Cálculo Diferencial Udec

El Sistema de los Números Reales y la Recta Real

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 

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O RV Q~ Q ~PHURV UH U HDOHV  \ (MHPSOR 6HDQ OR WDPELpQXQQ~PHURUHDO

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D Á   l  b  ~  b 

3DUDOR ORVQ VQ~P ~PHU HURV RVUH UHDO DOHV HV  \  VH VHF FXP XPSO SOH HTXH TXH   b  ~  b  (MHPSOR 3DUD

³ $VRFLDWLYD

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D Á Á   l Á  b  b  ~  b  b   

&RQVLG LGHU HUHHPRV PRVOR ORV VQ~ Q~PH PHUR URV VUH UHDO DOHHV Á (MHPSOR &RQV TXH

8  b  9 b

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³ (OHPHQWR1HXWUR E[   l WDO TXH D   l 

b~b~

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ÉÉ

Cálculo Diferencial

³ /D6XVWUDFFLyQ VHGHILQHFRPRODRSHUDFLyQLQYHUVDGHODDGLFLyQGHQ~PHURV UHDOHVHVGH UHDOHVHVGHFLUSD FLUSDUD UD FXDOTXLHU FXDOTXLHU Á   l

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c ~b c

VQ~PHURVU VUHDOHV  \  VH VH WLHQH TXH (MHPSOR 6HDQ ORVQ

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 c  ~  b c 

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D Á Á   l VHFXPSOH ³ &ODXVXUD

D Á   l Á  h   l

3DUDORVQ~PHURVUHDO RVUHDOHV HV \ VHFXPSOH VHFXPSOHTXH TXH  h HVWiHQOR HVWiHQORVUHDOH VUHDOHV V (MHPSOR 3DUDORVQ~PHU

³ &RQPXWDWLYD

D Á   l  h  ~  h 

'DGRVORVQ~PHURVUHD URVUHDOHV OHV \ VHFXPSOH VHFXPSOH  h  ~  h  (MHPSOR 'DGRVORVQ~PH

³ $VRFLDWLYD

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D Á  Á   l Á  h  h  ~  h  h 

(MHPSOR 6HD 6HDQ ORVQ VQ~PHURVU VUHDOHV

  Á \   HQWR QWRQFH QFHV VXF VXFHGH TXH        h h ~ h h    

8 9

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³ (OHPHQWR,QYHUVR0XOWLSOLFDWLYR

D   lÁ  £  E[ c ~

 c h  ~  h  c ~ 

  l WDO TXH  

(MHPSOR 3DUD 3DUDHO HOQ Q~P ~PHU HURU RUHD HDO O  VH VHFX FXPS PSOH OHT TXH XH 

 h

  ~ h  ~   

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el sistema de los números reales y la recta real

VQ~PHURVU VUHDOHV  \  VH VH WLHQH TXH (MHPSOR 6HDQ ORVQ

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3URSLH 3URSLHGDG GDG' 'LVW LVWULE ULEXWL XWLYD YDHQ HQl

D Á Á   l VHFXPSOH

 b ! h  ~  h  ! b  h  !  h  b  ! ~  h  ! b  h  !

6HDQO ORV RVU UHD HDOOHV   \ HQ HQWR WRQF QFHV HV (MHPSOR 6HDQ

 b ! h  ~  h ! b  h !  h  b ! ~  h ! b  h ! $XWRHYDOXDFLyQ

6LQUHVROYHULGHQWLILFDODVSURSLHGDGHVGHORVQ~PHURVUHDOHV

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³  b  ~  b 

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³  h c  ~ c  h 

³  h  ~  h 

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³ c  b  ~  b c  ~ 

5HVSXHVWDV ³ 3URSLHGDGFRQPXWDWLYDGHODDGLFLyQ

³ 3URSLHGDGGLVWULEXWLYD ³ 3URSLHGDGGHOHOHPHQWRQHXWURDGLWLYR ³ 'LYLVLyQRSHUDFLyQLQYHUVDGHODPXOWLSOLFDFLyQ ÉÉ


³ 3URSLHGDGDVRFLDWLYDGHODDGLFLyQ ³ 3URSLHGDGFRQPXWDWLYDGHODPXOWLSOLFDFLyQ ³ 6XVWUDFFLyQRSHUDFLyQLQYHUVDGHODDGLFLyQ ³ 3URSLHGDGGHOHOHPHQWRQHXWURPXOWLSOLFDWLYR ³ 3URSLHGDGGHOHOHPHQWRLQYHUVRPXOWLSOLFDWLYR ³ 3URSLHGDGGHOHOHPHQWRLQYHUVRDGLWLYR

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(MHPSOR 6HDQ ORV UHDOHV c  \  HQWRQFHV VyOR VH FXPSOH XQD GH ODV WUHV UHODFLRQHVDQ UHODFLRQHVDQWHULRU WHULRUHV HV

  c ³ 7UDQVLWLYLGDG D Á Á   l  (MHPSOR 3DUD  Á

 w  ¬ 

 \ c  VH WLHQH TXH     w  c ¬  c  

³ $GLFLyQ D Á Á   l  (MHPSOR 'DGRV   \

   ¬ b b

 VH FXPSOH       ¬  b   b  

³ 0XOWLSOLFDFLyQ D Á Á   l     w    ¬  h    h 

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(MHPSOR 6HDQ Á c  \  WDOHV TXH

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  ¯  v  ~

(MHPSOR

 ¯ v ~

0HQRUTXH

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(MHPSOR

   ¯   

0HQRURLJXDOTXH    ¯    (MHPSOR

   ¯   

/RV VtPERORV  Á  Á  Á  GHQRWDQ OD H[LVWHQFLD GH GHVLJXDOGDGHV



3URSLHGD 3URSLHGDGHV GHVGH GHOD ODV' V'HVLJ HVLJXDO XDOGDGH GDGHVV

³ 7UDQVLWLYD 6L    w    Á HQWRQFHV    (MHPSOR 6L    w   Á HQWRQFHV   

³ $GLWLYD 6L    w     HQWRQFHV  b    b  (MHPSOR 6L    w    HQWRQFHV  b    b 

³ 6L    w   l HQWRQFHV  b    b  (MHPSOR 6L    w   l HQWRQFHV  b    b 

³ 6L    w   l HQWRQFHV  c    c  (MHPSOR 6L c    w   l HQWRQFHV c  c    c 

³ 6L    w    HQWRQFHV  h    h  (MHPSOR 6L c    w    HQWRQFHV ² c ³ h    h 

³ 6L    w    HQWRQFHV  h    h 

 ! c !  ! c !

(MHPSOR 6L c    w c    HQWRQFHV c 

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³ 6L    \  HVWi HQWUH  \   HQWRQFHV    w     HV GHFLU      (MHPSOR 6L    w    HQWRQFHV     

³ 6L    HQWRQFHV c  c (MHPSOR 6L    HQWRQFHV c  c

 

,QWH ,QWHUY UYDO DORV RV5 5HD HDOH OHVV

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³ +DOODU OD HFXDFLyQ GH OD FLUFXQIHUHQFLD FRQ FHQWUR HQ * ²Á c ³ \ TXH HV WDQJHQWH QWH DO HMH @  ³ +DOODU OD HFXDFLyQ GH OD FLUFXQIHUHQFLD FRQ FHQWUR HQ * ²Á c ³ \ TXH HV WDQJHQWH QWH DO HMH ?  ³ 8QD FL FLUFXQIHUHQFLD SD SDVD SR SRU OR ORV SX SXQWRV ² c Á ³ \ ²Á ³  \ \ VX VX FH FHQWUR HV HVWi VREUH OD UHFWD % c & c  ~   +DOODU VX HFXDFLyQ ³ 8QD FXHUGD GH OD FLUFXQIHUHQFLD % b &  ~  HVWi VREUH OD UHFWD FX\D HFXDFLyQ HV % c & b  ~   +DOODU OD ORQJLWXG GH OD FXHUGD ³ 2E 2EWH WHQH QHUO UODH DHFX FXDF DFLy LyQG QGHO HODF DFLU LUFX FXQI QIHU HUHQ HQFL FLDV DVLW LWLH LHQH QHFH FHQW QWUR URHQ HQHO HOSX SXQW QWR R²Á ³ \ HV WDQJHQWH D OD UHFWD TXH FRQWLHQH ORV SXQWRV ²Á c ³ \ ²Á c ³  ³ 'HWHUPLQDUODHFXDFLyQGHODFLUFXQIHUHQFLDTXHSDVDSRUORVSXQWRV 7 ²Á ³Á 8²Á ³ \ 9² 9²Á c ³ ³ ¢3DUD TXp YDORUHV GH  SDVD OD FLUFXQIHUHQFLD ²% b  ³ b ²& c  ³  ~  SRU HO SXQWR ²Á³" ³ +DOODU HO YDORU GH  WDO TXH % b &  b % c & b  ~  VHD OD HFXDFLyQ GH XQD FLUFXQIHUHQFLDGHUDGLR 6ROXFLRQHV

³ ³

% b &  c % b & b  ~  % b &  c % b & b  ~  É  É


³ ³ ³ ³ ³ ³

% b &  c % b & c  ~  /RQJLWXGFXHUGD   % b &  c % c & c  ~  % b &  c % c & ~   ~ Á  ~ c   ~ c 

l

3DUiER 3DUiEROD OD

(VHOOXJDUJHRPpWULFRGHORVSXQWRVGHOSODQRTXHHTXLGLVWDQGHXQSXQWRILMR OODP OODPDG DGR R IRFR\G \GH HXQ XQD DUH UHFW FWD DIL ILMD MDO OOD ODPD PDGD GD GLUHFWUL]

6L - ²Á ³ HV HO IRFR GH OD SDUiERODÁ + ¢ & ~ c  HV OD HFXDFLyQ GH VX GLUHFWUL] \ 7 ²%Á &³ HVXQSXQWRGHHOODHQWRQFHVODHFXDFLyQGHODSDUiERODHV ¢ % ~ &

(O SXQWR = ²Á ³ VH GHQRPLQD YpUWLFH GH OD SDUiEROD OD UHFWD % ~  VH OODPD HMH GHVLPHWUtDGHODSDUiEROD (OWUD]RSHUSHQGLFXODUDOHMHGHODSDUiEROD\TXHSDVD SRUHOIRFRVHGHQRPLQDODGRUHFWR \VXORQJLWXGHV  À 6L    HQWRQFHV OD FXUYD TXHGD VLWXDGD VREUH HO HMH ?  HV QR QHJDWLYD \ VLPpW VLPpWUL ULFD FDU UHVS HVSHF HFWR WRD DO OHM HMH H@  VLWL 6L   HQ HQWR WRQF QFHV HVO OD DFX FXUY UYD DTX TXHG HGD DVL VLWX WXDG DGD DED EDMR MRH HO OHM HMH H? HV HVQ QR RSR SRVL WLYD YD\ \V VLP LPpW pWUL ULFD FD UHVS UHVSHF HFWR WRD DO OHM HMH H@ 

d d



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6L - ²Á ³ HV HO IRFR GH OD SDUiERODÁ + ¢ % ~ c  HV OD HFXDFLyQ GH VX GLUHFWUL] \ 7 ²%Á &³ HVXQSXQWRGHHOODHQWRQFHV VXHFXDFLyQFDQyQLFDHV ¢ & ~

 %

¢ = ²Á ²Á ³ 9pUWLFH ¢ -²Á³ )RFR ¢ + ¢ % ~ c  'LUHFWUL] ¢&~ (MHGHVLPHWUtD ¢  /RQJ ODGR UHFWR 6L   HQW HQWRQFHVOD VODFXU FXUYDTX YDTXHHGDVLW VLWXDGDDODGH DODGHUUHFKDGHO GHOHMH HMH @ \HVV \HVVLLP pWULFD UHVS UHVSHF HFWR WRD DO OHM HMH H?  6L   HQW HQWRQ RQFH FHVO VODF DFXU XUYD YDTX TXHG HGDV DVLW LWXD XDGD GDDO DODL DL]T ]TXL XLHU HUGD GDGH GHOH OHMH MH @ \HV \HVVL VLPp PpWU LFDD WULF UHVS UHVSHF HFWR WRD DO OHM HMH H? 

d d

(MHPSORV

²³ 'HWHUPLQHYpUWL 'HWHUPLQHYpUWLFHIRFRGLUHFW FHIRFRGLUHFWUL]ORQJLW UL]ORQJLWXGGHOODGRUHFWRHMHGHODSDUiE XGGHOODGRUHFWRHMHGHODSDUiEROD\ ROD\ JUiILFRSDUD ¢ ³ % ~ & ³ &  ~ c % 6ROXFLyQ

³ % ~ & & ¬   ~  ¬  ~  ¢ = ²Á ²Á ³ 9pUWLFH ¢ -²Á³ )RFR ¢ + ¢& ~ c 'LUHFWUL] (MHGHVLPHWUtD ¢%~ ¢ /RQJODGRUHFWR

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Ü ³&  ~ c % ¬  ~ c  ¬  ~ c

9pUWLFH )RFR 'LUHFWUL] (MHGHVLPHWUtD /RQJODGRUHFWR

¢ = ²Á ²Á ³  ¢ - ² c Á ³ ³   ¢+¢%~  ¢&~ ¢

 

Ü

6L = ²Á  ³HVHO HVHOYp YpUUWLFH\HO \HOHM HMHHGHVL GHVLP PHWUtDGHOD GHODSD SDUUiERODHVSD HVSDUUDOHORDOHM DOHMHH @  HQWRQFHV

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¢ ²% c  ³³ ~ ²& ²& c ³ (FXDFLyQ ¢ = ²Á ²Á ³ 9pUWLFH )RFR ¢ - ²Á ²Á  b ³ ¢ + ¢ & ~  c  'LUHFWUL] (MHGHVLPHWUtD ¢ % ~  /RQJOUHFWR ¢  6L   HQ HQWR WRQF QFHV HVO OD DFX FXUY UYD DHV HVF FyQ yQFD FDYD YDK KDF DFLD LDD DUU UULE LED D 6L   HQ HQWR WRQF QFHV HVO OD DFX FXUY UYD DHV HVF FyQ yQFD FDYD YDK KDF DFLD LDD DED EDMMR

d d

6L = ²Á ³HVHO HVHOYp YpUUWLFH\HO \HOHM HMHHGHVL GHVLP PHWUtDGHOD GHODSD SDUUiERODHVSD HVSDUUDOHORDO DOHMH HMH ?  HQWRQFHV

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¢ ²& c ³ ~ ²% ²% c ³ (FXDFLyQ 9pUWLFH ¢ = ²Á ²Á ³ ¢ - ² b Á Á ³ )RFR ¢ + ¢ % ~  c  'LUHFWUL] (MHGHVLPHWUtD ¢ & ~  /RQJODGRUHFWR ¢  6L   HQ HQWR WRQF QFHV HVO OD DFX FXUY UYD DHV HVD DEL ELHU HUWD WDK KDF DFLD LDO OD DGH GHUH UHFK FKD D 6L   HQ HQWR WRQF QFHV HVO OD DFX FXUY UYD DHV HVD DEL ELHU HUWD WDK KDF DFLD LDO OD DL] L]TX TXLH LHUG UGD D

d d

(MHPSORV

²³ 'HWHUPLQHYpUWL 'HWHUPLQHYpUWLFHIRFRGLUHFW FHIRFRGLUHFWUL]ORQJLW UL]ORQJLWXGGHOODGRUHFWRHMHGHODSDUiE XGGHOODGRUHFWRHMHGHODSDUiEROD\ ROD\

JUiILFRHQ ³ % b % b & b  ~  ³ & c % % c & & b  ~  6ROXFLyQ

 !  !

³ % b % b & b  ~  ¬ %  b % ~ c & c  ¬ % b % b  ~ c & c  b  ¬ %b  ~ c &b ¬  ~ c  ¬ ~ c

9pUWLFH )RFR

¢ = ² c Á c ³ ¢ - ² c Á c ³³

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!

 !


¢ + ¢& ~ c 'LUHFWUL] (MHGHVLPHWUtD ¢ % ~ c  /RQJODGRUHFWR ¢ 

Ü ³ & c % c & b  ~ 

 ! !  !  !

¬ &  c & ~ % c  ¬ &  c & b  ~ % c  b  ¬ &c  ~ %c ¬  ~  ¬~

9pUWLFH ¢ = ²Á ²Á ³ ¢ - ²Á³ )RFR ¢+¢%~ 'LUHFWUL] (MHGHVLPHWUtD ¢ & ~  /RQJODGRUHFWR ¢ 

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Ü À(MHU (MHUFL FLFL FLRV RV3U 3URSX RSXHV HVWR WRVV

³ +DOODUHOYpUWLFHIRFRGLUHFWUL]ORQJLWXGGHOODGRUHFWR\JUiILFRGH ³ &  b % ~   ³ & c  %  ~  ³ % b & ~  ³ +DO +DOODUO ODUODH DHFX FXDDFLyQ FLyQGH GHOD ODSD SDUi UiER EROD ODFX FX\\RYpU RYpUWL WLFH FH\ \IR IRFR FRVR VRQO QORV RVSX SXQW QWRV RV ²Á ³ \ ²Á ³ UHVSHFW UHVSHFWLYD LYDPHQ PHQWH WH +DOODU +DOODU DGHPiV DGHPiV ODHFXDFLyQ ODHFXDFLyQ GHVX GLUHFW GLUHFWUL]\ UL]\ ODORQJLWXG ODORQJLWXG

GHOODGRUHFWR ³ +DOODU OD HFXDFLyQ GH OD SDUiEROD GH IRFR HQHO SXQWR ²Á c ³ \ GLUHFWUL] OD UHFWD% ~ À ³ 'DGD OD SDUiEROD GH HFXDFLyQ & b & c % b  ~ Á KDOODU ODV FRRUGHQDGDV

GHOYpUWLFHGHOIRFR\ODHFXDFLyQGHODGLUHFWUL] ³ 'HWH 'HWHUP UPLQ LQDU DUOD ODHF HFXD XDFL FLyQ yQGH GHOD ODSD SDUD UDER EROD ODFR FRQY QYpU pUWL WLFH FHHQ HQHO HOSX SXQW QWR R²Á ³\GLU \GLUHF HFWU WUL] L] %~

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³ +DOO +DOODU DU OD HFXD HFXDFL FLyQ yQ GH OD SDUi SDUiERO ERODFRQYpUW DFRQYpUWLF LFHHQ HHQ HO RULJ RULJHQ HQ HMH HMH & FRPR FRPR HMHGH HMHGH ²Á  ³³ VLPHWUtD\TXHFRQWHQJDDOSXQWR²Á ³ 'HWHUPLQDUOD HFXDFLyQ GH OD SDUiEROD FRQ YpUWLFH HQ ²Á ³Á HMH GH VLPHWUtD SDUDOHORDOHMH&FRQWLHQHDOSXQWR²Á ³

+DOODU DU OD HFXD HFXDFL FLyQ yQ GH OD SDUi SDUiER EROD OD FRQ FRQ HMH HMH GH VLPH VLPHWU WUtD tD SDUD SDUDOH OHOR OR DO HMH HMH ? TXH ³ +DOO FRQWLHQHORVSXQWRV²Á ²Á ³Á ³Á ²Á ²Á ³Á ³Á ²Á ²Á ³ ³ 6L HO ODGR UHFWR GH XQD SDUiEROD HV HO VHJPHQWR TXH XQH ²Á ³ FRQ ²Á ³  GHWHUPLQDUODHFXDFLyQUHVSHFWLYDVLODSDUiERODSDVDSRUHOSXQWR²Á ²Á ³ ³ 6LX 6LXQD QDS SDU DUiE iERO ROD DSD SDVD VDS SRU RUO ORV RVS SXQ XQWR WRV V²Á ³ \ ²Á ³ FR FRQ QHM HMH HGH GHV VLP LPHW HWUt UtD DYH YHUW UWLF LFDO DO \ YpUWLFH VREUH OD UHFWD % c & ~  GHWHUPLQDU VX HFXDFLyQ 6ROXFLRQHV

³

³ 9pUWLFH

= ²Á ³ )RFR - ² c Á ³ 'LUHFWUL] % ~  /RQJODGRUHFWR  ~ 

d d

³ 9pUWLFH

= ²Á ³   )RFR -  

6 7

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'LUHFWUL] & ~ c /RQJODGRUHFWR

    ~ 

d d

³ 9pUWLFH

= ²Á ³  )RFR - Á c   'LUHFWUL] & ~   /RQJODGRUHFWR  ~ 

8

9

d d

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³

³ ³

% c % b & c  ~  (FXDFLyQ &~ 'LUHFWUL] /RQJODGRUHFWR  ~   (FXDFLyQ & b & c % b  ~  9pUWLFH = ² c Á c ³

d d

8

 

9

)RFR - c Á c  'LUHFWUL] % ~ c ³ ³ ³ ³ ³ ³

 

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derivadas de funciones

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%b

%

% c % % c %



c% % c

%

OLP  % b ! % %¦ ³

³

³

³

OLP

%¦B

OLP

%¦

¦B

¦B

6ROXFLRQHV

³

³

OLP

%¦

%

 % c  ²% c ³

OLP

OLP



% h !

8 9 b

8

 

 

  b    b 

9

% c %

%

~ 

89  

 c ²& b ³  OLP ~ c & &¦

É  É


³

³

³ ³ ³

³

³

³

OLP

%¦ OLP

%¦ OLP

%¦

l l l l l l l l c

%b

%

% c % % c %



l 



%

~ 

OLP

² % b ! %³ %³ ~ 

OLP

% h !

%¦ %¦B

67 

%



~

c% % c

~ c

~

 OLP ²% c ³ ²% c ³ ~  %¦

OLP

¦B

OLP

¦B

  ~

8 9 l 8 9 l b

 



  b   ~  b 





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Cálculo Diferencial Udec

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