Prepara tus exámenes y mejora tus resultados gracias a la gran cantidad de recursos disponibles en Docsity
Gana puntos ayudando a otros estudiantes o consíguelos activando un Plan Premium
Prepara tus exámenes
Prepara tus exámenes y mejora tus resultados gracias a la gran cantidad de recursos disponibles en Docsity
Prepara tus exámenes con los documentos que comparten otros estudiantes como tú en Docsity
Encuentra los documentos específicos para los exámenes de tu universidad
Estudia con lecciones y exámenes resueltos basados en los programas académicos de las mejores universidades
Responde a preguntas de exámenes reales y pon a prueba tu preparación
Consigue puntos base para descargar
Gana puntos ayudando a otros estudiantes o consíguelos activando un Plan Premium
Comunidad
Pide ayuda a la comunidad y resuelve tus dudas de estudio
Ebooks gratuitos
Descarga nuestras guías gratuitas sobre técnicas de estudio, métodos para controlar la ansiedad y consejos para la tesis preparadas por los tutores de Docsity
Asignatura: Matematicas I, Profesor: , Carrera: Derecho + Administración y Dirección de Empresas, Universidad: UC3M
Tipo: Ejercicios
1 / 26
Esta página no es visible en la vista previa
¡No te pierdas las partes importantes!
,QWURGXFFLyQ
'P ECRÈVWNQU CPVGTKQTGU UG JC GUVWFKCFQ NC KPVGITCN FG 4KGOOCP FG WPC HWPEKÎP H (^) CEQVCFC[FGHKPKFCGPWPKPVGTXCNQEGTTCFQ[CEQVCFQ C D (^) EQP C [ D PÕOGTQUTGCNGU 'P
GUVCUGEEKÎPKPVGPVCTGOQUIGPGTCNK\CTGUVGEQPEGRVQFGKPVGITCNRCTCHWPEKQPGUSWGPQXGTKHKECP CNIWPCFGNCUJKRÎVGUKUCPVGTKQTOGPVGEKVCFCUGUVQGU .CKPVGITCNFGWPCHWPEKÎPPQCEQVCFCFGHKPKFCGPWPKPVGTXCNQCEQVCFQEQOQRQT GLGORNQ
Z FZ^ Z F Z^ Z FZ
f f
³ (^) f^ f H^ Z^ F^ Z^ ³ (^) f^ E H^ Z^ F^ Z^ ³ (^) EfH^ Z^ FZ
UKPSWGKPHNW[CNCGNGEEKÎPFG E 'UVQ FGDG GPVGPFGTUG FGN UKIWKGPVG OQFQ 5K EQPXGTIGP NCU FQU KPVGITCNGU KORTQRKCU FGNUGIWPFQOKGODTQGPVQPEGUEQPXGTIG GZKUVG NCKPVGITCNKORTQRKCFGNRTKOGTOKGODTQ[ GU KIWCN C NC UWOC FG FKEJCU FQU KPVGITCNGU #JQTC DKGP EWCPFQ WPC FG NCU KPVGITCNGU FGN
EE [ [ [ E E E
^ E ^
G Y
Y (ODUHDGHHVWH FRQMXQWRQRDFRWDGR HVLQILQLWR
YpDVHODILJXUD FRQVLGHUDQGRSRUHMHPSOR F^ ^ \DTXH [ [ [ [ [ [ [ [
D D E
E
(^)
ff of of
f f
G G G G
OLP OLP
2TQRQUKEKÎP %QPFKEKQPGURCTCNCGZKUVGPEKCFGNCKPVGITCNFGWPCHWPEKÎPCEQVCFCUQDTGWP KPVGTXCNQPQCEQVCFQ
KPVGITCDNG GP C D RCTC VQFQ D t C EQP HWPEKÎP RTKOKVKXC ( VCN SWG
#UÈ RWGU NC GZKUVGPEKC FG WP PÕOGTQ TGCN M VCN SWG M (^) D NKO of ( D0 5 GU WPC EQPFKEKÎP
PGEGUCTKC [ UWHKEKGPVG RCTC NC EQPXGTIGPEKC FG NC KPVGITCN KORTQRKC FG WPC HWPEKÎP H SWG
CFOKVGRTKOKVKXC ( 'P GHGEVQ RQT GN UGIWPFQ VGQTGOC HWPFCOGPVCN FGN E¶NEWNQ KPVGITCN UG XGTKHKEC SWG RCTCVQFQ D t C
³ (^) C^ D H^ Z^ FZ^ (^ D^ (^ C
%QOQRQTFGHKPKEKÎP
³ (^) C^ f H^ Z^ F^ Z^ DNKO of^ ³ (^) C^ DH^ Z^ F^ Z^ DNKOofª¬^ (^ D^ ^ (^ Cº¼
UGT¶ UWHKEKGPVG GPVQPEGU SWG (^) DNKO of ( D0 5 M RCTC SWG NC KPVGITCN FG H GP Cf UGC
EQPXGTIGPVG[ (^) ³ (^) C^ f H Z FZ M ( C
.C EQPFKEKÎP VCODKÃPGUPGEGUCTKC[CSWGUK (^) ³ (^) C^ f H Z FZGUEQPXGTIGPVG[ ( GUNC
RTKOKVKXCFG H GPVQPEGU
³ (^) C^ f H^ Z^ F^ Z^ DNKO of^ ³ (^) C^ DH^ Z^ F^ Z^ DNKOof^ ª¬^ (^ D^ ^ (^ C^ º¼^.^ . ^ ^
5GXGTKHKECRWGU (^) EOLP of ) E1 6 / ) D1 6 N N
(MHPSOR (VWXGLDUVLHVFRQYHUJHQWHODLQWHJUDO G[ (^) [
f I \HQFDVRDILUPDWLYRFDOFXODUVXYDORU 6ROXFLyQ /DJUiILFDGH I 1 6[ (^) ^ [ HV G Y
Y
2EVpUYHVH TXH I HV XQD IXQFLyQ FRQWLQXD \ DFRWDGDHQ f 6 HQFRQFUHWR I 1 6[ d SDUDWRGR[ t SRUORWDQWR I HVLQWHJUDEOHHQWRGRLQWHUYDORGH ODIRUPD E FRQ E t
$GHPiVODSULPLWLYDGH I 1 6[ [^ HV ) [1 6 DUFWJ [SDUDODFXDOVHYHULILFD
EOLP of^ ) E1 6^ EOLP DUFWJof E S f 3RUORTXHWHQLHQGRHQFXHQWDHOUHVXOWDGRSUHFHGHQWHVHWLHQH G [ DUFWJ [ )
E I (^) ^ ^ ^
S (^) 1 6 S S (MHPSOR (VWXGLDUSDUDTXHYDORUHVGH D HVFRQYHUJHQWHODLQWHJUDOGHODIXQFLyQ I 1 6[ [HQ HOLQWHUYDOR D f 6 FRQ D!
RCTC D d RQT GN ETKVGTKQ FG EQORCTCEKÎP RQFGOQU FCT WPC EQPFKEKÎP UWHKEKGPVG FG
NCKPVGITCN (^) ³ (^) C^ f H Z FZGUEQPXGTIGPVG
NCKPVGITCN (^) ³ (^) C^ f H Z FZPQGUEQPXGTIGPVG
HVFRQYHUJHQWH\DTXHSDUDWRGR [ t
d (^) [ (^) H [ [
(MHPSOR /DLQWHJUDO [ [
G HVGLYHUJHQWHSXHVSDUDWRGR [ t [ [
[ [ [
t t
(^) ^ ^ VLHQGR^
G [ [
f
(MHPSOR 6HD I (^) 1 6[ VHQ[[ &RPR d VHQ[ ^ [d[ \ (^) I f (^) [ G HV FRQYHUJHQWH WHQGUHPRV TXH[
VHQ [ [ [
f I G^ HVFRQYHUJHQWHGHGRQGHVHGHGXFHTXH^ VHQ [ [ [
f I G^ HVWDPELpQFRQYHUJHQWH
[UGFGPQVCRQT (^) ³ (^) C^ D H Z FZ
³ (^) C^ D H^ Z^ F^ Z^ ³ (^) CE H^ Z^ F^ Z^ ³ (^) EDH^ Z^ FZ
&G PWGXQ GP GUVG ECUQ NC KPVGITCN (^) ³ (^) C^ D H Z FZ GU EQPXGTIGPVG UK NQ UQP NCU FQU KPVGITCNGU
KORTQRKCUSWGCRCTGEGPGPGNUGIWPFQOKGODTQ'PEWCPVQWPCFGGUVCUÕNVKOCUKPVGITCNGUPQ GZKUVCVCORQEQGZKUVKT¶NCKPVGITCNFGNRTKOGTOKGODTQ
)GQOÃVTKECOGPVG (^) ³ (^) C^ D H Z FZTGRTGUGPVCGN¶TGCFGNEQPLWPVQ $
Y
G Y
B D C
$
5GKPFKECCEQPVKPWCEKÎPWPCEQPFKEKÎPRCTCNCGZKUVGPEKCFGNCKPVGITCNFGWPCHWPEKÎP
³ (^) C^ D H^ Z^ FZ^ (^ D^ M
FDOFXODUVXYDORU
G Y
Y
(VFODURTXH I HVXQDIXQFLyQFRQWLQXD\DFRWDGD HQ H SDUD WRGR H! $Vt SXHV I HV LQWHJUDEOHHQ H \VXSULPLWLYDHV ) [1 6 [ OQ [ [ \DTXH ) c (^) 1 6[ OQ [ [ [ OQ[ I (^) 1 6[
$GHPiVSDUDODSULPLWLYDVHYHULILFD OLP H o )1 6H OLPH o 1 H OQ H H 6 OLPH o H 1 OQ H 6 OLP HoOQHH
\DSOLFDQGR/C+{SLWDOVHREWLHQH OLP H o )1 6H OLPH o (^) HH OLPHoH 7HQLHQGRHQFXHQWDHOWHRUHPDDQWHULRUUHVXOWD
I OQ^ [^ G[^ )^ 1 6^ ^ ^ ^ OQ^ ^ ^ ^ ^ (MHPSOR (VWXGLDU SDUD TXp YDORUHV GH D HV FRQYHUJHQWH OD LQWHJUDO GH OD IXQFLyQ I 1 6 [ (^) [ D 1 6
D HQHOLQWHUYDOR^1 D E^ 6ROXFLyQ /DJUiILFDGH I VHJ~QORVYDORUHVGH D HVGHOWLSR D
D D B C B C
D
D D!
B^ C^ ^ B C (^)
3DUDWRGR D I 1 6 [ (^) [ D 1 6
D HVFRQWLQXD\DFRWDGDHQORVLQWHUYDORVGHODIRUPD^ D^ H^ E^ H^! ^ \ SRUFRQVLJXLHQWHLQWHJUDEOHVREUHGLFKRVL QWHUYDORV/DSULPLWLYDGH I HV
) [ [^ D ) [ [ D
0 5^0 0 5
z
D D D D
VL OQ VL
3DUDFDGDXQRGHHVWRVFDVRVVHYHULILFD
x 6L D z OLP H OLPH OLP
D H
D o H^ o H^ H D DD
o
^ ^ %&f ! '
) D (^) D D (^0 5) [ 0 5 VL VL x 6L D OLP H o ) D 0 H 5 OLP OQH o D H D OLP OQHo H f $VtSXHVGHDFXHUGRFRQHOFULWHULR
(MHPSOR 6HD I (^) 1 6 [ (^1) [ [^ 6 GHILQLGDHQ 1 /DLQWHJUDOGHHVWDIXQFLyQQRHVFRQYHUJHQWH
SXHV d (^) [ ^ d (^1) [ 6 [SDUDWRGR [ 1
(MHPSOR 6HD I [ [ [ 1 6
^
GHILQLGDHQ 1 /DLQWHJUDOGHHVWDIXQFLyQHVFRQYHUJHQWHSXHV
d [ [ d [ \
I (^) [ G[^ HVFRQYHUJHQWH
(MHPSOR /DLQWHJUDO , (^) I (^) [[ [
VHQ
G HVFRQYHUJHQWHSXHVSDUDWRGR [ 1
d VHQ^ [ d [ [ VLHQGR^
I (^) [ HVFRQYHUJHQWH
SRUORTXH VHQ
(^) [ I (^) [ G[^
HVFRQYHUJHQWH\SRUWDQWR , HVFRQYHUJHQWHWDPELpQ
,QWHJUDOHV(XOHULDQDV
8COQU C GUVWFKCT FQU GLGORNQU FG KPVGITCNGU KORTQRKCU SWG FGRGPFGP FG WPQ Q FQU RCT¶OGVTQU TGURGEVKXCOGPVG UQP NCU KPVGITCNGU $GVC [ )COOC VCODKÃP NNCOCFCU KPVGITCNGU 'WNGTKCPCU &GHKPKEKÎP 5GFGHKPGNCHWPEKÎP$GVCFGRCT¶OGVTQUR[SEQOQNCKPVGITCN
R S
S
S
R
UGJCEGEGTQUK^ S^ ^ ^
2TQRQUKEKÎP 'ZKUVGPEKC 5K R! [ S! NCKPVGITCN E R S^ GUEQPXGTIGPVG &GOQUVTCEKÎP8COQU C GUVWFKCT NQU XCNQTGU FG R [ S RCTC NQU EWCNGU GUVC KPVGITCN GU
EQPXGTIGPVG'NGUVWFKQNQXCOQUCJCEGTRQTEQORCTCEKÎP[RQTNQVCPVQQDVGPFTGOQUWPC EQPFKEKÎPUWHKEKGPVGFGGZKUVGPEKCQEQPXGTIGPEKC'PRTKOGTNWICTEQPUKFGTCOQU
d (^) (^) d
S R R
d (^) d (^)
R
d ^ (^) d
Z R^ G Z^ ZG ZG Z Z
R Z
R Z
2QT NCU RTQRKGFCFGU [ [ RQT GUVCT VCDWNCFC NC HWPEKÎP * 0 5R UG FGURTGPFG NC
KORQTVCPEKC FG NC RTQRKGFCF SWG PQU RGTOKVG ECNEWNCT EQP EKGTVC HCEKNKFCF GN XCNQT FG CNIWPCUKPVGITCNGUKORTQRKCU
(MHPSOR &RPR E S
S 1 ^6 ^ I GW^ \^ E^ ^
1 6 *^1 *^ 1 6 6 REWHQHPRVTXH^ *^1 6 S^
(MHPSOR +DOODU (^) I f^ [ H [G[
6ROXFLyQ (^) I f^ [ H [G[ (^) * 1 6 * * 1 6 * 1 6
(MHPSOR +DOODU (^) I fOQ[ [G[
6ROXFLyQ (IHFWXDQGRHOFDPELRGHYDULDEOH
OQ [ W [ H [[ [^ WW^ [^ [^ W^ H^ W [ W
W W
o f o f
% &
KK
'
KK
G G G G G
VHWLHQHTXH
OQ [ [ [^ W H^ W W ^
f f I G^ I G^ *1 6^ ^ ^
C C
C C
E 5GVTCVCFGWPCKPVGITCNKORTQRKCRWGUGNKPVGTXCNQFGKPVGITCEKÎP= f PQGUCEQVCFQ'N
KPVGITCPFQ PQ EWORNG NC EQPFKEKÎP PGEGUCTKC FG KPVGITCEKÎP [C SWG NKO Z NKO
Z Z
of of (^) f WVKNK\CPFQ.O*ÏRKVCN [RQTVCPVQNCKPVGITCNGUFKXGTIGPVG 6CODKÃP RQFTÈCOQU JCDGT RTQEGFKFQ WVKNK\CPFQ ETKVGTKQU FG EQORCTCEKÎP Z GZ Z
Z
Z
f
F 5GVTCVCFGWPCKPVGITCNKORTQRKCRWGUGNKPVGTXCNQFGKPVGITCEKÎP f f PQGUCEQVCFQ .CHWPEKÎPCKPVGITCTGUCEQVCFC[UGEWORNGNCEQPFKEKÎPPGEGUCTKCFGKPVGITCEKÎP[CSWG
ZNKO of^ Z ^ ^ Z [ NKO Z of^ Z ^ ^ Z [RQTVCPVQRQFGOQUFGUEQORQPGTNCKPVGITCNGPUWOCFGQVTCUFQUKPVGITCNGUFQPFGUÎNQWPQ FGNQUGZVTGOQUFGNKPVGTXCNQFGKPVGITCEKÎPUGCPQCEQVCFQRCTCFGURWÃUCRNKECTNCFGHKPKEKÎP
E D C C D E E D (^) E D C C D E C C^ D E E C C D
f f of of of of of of
of of
³ ³ ³ ³ ³
^ ^
D D
of of of of
f ³ (^) GPUWOCFGKPVGITCNGU
³ (^) Z
[
f ³ (^) .CHWPEKÎPCKPVGITCTGUEQPVKPWCGPGNKPVGTXCNQ=?[RQTVCPVQDCUVC
GUVWFKCTNCEQPXGTIGPEKCFG
f ³ (^) 7VKNK\COQUETKVGTKQUFGEQORCTCEKÎP
2QT UGT NC HWPEKÎP CTEVIZ WPC HWPEKÎP ETGEKGPVG GP GN KPVGTXCNQ = f [ DNKO CTE VI of^ Z S^ VGPGOQU ^
Z=^ f^
#FGO¶UUGXGTKHKECSWG Z ^! Z (^) Z Z NWGIQ
Z=^ f^
^
f (^) S f f ³ (^) ³ RQT ETKVGTKQU FG EQORCTCEKÎP VGPGOQU SWG
f ³ (^) VCODKÃPGUEQPXGTIGPVG[EQPENWKOQUSWG
f ³ (^) NQGU %CNEWNCOQUGNXCNQTFGNCKPVGITCNWVKNK\CPFQNCFGHKPKEKÎP (^) ^ ^
D^ D D D
f of of
³ ³ (^) © ¹
KPVGITCEKÎP[CSWG NKO Z o (^) Z^ f
#RNKECOQUNCFGHKPKEKÎP FZ Z
H H H (^) H
^ ^
KPVGITCEKÎP[CSWG NKO Z o (^) Z f8COQUCCRNKECTETKVGTKQUFGEQORCTCEKÎPRCTCGUVWFKCT
UKUGVTCVCFGWPCKPVGITCNEQPXGTIGPVG%QOQ Z d Z d Z
UG FGFWEG SWG Z Z Z Z d Z Z Z (^) > @ [
