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/͘ ͘^͘ ,ƵĞƌƚĂ ZŽƐĂƌŝŽ
EŽŵďƌĞ LJ ĂƉĞůůŝĚŽƐ͗ͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺ ƵƌƐŽ͗ͺͺͺͺͺͺ &ĞĐŚĂ͗ͺͺͺͺͺͺͺ ŶǑ ĚĞ ĨŽůŝŽƐ͗ͺͺͺͺ
'DGR HO PRQRPLR Ȃ ‫ ݔ‬ଵଶ
H &DOFXOD VX YDORU QXPpULFR FXDQGR ‫ ݔ‬ൌ ͳ
D 'L FXiO HV VX SDUWH OLWHUDO E ¢&XiO HV VX FRHILFLHQWH" F ¢&XiO HV VX JUDGR"
G 'D XQ PRQRPLR VHPHMDQWH
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5HDOL]D ODV VLJXLHQWHV RSHUDFLRQHV FRQ PRQRPLRV
'DGR HO SROLQRPLR ܲሺ‫ݔ‬ሻ ൌ ͷ‫ ݔ‬ଶ ൅ Ͷ‫ ݔ‬ସ െ Ͷ െ ʹ‫ ݔ‬ଶ െ ͵‫ ݔ‬ସ ൅ ͳ
D 5HG~FHOR E 'L FXiO HV HO WpUPLQR LQGHSHQGLHQWH F ¢&XiO HV VX JUDGR" G &DOFXOD ܲሺͲሻ
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ଶ௫ య ି଻௫ మ ିଵଵ௫ାଵଷ
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ଶ௫ାଷ
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ሺʹ‫ ݕݔ‬ଶ െ ͵‫ ݔ‬ଶ ሻଶ
ଵ
D 'HVDUUROOD OD VLJXLHQWH H[SUHVLyQ XWLOL]DQGR ODV LGHQWLGDGHV QRWDEOHV
‫ݔ‬ଶ െ
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E ([SUHVD HO VLJXLHQWH SROLQRPLR FRPR SURGXFWR GH GRV IDFWRUHV
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ଶ௫ିଷ ௫ାଷ ௫ିଵ
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ଶ ସ ଶ
6LPSOLILFD
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ଶ ଷ ଷ
6LPSOLILFD
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ଵ ଵ ଵ
௫ିଵ ௫ାଵ ௫ మ ିଵ
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(QFXHQWUD XQD VROXFLyQ DSUR[LPDGD FRQ XQ GHFLPDO GH ‫ ݔ‬ସ ൌ ͷͲͲͲ QR YDOH FDOFXODU GLUHFWDPHQWH
OD UDt] FXDUWD GHO QXPHUR HQ FXHVWLyQ FRQ OD FDOFXODGRUD
5HVXHOYH ODV VLJXLHQWHV HFXDFLRQHV
͵ሺ‫ ݔ‬െ ʹሻ ൅ ͷሺ‫ ݔ‬െ ͳሻ ൌ ʹ‫ ݔ‬െ ʹሺ‫ ݔ‬൅ ͵ሻ ൅ ͳͳ
௫ାଵ ௫ିଷ ௫ି଼
൅ െʹ‫ ݔ‬ൌ െ͸
ଶ ହ ହ
‫ݔ‬ሺʹ െ ‫ݔ‬ሻሺ‫ ݔ‬െ ͹ሻሺ‫ ݔ‬ଶ െ ͻሻ ൌ Ͳ
‫ ݔ‬െ ξ‫ ݔ‬ൌ ʹ
ଵ ଷି௫
െʹ ൌ
௫మ ଷ௫ మ
(QFXHQWUD OD VROXFLyQ R VROXFLRQHV GH ORV VLJXLHQWHV SUREOHPDV FRPSUXHED TXH VRQ ODV YHUGDGHUDV
&DOFXOD OD HGDG GH $OEHUWR VDELHQGR TXH GHQWUR GH DxRV WHQGUi HO WULSOH GH VX HGDG DFWXDO
(O SURGXFWR GH GRV Q~PHURV FRQVHFXWLYRV HV ¢4Xp Q~PHURV VRQ"

2. 5HVXHOYH ORV VLJXLHQWHV VLVWHPDV
Ͷ‫ ݔ‬െ ͷ‫ ݕ‬ൌ െʹ
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SRU HO PpWRGR GH VXVWLWXFLyQ
͸‫ ݔ‬൅ ʹ‫ ݕ‬ൌ Ͷ
൜
͵‫ ݔ‬൅ ‫ ݕ‬ൌ ʹ
SRU HO PpWRGR GH LJXDODFLyQ
ͷ‫ ݔ‬െ ͵‫ ݕ‬ൌ ͳ
൜
Ͷ‫ ݔ‬൅ ʹ‫ ݕ‬ൌ ͳͶ
SRU HO PpWRGR GH UHGXFFLyQ
&ODVLILFD HO VLJXLHQWH VLVWHPD HQ FRPSDWLEOH H LQFRPSDWLEOH HQ FDVR GH VHU FRPSDWLEOH HQ
GHWHUPLQDGR R LQGHWHUPLQDGR MXVWLILFDQGR OD UHVSXHVWD ¢&XiO HV OD SRVLFLyQ GH ODV UHFWDV TXH OR
‫ݔ‬൅‫ ݕ‬ൌʹ
IRUPDQ"
൜
ʹ‫ ݔ‬൅ ʹ‫ ݕ‬ൌ ͵
5HVXHOYH HO VLJXLHQWH VLVWHPD SRU HO PpWRGR TXH SUHILHUDV
‫ݕ‬െͶ
‫ݔ‬ൌ ൅ͳ
൞ ͵
ͳ ‫ݔ‬൅Ͷ
‫ݕ‬൅ ൌ
͵ ͵
5HVXHOYH HO VLJXLHQWH VLVWHPD SRU HO PpWRGR TXH SUHILHUDV
ʹ‫ ݕݔ‬ൌ ͵
൜
‫ ݔ‬൅ ʹ‫ ݕ‬ൌ Ͷ
1RWDV
• 1R VH SXHGH XVDU QL OiSL] QL EROtJUDIR URMR QL WtSH[ (O XVR GH FXDOTXLHUD GH HOORV DQXODUi OD
SUHJXQWD GRQGH DSDUH]FDQ
• +DEODU DQWHV GH TXH WRTXH HO WLPEUH VXSRQGUi XQ VXVSHQVR LQPHGLDWR GHO H[DPHQ
• 1R VH SXHGH LQWHUFDPELDU PDWHULDO (O KDFHUOR VXSRQGUi XQ VXVSHQVR LQPHGLDWR GHO H[DPHQ
• 7RGDV ODV SUHJXQWDV WLHQHQ HO PLVPR YDORU
3UHJXQWDV D UHDOL]DU
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