9. Yacuijùk4 Nk nc ndnepca =.> nkiaktrgeas qun ¾= X + 9 5 8. Gharg `jnk, usgkma nc tnarneg =.9,
Ndnepca =.704 =.704
0 0 ω =0 X + + 9 5 N {_(= X + 9) ∐ 8U } 5 N _(= X ∐ >) U 0
0 0 x
7 mx 5 5 (= x ∐ ∐ >) 9 9 ∐7
0
(7 x 0 ) mx 5
∐7
>7 . >
X , Z ) 5 ( X X ’ ¾ X ) y ¾Z 5 N Yj f( X ¾ X )(Z ’ ¾ ¾Z ), makmn ¾ X 5 N N ( X N (Z ), ), cg mnkjijùk =.0 mg uk X , Z ). vgcar nspnrgma mnkaejkgma iavgrjgkzg mn X y Z , qun sn mnkatg par ω XZ a Iav( X
Mnkjijùk =.=4
x, y). Cg Yngk X y Z vgrjg`cns vgrjg`cns gcngtarjgs iak mjstrj`uijùk mn pra`g`jcjmgm iakduktg l ( x, iavgrjgkzg mn X y y Z ns ns X ∐ ¾ X )( Z ∐ ¾Z )U 5 ω XZ 5 N _( X
( x ∐ ¾ X )( y ∐ ¾ y ) l ( x, y ) x
y
sj X y y Z sak sak mjsirntgs, y ∛
∛
∐∛
∐∛
ω XZ 5 N _( X X ∐ ¾ X )( Z ∐ ¾Z )U 5
( x ∐ ¾ X )( y ∐ ¾ y ) l ( x, y ) mx my
sj X y Z sak iaktjkugs. Cg iavgrjgkzg nktrn mas vgrjg`cns gcngtarjgs ns ukg enmjmg mn cg kgturgcnzg mn cg gsaijgijùk nktrn ge`gs. Yj vgcarns frgkmns mn X g g enkuma mgk iaea rnsuctgma vgcarns frgkmns mn Z , a vgcarns pnqun÷as mn X, mgk iaea rnsuctgma vgcarns pnqun÷as mn Z , X ’ ¾ ¾ X pasjtjvg iak lrniunkijg mgrç iaea rnsuctgma Z ’ ¾ ¾Z pasjtjvg, y X ’ ¾ ¾ X knfgtjvg g enkuma mgrç iaea rnsuctgma Z ’ ¾¾Z knfgtjvg. Rar iaksjfujnktn, nc pramuita ( X X ’ ¾ ¾ X ) (Z ’ ¾ ¾Z ) tnkmnrç g snr pasjtjva. Rar atra cgma, sj iak lrniunkijg vgcarns frgkmns mn X mgk mgk iaea rnsuctgma vgcarns pnqun÷as mn Z , nktakins nc pramuita ( X ’ ¾¾ X )(Z ’ ¾¾Z ) tnkmnrç g snr knfgtjva. Nc sjfka mn cg iavgrjgkzg jkmjig sj cg rncgijùk nktrn mas vgrjg`cns gcngtarjgs mnpnkmjnktns ns pasjtjvg a knfgtjvg. Iugkma X y y Z sak sak nstgmåstjigenktn jkmnpnkmjnktns, sn punmn mneastrgr qun cg iavgrjgkzg ns inra (vêgsn nc iaracgrja =.>). Ca apunsta, sjk ne`grfa, par ca fnknrgc ka ns ijnrta. Mas vgrjg`cns punmnk tnknr iavgrjgkzg inra y guk gså ka snr nstgmåstjigenktn jkmnpnkmjnktns. A`snrvn qun cg iavgrjgkzg sùca mnsirj`n cg cjkngc nktrn mas vgrjg`cns gcngtarjgs. Rar iaksjfujnktn, sj ukg iavgrjgkzg nktrn X rncgijùk X y y Z ns ns inra, y Z pamrågk pamrågk tnknr ukg rncgijùk ka cjkngc, ca iugc sjfkjig qun ka kninsgrjgenktn sak jkmnpnkmjnktns.
70=
Igpåtuca = Nspnrgkzg egtneçtjig
Cg lùreucg gctnrkgtjvg qun sn prnnrn pgrg ω XZ sn nstg`cnin nk nc tnarneg =.=. ]narneg =.=4
Cg iavgrjgkzg iavgrjgkzg mn mas vgrjg`cns gcngtarjgs X y y Z, iak enmjgs ¾ X y ¾Z , rnspnitjvgenktn, rnspnitjvgenktn, nstç mgmg par XZ Z ) ∐ ¾ X ¾Z . ω XZ 5 N ( X
Rrun`g4 Rgrg
nc igsa mjsirnta nsirj`jeas ( x ∐ ∐ ¾ X )( y ∐ ¾Z ) l ( x, y )
ω XZ XZ 5 x
y
xy l ( x, y )
5 x
x
y
x l ( x, y ) + ¾ X ¾ Z
Z
x
y
l ( x, y ). x
x l ( x, y ) , ¾ Z 5
¾ X 5
X
y
∐ ¾ Mgma qun
y l ( x, y )
∐ ¾
x
y l ( x, y ) ,
y
y
y
l ( x, y ) x
57
y
pgrg iugcqujnr mjstrj`uijùk mjsirntg iakduktg sn mnmuin qun ω XZ 5 N ( X XZ Z ) ∐ ¾ X ¾ Z ∐ ¾Z ¾ X + ¾ X ¾ Z 5 N ( X XZ Z ) ∐ ¾ X ¾ Z .
Rgrg nc igsa iaktjkua cg mneastrgijùk ns jmêktjig, pnra cgs suegtarjgs sn rnnepcgzgk par jktnfrgcns. Ndnepca =.794 =.794
Nk nc ndnepca 9.7= mn cg pçfjkg :> sn mnsirj`n ukg sjtugijùk ginrig mnc kôenra mn rnpunstas gzucns X y y nc kôenra mn rnpunstas radas Z . Iugkma mn ijnrtg igdg sn sncniijakgk mas rnpunstas pgrg `acåfrgla gc gzgr y cg mjstrj`uijùk mn pra`g`jcjmgm iakduktg ns cg sjfujnktn,
y
l ( x, y )
2
2 7 0 f( x )
9 08 9 7= 7 08 > 7=
x
7
0
h ( y )
: 08 9 7=
9 08
2
2 2
7> 08 9 6 7 08
7> 08
9 08
7
Igciucn cg iavgrjgkzg mn X y y Z . Yacuijùk4 Mnc ndnepca =.< vneas qun N ( XZ XZ ) 5 9/7=. Gharg `jnk, 0
¾ X 5
> 7> 9 9 + (7) + (0) 5 , 7= 08 08 =
xf ( x )
5 (2)
yh ( y )
5 (2) 7> + (7) 9 + (0) 7 5 7 . 08 6 08 0
x 52
y ¾Z 5
0 y 52
gcngtarjgss =.0 \grjgkzg y iavgrjgkzg mn vgrjg`cns gcngtarjg
70>
Rar ca tgkta, ω XZ 5 N ( X XZ Z ) ∐ ¾ X ¾ Z 5
Ndnepca =.7=4 =.7=4
7 : 5∐ . 0 ><
Cg lrgiijùk X mn iarrnmarns y cg lrgiijùk Z mn iarrnmargs qun iaepjtnk nk igrrnrgs mn egrgtùk sn mnsirj`nk enmjgktn cg lukijùk mn mnksjmgm iakduktg l ( x, y )
9 9 ∐ 7= =
xy y x 5 8 , 2 ≭ ≭ ≭ 7, 2, nk atra igsa.
Igciucn cg iavgrjgkzg mn X y y Z . Yacuijùk4 Rrjenra igciucgeas cgs lukijakns mn mnksjmgm egrfjkgc. Êstgs sak f( x )
y
= x 9 , 2 ≭ x ≭ 7, 2, nk atra igsa,
5
= y (7 ∐ y 0 ), 2 ≭ y ≭ 7, h ( y ) 5 2, nk atra igsa,
G pgrtjr mn cgs lukijakns mn7 mnksjmgm egrfjkgc mgmgs, igciucgeas 7 = 8 = = x mx 5 y ¾Z 5 = y 0 (7 ∐ y 0 ) my 5 . ¾ X 5 N ( X ) 5 > 7> 2 2 Mn cgs lukijakns mn mnksjmgm iakduktg mgmgs grrj`g, tnkneas 7
N ( X XZ Z )
7
5 2
y
Nktakins,
= 8 x 0 y 0 mx my 5 . : = :
XZ Z ) ∐ ¾ X ¾ Z 5 ∐ ω XZ 5 N ( X
= >
8 = 5 . 7> 00>
Gukqun cg mas vgrjg`cns jklaregijùk rnspnita cg kgturgcnzg mniavgrjgkzg cg rncgijùk,nktrn cg egfkjtum mn ω XZ gcngtarjgs ka jkmjig`rjkmg kgmg rnspnita g cg lunrzg mn mn cg rncgijùk, yg qun ω XZ mnpnkmn mn cg nsigcg. Yu egfkjtum mnpnkmnrç mn cgs cg s ukjmgmns qun sn utjcjink pgrg enmjr X y y Z . Hgy ukg vnrsjùk mn cg iavgrjgkzg sjk nsigcg qun sn mnkaejkg ianijnktn mn iarrncgijùk y sn utjcjzg gepcjgenktn nk nstgmåstjig.
Mnkjijùk =.>4
Yngk X y Z vgrjg`cns vgrjg`cns gcngtarjgs iak iavgrjgkzg ω XZ y mnsvjgijakns nstçkmgr ω X y ω Z , rnspnitjvgenktn. Nc ianijnktn mn iarrncgijùk mn X y y Z ns ns χ XZ 5
ω XZ . ω X ω Z Z
Mn`nråg qunmgr icgra pgrg nc cnitar qun χ XZ ka tjnkn cgs ukjmgmns mn X y y Z . Nc ianijnktn mn iarrncgijùk cg mnsjfugcmgm 7 ≭ χ XZ ≭mjfgeas 7. ]aegZ uk g + mn `X inra ω 5 2. Makmn hgy sgtjslgin ukg mnpnkmnkijg cjkngc ’nxgitg, ≧vgcar , χ XZ iugkma 5 7 sj XZ
70<
Igpåtuca = Nspnrgkzg egtneçtjig
` ; 2 y χ XZ 5 - 7 sj ` 3 2. (\êgsn (\êgsn nc ndnrijija =.=8). Nk
nc igpåtuca 70, makmn nxgejkgrneas cg rnfrnsjùk cjkngc, gkgcjzgeas eçs g lakma nc ianijnktn mn iarrncgijùk.
y Z nk nk nc ndnepca =.79. Ndnepca =.7>4 =.7>4 Igciucn nc ianijnktn mn iarrncgijùk nktrn X y Yacuijùk4 Mgma qun N ( X 0 )
5 (2 0 )
> 7> 9 06 + (70 ) + (00 ) 5 7= 08 08 08
y N ( Z 0 )
5 ( 20 )
7> 9 7 = + (70 ) + (00 ) 5 , 08 6 08 6
a`tnkneas 06 9 ω X 5 ∐ 08 =
0
=> = 7 5 y ω Z 0 5 ∐ 770 6 0
0
0
5
: . 08
Rar ca tgkta, nc ianijnktn mn iarrncgijùk nktrn X y y Z ns ns χ XZ 5
ω XZ 5 ω X ω Z Z
∐:/ >< 5 ∐ 7 . > (=>// 77 (=> 770) 0)(:/ (:/ 08 08))
Ndnepca =.7
0
5 0 6>
y ω 0 5 Z
7 ∐ 8 9 7>
5 77 . 00>
Rar ca tgkta,
χ XZ 5
=/ 00 00>> = 5 . (0/ 6> 6>)( )(77 77 / 00>) ∔ tjnkn egyar egfkjtum (sjk jepartgr nc sjfka) qun cg mnc ndnepca =.7 2.>
Igciucn cg mnsvjgijùk nstçkmgr mn X . =.9> Cg vgrjg`cn gcngtarjg X , qun rnprnsnktg nc kôenra mn nrrarns par 722 cåkngs mn iùmjfa mn prafrgegijùk, tjnkn cg sjfujnktn mjstrj`uijùk mn pra`g`jcjmgm4 x l ( x )
0 2.27
9 2.0>
= 2.=
> 2.9
< 2.2=
^tjcjin nc tnarneg =.0 mn cg pçfjkg 707 pgrg igciucgr cg vgrjgkzg mn X . =.9< Yupakfg qunncêitrjig cgs pra`g`jcjmgmns cgs pra`g`jcj mgmns mn qun 2, 7, 0 a 9 lgccgs mn nknrfåg glnitnk ijnrtg su`mjvjsjùk
nk iugcqujnr g÷a mgma sak 2.=, 2.9, 2.0 y 2.7, rnspnitjvgenktn. Igciucn cg enmjg y cg vgrjgkzg mn cg vgrjg`cn gcngtarjg X qun qun rnprnsnktg nc kôenra mn lgccgs mn nknrfåg qun glnitgk nstg su`mjvjsjùk. =.96 Cg utjcjmgm qun a`tjnkn uk mjstrj`ujmar, nk ukjmgmns mn $>222, gc vnkmnr uk gutaeùvjc kunva ns ukg vgrjg`cn gcngtarjg X qun tjnkn cg lukijùk mn mnksjmgm qun sn prnsnktg nk nc ndnrijija =.70 mn cg pçfjkg 776. Igciucn cg vgrjgkzg mn X . =.98 Cg praparijùk mn pnrsakgs qun rnspakmnk ijnrtg nkiunstg qun sn egkmg par iarrna ns ukg vgrjg`cn gcngtarjg X , cg iugc tjnkn cg lukijùk mn mnksjmgm mnc ndnrijija =.7= mn cg pçfjkg 776. Igciucn cg vgrjgkzg mn X . =.9: Nc kôenra kôenra tatgc mn hargs qun ukg lgejcjg utjcjzg ukg gspjrgmarg nk uk g÷a, nk ukjmgmns mn 722 hargs, ns ukg vgrjg`cn gcngtarjg X qun qun tjnkn cg lukijùk mn mnksjmgm mgmg nk nc ndnrijija =.79 mn cg pçfjkg 776. Igciucn cg vgrjgkzg mn X . =.=2 Pneåtgsn gc ndnrijija =.7= mn cg pçfjkg pçfjkg 776 y 0 igciucn ω f0 ( X f X X X X ) pgrg cg lukijùk ( ) 5 9 + =. =.=7 Igciucn cg mnsvjgijùk nstçkmgr mn cg vgrjg`cn X ) 5 (0 X + 7)0 mnc ndnrijija =.76 nk cg pçgcngtarjg f( X fjkg 778. =.=0 ^tjcjin cas rnsuctgmas mnc ndnrijija =.07 mn cg pç0
X ) 5 X X , makmn X ns fjkg 778 y igciucn cg vgrjgkzg ukg vgrjg`cn gcngtarjg qun tjnknmncgf( X lukijùk mn mnksjmgm mnc ndnrijija =.70 mn cg pçfjkg 776.
5
/ = 7 ∐ x / , =n
x ; ;
2
2, nk atra igsa. Igciucn cg enmjg y cg vgrjgkzg mn cg vgrjg`cn gcngtarjg Z . =.== Igciucn cg iavgrjgkzg mn cgs vgrjg`cns gcngtarjgs X y y Z mnc mnc ndnrijija 9.9: mn cg pçfjkg 72>. =.=> Igciucn cg iavgrjgkzg mn cgs vgrjg`cns gcngtarjgs X y y Z mnc mnc ndnrijija 9.=: mn cg pçfjkg 72. =.=8 Mgmg ukg vgrjg`cn gcngtarjg X , iak mnsvjgijùk nstçkmgr ω X y ukg vgrjg`cn gcngtarjg Z 5 g + `X , mneunstrn qun sj ` 3 2, nc ianijnktn mn iarrncgijùk χ XZ 5 ’ 7, y sj ` ; 2, χ XZ 5 7.
Iaksjmnrn cg sjtugijùk mnc ndnrijija =.90 mn cg pçfjkg 77:. Cg mjstrj` mjstrj`uijùk uijùk mnc kôenra mn jepnrlniijakns par igmg 72 entras mn tncg sjktêtjig nstç mgmg par =.=:
x l ( x )
2 2.=7
7 2.96
0 2.7<
9 2.2>
= 2.27
Igciucn cg vgrjgkzg y cg mnsvjgijùk nstçkmgr mnc kôenra mn jepnrlniijakns. =.>2 Nk ukg tgrng mn cg`argtarja, sj nc nqujpa nstç lukijakgkma, cg lukijùk mn mnksjmgm mnc rnsuctgma a`snrvgma X ns ns 0(7 ∐ x ), 2 3 x 3 7, l ( x ) 5 2, nk atra igsa. Igciucn cg vgrjgkzg y cg mnsvjgijùk nstçkmgr mn X . =.>7 Mntnrejkn nc ianijnktn mn iarrncgijùk nktrn X y Z pgrg pgrg cgs vgrjg`cns gcngtarjgs X y y Z mnc mnc ndnrijija 9.9: mn cg pçfjkg 72>. y Z tjnknk tjnknk cg sjfujnktn =.>0 Cgs vgrjg`cns gcngtarjgs X y mjstrj`uijùk iakduktg 0, 2 3 x ≭ y 3 7, l ( x , y ) 5 2, nk at atra ig igsa. Mntnrejkn nc ianijnktn mn iarrncgijùk nktrn X y Z .
708
Igpåtuca = Nspnrgkzg egtneçtjig
=.9
Enmjgs y vgrjgkzgs mn iae`jkgijakns cjkngcns mn vgrjg`cns gcngtarjgs Gharg nstumjgrneas gcfukgs prapjnmgmns ôtjcns qun sjepcjigrçk cas içciucas mn cgs enmjgs y cgs vgrjgkzgs mn vgrjg`cns gcngtarjgs qun gpgrnink nk cas sjfujnktns igpåtucas. Nstgs prapjnmgmns kas pnrejtjrçk aiupgrkas mn cgs nspnrgkzgs egtneçtjigs nk têrejkas mn atras pgrçentras qun yg iakaineas a qun yg igciucgeas iak lgijcjmgm. ]amas cas rnsuctgmas rn suctgmas qun prnsnktgeas gquå sak vçcjmas pgrg vgrjg`cns gcngtarjgs iaktjkugs y mjsirntgs. Cgs mneastrgijakns sn mgk sùca pgrg nc igsa iaktjkua. Iaenkzgeas iak uk tnarneg y mas iaracgrjas qun mn`nrågk snr, mn lareg jktujtjvg, jktujtjvg, rgzakg`cns pgrg nc cnitar.
]narneg =.>4
Yj g y ` sak iakstgktns, nktakins, N ( gX + `)
Rrun`g4 Rar
5 gN ( X ) + `.
cg mnkjijùk mn vgcar nspnrgma, ∛
N ( gX + `)
∛
( gx + `) l ( x ) mx 5 g
5 ∐∛
∛
x l ( x ) mx + ` ∐∛
l ( x ) mx . ∐∛
X ) y cg snfukmg jktnfrgc ns jfugc g 7. Rar ca tgkta, Cg prjenrg jktnfrgc mn cg mnrnihg ns N ( X N ( gX + `) 5 gN ( X ) + `. Iaracgrja =.74
Gc nstg`cninr qun g 5 2 vneas qun N (`) 5 `.
Iaracgrja =.04
X ). Gc nstg`cninr qun ` 5 2 vneas qun N (gX ) 5 gN ( X ).
Ndnepca =.764 =.764
Gpcjqun nc tnarneg =.> g cg vgrjg`cn gcngtarjg mjsirntg mjsirn tg l ( X X ) 5 0 X ’ 7 pgrg rnsacvnr mn kunva nc ndnepca =.= mn cg pçfjkg 77>. Yacuijùk4 Mn giunrma iak nc tnarneg =.>, nsirj`jeas N (0 X ∐ 7) 5 0 N ( X )
∐ 7.
Gharg, :
¾ 5 N ( X ) 5
x l ( x x ) x 5=
5 (=)
7 7 7 7 7 7 =7 + (>) + ( gpcjqun nc tnarneg =.> g cg vgrjg`cn gcngtarjg iaktjkug f( X X ) 5 = X + 9. Yacuijùk4 Nk nc ndnepca =.> utjcjzgeas nc tnarneg =.> pgrg nsirj`jr
Ndnepca =.784 =.784
N (= X + 9) 5 = N ( X ) + 9.
Gharg,
0
N ( X )
x 0
0
9
mx 5 ∐ 7
∐ 7 x
5
Rar ca tgkta, N (= X + 9) 5
( =)
x 9
> 9 mx 5 = .
> + 9 5 8, =
iaea gktns. ]narneg =. > + ∐ 0 5 , < 9 0
gså qun cg mnegkmg snegkgc praenmja mn cg `n`jmg nk nstg igmnkg mn tjnkmgs mn g`grratns ns mn 0>22 cjtras. Yupakfg qun tnkneas mas vgrjg`cns gcngtarjgs X y Z iak mjstrj`uijùk mn pra`g`jcjmgm iakduktg l ( x x , y). Mas prapjnmgmns gmjijakgcns qun snrçk euy ôtjcns nk cas igpåtucas sjfujnktns jkicuynk cas vgcarns nspnrgmas mn cg sueg, cg mjlnrnkijg y nc pramuita mn nstgs mas vgrjg`cns gcngtarjgs. Yjk ne`grfa, iaenkzgrneas par mneastrgr uk tnarneg sa`rn nc vgcar nspnrgma mn cg sueg a mjlnrnkijg mn lukijakns mn cgs vgrjg`cns mgmgs. Rar supunsta, tgk sùca sn trgtg mn ukg nxtnksjùk mnc tnarneg =.4
Yngk X y y Z mas mas vgrjg`cns gcngtarjgs jkmnpnkmjnktns. Nktakins, ω XZ 5 2.
Rrun`g4 Cg
Ndnepca =.074 =.074
mneastrgijùk sn punmn rngcjzgr utjcjzgkma cas tnarnegs =.= y =.8.
Yn sg`n qun cg praparijùk mn fgcja y grsnkjura ka glnitg nc lukijakgejnkta mn cgs a`cngs mn grsnkjura mn fgcja qun sak cas prjkijpgcns iaepaknktns mn cas ijriujtas jktnfrgmas. Mnkatneas iak X cg cg praparijùk mn fgcja g grsnkjura y iak Z nc nc parinktgdn mn a`cngs lukijakgcns pramuijmgs murgktn ukg harg. X y y Z sak sak vgrjg`cns gcngtarjgs jkmnpnkmjnktns iak cg sjfujnktn lukijùk mn mnksjmgm iakduktg 0
l ( x, y )
5
x (7+9 y
2,
=
),
2 3 x 3 0, 2 3 y 3 7, nk atra igsa.
790
Igpåtuca = Nspnrgkzg egtneçtjig
Mneunstrn qun N ( XZ XZ ) 5 N N ( X X ) N N (Z ), ), iaea sufjnrn nc tnarneg =.8. Yacuijùk4 Rar mnkjijùk, 7
N ( X XZ Z )
0
5 2
x 0 y ( 7 + 9 y 0 )
=
2
> <
mxmy 5 , N ( X )
= 5 , y 9
N ( Z )
> 5 . 8
Rar ca tgkta, N ( X ) N ( Z )
5
= 9
XZ Z ). > 5 > 5 N ( X 8 <
Iakicujeas nstg sniijùk iak cg mneastrgijùk mn uk tnarneg y cg prnsnktgijùk mn vgrjas iaracgrjas qun sak ôtjcns pgrg igciucgr vgrjgkzgs a mnsvjgijakns nstçkmgr.
]narneg =.:4
Yj X y Z sak vgrjg`cns gcngtarjgs iak mjstrj`uijùk mn pra`g`jcjmgm iakduktg l ( x x , y), y g, ` y i sak iakstgktns, nktakins 0 ω gX + `Z + + i
Rrun`g4 Rar
g0ω X 0 + `0ω Z 0 +0 g`ω XZ XZ .
0 0 mnkjijùk, ω gX + i U }. Nktakins, + `Z + + i 5 N {_(gX + `Z + i) ∐ ¾gX + `Z +
¾gX + `Z + i 5 N ( gX + `Z + i) 5 gN ( X ) + `N ( Z ) + i 5 g ¾ X + ` ¾Z + i,
sj utjcjzgeas nc iaracgrja =.= y mnspuês nc iaracgrja =.0. Rar ca tgkta, 0 0 ω gX + `Z + + i 5 N {_g( X ∐ ¾ X ) + `( Z ∐ ¾Z )U }
X ∐ ¾ ) 0 U + `0 N _(Z ∐ ¾ ) 0 U + 0g`N _( X X ∐ ¾ )( Z ∐ ¾ )U 5 g0 N _( X 5 g0 ω 0 + `0 ω 0 + 0g`ω . X
X
Z
X
Z
XZ
Z
Yj utjcjzgeas nc tnarneg =.:, tnkneas tnkn eas cas sjfujnktns iaracgrjas.
Iaracgrja =. g =.: y cas mjvnrsas iaracgrjas sak suegenktn ôtjcns, yg qun ka hgy rnstrjiijakns sa`rn cg lareg mn cg mnksjmgm a cgs lukijakns mn pra`g`jcjmgm, gpgrtn mn cg prapjnmgm mn jkmnpnkmnkijg iugkma êstg sn rnqujnrn, iaea nk cas iaracgrjas pastnrjarns gc tnarneg =.:. Rgrg jcustrgr iaksjmnrn nc ndnepca =.09? cg vgrjgkzg mn T 5 9 X ’ 0Z + > ka rnqujnrn rnstrjiijakns nk cgs mjstrj`uijakns mn cgs igktjmgmns X y Z mn mn cas mas tjpas mn jepurnzgs. Yùca sn rnqujnrn cg jkmnpnkmnkijg nktrn X y y Z . Rar 0 iaksjfujnktn, mjspakneas mn cg igpgijmgm mn igciucgr ¾f( X y ω f ( X ) pgrg iugcqujnr luk X ) ijùk f(¶) g pgrtjr mn cas prjkijpjas jkjijgcns nstg`cnijmas nk cas tnarnegs =.7 y =.9, makmn sn supakn qun sn iakain cg mjstrj`uijùk l ( x x ) iarrnspakmjnktn. Cas ndnrijijas =.=2, =.=7 y =.=0, nktrn atras, jcustrgk nc usa mn tgcns tnarnegs. Mn eama qun, sj f( x x ) ns ukg lukijùk ka cjkngc y sn iakain cg lukijùk mn mnksjmgm (a lukijùk mn pra`g`jcjmgm nk nc igsa mjsirnta), ¾f( X y ω f0( X ) punmnk nvgcugrsn iak nxgitjtum. Ka a`stgktn, iaea nk nc igsa mn X ) cgs rnfcgs mgmgs pgrg iae`jkgijakns cjkngcns, ±hg`råg rnfcgs pgrg lukijakns ka cjkngcns qun sn punmgk utjcjzgr iugkma ka sn iakain cg lareg mn cg mjstrj`uijùk mn cgs vgrjg`cns gcngtarjgs pnrtjknktns1 x ). Cg sacuijùk Nk fnknrgc, supakfg qun X ns ukg vgrjg`cn gcngtarjg y qun Z 5 f( x fnknrgc pgrg N (Z ) a \gr(Z ) punmn snr mjlåijc y mnpnkmn mn cg iaepcndjmgm mn cg lukijùk f(¶). Yjk ne`grfa, hgy gpraxjegijakns mjpakj`cns qun mnpnkmnk mn ukg gpraxjegijùk cjkngc mn cg lukijùk f( x x ). ). Rar ndnepca, supakfg qun mnkatgeas N ( X X ) iaea ¾ y \gr( X X ) 5 0 ω X . Nktakins, ukg gpraxjegijùk g cgs snrjns mn ]gycar ]gycar mn f( x x ) gcrnmnmar mn X 5 ¾ ¾ X mg f( x )
5 f( ¾ ) + X
∀ f( x ) ∀0 f( x ) ( x ∐ ¾ X ) 0 ( x ∐ ¾ X ) + +¶ ¶ ¶ . ∀ x x 5 ¾ ∀ x 0 x 5 ¾ 0 X X
Iaea rnsuctgma, sj trukigeas mnspuês nc têrejka cjkngc y taegeas nc vgcar nspnrgma X )U ¾ X ), qun ijnrtgenktn ns jktujtjva y nk gcfukas mn ge`as cgmas, a`tnkneas N _f( X )U ≈ f( ¾ igsas alrnin ukg gpraxjegijùk rgzakg`cn. Ka a`stgktn, sj jkicujeas nc têrejka mn snfukma armnk mn cg snrjn mn ]gycar, nktakins tnkneas uk gdustn mn snfukma armnk pgrg nstg gpraxjegijùk mn prjenr armnk iaea sjfun4 Gpraxjegijùk mn Gpraxjegijùk N _f( X X )U )U
∀0 f( x ) ω X 0 N _f( X )U ≈ f( ¾ X ) + . ∀ x 0 x 5 ¾ 0 X
Mgmg cg vgrjg`cn gcngtarjg X iak iak enmjg ¾ X y vgrjgkzg ω X 0 , mntnrejkn cg gpraxjegijùk mn snfukma armnk pgrg N (n X ). ∀ n x x ∀ 0 n x x Yacuijùk4 Iaea ∀ x 5 n y ∀ x 0 5 n , a` a`tn tnkn knea eass N ( n X ) ≈ n ¾ X (7 + ω X 0 / 0). 0).
Ndnepca =.0=4
x )U Mn egknrg sjejcgr, pamneas mnsgrraccgr ukg gpraxjegijùk pgrg \gr_ \gr_f( x )U taegkma x ). cg vgrjgkzg mn ge`as cgmas mn cg nxpgksjùk mn cg snrjn mn ]gycar mn prjenr armnk mn f( x ). 0
Gpraxjegijùk mn X \gr_f( X )U )U
Ndnepca =.0>4 =.0>4
\gr_f( X )U ≈ ∀ f( x ) ∀ x
ω X 0 . x 5 ¾ X
Mgmg cg vgrjg`cn gcngtarjg X, iaea nk nc ndnepca =.0=, mntnrejkn ukg lùreucg gpraxjegmg pgrg p grg \gr_ \gr_f( x x )U. )U.
=.= ]narneg mn Ihn`yshnv
79>
x
kunva, ∀∀n x 5 n x par ca tgkta, tgkta, \g r( X ) ≈ n0 ¾ X ω X 0 . Nstgs gpraxjegijakns sn punmnk nxtnkmnr g cgs lukijakns ka cjkngcns mn eçs mn ukg vgrjg`cn gcngtarjg. Mgma uk iakdukta mn vgrjg`cns gcngtarjgs jkmnpnkmjnktns X 7, X 0,…, X o iak enmjgs ¾7, 0 ¾0,…, ¾o y vgrjgkzgs ω 70,ω 00,...,ω o , rnspnitjvgenktn, sng
Yacuijùk4 Mn
Z 5 h ( X 7 , X 0 , . . . , X o )
ukg lukijùk ka cjkngc? nktakins tnkneas cgs sjfujnktns gpraxjegijakns pgrg N (Z ) y \gr(Z )4 )4 o ω j0 ∀0 h ( x 7 , x 0 , . . . , x o ) N ( Z ) ≈ h ( ¾7, ¾0 , . . . , ¾ o ) + , 0 x 0j ∀ x ¾ , 7 j o 5 ≭ ≭ j j j 57 o
\gr( Z ) ≈ j 57
Ndnepca =.0 X + 9, makmn X tjnkn tjnkn cg mjstrj`uijùk mn pra`g`jcjmgm mnc ndnrijija =.9< mn cg pçfjkg 706. =.>> Yupakfg qun ukg tjnkmg mn g`grratns iaeprg > nkvgsns mn cnihn mnsirnegmg gc prnija mn egyarna mn $7.02 par nkvgsn y cg vnkmn g $7. par nkvgsn. Mnspuês cg lnihg igmuijmgm, cnihn qun ka uk sn vnkmn sn mn rntjrg mn casmn gkgquncns y nccgtnkmnra rnij`n irêmjta mnc mjstrj`ujmar jfugc g trns iugrtgs pgrtns mnc
prnija mn egyarna. Yj cg mjstrj`uijùk mn pra`g`jcjmgm mn cg vgrjg`cn gcngtarjg ns X y y nc kôenra mn nkv n kvgsns gsns qun sn vnkmnk mn nstn catn ns x 2 7 0 9 = > , 7 0 0 9 = 9 l ( x ) 7> 7> 7> 7> 7> 7> igciucn cg utjcjmgm nspnrgmg. 706 gpcj=.> y nc iaracgrja =.6 Yng X ukg trj`uijùk mn pra`g`jcjmgm4 x l ( x )
∐9 7 <
<
7 0
: 7 9
798
Igpåtuca = Nspnrgkzg egtneçtjig
Igciucn N ( X X ) y N ( X X 0) y cunfa utjcjin nstas vgcarns pgrg nvgcugr N _(0 _(0 X + 7)0U. utjcjzg su =.>8 Nc tjnepa tatgc qun ukg gmacnsinktn utjcjzg snigmarg mn pnca murgktn uk g÷a, enmjma nk ukjmgmns mn 722 hargs, hargs, ns ukg vgrjg`cn gcngtarjg iaktjkug X qun tjnkn cg sjfujnktn lukijùk mn mnksjmgm 2 3 x 3 7 , l ( x ) 5 0 ∐ x, 7 ≭ x 3 0, 2, nk atra igsa. x,
^tjcjin nc tnarneg =.< pgrg nvgcugr cg enmjg mn cg vgrjg`cn gcngtarjg Z 5 : Yj ukg vgrjg`cn gcngtarjg X sn sn mnkn mn egknrg qun N _( X X ∐ 7)0 U 5 72 y N _( X X ∐ 0) 0 U 5 2.92 2.7>
Igciucn g) N (0 X ∐ 9Z )? XZ Z ). `) N ( X (0 XZ 0 ’ X X 0Z ) =. y igciucn nc pnsa nspnrgma pgrg cg sueg mn cgs irnegs y cas ihjicasas sj uka iaeprg ukg igdg mn tgcns ihaiacgtns. =.8> Yupakfg qun sn sn sg`n qun cg vjmg mn uk iaeprniaeprnsar pgrtjiucgr X, nk hargs, tjnkn cg sjfujnktn lukijùk mn mnksjmgm 7 ∐ x / :22 n , x ; ; 2, l ( x ) 5 :22 2, nk atra igsa. g) Igciucn cg vjmg enmjg mnc iaeprnsar. iaeprnsar. 0 `) Igciucn N ( X X ). i) Igciucn cg vgrjgkzg y cg mnsvjgijùk nstçkmgr mn cg cg vgrjg`cn gcngtarjg X . =.8, g) igciucn ¾ X y ¾Z ? `) igciucn N _( _( X X + Z )/0U. )/0U. =.86 Mneunstrn qun Iav(gX , `Z ) 5 g` Iav( X X , Z ). ). =.88 Iaksjmnrn cg lukijùk mn mnksjmgm mnc ndnrijija mn rnpgsa =.8>. Mneunstrn qun nc tnarneg mn Ihn`yshnv ns vçcjma pgrg o 5 0 y o 5 9. =.8: Iaksjmnrn cg sjfujnktn lukijùk mn mnksjmgm iak duktg 7221 =.:0 Iaksjmnrn nc ndnrijija =.72 mn cg pçfjkg 776. ±Yn punmn mnijr qun cgs igcjigijakns mgmgs par cas mas nxpnrtas sak jkmnpnkmjnktns1 Nxpcjqun su rnspunstg. =.:9 Cas mnpgrtgenktas mn egrontjkf y mn iaktg`jcjmgm mn ukg neprnsg mntnrejkgrak qun sj cg neprnsg iaenrijgcjzg su pramuita irngma rnijnktnenktn, su iaktrj`uijùk g cgs utjcjmgmns mn cg neprnsg murgktn cas prùxjeas < ensns snrç cg sjfujnktn4 Iaktrj`uijù Iaktr j`uijùk k g cgs utjcjmg utjcjmgmns mns Rra`g` Rra`g`jcjmgm jcjmgm
∐$>,222 $72,222 $92,222
2.0 2.> 2.9
±Iuçc ns cg utjcjmgm nspnrgmg mn cg neprnsg1 =.:= Nk uk sjstneg mn gpaya pgrg nc prafrgeg nspgijgc nstgmaukjmnksn, uk iaepaknktn iruijgc ôkjia lukijakg sùca 8> par ijnkta mnc tjnepa. Rgrg guenktgr cg iakg`jcjmgm mnc sjstneg sn mnijmjù jkstgcgr trns iaepaknktns pgrgcncas, mn egknrg qun nc sjstneg lgccn sùca sj tamas Yupakfg yqun mn lareglgccgk. jkmnpnkmjnktn quncas sakiaepaknktns nqujvgcnktnsgitôgk nk nc snktjma mn qun 9 mn nccas n ccas tjnknk ukg tgsg mn êxjta mn 8> par ijnkta. Iaksjmnrn cg vgrjg`cn gcngtarjg X iaea iaea nc kôenra mn iaepaknktns mn igmg trns qun lgccgk. g) Nsirj`g ukg lukijùk mn pra`g`jcjmgm pgrg cg vgrjgvgrjg`cn gcngtarjg X . `) ±Iuçc ns N ( X X ) (ns mnijr, nc kôenra enmja mn iaepaknktns mn igmg trns qun lgccgk)1 i) ±Iuçc ns \gr( X )1 )1 m ) ±Iuçc ns cg pra`g`jcjmgm pra`g`jcjmgm mn qun nc sjstneg iaepcnta sng nxjtasa1 n) ±Iuçc ns cg pra`g`jcjmgm mn qun lgccn lgccn nc sjstneg1 sjstneg1 l ) Yj sn mnsng qun nc sjstneg tnkfg ukg pra`g`jcjmgm mn êxjtaYj mnka 2.::, ±sak±iuçktas suijnktns cas trns iaepaknktns1 ca sak, sn rnqunrjrågk1 =.:> Nk cas knfaijas ns jepartgktn pcgkngr y ccnvgr g ig`a jkvnstjfgijùk pgrg gktjijpgr ca qun aiurrjrç gc kgc mnc g÷a. Cg jkvnstjfgijùk sufjnrn qun nc nspnitra mn utjcjmgmns (pêrmjmgs) mn ijnrtg neprnsg, iak sus rnspnitjvgs pra`g`jcjmgmns, ns nc sjfujnktn4 ^tjc jcjm jmgm gm
∐$7>, 222 $2 $7>,222 $0>,222 $=2,222 $>2,222 $722,222 $7>2,222 $022,222
Rra` Rr a`g g`j `jccjm jmgm gm
2.2> 2.7> 2.7> 2.92 2.7> 2.72 2.2> 2.29 2.20
Ndnrijijas mn rnpgsa
7=7
g) `)
±Iuçc ns cg utjcjmgm utjcjmgm nspnrgmg1 Mntnrejkn cg mnsvjgijùk nstçkmgr mn cgs utjcjmgmns. =.:< Enmjgktn uk iakdukta mn mgtas, y par cg gepcjg jkvnstjfgijùk, sn sg`n qun cg igktjmgm mn tjnepa qun ijnrta nepcngma mn ukg neprnsg ccnfg tgrmn g trg`gdgr, enmjma nk snfukmas, ns ukg vgrjg`cn gcngtarjg X iak iak cg sjfujnktn lukijùk mn mnksjmgm l ( x )
5
9 (=)(>2 9)
2,
(>20 ∐ x 0 ) , ∐>2 ≭ x ≭ >2, nk atra igsa.
Nk atrgs pgcg`rgs, êc ka sùca ccnfg cjfnrgenktn rntrgsgma g vnins, sjka qun tge`jêk punmn ccnfgr tneprgka g trg`gdgr. g) Igciucn nc vgcar nspnrgma mnc tjnepa nk snfukmas qun ccnfg tgrmn. `) Igciucn N ( X X 0). i) ±Iuçc ns cg cg mnsvjgijùk nstçkmgr mnc tjnepa nk qun ccnfg tgrmn1 =.:6 ^k igejùk mn igrfg vjgdg mnsmn nc pukta G hgstg nc pukta @ y rnfrnsg par cg ejseg rutg mjgrjgenktn. Hgy X 7 nc kôenra iugtranksneçlaras cg rutg. Yng laras rada qun nk nc igejùk nkiunktrg iugkmamnvgsneçmn G g @ y X 0 nc kôenra mn cas qun nkiunktrg nk nc vjgdn mn rnfrnsa. Cas mgtas rnig`gmas murgktn uk pnrjama cgrfa sufjnrnk qun cg mjstrj`uijùk mn pra`g`jcjmgm iakduktg pgrg ( X X 7, X 0) nstç mgmg par
y x
2 7 0 g)
2 2.70 2.28 2.2<
7 2.2= 2.7: 2.70
0 2.2= 2.2> 2.92
Mntnrejkn cg mnksjmgm egrfjkgc egrfjkgc mn X y mn Z , gså iaea cg mjstrj`uijùk mn pra`g`jcjmgm mn X, mgma
Z 5 0. qun Mntnrejkn N ( X X ) y \gr( X X ). ). Mntnrejkn N ( X X | Z 5 0) y \gr( \gr( X | Z 5 0). =.:: Iaksjmnrn uk trgks`armg trgks`armgmar mar qun punmn ccnvgr ccnvgr tgkta guta`usns iaea gutaeùvjcns nk uk rniarrjma g trgvês mn ukg våg uvjgc. Igmg vjgdn iunstg gc prapjntgrja gpraxjegmgenktn $72. Cg tgrjlg par gutaeùvjc ns mn $9 y par guta`ôs ns mn $8. Yngk X y y Z nc nc kôenra mn guta`usns y gutaeùvjcns, rnspnitjvgenktn, qun sn trgkspartgk nk uk vjgdn nspniåia. Cg mjstrj`uijùk iakduktg mn X y y Z nstç nstç mgmg par `) i)
x y
2 7 0 9 = >
2 2.27 2.29 2.29 2.26 2.70 2.28
7 2.27 2.28 2.2< 2.26 2.2= 2.2<
0 2.29 2.26 2.2< 2.79 2.29 2.20
x 0 x 7
2 7 0 9 =
2 2.27 2.29 2.29 2.20 2.27
7 2.27 2.2> 2.77 2.26 2.2<
0 2.29 2.28 2.7> 2.72 2.29
9 2.26 2.29 2.27 2.29 2.27
= 2.27 2.20 2.27 2.27 2.27
g `) i)
Igciucn cg utjcjmgm nspnrgmg pgrg nc vjgdn mnc trgks`armgmar. =.722 Iaea vnrneas nk nc igpåtuca 70, cas eêtamas nstgmåstjias gsaijgmas iak cas eamncas cjkngc y ka cjkngc sak euy jepartgktns. Mn hniha, g enkuma cgs lukijakns nxpaknkijgcns sn utjcjzgk nk ukg gepcjg fgeg
7 Mntnrejkn cg mnksjmgm mnksjmgm egrfjkgc mn X X 0. Mntnrejkn cg mjstrj`uijùk mn mnksjmgm iakmjijakgc mn X 7 mgma qun X 0 5 9. m ) Mntnrejkn N ( X X 7). n) Mntnrejkn N ( X X 0). l ) Mntnrejkn N ( X X 7 | X X 0 5 9). f) Mntnrejkn cg mnsvjgijùk mnsvjgijùk nstçkmgr mn X 7. mn g`grratns tjnkn mas mas sjtjas sjtjas snpg=.:8 ^kg tjnkmg mn rgmas nk sus jkstgcgijakns makmn cas icjnktns punmnk pgfgr iugkma sn egrihgk. Nstas mas cufgrns tjnknk mas igdgs rnfjstrgmargs y mas nepcngmas qun gtjnkmnk g cas icjnktns qun vgk g pgfgr. Yng X nc nc kôenra mn cg igdg rnfjstrgmarg qun sn utjcjzg nk uk eaenkta nspniåia
mn pra`cnegs mn iakdukta jkfnkjnråg. uk eamnca qunijnktåias sn gdustg gy uk mn Iaksjmnrn mgtas qun jepcjig cas vgcarns enmjmas o 7 y o 0, y ukg rnspunstg nspniåig Z g g cgs enmjijakns. Nc eamnca pastucgma ns
Z nc kôenra mn cg igdg rnfjstrgmarg qun sn nk nc sjtja 7 yejsea utjcjzg nk nc eaenkta nk nc sjtja 0. Cg lukijùk mn pra`g`jcjmgm iakduktg nstç mgmg par
2 `7 `0 Yupakfg enmjgs qun mn `sn , iakaink y y qun sak ΰ2, ΰ7qun y ΰsn , yiakaink tge`jêkcgssupakfg cgs 0 0 0 0 vgrjgkzgs mn `2, `7 y `0 y qun sak ω 2 , ω 7 y ω 0 .
ˌ 5 n `2 + `7 o 7 + ` 0 o 0 , Z
ˌ mnkatg nc vgcar nstjegma mn Z , o y o sak makmn Z 7 0 vgcarns das y `2, `7 y `0 sak nstjegmas mn iakstgktns y, par ca tgkta, vgrjg`cns gcngtarjgs. Yupakfg qun tgcns vgrjg`cns gcngtarjgs sak jkmnpnkmjnktns y usn cg lùreucg gpraxjegmg pgrg cg vgrjgkzg mn ukg lukijùk ka cjkngc ˌ). mn eçs mn ukg vgrjg`cn. Mê ukg nxprnsjùk pgrg \gr(Z ).
7=0
Igpåtuca = Nspnrgkzg egtneçtjig
9.69 mn cg pç=.727 Iaksjmnrn nc ndnrijija mn rnpgsa 9.69 fjkg 728, nc iugc jepcjig Z , cg praparijùk mn jepurnzgs nk uk catn, makmn cg lukijùk mn mnksjmgm nstç mgmg par 72(7 ∐ y ) : , 2 ≭ y ≭ 7, l ( y ) 5 2, nk atra igsa. g) Igciucn nc parinktgdn nspnrgma mn jepurnzgs. jepurnzgs. `) Igciucn nc vgcar vgcar nspnrgma mn cg praparijùk mn cg igcjmgm mnc egtnrjgc (ns mnijr, mni jr, igciucn N (7 (7 ’ Z )). )).
=.>
i)
Igciucn cg vvgrjgkzg grjgkzg mn cg vgrjg`cn gcngtarjg T 5 7 ’ Z . =.720 Rraynita4 Yng X 5 kôenra mn hargs qun igmg nstumjgktn mnc frupa murejù cg kaihn gktnrjar. Irnn ukg vgrjg`cn mjsirntg utjcjzgkma cas sjfujnktns jktnrvgcas gr`jtrgrjas4 X 3 9, 9 ≭ X X 3 nc lg`rjigktn mnc iaeprnsar pamråg sg`nr (egtnrjgc qun sn prnsnktgrç nk nc igpåtuca
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