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inedinjvefnivefinvefinvefin, Apuntes de Matemáticas

oimxdwiuc few fe ivefw hvfij vf ivfe

Tipo: Apuntes

2021/2022

Subido el 15/05/2022

jhonatan_melo
jhonatan_melo 🇨🇴

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Cmrm smnrg ahofbhb y nfgdgsthr gd ohs gjprgshsCmrm smnrg ahofbhb y nfgdgsthr gd ohs gjprgshs
½Auèogs amdsfbgrh qug smd ohs ahushs bgo æxftm bg oh½Auèogs amdsfbgrh qug smd ohs ahushs bgo æxftm bg oh auoturh mrihdfzhafmdho bg Hppogauoturh mrihdfzhafmdho bg Hppog==
Oh auoturh mrihdfzhafmdho amjprgdbg Oh auoturh mrihdfzhafmdho amjprgdbg ohs gxpgrfgdafhs thdtm pgrsmdhogs amjm auoturhogs bgohs gxpgrfgdafhs thdtm pgrsmdhogs amjm auoturhogs bg
udh mrihdfzhafñd, go aofjh mrihdfzhafmdho smd ohs pgragpafmdgs qug go trhnhkhbmr tfgdgudh mrihdfzhafñd, go aofjh mrihdfzhafmdho smd ohs pgragpafmdgs qug go trhnhkhbmr tfgdg
crgdtg h oh gjprgsh, jgbfhdtg su pcrgdtg h oh gjprgsh, jgbfhdtg su prmagsm, su gkgauafñd y hbjfdfstrhafñd. Gstm thjnfædrmagsm, su gkgauafñd y hbjfdfstrhafñd. Gstm thjnfæd
bgpgdbg bg ohs fdtgrhaafmdgs qug tfgdg go trhnhkhbmr gd su gdtmrdm ohnmrho, smafhofzhr amd susbgpgdbg bg ohs fdtgrhaafmdgs qug tfgdg go trhnhkhbmr gd su gdtmrdm ohnmrho, smafhofzhr amd sus
amjphøgrms bg trhnhkm y supgrfmrgs. Mntgdgjms ud nugd trhtm y cmrjhafñd mrihdfzhafmdho yamjphøgrms bg trhnhkm y supgrfmrgs. Mntgdgjms ud nugd trhtm y cmrjhafñd mrihdfzhafmdho y
smafho.smafho.
\mbgjms gdamdtrhr qug oh au\mbgjms gdamdtrhr qug oh auoturh bg Hpoog gs oh ûdfah gd oturh bg Hpoog gs oh ûdfah gd bgshrrmoohr fdfafhtfvhs, prmygatms ybgshrrmoohr fdfafhtfvhs, prmygatms y
bgjès, phrh hsê jfsjm arghr tgadmomiêh g bgjès, phrh hsê jfsjm arghr tgadmomiêh g fddmvhafñd phrh oh smafgbhb, sfgdbm fddmvhafñd phrh oh smafgbhb, sfgdbm jès htrhatfvhjès htrhatfvh
phrh oms usuhrfms qug bgsghd gsthr gd amdsthdtg hatuhofzhafñd amd oh gjprgsh gd auhdtm hphrh oms usuhrfms qug bgsghd gsthr gd amdsthdtg hatuhofzhafñd amd oh gjprgsh gd auhdtm h
sus prmbuatms.sus prmbuatms.
Oh arghtfvfbhb, fddmvhafñd gd auhdtm ho bgshrrmoom bgo smctwhrg y `hrbwhrg bg susOh arghtfvfbhb, fddmvhafñd gd auhdtm ho bgshrrmoom bgo smctwhrg y `hrbwhrg bg sus
prmbuatms, smd jmtfvms bg fddmvhafñd phrh qug sus usuhrfms y smafgbhb hbqufgrhd susprmbuatms, smd jmtfvms bg fddmvhafñd phrh qug sus usuhrfms y smafgbhb hbqufgrhd sus
sgrvfafms y prmbuatms.sgrvfafms y prmbuatms.
Gd su fjhigd dmthjms udh jhdzhdh jmrbfbh, hoim mrfifdho y ûdfah qug bfstfdiug y bh h ohGd su fjhigd dmthjms udh jhdzhdh jmrbfbh, hoim mrfifdho y ûdfah qug bfstfdiug y bh h oh
gjprgsh y h sus prmbuatms afgrth gogihdafh, gd sus bfsgøms `hy mrfifdhofbhb, pugbg qug omsgjprgsh y h sus prmbuatms afgrth gogihdafh, gd sus bfsgøms `hy mrfifdhofbhb, pugbg qug oms
fjftgd cêsfahjgdtg, go `hrbwhrg, pgrm go smctwhrg dm om fjftgd cêsfahjgdtg, go `hrbwhrg, pgrm go smctwhrg dm om pugbgd ampfhr yh qug pugbgd ampfhr yh qug Hpoog pmsggHpoog pmsgg
udh irhd sgiurfbhb gd auhdtm h bupofahbms bg fjgf, yh qug ahbh prmbuatm gs ûdfam yudh irhd sgiurfbhb gd auhdtm h bupofahbms bg fjgf, yh qug ahbh prmbuatm gs ûdfam y
rgifstrhbm pmr oh gjprgsh ho jmjgdtm bg rgifstrhbm pmr oh gjprgsh ho jmjgdtm bg sgr hatfvhbm y rgifstrhbm, sus fdvgdtmrgs sfgjprgsgr hatfvhbm y rgifstrhbm, sus fdvgdtmrgs sfgjprg
gstèd h oh bfspmsfafñd bg hatuhofzhafñd gstèd h oh bfspmsfafñd bg hatuhofzhafñd bg sfstgjhs mpgrhtfvms gd bgshrrmoom phrh sgrbg sfstgjhs mpgrhtfvms gd bgshrrmoom phrh sgr
ohdzhbms ho jgrahbm.ohdzhbms ho jgrahbm.
½Xuæ phpgo kugih go oêbgr bg oh mrihdfzhafñd gd oh shtfschaafñd ohnmrho, oh ahofbhb bg½Xuæ phpgo kugih go oêbgr bg oh mrihdfzhafñd gd oh shtfschaafñd ohnmrho, oh ahofbhb bg
vfbh y go nfgdgsthr bg oms gjpoghbms=vfbh y go nfgdgsthr bg oms gjpoghbms=
Amjm su phohnrh om bfag, gs go gdahrihbm bg vfifohr qug tmbms oms jgahdfsjms bg ohAmjm su phohnrh om bfag, gs go gdahrihbm bg vfifohr qug tmbms oms jgahdfsjms bg oh
gjprgsh sg gdaugdtrgd cudafmdhdbm, gd gstg ahsm oms gdirhdhkgs phrh go nugdgjprgsh sg gdaugdtrgd cudafmdhdbm, gd gstg ahsm oms gdirhdhkgs phrh go nugd
cudafmdhjfgdtm bg oh gjprgsh5 sgrêhd oms trhnhkhbmrgs. Om vfsuhofzhjms amjm udh pgrsmdhcudafmdhjfgdtm bg oh gjprgsh5 sgrêhd oms trhnhkhbmrgs. Om vfsuhofzhjms amjm udh pgrsmdh
gdahrihbh bg jgkmrhr oh prmbuatfvfbhb bg oms bfcgrgdtgs sgatmrgs qug tgdih oh gjprgsh,gdahrihbh bg jgkmrhr oh prmbuatfvfbhb bg oms bfcgrgdtgs sgatmrgs qug tgdih oh gjprgsh,
rgohafmdèdbmsg juy nfgd amd oms aofgdtgs, irhafhs h oh gxpgrfgdafh hbqufrfbh pmr phrtg bgorgohafmdèdbmsg juy nfgd amd oms aofgdtgs, irhafhs h oh gxpgrfgdafh hbqufrfbh pmr phrtg bgo
trhnhkm gd rgohafñd amd sus trhnhkhbmrgs, jgkmrhdbm hsê oh shtfschaafñd ohnmrho.trhnhkm gd rgohafñd amd sus trhnhkhbmrgs, jgkmrhdbm hsê oh shtfschaafñd ohnmrho.
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Cmrm smnrg ahofbhb y nfgdgsthr gd ohs gjprgshsCmrm smnrg ahofbhb y nfgdgsthr gd ohs gjprgshs

½Auèogs amdsfbgrh qug smd ohs ahushs bgo æxftm bg oh½Auèogs amdsfbgrh qug smd ohs ahushs bgo æxftm bg oh auoturh mrihdfzhafmdho bg Hppogauoturh mrihdfzhafmdho bg Hppog==

Oh auoturh mrihdfzhafmdho amjprgdbgOh auoturh mrihdfzhafmdho amjprgdbg ohs gxpgrfgdafhs thdtm pgrsmdhogs amjm auoturhogs bgohs gxpgrfgdafhs thdtm pgrsmdhogs amjm auoturhogs bg udh mrihdfzhafñd, go aofjh mrihdfzhafmdho smd ohs pgragpafmdgs qug go trhnhkhbmr tfgdgudh mrihdfzhafñd, go aofjh mrihdfzhafmdho smd ohs pgragpafmdgs qug go trhnhkhbmr tfgdg crgdtg h oh gjprgsh, jgbfhdtg su pcrgdtg h oh gjprgsh, jgbfhdtg su prmagsm, su gkgauafñd y hbjfdfstrhafñd. Gstm thjnfædrmagsm, su gkgauafñd y hbjfdfstrhafñd. Gstm thjnfæd bgpgdbg bg ohs fdtgrhaafmdgs qug tfgdg go trhnhkhbmr gd su gdtmrdm ohnmrho, smafhofzhr amd susbgpgdbg bg ohs fdtgrhaafmdgs qug tfgdg go trhnhkhbmr gd su gdtmrdm ohnmrho, smafhofzhr amd sus amjphøgrms bg trhnhkm y supgrfmrgs. Mntgdgjms ud nugd trhtm y cmrjhafñd mrihdfzhafmdho yamjphøgrms bg trhnhkm y supgrfmrgs. Mntgdgjms ud nugd trhtm y cmrjhafñd mrihdfzhafmdho y smafho.smafho.

\mbgjms gdamdtrhr qug oh au\mbgjms gdamdtrhr qug oh auoturh bg Hpoog gs oh ûdfah gdoturh bg Hpoog gs oh ûdfah gd bgshrrmoohr fdfafhtfvhs, prmygatms ybgshrrmoohr fdfafhtfvhs, prmygatms y bgjès, phrh hsê jfsjm arghr tgadmomiêh gbgjès, phrh hsê jfsjm arghr tgadmomiêh g fddmvhafñd phrh oh smafgbhb, sfgdbmfddmvhafñd phrh oh smafgbhb, sfgdbm jès htrhatfvhjès htrhatfvh phrh oms usuhrfms qug bgsghd gsthr gd amdsthdtg hatuhofzhafñd amd oh gjprgsh gd auhdtm hphrh oms usuhrfms qug bgsghd gsthr gd amdsthdtg hatuhofzhafñd amd oh gjprgsh gd auhdtm h sus prmbuatms.sus prmbuatms.

Oh arghtfvfbhb, fddmvhafñd gd auhdtm ho bgshrrmoom bgo smctwhrg y hrbwhrg bg susOh arghtfvfbhb, fddmvhafñd gd auhdtm ho bgshrrmoom bgo smctwhrg yhrbwhrg bg sus prmbuatms, smd jmtfvms bg fddmvhafñd phrh qug sus usuhrfms y smafgbhb hbqufgrhd susprmbuatms, smd jmtfvms bg fddmvhafñd phrh qug sus usuhrfms y smafgbhb hbqufgrhd sus

sgrvfafms y prmbuatms.sgrvfafms y prmbuatms. Gd su fjhigd dmthjms udh jhdzhdh jmrbfbh, hoim mrfifdho y ûdfah qug bfstfdiug y bh h ohGd su fjhigd dmthjms udh jhdzhdh jmrbfbh, hoim mrfifdho y ûdfah qug bfstfdiug y bh h oh gjprgsh y h sus prmbuatms afgrth gogihdafh, gd sus bfsgøms hy mrfifdhofbhb, pugbg qug omsgjprgsh y h sus prmbuatms afgrth gogihdafh, gd sus bfsgømshy mrfifdhofbhb, pugbg qug oms fjftgd cêsfahjgdtg, go hrbwhrg, pgrm go smctwhrg dm omfjftgd cêsfahjgdtg, gohrbwhrg, pgrm go smctwhrg dm om pugbgd ampfhr yh qugpugbgd ampfhr yh qug Hpoog pmsggHpoog pmsgg udh irhd sgiurfbhb gd auhdtm h bupofahbms bg fjgf, yh qug ahbh prmbuatm gs ûdfam yudh irhd sgiurfbhb gd auhdtm h bupofahbms bg fjgf, yh qug ahbh prmbuatm gs ûdfam y rgifstrhbm pmr oh gjprgsh ho jmjgdtm bgrgifstrhbm pmr oh gjprgsh ho jmjgdtm bg sgr hatfvhbm y rgifstrhbm, sus fdvgdtmrgs sfgjprgsgr hatfvhbm y rgifstrhbm, sus fdvgdtmrgs sfgjprg gstèd h oh bfspmsfafñd bg hatuhofzhafñdgstèd h oh bfspmsfafñd bg hatuhofzhafñd bg sfstgjhs mpgrhtfvms gd bgshrrmoom phrh sgrbg sfstgjhs mpgrhtfvms gd bgshrrmoom phrh sgr ohdzhbms ho jgrahbm.ohdzhbms ho jgrahbm.

½Xuæ phpgo kugih go oêbgr bg oh mrihdfzhafñd gd oh shtfschaafñd ohnmrho, oh ahofbhb bg½Xuæ phpgo kugih go oêbgr bg oh mrihdfzhafñd gd oh shtfschaafñd ohnmrho, oh ahofbhb bg vfbh y go nfgdgsthr bg oms gjpoghbms=vfbh y go nfgdgsthr bg oms gjpoghbms=

Amjm su phohnrh om bfag, gs go gdahrihbm bg vfifohr qug tmbms oms jgahdfsjms bg ohAmjm su phohnrh om bfag, gs go gdahrihbm bg vfifohr qug tmbms oms jgahdfsjms bg oh gjprgsh sg gdaugdtrgd cudafmdhdbm, gd gstg ahsm oms gdirhdhkgs phrh go nugdgjprgsh sg gdaugdtrgd cudafmdhdbm, gd gstg ahsm oms gdirhdhkgs phrh go nugd cudafmdhjfgdtm bg oh gjprgsh5 sgrêhd oms trhnhkhbmrgs. Om vfsuhofzhjms amjm udh pgrsmdhcudafmdhjfgdtm bg oh gjprgsh5 sgrêhd oms trhnhkhbmrgs. Om vfsuhofzhjms amjm udh pgrsmdh gdahrihbh bg jgkmrhr oh prmbuatfvfbhb bg oms bfcgrgdtgs sgatmrgs qug tgdih oh gjprgsh,gdahrihbh bg jgkmrhr oh prmbuatfvfbhb bg oms bfcgrgdtgs sgatmrgs qug tgdih oh gjprgsh, rgohafmdèdbmsg juy nfgd amd oms aofgdtgs, irhafhs h oh gxpgrfgdafh hbqufrfbh pmr phrtg bgorgohafmdèdbmsg juy nfgd amd oms aofgdtgs, irhafhs h oh gxpgrfgdafh hbqufrfbh pmr phrtg bgo trhnhkm gd rgohafñd amd sus trhnhkhbmrgs, jgkmrhdbm hsê oh shtfschaafñd ohnmrho.trhnhkm gd rgohafñd amd sus trhnhkhbmrgs, jgkmrhdbm hsê oh shtfschaafñd ohnmrho.

H oh ouz bg ohs ogaturhs trhnhH oh ouz bg ohs ogaturhs trhnhkhbhs, ½Argg ustgb qug oms trhnhkhbmrgs bkhbhs, ½Argg ustgb qug oms trhnhkhbmrgs bg Hppogg Hppog bgngrêhd tgdgr ud sfdbfahtm=bgngrêhd tgdgr ud sfdbfahtm=

\fgdsm qug\fgdsm qug sfsf bgngrêhd bg tgdgr ud sfdbfahtm, yh qug gstg mrihdfsjm vgoh pmr go nugdbgngrêhd bg tgdgr ud sfdbfahtm, yh qug gstg mrihdfsjm vgoh pmr go nugd cudafmdhjfgdtm bg oh gjprgsh, gd auhdtm h oh dmrjhtfvfbhb qug bgngd bg aujpofr omscudafmdhjfgdtm bg oh gjprgsh, gd auhdtm h oh dmrjhtfvfbhb qug bgngd bg aujpofr oms trhnhkhbmrgs, qug oms trhnhkhbmrgs gstæd ohnmrhdbm amdtrhnhkhbmrgs, qug oms trhnhkhbmrgs gstæd ohnmrhdbm amd tmbms oms ngdgcfafms qug ogs nrfdbh ohtmbms oms ngdgcfafms qug ogs nrfdbh oh gjprgsh pmr ogy, gstg sg gdaugdtrhgjprgsh pmr ogy, gstg sg gdaugdtrh ho thdtm bgo nfgdgs bg sus gjpoghbms,ho thdtm bgo nfgdgs bg sus gjpoghbms, jgkmrhdbm sujgkmrhdbm su rgjudgrhafñd shohrfho, huxfofms bg vfvfgdbh gdtrg mtrhs hyubhs.rgjudgrhafñd shohrfho, huxfofms bg vfvfgdbh gdtrg mtrhs hyubhs.

½Auèogs smd sus amdaousfmdgs y hprgdbfzhkgs hagrah bg oh fifgdg y oh sgiurfbhb smafho½Auèogs smd sus amdaousfmdgs y hprgdbfzhkgs hagrah bg ohfifgdg y oh sgiurfbhb smafho gd Hppog= Kustfcfqug su rgspugsth y prgsgdtggd Hppog= Kustfcfqug su rgspugsth y prgsgdtg trgs gkgjpoms.trgs gkgjpoms.

\fgdsm qug oh gjprgsh bgngrêh bg ihrhdtfzhrog ho gjpoghbm ud trhnhkm gsthnog, sgiurm y\fgdsm qug oh gjprgsh bgngrêh bg ihrhdtfzhrog ho gjpoghbm ud trhnhkm gsthnog, sgiurm y shoubhnog, thdtm phrh su fdtgirfbhb cêsfah amjmshoubhnog, thdtm phrh su fdtgirfbhb cêsfah amjm psfamoñifah, nrfdbhrog ho gjpoghbm bfcgrgdtgspsfamoñifah, nrfdbhrog ho gjpoghbm bfcgrgdtgs mpmrtudfbhbgs bg supgrhrsg, ahphaftèdbmsg jfgdtrhs sg gdaugdtrh ohnmrhdbm gd oh gjprgsh,mpmrtudfbhbgs bg supgrhrsg, ahphaftèdbmsg jfgdtrhs sg gdaugdtrh ohnmrhdbm gd oh gjprgsh,

yh qug gstm hyubh ho argafjfgdtm y cmrthogafjfgdtm bg oh gjprgsh.yh qug gstm hyubh ho argafjfgdtm y cmrthogafjfgdtm bg oh gjprgsh.

8 8.. GGkkggjjppoomm;; Xug oms gjpoghbms sg gdaugdtrgd ahphafthbms gd auhdtm h dutrfafñd,Xug oms gjpoghbms sg gdaugdtrgd ahphafthbms gd auhdtm h dutrfafñd, prmirhjhs bg prgvgdafñd y prgsgdthr ud nugd auhbrm bg prmtgaafñd pgrsmdho.prmirhjhs bg prgvgdafñd y prgsgdthr ud nugd auhbrm bg prmtgaafñd pgrsmdho.

.. GGkkggjjppoomm;; bgtgrjfdhr ohs bfcgrgdtgs pmsturhs grimdñjfahs bg oms trhnhkhbmrgs, hsê bgbgtgrjfdhr ohs bfcgrgdtgs pmsturhs grimdñjfahs bg oms trhnhkhbmrgs, hsê bg gsth jhdgrh chafofthr oh vfbh bg gooms hdtg susgsth jhdgrh chafofthr oh vfbh bg gooms hdtg sus pugstms bg trhnhkm, bg gsth jhdgrh sgpugstms bg trhnhkm, bg gsth jhdgrh sg fdargjgdth oh prmbuatfvfbhb bg oh gjprgsh yh qug go gjpoghbm sg gdaugdtrh gdfdargjgdth oh prmbuatfvfbhb bg oh gjprgsh yh qug go gjpoghbm sg gdaugdtrh gd ñptfjhs amdbfafmdgs phrh ohnmrhr.ñptfjhs amdbfafmdgs phrh ohnmrhr.

1 1.. GGkkggjjppoomm;; \rgsgdthr jgdsuhojgdtg oh dñjfdh y ud amdtrmo bg smpmrtg gd bmdbg\rgsgdthr jgdsuhojgdtg oh dñjfdh y ud amdtrmo bg smpmrtg gd bmdbg mnsgrvgjms go phim bg oh sgiumnsgrvgjms go phim bg oh sgiurfbhb smafho bg tmbms oms gjpoghbms, qugrfbhb smafho bg tmbms oms gjpoghbms, qug sgsg gdaugdtrgd vgrbhbgrhjgdtg vfdauohbms amd oh gjprgsh.gdaugdtrgd vgrbhbgrhjgdtg vfdauohbms amd oh gjprgsh.

H oh ouz bgo jhtgrfho y bgo ahsm trhnhkhbm, prmpmdih ud pohd phrh jgkmrhr oh ahofbhb yH oh ouz bgo jhtgrfho y bgo ahsm trhnhkhbm, prmpmdih ud pohd phrh jgkmrhr oh ahofbhb y go nfgdgsthr bg oms amohnmrhbmrgs bg udh gjprgsh (pugbg sgr oh gjprgsh gd oh quggo nfgdgsthr bg oms amohnmrhbmrgs bg udh gjprgsh (pugbg sgr oh gjprgsh gd oh qug ustgb ohnmrh). Gd gstg, gs fjpmrthdtg go bgshrrmustgb ohnmrh). Gd gstg, gs fjpmrthdtg go bgshrrmoom bg gkgjpoms y bg mnkgtfvmsoom bg gkgjpoms y bg mnkgtfvms mnsgrvhnogs y jgbfnogs.mnsgrvhnogs y jgbfnogs.

go sgatmr bg Hppog amjm omgo sgatmr bg Hppog amjm om pmbgjms gvfbgdafhrpmbgjms gvfbgdafhr pmr su omim qug fdbfah fddmvhafñd ypmr su omim qug fdbfah fddmvhafñd y arghtfvfbhb phrh oh smafgbhb, gstè gd amdsthdtg nûsqugbh bg tgadmomiêh y hrbwhrg phrh susarghtfvfbhb phrh oh smafgbhb, gstè gd amdsthdtg nûsqugbh bg tgadmomiêh yhrbwhrg phrh sus dugvhs oêdghs tgadmoñifahs, fdamrpmrhdbm ahbh bêh jès gqufpms tgadmoñifams gd oh vfbhdugvhs oêdghs tgadmoñifahs, fdamrpmrhdbm ahbh bêh jès gqufpms tgadmoñifams gd oh vfbh amtfbfhdh bg ohs pgrsmdhs.amtfbfhdh bg ohs pgrsmdhs.