TEMA 1 Conceptos previos

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&RQFHSWRVSUHYLRV
&RQWHQLGRV
1~PHURVUHDOHV/DUHFWDUHDO
'HVLJXDOGDGHV,QWHUYDORVGHODUHFWDUHDO
9DORUDEVROXWR3URSLHGDGHV
(MHUFLFLRVUHVXHOWRV
(MHUFLFLRVSURSXHVWRV
1~PHURVUHDOHV/DUHFWDUHDO
.QUFKHGTGPVGUEQPLWPVQUPWOÃTKEQUJCPUWTIKFQRQTPGEGUKFCFGUTGCNGUSWGGNJQODTG
JC FGDKFQ FG CHTQPVCT C NQ NCTIQ FG NC JKUVQTKC .C RTKOGTC PGEGUKFCF PWOÃTKEC NC VWXKGTQP
PWGUVTQU CPVGRCUCFQU EQOQ EQPUGEWGPEKC FGN TGEWGPVQ FG QDLGVQU CPKOCNGU [ QVTCU
RGTVGPGPEKCUFGGUVCHQTOCUWTIKGTQPNQUSWGJQ[FGPQOKPCOQUPÕOGTQUPCVWTCNGU
&GHKPKEKÎP 0ÕOGTQUPCVWTCNGU
'N EQPLWPVQ FG NQU PÕOGTQU PCVWTCNGU SWG UG FGPQVC RQT GUV¶ HQTOCFQ RQT NQU
PÕOGTQU
%QP GUVG EQPLWPVQ PWOÃTKEQ UG RQFÈCP TGCNK\CT UWOCU [ RTQFWEVQU FG PÕOGTQU
PCVWTCNGU [C SWG UG VTCVCDC FG QRGTCEKQPGU DKGP FGHKPKFCU FCFQU FQU PÕOGTQU PCVWTCNGU NC
UWOCQRTQFWEVQFGÃUVQUUKGORTGFCEQOQTGUWNVCFQQVTQPÕOGTQPCVWTCN5KPGODCTIQNCU
FKHGTGPVGU QRGTCEKQPGU EQOGTEKCNGU FG EQORTCXGPVC PGEGUKVCTQP EQPUVTWKT WPC PWGXC
QRGTCEKÎP NC FKHGTGPEKC 'UVC QRGTCEKÎP PQ RTGUGPVCDC KPKEKCNOGPVG PKPIÕP RTQDNGOC RQTSWG
PQ GTC NÎIKEQ ICUVCT O¶U FG NQ SWG WPQ RQUGÈC UKP GODCTIQ NCU QRGTCEKQPGU OGTECPVKNGU UÈ
RTQXQECTQPNCPGEGUKFCFFGFGHKPKTFKHGTGPEKCUGPVTGFQUPÕOGTQUPCVWTCNGUEWCNGUSWKGTC'UVC
EKTEWPUVCPEKCRTQRKEKÎNCPGEGUKFCFFGKPVTQFWEKTNQUPÕOGTQUPCVWTCNGUPGICVKXQU#WPSWGNQU
RTKOGTQU GP WVKNK\CT NQU PÕOGTQU PGICVKXQU HWGTQP NQU EJKPQU JCEG [C O¶U FG UKINQU
PWGUVTC EKXKNK\CEKÎP QEEKFGPVCN UG TGUKUVKÎ FG VCN HQTOC C OCPGLCT FG OCPGTC PCVWTCN NCU
ECPVKFCFGUPGICVKXCUSWGCÕPGPGNUKINQ:8++&GUECTVGU
GPVTGQVTQUCHKTOCDCSWGQEWCPFQ
WPC GEWCEKÎP VGPÈC UQNWEKÎP PGICVKXC NC UQNWEKÎP GTC HCNUCR .QU PÕOGTQU PGICVKXQU
CFSWKTKGTQPUWXGTFCFGTCPCVWTCNG\CEWCPFQUGGORG\CTQPCXKUWCNK\CTGPWPCTGEVC
'NUKIPQOGPQU
RTQEGFGPVGFGWPCQORFGHQTOCFCQOKPWURUGGORNGÎRQTRTKOGTC
XG\EQOQFKUVKPVKXQFGNQUPÕOGTQUPGICVKXQUGPNC#NGOCPKCFGNUKINQ:8+
(YROXFLyQGHOVLJQRPHQRV
.C CRCTKEKÎP FG NQU PÕOGTQU PCVWTCNGU PGICVKXQU [ GN EGTQ LWPVQ C NQU PÕOGTQU
PCVWTCNGUEQPHQTOCTQPGNEQPLWPVQFGNQUPÕOGTQUGPVGTQU
&GHKPKEKÎP 0ÕOGTQUGPVGTQU
'NEQPLWPVQFGNQUPÕOGTQUGPVGTQUSWGUGTGRTGUGPVCEQPNCNGVTC ' GUV¶HQTOCFQ
RQT NQU PÕOGTQU PCVWTCNGU
05 GN EGTQ 05 [ NQU PÕOGTQU PCVWTCNGU PGICVKXQU
2GTQ PWGXCOGPVG UWTIKGTQP PWGXCU PGEGUKFCFGU 'HGEVKXCOGPVG EQP NC EQORNGLKFCF
FGN EQOGTEKQ [ FG NC UQEKGFCF CRCTGEKGTQP EWGUVKQPGU TGNCEKQPCFCU EQP NQU TGRCTVQU
FKUVTKDWEKQPGUFGJGTGPEKCUGVENQSWGEQPFWLQFGHQTOCKPGXKVCDNGCNEQPEGRVQFGHTCEEKÎP
.CUHTCEEKQPGU[CHWGTQPEQPUKFGTCFCUFGOCPGTCEQPEGRVWCNRQTNQUGIKREKQUGPGNUKINQ::
C% [ O¶U VCTFG NC EQOWPKFCF RKVCIÎTKEC GPEQPVTÎ PWOGTQUCU TGNCEKQPGU SWG XKPEWNCDCP NC
OÕUKEC EQP NQU PÕOGTQU HTCEEKQPCTKQU C VTCXÃU FG NQU CTOÎPKEQU 'P TGCNKFCF GN RTQDNGOC
XKGPG OQVKXCFQ RQT NC KPUWHKEKGPEKC FGN EQPLWPVQ FG NQU PÕOGTQU GPVGTQU RCTC GPEQPVTCT
UQNWEKQPGUCGEWCEKQPGUEQOQ Z
'NEQPLWPVQFGNQUPÕOGTQUHTCEEKQPCTKQUFKQNWICTCN
EQPLWPVQFGNQUPÕOGTQUTCEKQPCNGUSWGUGFGHKPGPFGNCUKIWKGPVGHQTOC
&GHKPKEKÎP 0ÕOGTQUTCEKQPCNGU
0
5
&KTGOQUSWG C GUWPPÕOGTQTCEKQPCN C  UKUGRWGFGGZRTGUCTEQOQEQEKGPVGFG
FQUPÕOGTQUGPVGTQUGUFGEKTUKGZKUVGP R S  ' VCNGUSWG C
R S S z 'UVGEQPLWPVQFGPÕOGTQUVKGPGWPCRGEWNKCTKFCFGURGEKCN[GUSWGWPOKUOQPÕOGTQ
TCEKQPCN UG RWGFG GZRTGUCT RQT OGFKQ FG KPHKPKVQU PÕOGTQU HTCEEKQPCTKQU VQFQU GNNQU
GSWKXCNGPVGUGPVTGUK#UÈRQTGLGORNQ
&G GPVTG VQFCU NCU HTCEEKQPGU SWG TGRTGUGPVCP WP PÕOGTQ TCEKQPCN GZKUVG WPC SWG VKGPG NC
RTQRKGFCF FG UGT KTTGFWEKDNG GU FGEKT GUV¶ HQTOCFC RQT FQU PÕOGTQU RTKOQU GPVTG UK GP GN
GLGORNQCPVGTKQT
%WCPFQ GHGEVWCOQU NC FKXKUKÎP GPVTG FQU PÕOGTQU GPVGTQU GP QECUKQPGU RQFGOQU
QDVGPGTQDKGPWPPÕOGTQFGEKOCNGZCEVQ
EQPWPPÕOGTQHKPKVQFGEKHTCUFGEKOCNGUQDKGPWP
PÕOGTQ FGEKOCN RGTKÎFKEQ EQP WP PÕOGTQ KPHKPKVQ FG EKHTCU FGEKOCNGU IGPGTCFCU RQT NC
TGRGVKEKÎPRGTKÎFKECFGWPCECPVKFCFHKPKVCFGGNNCU'UVCEKTEWPUVCPEKCPQURGTOKVGCUGIWTCTNC
UKIWKGPVGRTQRKGFCF
2TQRQUKEKÎP6QFQPÕOGTQTCEKQPCNUGRWGFGGZRTGUCTGPHQTOCFGEKOCNGZCEVCQRGTKÎFKEC
2QT GLGORNQ NQU PÕOGTQU TCEKQPCNGU
UG RWGFGP GZRTGUCT FG HQTOC
FGEKOCNEQOQ
PÕOGTQFGEKOCNRGTKÎFKEQ
PÕOGTQFGEKOCNGZCEVQ
PÕOGTQFGEKOCNRGTKÎFKEQOKZVQ
PÕOGTQFGEKOCNGZCEVQ
%QOQ CECDCOQU FG QDUGTXCT GP GUVG GLGORNQ NC GZRTGUKÎP FGEKOCN FG WPC HTCEEKÎP
RWGFGUGTFGFQUVKRQU
'ZRTGUKÎPFGEKOCNGZCEVC
'ZRTGUKÎPFGEKOCNRGTKÎFKEC
x GZRTGUKÎPFGEKOCNRGTKÎFKECRWTC
x GZRTGUKÎPFGEKOCNRGTKÎFKECOKZVC
%QP NC EQPUVTWEEKÎP FG NQU PÕOGTQU TCEKQPCNGU UG RGPUÎ SWG EWCNSWKGT NQPIKVWF GTC
UWUEGRVKDNGFGUGTOGFKFCWVKNK\CPFQGUVGVKRQFGPÕOGTQU5KPGODCTIQKPENWUQGPNC)TGEKC
EN¶UKECUGFGUEWDTKGTQPNCUFGPQOKPCFCUECPVKFCFGURKPEQPOGPUWTCDNGURGUFGEKTECPVKFCFGU
SWG PQ RQFÈCP GZRTGUCTUG RQT OGFKQ FG HTCEEKQPGU Q NQ SWG GU NQ OKUOQ ECPVKFCFGU EW[C
GZRTGUKÎP FGEKOCN PQ GTC PK GZCEVC PK RGTKÎFKEC #UÈ RWGU UG KPVTQFWLQ WP VGTEGT VKRQ FG
GZRTGUKQPGUFGEKOCNGUNCUSWGPQGTCPPKFGEKOCNGUGZCEVCUPKFGEKOCNGURGTKÎFKECU
7PQFGNQUGLGORNQUEN¶UKEQUFGGUVCUQECPVKFCFGUKPEQPOGPUWTCDNGURGUGNPÕOGTQ S [ SWG FGDG UW PQODTG C NC EQOWPKFCF RKVCIÎTKEC
EW[C GZRTGUKÎP FGEKOCN GU 1VTQ GLGORNQ EN¶UKEQ GU GN PÕOGTQ
'HGEVKXCOGPVG NC GZRTGUKÎP FGEKOCN FG
GU
SWG C RTKOGTC XKUVC PQ VKGPG GN CURGEVQ FG UGT WPC GZRTGUKÎP
FGEKOCN RGTKÎFKEC [ OWEJQ OGPQU WPC GZRTGUKÎP FGEKOCNGZCEVC2GTQRCTCEGTEKQTCTPQUFGN
JGEJQFGSWG GUWPCFGGUVCUECPVKFCFGUKPEQPOGPUWTCDNGUXGCOQUWPCFGOQUVTCEKÎPSWG
PQUICTCPVK\CSWG PQGUWPPÕOGTQTCEKQPCN
2TQRQUKEKÎP PQGUWPPÕOGTQTCEKQPCN
&GOQUVTCEKÎP2CTC FGOQUVTCT GUVC RTQRQUKEKÎP WVKNK\CTGOQU GN OÃVQFQ FG TGFWEEKÎP CN
CDUWTFQGUFGEKTRCTVKGPFQFGNCJKRÎVGUKUFGSWG GUWPPÕOGTQTCEKQPCNNNGICTGOQUCWP
TGUWNVCFQEQPVTCFKEVQTKQEQPGUCJKRÎVGUKUFGRCTVKFC
'P GHGEVQ UWRQPICOQU SWG
GU WP PÕOGTQ TCEKQPCN 'UVQ SWKGTG FGEKT SWG
GZKUVKT¶P R S  ' PÕOGTQURTKOQU
UKPHCEVQTGUEQOWPGUVCNGUSWG
EWCFTCFQCODQUOKGODTQUFGNCKIWCNFCFVGPGOQUSWG
R
S
R
S
5KGNGXCOQUCN
GUFGEKT R S 5KRGU
KORCT GUVC KIWCNFCF ECTGEG FG UGPVKFQ FCFQ SWG GN EWCFTCFQ FG WP PÕOGTQ KORCT GU QVTQ
PÕOGTQ KORCT [ S JC FG UGT PGEGUCTKCOGPVG RCT &GFWEKOQU RQT NQ VCPVQ SWG R
PGEGUCTKCOGPVG UGT¶ WP PÕOGTQ RCT GP EW[Q ECUQ R
EQPUGEWGPEKC S 0 M 5
M GU FGEKT S M RCTC EKGTVQ GPVGTQ M 'P
M FG FQPFG UG FGFWEG SWG S VCODKÃP
VKGPGSWGUGTRCT#UÈRWGU R [ S JCPFGUGTRCTGUNQEWCNGUEQPVTCFKEVQTKQEQPGNJGEJQSWG
KORQPÈCOQUCNRTKPEKRKQFGSWG R [ S PQVGPÈCPHCEVQTGUGPEQOÕP2QTVCPVQ PQGUWP
PÕOGTQTCEKQPCN
'UVCU ECPVKFCFGU KPEQPOGPUWTCDNGU FKGTQP NWICT C NQU FGPQOKPCFQU PÕOGTQU
KTTCEKQPCNGU'PGUVGECUQNCKPUWHKEKGPEKCFG RCTCTGUQNXGTWPCGEWCEKÎPVCPUGPEKNNCEQOQ
Z
RTQRKEKÎNCCRCTKEKÎPFGGUVQUPÕOGTQUKTTCEKQPCNGU
&GHKPKEKÎP 0ÕOGTQUKTTCEKQPCNGU
&KTGOQU SWG R GU WP PÕOGTQ KTTCEKQPCN UK UW GZRTGUKÎP FGEKOCN PQ GU RGTKÎFKEC PK
GZCEVC'NEQPLWPVQFGNQUPÕOGTQKTTCEKQPCNGUUGUWGNGFGPQVCTRQTGNUÈODQNQ # RCTVKT FGN EQPLWPVQ FG NQU PÕOGTQU TCEKQPCNGU [ FGN EQPLWPVQ FG NQU PÕOGTQU
KTTCEKQPCNGUUGEQPUVTW[ÎWPPWGXQEQPLWPVQGNEQPLWPVQFGNQUPÕOGTQUTGCNGU
&GHKPKEKÎP 0ÕOGTQTGCNGU
0
5
0
5
&KTGOQUSWGRGUWPPÕOGTQTGCN R  UKRGUQDKGPWPPÕOGTQTCEKQPCN R  0
5
QDKGPGUWPPÕOGTQKTTCEKQPCN R  &GGUVCHQTOCRQFGOQUCHKTOCTSWG ‰ # NQ NCTIQ FG GUVC UGEEKÎP JGOQU RQFKFQ EQPUVCVCT NCU UWEGUKXCU CORNKCEKQPGU FG NQU
EQPLWPVQUPWOÃTKEQUSWGJCPVGPKFQNWICTCNQNCTIQFGNCJKUVQTKCEQOQEQPUGEWGPEKCFGNCU
PGEGUKFCFGUSWGGNJQODTGJCKFQVGPKGPFQRCTCOGFKTNCUOCIPKVWFGU&GGUVCHQTOCRQFGOQU
TGUWOKT NC UKIWKGPVG ECFGPC FG KPENWUKQPGU  '   6CN EQOQ QDUGTXCOQU GP GUVC
ECFGPCEQPLWPVKUVCGNEQPLWPVQFGNQUPÕOGTQUPCVWTCNGUGUGNQTKIGPFGNQUPÕOGTQUUKVWCEKÎP
SWGJCRTQXQECFQCHKTOCEKQPGUEQOQNCFG.-TQPGEMGTQ&KQUETGÎNQUPÕOGTQUPCVWTCNGU
'NTGUVQGUQDTCFGNJQODTGR
.C QPCVWTCNR QTFGPCEKÎP FG NQU PÕOGTQU PCVWTCNGU RWGFG GZVGPFGTUG CN TGUVQ FG
EQPLWPVQU'PEQPETGVQGPGNEQPLWPVQFGNQUPÕOGTQUTGCNGURQFGOQUGUVCDNGEGTWPCTGNCEKÎP
FGQTFGP d QOGPQTQKIWCNRFGNCUKIWKGPVGHQTOC
&CFQU C D  FKTGOQUSWG C d D œ D C t 6GPKGPFQ GP EWGPVC GUVC QTFGPCEKÎP UK FKDWLCOQU WPC TGEVC UQDTG NC SWG OCTECOQU
EQP QR WP RWPVQ EWCNSWKGTC CN SWG NNCOCTGOQU QTKIGP [ EQP QR QVTQ RWPVQ CN SWG
NNCOCTGOQUWPKFCFGURQUKDNGTGRTGUGPVCTEWCNSWKGTPÕOGTQTGCNUQDTGFKEJCTGEVCUKVWCPFQC
NCFGTGEJCFGNQRNQUPÕOGTQUTGCNGURQUKVKXQU[CUWK\SWKGTFCNQUPÕOGTQUTGCNGUPGICVKXQU
2CTCTGRTGUGPVCTEWCNSWKGTPÕOGTQTGCNVCPUÎNQJC[SWGGURGEKHKECTGNNWICTSWGQEWRC
ECFCPÕOGTQGPNCTGEVCNNCOCFCTGEVCTGCN2QTGLGORNQUKFGUGCOQUTGRTGUGPVCTGNPÕOGTQ
DCUVCEQPUKFGTCTSWG d d 

FG VCN HQTOC SWG UK FKXKFKOQU NC FKUVCPEKC GPVTG [ GP RCTVGU KIWCNGU GPVQPEGU UG
UKVÕCGPNCUGIWPFC
5K CJQTC FGUGCOQU TGRTGUGPVCT VGPKGPFQ GP EWGPVC SWG d d UK
TGRTGUGPVCOQU[
[ FKXKFKOQU CJQTC GN UGIOGPVQ GPVTG [ GP RCTVGU KIWCNGU UKVWCTGOQU GP NC
VGTEGTC&GJGEJQUGRWGFGCUGIWTCTSWGCVQFQPÕOGTQTGCNNGEQTTGURQPFGWPÕPKEQNWICTGP
NC TGEVC TGCN TGRTGUGPVCEKÎP SWG PQU RGTOKVG XKUWCNK\CT NC RQUKEKÎP SWG QEWRC EWCNSWKGT
PÕOGTQTGCNUGCÃUVGTCEKQPCNQKTTCEKQPCN
'UVCEQPUKFGTCEKÎPPQURGTOKVGRNCPVGCTPQUFQUEWGUVKQPGUOW[KPVGTGUCPVGU‹%W¶PVQU
PÕOGTQU TCEKQPCNGU JC[ GPVTG FQU PÕOGTQU TGCNGU EWCNSWKGTC! ‹[ EW¶PVQU KTTCEKQPCNGU! .C
TGURWGUVC C CODCU RTGIWPVCU GU NC OKUOC GPVTG FQU PÕOGTQU TGCNGU C [ D FKUVKPVQU
EWCNGUSWKGTCGZKUVGPKPHKPKVQUTCEKQPCNGUGKTTCEKQPCNGU
2QT GLGORNQ GPVTG [
GUV¶P VQFQU NQU TCEKQPCNGU FG NC
HQTOC [ GPVTG
KTTCEKQPCNGU [ GUV¶P NQU
'HVLJXDOGDGHV,QWHUYDORVGHODUHFWDUHDO
'PGNCRCTVCFQCPVGTKQTUGJCEQOGPVCFQSWGNCTGNCEKÎPQ d RGUVCDNGEGWPCTGNCEKÎPFG
QTFGP GP GN EQPLWPVQ FG NQU PÕOGTQU TGCNGU 'UVC TGNCEKÎP FG QTFGP Q d R EWORNG CNIWPCU
RTQRKGFCFGUKORQTVCPVGUGPTGNCEKÎPCNCUWOCGNRTQFWEVQ[GNEQEKGPVGFGPÕOGTQUTGCNGU
2TQRQUKEKÎP
K 5KCNQUFQUOKGODTQUFGWPCFGUKIWCNFCFUGNGUUWOCQTGUVCWPOKUOQPÕOGTQUG
QDVKGPGQVTCFGUKIWCNFCFGPGNOKUOQUGPVKFQGUFGEKT
C D E  C d D œ CrE d DrE
KK 5KNQUFQUOKGODTQUFGWPCFGUKIWCNFCFUGOWNVKRNKECPQFKXKFGPRQTWPPÕOGTQTGCN
OC[QTSWGEGTQUGQDVKGPGWPCFGUKIWCNFCFFGNOKUOQUGPVKFQGUFGEKT
C D  [E  E ! C d D œ C ˜ E d D ˜ E œ
C D
d E E
KKK 5KCNQUFQUOKGODTQUFGWPCFGUKIWCNFCFUGNGUOWNVKRNKECQFKXKFGRQTWPPÕOGTQ
TGCNOGPQTSWGEGTQUGQDVKGPGWPCFGUKIWCNFCFGPUGPVKFQEQPVTCTKQGUFGEKT
C D  [E  E C d D œ C ˜ E t D ˜ E œ
C D
t E E
'UVCU RTQRKGFCFGU UQP HWPFCOGPVCNGU C NC JQTC FG TGUQNXGT FGUKIWCNFCFGU 2QT
GLGORNQ UK FGUGCOQU TGUQNXGT WPC KPGEWCEKÎPGUVQGUJCNNCTNQUPÕOGTQU ZSWGEWORNGPNC
FGUKIWCNFCF
Z Z t
RTQEGFGOQUFGNCUKIWKGPVGHQTOC
/WNVKRNKECPFQRQT
SWGGUGNOÈPKOQEQOÕPOÕNVKRNQFG[SWGFC
5 0
0
0
5
0
5
Z Z t
5
GUFGEKT ˜ Z t ˜ Z QGSWKXCNGPVGOGPVG Z t Z 4GUVCPFQCCODQUOKGODTQUTGUWNVC
Z t o Z t Z 4GUVCPFQCJQTC Z UGQDVKGPG
t Z Z FGFQPFG t Z RQTNQSWGNCUQNWEKÎPDWUECFCGUGNEQPLWPVQFGNQUPÕOGTQUTGCNGU
Z  VCNGUSWG Z d .C TGNCEKÎP FG QTFGP RGTOKVG FGHKPKT CNIWPQU UWDEQPLWPVQU FG PÕOGTQU TGCNGU SWG
VKGPGP WPC KPVGTRTGVCEKÎP IGQOÃVTKEC OW[ UGPEKNNC NQU KPVGTXCNQU .QU KPVGTXCNQU FG NC TGEVC
TGCN GUV¶P FGVGTOKPCFQU RQT FQU PÕOGTQU TGCNGU [ GUV¶P HQTOCFQU RQT NQU PÕOGTQU
EQORTGPFKFQUGPVTGCODQU5GIÕPUGEQPUKFGTGPNQUGZVTGOQUKPENWKFQUQPQGPGNKPVGTXCNQUG
VKGPGPNQUUKIWKGPVGUVKRQUFGKPVGTXCNQU
CKPVGTXCNQUCDKGTVQU
0CD5 :Z  C Z D? B
C
:Z  C d Z d D? EKPVGTXCNQUCDKGTVQURQTNCFGTGEJC[EGTTCFQURQTNCK\SWKGTFC
C D
C
DKPVGTXCNQUEGTTCFQU
C D
B
5 :Z  C d Z D? B
C
FKPVGTXCNQUCDKGTVQURQTNCK\SWKGTFC[EGTTCFQURQTNCFGTGEJC
0CD :Z  C Z d D? B
C
5K EQPUKFGTCOQU NQU UÈODQNQU f[ f EQOQ GNGOGPVQU SWG RWGFGP FGVGTOKPCT
KPVGTXCNQUFGNCTGEVCTGCN
GUVQUUÈODQNQUPQTGRTGUGPVCPWPPÕOGTQTGCNUQPWPCOCPGTCFG
TGRTGUGPVCTGNEQPEGRVQFGN QKPHKPKVQRQUKVKXQR[FGNQKPHKPKVQPGICVKXQRGPVQPEGURQFTÈCOQU
EQPUKFGTCT UGOKTTGEVCU FGVGTOKPCFCU RQT WP PÕOGTQ [ HQTOCFCU RQT VQFQU NQU PÕOGTQU
OC[QTGU Q OGPQTGU SWG ÃN 5GIÕP SWG GN PÕOGTQ RGTVGPG\EC Q PQ C NC UGOKTTGEVC UG VKGPGP
FKUVKPVQUVKRQUFGUGOKTTGEVCUQKPVGTXCNQUKPHKPKVQU
CUGOKTTGEVCRQUKVKXC
Cf
5 :Z  Z t C? B
DUGOKTTGEVCPGICVKXC
0fC :Z  Z d C? B
FUGOKTTGEVCPGICVKXCCDKGTVC
0fC5 :Z  Z C? B
EUGOKTTGEVCRQUKVKXCCDKGTVC
0Cf5 :Z  Z ! C? B
'P CNIWPCU QECUKQPGU NQU UWDEQPLWPVQU FG UG UWGNGP GZRTGUCT GP HQTOC FG
FGUKIWCNFCFGUSWGUGVTCFWEGPGPKPVGTXCNQUQUGOKTTGEVCU
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<[  1 ; 6 \1[ 6 1[ 6A GHEHPRV
6L LQWHQWDPRV GHWHUPLQDU %
UHVROYHUGRVGHVLJXDOGDGHV'HODSULPHUDUHVXOWD 1
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ODGHVLJXDOGDG
1
6
1
6
;[  [ ! \[ ! @ 1 f6 ˆ 1 f6 6HFRQVLGHUDDKRUDHOFRQMXQWR &
;[  [ d [ @ (QHVWHFDVRVLUHVROYHPRV
[ d [ œ [ d œ [ d œ [ d
VHFRQFOX\HTXH &
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1f 'P CNIWPCU QECUKQPGU UWGNGP CRCTGEGT EQPLWPVQU FGHKPKFQU RQT FGUKIWCNFCFGU FG VKRQ
EWCFT¶VKEQGUFGEKTGPNCUSWGKPVGTXKGPGWPRQNKPQOKQFGITCFQ
(MHPSOR
6HD '
RSHUDQGRWHQHPRV
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