Preparaduria Viernes 6-2-2009

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35(3$5$'85,$9,(51(6 1RWD(OYLHUQHVGHODVHPDQDVtKXERSUHSDSRUORWDQWRDODVSHUVRQDVTXHQR DVLVWLHURQDTXtOHFRSLRODFODVH  6HD



݊ሺ݊ ൅ ͳሻ ෍݅ ൌ ቆ ቇ  ʹ ଶ



௜ୀଵ

'HPXHVWUHSRULQGXFFLyQPDWHPiWLFD 3ULQFLSLRGHLQGXFFLyQ

6HDሼܲ௡ ሽXQDVHULHGHSURSRVLFLyQ HQXQFLDGRV TXHVDWLVIDFHQHVWDVGRVFRQGLFLRQHV

 ܲ௡ HVYHUGDGHUD SRUORJHQHUDOHYDOXDPRV1    /DFHUWH]DGHܲ௜ LPSOLFDODYHUDFLGDGGHܲ௜ାଵ ݅ ൒ ܰ (QWRQFHVܲ௡ HVYHUGDGHURSDUD݊ ൒ ܰ

6ROXFLyQ

3DUD݊ ൌ ͳܲଵ ǣͳଷ ൌ ቀ

ሺଵሻሺଵାଵሻ ଶ

ሺ݅ ሻሺ݅ ൅ ͳሻ ቇ  ܲ௜ ǣͳ ൅ ʹ ൅ ͵ ൅ ‫ ڮ‬൅ ݅ ൌ  ቆ ʹ ଷ









ܲ௜ାଵ ǣͳଷ ൅ ʹଷ ൅ ͵ଷ ൅ ‫ ڮ‬൅ ሺ݅ ൅ ͳሻଷ ൌ ቆ

ቁ 



ሺ݅ ൅ ͳሻሺ݅ ൅ ʹሻ ቇ  ʹ ଶ

¢6HUDTXHܲ௜ LPSOLFDODYHUDFLGDGGHܲ௜ାଵ " 9HDPRVHQWRQFHV ܲ௜ ǣ ሺᇣᇧᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇧᇥ ͳ ൅ ʹ ൅ ͵ ൅ ‫ ڮ‬൅ ݅ ൅ ሺ݅ ൅ ͳ ଷ





௉೔

ଷሻ

ሻଷ

ሺ݅ ሻሺ݅ ൅ ͳሻ ቇ ൅ ሺ݅ ൅ ͳሻଷ  ൌ  ቆ ʹ ଶ

'HVDUUROODPRVHOODGRGHUHFKRGHODLJXDOGDG

ሺ݅ ሻሺ݅ ൅ ͳሻ ሺ݅ ൅ ͳሻଶ ሺ݅ ൅ ʹሻଶ ݅ଶ ݅ ଶ ൅ Ͷ݅ ൅ Ͷ ଷ ଶ ଶ ሺ ሻ ሺ ሻ ሺ ሻ ሺ ሻ ቆ ቇ ൅ ݅൅ͳ ൌ ݅൅ͳ ൭ ൅ ݅൅ͳ ൱ ൌ ݅൅ͳ ቆ ቇൌ  Ͷ Ͷ ʹ Ͷ ଶ

2EVHUYHTXHܲ௜ LPSOLFDܲ௜ାଵ  ൌ൐3RUORFXDOSRULQGXFFLyQODLJXDOGDGGDGDHV 9(5'$'(52

'HPXHVWUHSRULQGXFFLyQTXH

ͳ െ ‫ ݎ‬௡ାଵ ͳ ൅ ‫ ݎ‬൅ ‫ ݎ‬൅ ‫ ݎ‬൅ ‫ ݎ‬൅ ‫ڮ‬൅ ‫ ݎ‬ൌ  ͳെ‫ݎ‬ ଶ







1RWHTXHODVXPDWRULDDOODGRL]TXLHUGRVHUHHVFULEHFRPR ௡

෍ ‫ݎ‬௡  ௜ୀ଴

(QWRQFHVVLJXLHQGRODPLVPDLGHDGHOHMHUFLFLRDQWHULRUVHWLHQHTXH  3DUD݊ ൌ Ͳ‫ ݎ‬଴ ൌ

ଵି௥ ଵି௥



ܲ௜ ǣͳ ൅ ‫ ݎ‬൅ ‫ ݎ‬ଶ ൅ ‫ ݎ‬ଷ ൅ ‫ ݎ‬ସ ൅ ‫ ڮ‬൅ ‫ ݎ‬௜ ൌ

ͳ െ ‫ ݎ‬௜ାଵ  ͳെ‫ݎ‬

ܲ௜ାଵ ǣͳ ൅ ‫ ݎ‬൅ ‫ ݎ‬ଶ ൅ ‫ ݎ‬ଷ ൅ ‫ ݎ‬ସ ൅ ‫ ڮ‬൅ ‫ ݎ‬௜ାଵ ൌ

¢6HUDTXHܲ௜ LPSOLFDODYHUDFLGDGGHܲ௜ାଵ "

ܲ௜ ǣ൫ͳ ൅ ‫ ݎ‬൅‫ ݎ‬൅ ‫ ݎ‬൅ ‫ ݎ‬൅ ‫ڮ‬൅ ‫ ݎ‬൯ ൅ ‫ݎ‬ ᇣᇧᇧᇧᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇧᇧᇧᇥ ଶ



௉೔





ͳ െ ‫ ݎ‬௜ାଶ  ͳെ‫ݎ‬

௜ାଵ

ͳ െ ‫ ݎ‬௜ାଵ ൌ ൅ ‫ ݎ‬௜ାଵ  ͳെ‫ݎ‬

'HLJXDOIRUPDGHVDUUROODPRVHOODGRGHUHFKR

ͳ െ ‫ ݎ‬௜ାଵ ൅ ሺͳ െ ‫ݎ‬ሻ൫‫ ݎ‬௜ାଵ ൯ ൫ͳ െ ‫ ݎ‬௜ାଶ ൯ ͳ െ ‫ ݎ‬௜ାଵ ௜ାଵ ൅‫ݎ‬ ൌ ൌ  ͳെ‫ݎ‬ ͳെ‫ݎ‬ ͳെ‫ݎ‬

2EVHUYHTXHܲ௜ LPSOLFDܲ௜ାଵ  ൌ൐3RUORFXDOSRULQGXFFLyQODLJXDOGDGGDGDHV 9(5'$'(52 5HVXHOYDODLQWHJUDO න

݀‫ݔ‬

ξ͹ െ Ͷ‫ ݔ‬ଶ



3RUSULPHUDDSUR[LPDFLyQ8GSXHGHSHQVDUTXHHVWDHVODLQWHJUDOGHODDUFRVLQ 7LHQHUD]yQSHURQRGHOWRGR6(68*,(5(5($/,=$572'26/263$62648( 6,*8(1$&217,18$&,213$5$68'(02675$&,21 'DWR/DLQWHJUDOGHODUFRVHQRWLHQHODIRUPDGH න

݀‫ݔ‬

ξͳ ൅ ‫ ݔ‬ଶ

ൌ ƒ”…•‹ሺ‫ ݔ‬ሻ ൅ ‫ܥ‬

7HQHPRV න

݀‫ݔ‬

ξ͹ െ Ͷ‫ ݔ‬ଶ

ൌන

݀‫ݔ‬

ට͹ ቀͳ ൅ Ͷ ‫ ݔ‬ଶ ቁ ͹

ൌ ณන ଵ

݀‫ݔ‬



Ͷ ඨ͹ ൭ͳ ൅ ቆට ‫ݔ‬ቇ ൱ ͹

ൌ ณ ଶ

ͳ

ξ͹



݀‫ݔ‬

ξͳ ൅ ܽଶ



'HEHHQWHQGHUTXHVHKL]RHQODLJXDOGDG  ¢2EVHUYRELHQ" ¢(VWDFRUUHFWDODLJXDOGDG  "12¢3RUTXp" 6HUHDOL]RXQFDPELRGHYDULDEOH Ͷ Ͷ ͹ ܽ ൌ ඨ ‫ ݔ‬ൌ൐ ݀ܽ ൌ ඨ ݀‫ ݔ‬ൌ൐  ඨ ݀ܽ ൌ ݀‫ݔ‬ ͹ ͹ Ͷ 3RUORFXDOIDOWDFDPELDUHOGLIHUHQFLDO ͹ ݀ܽ ͳ ݀ܽ ͳ ૚ ૝ නඨ ൬ ൰ൌ න ൌ ƒ”…•‹ሺܽሻ ൅ ‫ ܥ‬ൌ ‫ ܖܑܛ܋ܚ܉‬ቌඨ ࢞ቍ ൅ ࡯ Ͷ ξͳ ൅ ܽଶ ૠ ξ͹ ξͶ ξͳ ൅ ܽଶ ξͶ ξ૝ ͳ

'((67$0$1(5$'(%(5,$5($/,=$5(67$6,17(*5$/(6 5HVXHOYD න 62/8&,21 න

‫ݔ݀ݔ‬

ܽ ଶ ටͳ െ

‫ݔ‬ସ  ܽସ



ͳ න ܽଶ

‫ݔ݀ݔ‬

ଶ ඨͳ െ ൬‫ ݔ‬ଶ ൰ ܽ





න න

݀‫ݔ‬

‫ ݔ‬ටͶ ൬

‫ݔ‬ଶ െ ͳ൰ ʹ

ൌන

ʹ‫ ݔ‬ඨ൬

݀‫ݔ‬

‫ ݔ‬ଶ ൰ െͳ ξʹ

ξܽସ െ ‫ ݔ‬ସ



૚ ࢞૛ ‫ ܖܑܛ܋ܚ܉‬ቆ ૛ ቇ ൅ ࡯ ૛ ࢇ

5HVXHOYD

62/8&,21

‫ݔ݀ݔ‬



݀‫ݔ‬

‫ݔ‬ξʹ‫ ݔ‬ଶ െ Ͷ



૚ ࢞ ‫ ܋܍ܛ܋ܚ܉‬൬ ൰ ൅ ࡯ ૛ ξ૛

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ʹെͲ  ݊

6HWLHQH ‫ݔ‬ଵ ൌ Ͳ ൅

ʹ ʹ ൌ  ݊ ݊

‫ݔ‬ଶ ൌ

ʹ ʹ Ͷ ൅ ൌ  ݊ ݊ ݊

‫ݔ‬ଷ ൌ

Ͷ ʹ ͸ ൅ ൌ  ݊ ݊ ݊

« ¢&yPRSRGHPRVGHILQLUSDUDXQJHQpULFR" ‫ݔ‬௜ ൌ

ଶ‫כ‬௜ ௡

9HULILFDPRVTXHSDUDL QVHWLHQH‫ݔ‬௡ ൌ ʹ

1RWHTXHHVWDH[SUHVLyQJHQHUDOL]DGDUHSUHVHQWDHOSXQWR'(5(&+2¢3RUTXp" 8QUHFWiQJXORGHEDMRGHOiUHDGHODJUDILFDWHQGUiiUHD ‫ܣ‬௜ ൌ ሺο‫ ݔ‬ሻ൫݂ ሺ‫ݔ‬௜ ሻ൯ 1RWHTXHο‫ݔ‬HVODEDVHGHOUHFWiQJXOR\݂ሺ‫ݔ‬௜ ሻHVODDOWXUDGHOUHFWiQJXOR

3RUORFXDOVLVXPDPRVHVWDViUHDVFRQQFDQWLGDGGHUHFWiQJXORV ௡

෍ ο‫݂ݔ‬ሺ‫ݔ‬௜ ሻ ൌ ܴ݁ ௜ୀଵ

'HILQLFLyQGHODVXPDGH5LHQPDQQ  3RUHOSXQWRGHUHFKRVHGHILQH ‫ݔ‬௜ ൌ

ʹ݅ ʹ݅ ଷ ͺ  ൌ൐ ݂ሺ‫ݔ‬௜ ሻ ൌ ൬ ൰  ൌ൐ ݂ሺ‫ݔ‬௜ ሻ ൌ ଷ ݅ ଷ  ݊ ݊ ݊

3RUORFXDO ௡



௜ୀଵ

௜ୀଵ



ʹ ͺ ͳ͸ ͳ͸ ሺ݊ሻሺ݊ ൅ ͳሻ ෍ ൬ ൰ ൬ ଷ ݅ ଷ ൰ ൌ ସ ෍ ݅ ଷ ൌ൐ ቇ  ต ‫ܣ‬௡ ൌ ସ ቆ ݊ ݊ ʹ ݊ ݊ ா௝௘௥௖௜௖௜௢ଵ $TXtVHREWXYRXQDH[SUHVLyQHQIXQFLyQGHODFDQWLGDGGHUHFWiQJXORVTXH FRQVLGHUDPRVGHEDMRGHODJUDILFD $KRUDVLD8GVHOHGDXQQXPHUR),1,72GHUHFWiQJXORWDQVROR6867,78<$GLFKR QXPHURHQHVWDH[SUHVLyQRELHQ5($/,&(FDGDSDUWLFLyQSRUVHSDUDGR 3DUDHQHVWHFDVRGHOHMHUFLFLRGHEHPRVFRQVLGHUDUXQQ~PHUR,1),1,72GHUHFWiQJXOR SDUDTXHDVtHOiUHDVHDORPiVDSUR[LPDGDDODUHDO 3RUORFXDOGHEHPRVHYDOXDU ͳ͸ ସ ሺ݊ ൅ ʹ݊ ଷ ൅ ݊ ଶ ሻ ൌ Ͷ ௡՜ஶ ݊ ସ Ž‹

 3RUHOSXQWRL]TXLHUGR6HGHILQHFRPRXQRDQWHULRUDOGHUHFKR





௜ୀଵ

௜ୀଵ

‫ݔ‬௜ିଵ ൌ ௡

ʹ ሺ݅ െ ͳሻ ݊

ʹ ͺ ͳ͸ ͳ͸ ෍ ൬ ൰ ൬ ଷ ሺ݅ െ ͳሻଷ ൰ ൌ ସ ෍ሺ݅ െ ͳሻଷ ൌ ସ ෍ሺ݅ ଷ െ ͵݅ ଶ ൅ ͵݅ െ ͳሻ ݊ ݊ ݊ ݊ ௜ୀଵ

3RUSURSLHGDGHVGHVXPDWRULDVLJPDVHREWLHQH ଶ

ͳ͸ ݊ሺ݊ ൅ ͳሻ ݊ሺ݊ ൅ ͳሻሺʹ݊ ൅ ͳሻ ݊ሺ݊ ൅ ͳሻ ቇ െ ͵ቆ ቇ൅ ͵ቆ ቇ െ ݊൱ ‫ܣ‬௡ ൌ ൬ ସ ൰ ൭ቆ ݊ ʹ ͸ ʹ

(YDOXDPRVOtPLWHŽ‹௡՜ஶ ‫ܣ‬௡ ൌ Ͷ

 3RUSXQWRLQWHUPHGLRVHGHILQHHO‫ݔ‬௠ ൌ

௫೔ ା௫೔షభ ଶ



/RTXHUHVXOWD





௜ୀଵ

௜ୀଵ

ʹ ͳ ‫ݔ‬௠ ൌ ൬ ൰ ൬݅ െ ൰ ݊ ʹ ௡

ʹ ͺ ͳ ଷ ͳ͸ ͳ ଷ ͳ͸ ͵ ͵ ͳ ෍ ൬ ൰ ቆ ଷ ൬݅ െ ൰ ቇ ൌ ସ ෍ ൬݅ െ ൰ ൌ ସ ෍ ൬݅ ଷ െ ݅ ଶ ൅ ݅ െ ൰ ݊ ݊ ʹ ݊ ʹ ݊ ʹ Ͷ ͺ ௜ୀଵ

3RUSURSLHGDGHVGHVXPDWRULDVLJPD ଶ

ͳ͸ ͳ ݊ሺ݊ ൅ ͳሻ ݊ሺ݊ ൅ ͳሻሺʹ݊ ൅ ͳሻ ݊ሺ݊ ൅ ͳሻ ቇ െ ͳʹ ቆ ቇ ൅ ͸ቆ ቇ െ ݊൱ ‫ܣ‬௡ ൌ ൬ ସ ൰ ൭ͺ ቆ ݊ ͺ ʹ ͸ ʹ (YDOXDPRVOtPLWH Ž‹ ‫ܣ‬௡ ൌ Ͷ

௡՜ஶ

 5HVXHOYDODHFXDFLyQGLIHUHQFLDO ଵ

݂ ᇱᇱ ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬ଷ െ ͷ &RQFRQGLFLyQ ݂ ᇱ ሺͲሻ ൌ ʹǢ ݂ ሺͳሻ ൌ ͺ 5HHVFULELPRVODHFXDFLyQHQRWURIRUPDGLIHUHQFLDO ଵ ଵ ݀൫݂ሺ‫ ݔ‬ሻ൯ ݀ ݂݀ሺ‫ݔ‬ሻ ቆ ቇ ൌ ‫ ݔ‬ଷ െ ͷ ൌ൐ ݀ ቆ ቇ ൌ ൬‫ ݔ‬ଷ െ ͷ൰ ݀‫ݔ‬ ݀‫ݔ݀ ݔ‬ ݀‫ݔ‬

,QWHJUDPRV න݀ቆ

ଵ ݂ሺ݀‫ݔ‬ሻ ݂݀ሺ‫ ݔ‬ሻ ͵ ସ ቇ ൌ න ൬‫ ݔ‬ଷ െ ͷ൰ ݀‫  ݔ‬ൌ൐ ൌ ‫ ݔ‬ଷ െ ͷ‫ ݔ‬൅ ‫ܥ‬ ݀‫ݔ‬ ݀‫ݔ‬ Ͷ

$SOLFDQGRFRQGLFLyQLQLFLDO ݂݀ሺͲሻ ൌ ‫ ܥ‬ൌ൐ ‫ ܥ‬ൌ ʹ ݀‫ݔ‬ 3RURWURODGR

͵ ସ ͵ ସ ݀൫݂ ሺ‫ݔ‬ሻ൯ ൌ ൬ ‫ ݔ‬ଷ െ ͷ‫ ݔ‬൅ ʹ൰ ݀‫ ݔ‬ൌ൐  න ݂݀ ሺ‫ݔ‬ሻ ൌ න ൬ ‫ ݔ‬ଷ െ ͷ‫ ݔ‬൅ ʹ൰ ݀‫ݔ‬ Ͷ Ͷ ݂ ሺ‫ݔ‬ሻ ൌ

ͻ ଻ ͷ ଶ ‫ ݔ‬ଷ െ ‫ ݔ‬൅ ʹ‫ ݔ‬൅ ‫ܥ‬ ʹͺ ʹ

$SOLFDQGRFRQGLFLyQLQLFLDO ʹʹͻ ͻ ͷ െ ൅ ʹ ൅ ‫ ܥ‬ൌ ͺ ൌ൐ ‫ ܥ‬ൌ  ʹͺ ʹͺ ʹ

݂ሺͳሻ ൌ  6HD

గ గ గ ߨ න •‹ଶ ሺ‫ݔ‬ሻ ݀‫ ݔ‬ൌ Ǣන •‹ሺ‫ ݔ‬ሻ ݀‫ ݔ‬ൌ Ͳ Ǣන …‘•ሺ‫ ݔ‬ሻ ݀‫ ݔ‬ൌ ʹ ʹ ଴ ଴ ଴

'HWHUPLQH గ



D‫׬‬଴ ሺ…‘•ሺ‫ݔ‬ሻ ൅ Ͷሻଶ ݀‫ݔ‬E‫׬‬଴ ሺʹ •‹ሺ‫ݔ‬ሻ ൅ ͵ …‘•ሺ‫ ݔ‬ሻ ൅ ͳሻ݀‫ ݔ‬ 6ROXFLRQ D గ







න ሺ…‘• ଶ ሺ‫ ݔ‬ሻ ൅ ͺ …‘•ሺ‫ݔ‬ሻ ൅ ͳ͸ሻ݀‫ ݔ‬ൌ න …‘• ଶ ሺ‫ݔ‬ሻ ݀‫ ݔ‬൅ ͺ න …‘•ሺ‫ ݔ‬ሻ ݀‫ ݔ‬൅ ͳ͸ න ݀‫ݔ‬ ଴













න ሺͳ െ •‹ ‫ ݔ‬ሻሻ݀‫ ݔ‬൅ ͺሺʹሻ ൅ ͳ͸ሺߨሻ ൌ න ݀‫ ݔ‬െ න •‹ଶ ሺ‫ݔ‬ሻ ݀‫ ݔ‬൅ ͳ͸ ൅ ͳ͸ߨ ଶሺ



ൌߨെ



ߨ ૜૜ ൅ ͳ͸ ൅ ͳ͸ߨ ൌ ૚૟ ൅ ࣊ ʹ ૛



E గ





ʹ න •‹ሺ‫ݔ‬ሻ݀‫ ݔ‬൅ ͵ න …‘•ሺ‫ݔ‬ሻ ݀‫ ݔ‬൅ න ݀‫  ݔ‬ൌ൐ ʹሺͲሻ ൅ ͵ሺʹሻ ൅ ߨ ൌ ૟ ൅ ࣊ ଴






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