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Matematica Basica 12, Apuntes de Matemáticas

Matematica basica, para estudiantes de universidad

Tipo: Apuntes

2020/2021

Subido el 06/06/2021

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bg1

LB]NLÂ]OGB HÂWOGBLB]NLÂ]OGB HÂWOGB
G]OQOM M G]OQOM M G G COAOG M COAOG M
]7]7
]BXNB]BXNB
O.
O. MB]EW OKAEXLB]OQEW4MB]EW OKAEXLB]OQEW4
]ìttuuccee 44 Ndnrgogoes sehrn okngubgoekns coknbcns, pre`rblbgo÷kNdnrgogoes sehrn okngubgoekns coknbcns, pre`rblbgo÷k
coknbc, cìlotns, oktnrprntbgo÷k `râagb y lïtemes mn gâcguce.coknbc, cìlotns, oktnrprntbgo÷k `râagb y lïtemes mn gâcguce.
]]ooppe e mmn n ppbbrrttooggooppbbggo÷kk 44 `rupbc (lâxole mn = pbrtogopbktns)`rupbc (lâxole mn = pbrtogopbktns)
ZZccbbzze e mmn n nnkkttrrnn``bb 44 Mïgolb snlbkb mn gcbsn (Wnlbkb 2:)Mïgolb snlbkb mn gcbsn (Wnlbkb 2:)
LLnnmmooe e mmn n pprrnnssnnkkttbbggo÷kk 44 Bucb vortubc / lnkô prokgopbc / ]7Bucb vortubc / lnkô prokgopbc / ]7
GGbbccooaaggbbggo÷kk 44 : b 8: ‗ 2>% mnc prelnmoe akbc: b 8: ‗ 2>% mnc prelnmoe akbc
OO.
OO. NQOMNKGOB MN BZXNKMOTBDN4NQOMNKGOB MN BZXNKMOTBDN4
MnsbrMnsbrreccb uk trbhbde prâgtoge nk reccb uk trbhbde prâgtoge nk nc qun nc qun sn rnsuncvn ndnrgogsn rnsuncvn ndnrgogoes y oes y prehprehcnlbs sehrn Wostnlb mncnlbs sehrn Wostnlb mn
OknOkngubgubgoegoekns kns CokCoknbcnnbcns, s, EptEptoloolozbgzbgo÷k o÷k y y cìlcìlotns otns mn mn ukukgoegoekns kns ncnncnlnklnktbcntbcns s vokvokgucgucbmbbmbs s b b susu
nspngobcombm y bakns.nspngobcombm y bakns.
OOO.
OOO. OKMOGBGOEKNWOKMOGBGOEKNW
Zbrb nc mnsbrrecce mnc trbhbde prâgtoge sn mnhn geksomnrbr4Zbrb nc mnsbrrecce mnc trbhbde prâgtoge sn mnhn geksomnrbr4
2.2. Nc geNc gektnkomktnkome mn e mn ces l÷ces l÷muces muces 0, 6 0, 6 y ? rny ? rnvosbmevosbmes nk s nk cb ukcb ukombm.ombm.
8.8. GeGekmkmogogoeoeknkns pbs pbrb nrb nc nkc nkvìvìe4e4
Nc megulnkte mnhn snr prnsnktbme nk brgfove mn Ls. Perm (.meg).Nc megulnkte mnhn snr prnsnktbme nk brgfove mn Ls. Perm (.meg).
@rbhb nc brgfove gek nc so`uonktn erlbte4@rbhb nc brgfove gek nc so`uonktn erlbte4
]7U(kelhrn mnc gurse)UBpnccomes y kelhrns gelpcntes]7U(kelhrn mnc gurse)UBpnccomes y kelhrns gelpcntes
Ndnlpce4Ndnlpce4 ]7ULbtnlâtogb hâsogb ULLL]7ULbtnlâtogb hâsogb ULLL
7.7. NxNxtntnkskso÷o÷k mk mnc nc trtrbhbhbdbde4e4
Cb nxtnkso÷k lìkolb snrâ mn 8 pâ`okbs (gbrbs).Cb nxtnkso÷k lìkolb snrâ mn 8 pâ`okbs (gbrbs).
=.=. BsnBsn`ôrn`ôrnsn mn nkvobr nc brgfsn mn nkvobr nc brgfove gerove gerrngrngte y te y gulgulpcopcor gek cbs gekmr gek cbs gekmogoogoeknekns mn s mn nkvnkvìe, mn ceìe, mn ce
gektrbroe, ke fbhrâ epgo÷k b rngcbles pestnroerns.gektrbroe, ke fbhrâ epgo÷k b rngcbles pestnroerns.
KE]B4 Wo nc/cb nstumobktn KE]B4 Wo nc/cb nstumobktn gelntn gubcquonr tope mn pcb`oe su puktubgo÷k butelâtogb snrâ gnregelntn gubcquonr tope mn pcb`oe su puktubgo÷k butelâtogb snrâ gnre
(:).(:).
OQ.
OQ. BKNVEW4BKNVEW4
2.2. Nc trbhNc trbhbde prbde prâgtoge âgtoge sn mnsbsn mnsbrreccb rreccb sn`ôk sn`ôk nc ernc erlbte nslbte nstbhcngtbhcngome4ome4
Nc mnsbrrecce mn cb secugo÷k mn gbmb prehcnlb mnhn snr gek ermnk y gcbrombmNc mnsbrrecce mn cb secugo÷k mn gbmb prehcnlb mnhn snr gek ermnk y gcbrombm
ukmblnktbme gek ces gekegolonktes bmquoromes.ukmblnktbme gek ces gekegolonktes bmquoromes.
Zâ`. 2
Zâ`. 2
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G]OQOMG]OQOM MM GG COAOGCOAOG MM ‗‗ ]7]

]BXNB]BXNB

O.O. MB]EW OKAEXLB]OQEW4MB]EW OKAEXLB]OQEW

]]ììttuuccee 44 Ndnrgogoes sehrn okngubgoekns coknbcns, prerblbgo÷kNdnrgogoes sehrn okngubgoekns coknbcns, prerblbgo÷k

coknbc, cìlotns, oktnrprntbgo÷k râagb y lïtemes mn gâcguce.coknbc, cìlotns, oktnrprntbgo÷krâagb y lïtemes mn gâcguce.

]]ooppee mmnn ppbbrrttooggooppbbggoo÷÷kk 44 rupbc (lâxole mn = pbrtogopbktns)rupbc (lâxole mn = pbrtogopbktns)

ZZccbbzzee mmnn nnkkttrrnn``bb 44 Mïgolb snlbkb mn gcbsn (Wnlbkb 2:)Mïgolb snlbkb mn gcbsn (Wnlbkb 2:)

LLnnmmooee mmnn pprrnnssnnkkttbbggoo÷÷kk 44 Bucb vortubc / lnkô prokgopbc / ]7Bucb vortubc / lnkô prokgopbc / ]

GGbbccooaaggbbggoo÷÷kk 44 : b 8: ‗ 2>% mnc prelnmoe akbc: b 8: ‗ 2>% mnc prelnmoe akbc

OO.OO. NQOMNKGOB MN BZXNKMOTBDN4NQOMNKGOB MN BZXNKMOTBDN

MnsbrMnsbrreccb uk trbhbde prâgtoge nkreccb uk trbhbde prâgtoge nk nc qunnc qun sn rnsuncvn ndnrgogsn rnsuncvn ndnrgogoes yoes y prehprehcnlbs sehrn Wostnlb mncnlbs sehrn Wostnlb mn
OknOkngubgubgoegoeknskns CokCoknbcnnbcns,s, EptEptoloolozbgzbgo÷ko÷k yy cìlcìlotnsotns mnmn ukukgoegoeknskns ncnncnlnklnktbcntbcnss vokvokgucgucbmbbmbss bb susu
nspngobcombm y bakns.nspngobcombm y bakns.

OOO.OOO. OKMOGBGOEKNWOKMOGBGOEKNW

Zbrb nc mnsbrrecce mnc trbhbde prâgtoge sn mnhn geksomnrbr4Zbrb nc mnsbrrecce mnc trbhbde prâgtoge sn mnhn geksomnrbr
2.2. Nc geNc gektnkomktnkome mne mn ces l÷ces l÷mucesmuces 0, 60, 6 y? rny? rnvosbmevosbmes nks nk cb ukcb ukombm.ombm.
8.8. GeGekmkmogogoeoeknkns pbs pbrb nrb nc nkc nkvìvìe4e
 Nc megulnkte mnhn snr prnsnktbme nk brgfove mn Ls. Perm (.meg).Nc megulnkte mnhn snr prnsnktbme nk brgfove mn Ls. Perm (.meg).
 @rbhb nc brgfove gek nc souonktn erlbte4@rbhb nc brgfove gek nc souonktn erlbte
]7U(kelhrn mnc gurse)UBpnccomes y kelhrns gelpcntes]7U(kelhrn mnc gurse)UBpnccomes y kelhrns gelpcntes

Ndnlpce4Ndnlpce4 ]7ULbtnlâtogb hâsogb ULLL]7ULbtnlâtogb hâsogb ULLL

7.7. NxNxtntnkskso÷o÷k mk mncnc trtrbhbhbdbde4e
Cb nxtnkso÷k lìkolb snrâ mn 8 pâokbs (gbrbs).Cb nxtnkso÷k lìkolb snrâ mn 8 pâokbs (gbrbs).
=.=. BsnBsnôrnôrnsn mn nkvobr nc brgfsn mn nkvobr nc brgfove gerove gerrngrngte yte y gulgulpcopcor gek cbs gekmr gek cbs gekmogoogoeknekns mns mn nkvnkvìe, mn ceìe, mn ce
gektrbroe, ke fbhrâ epgo÷k b rngcbles pestnroerns.gektrbroe, ke fbhrâ epgo÷k b rngcbles pestnroerns.
KE]B4 Wo nc/cb nstumobktnKE]B4 Wo nc/cb nstumobktn gelntn gubcquonr tope mn pcboe su puktubgo÷k butelâtogb snrâ gnregelntn gubcquonr tope mn pcboe su puktubgo÷k butelâtogb snrâ gnre

OQ.OQ. BKNVEW4BKNVEW

2.2. Nc trbhNc trbhbde prbde prâgtogeâgtoge sn mnsbsn mnsbrreccbrreccb snôksnôk nc ernc erlbte nslbte nstbhcngtbhcngome4ome
Nc mnsbrrecce mn cb secugo÷k mn gbmb prehcnlb mnhn snr gek ermnk y gcbrombmNc mnsbrrecce mn cb secugo÷k mn gbmb prehcnlb mnhn snr gek ermnk y gcbrombmukmblnktbme gek ces gekegolonktes bmquoromes.ukmblnktbme gek ces gekegolonktes bmquoromes.

. 2Zâ. 2

MuMurbrbktktnn ncnc mnmnsbrsbrrereccccee mnmn sesecucugo÷go÷kk mnmn gbgbmbmb prprehehcncnlblb mnmnhnhn snsnrr prprngngosose,e,
gefnrnktn, honk erbkozbme, âgoc mn gelprnkmnr y guombmese nk cb erterbìbgefnrnktn, honk erbkozbme, âgoc mn gelprnkmnr y guombmese nk cb erterbìb
y rnmbggo÷k.y rnmbggo÷k.
CbCb rnsrnspunpunstbstb mnmn gbmgbmbb prnprnukuktbtb y/e ìtnly/e ìtnl snsn lunlunstrstrbb mnmn ererlblb nxpnxpcìgcìgotb,otb,
gefnrnktn gek nc mnsbrrecce mn gbmb prehcnlb.gefnrnktn gek nc mnsbrrecce mn gbmb prehcnlb.

Q.Q. XÔHXOGB MN NQBC^BGO×K4XÔHXOGB MN NQBC^BGO×K

Cb bsokbgo÷k mnc puktbdn lâxole b gbmb grotnroe ns bpcogbhcn so nstn sn gulpcn b kovncCb bsokbgo÷k mnc puktbdn lâxole b gbmb grotnroe ns bpcogbhcn so nstn sn gulpcn b kovnc
sbtosbgteroe. Nc megnktn mnc gurse mntnrlokb nc puktbdn mn gbmb ìtnl mn bgunrme b su duogoesbtosbgteroe. Nc megnktn mnc gurse mntnrlokb nc puktbdn mn gbmb ìtnl mn bgunrme b su duogoe
mn nxpnrte.mn nxpnrte.
Gbmb prnuktb prnsnktb su rnspngtovb rôhrogbGbmb prnuktb prnsnktb su rnspngtovb rôhrogb
ZXN@^K]B K´ :2ZXN@^K]B K´ :

@@rrbbaaggbb ccbb secugo÷k mn gbmbsecugo÷k mn gbmb okokngngububgogo÷k÷k mnmncc WWOOCC ((ppeerr

snpbrbme)snpbrbme)usbkme uk pukteusbkme uk pukte mn prunhb y ukbmn prunhb y ukb tbhucbgo÷ktbhucbgo÷k rnspngtovbrnspngtovb (2 pukte)(2 pukte)

NNccbbhheerrbb ccbb rnorno÷k÷k secusecugo÷kgo÷k mnc WOC lnmobktnmnc WOC lnmobktn ukuk rrâaâagege yy cece

selhrnb.selhrnb.

(2 pukte)(2 pukte)

DustoagbDustoagb lbtnlâtoglbtnlâtogblnktnblnktn ncnc prepregngnsese pbpbrbrb fbcfbccbcbrr cbcbss gegeermermnknkbmbbmbss mnmn

ccerno÷k secugo÷krno÷k secugo÷kess vïvïrrtotogngnss mnmn ccbb

(2 pukte)(2 pukte)

MMnntntnrlrlookkbb nnkk ukukbb ttbbhhccbb ccbbss ggeeeerrmmnnkkbbmmbbss mmnn tetememess cecess vïvïrtrtogognsns

ququnrnrno÷kno÷k gegek secsecugokererlbugo÷klbk÷kk mncmnccbcb WOCWOC

(2 pukte)(2 pukte) ììttnnll bb bb hh hh

ZXN@^K]B :84ZXN@^K]B : @rba@rbagbgb cbcb secusecugo÷go÷kk mn gbmb rnstroggo÷kmn gbmb rnstroggo÷k ((ppeerr ssnnppbbrrbbmmee)) usbkme uk pukte mnusbkme uk pukte mn pprruunnhhbb yy uukkbb tbhucbgo÷ktbhucbgo÷k rnspngtovb.rnspngtovb. (2 pukte)(2 pukte)

Aerlb cb rno÷kAerlb cb rno÷k bgtohcnbgtohcn llnnmmoobbkkttnn uukk ``rrââaaggee yy ccee selhrnb.selhrnb.

(2 pukte)(2 pukte)

DustoagbDustoagb lblbtntnlâlâtotogbgblnlnktktnn ncnc pregpregnsense pbrb fbccbpbrb fbccbrr cbscbs ggeeerermnmnkkbbmbmbss mnmn cecess vïvïrtrtogognsns mnmn cbcb rnrnoo÷k÷k bgtohcnbgtohcn

(2 pukte)(2 pukte)

MMnnttnnrrllookkbb ccbbss gegeerermnmnkbkbmbmbss mnmncc vvïïrrttooggnn qquunn llbbxxoolloozzbb ccbb ukgo÷k ehdntoveukgo÷k ehdntove

(2 pukte)(2 pukte) ììttnnll bb bb hh hh

ZXN@^K]BZXN@^K]B :74: ^s^sbb cbcb keketbtbgogo÷k÷k mnmn cìlocìlotntn yy ukmukmblnkblnktbtb cbcb okokmnmntntnrlrlokokbgbgo÷o÷kk :/::/: (2 pukte)(2 pukte)

NkNkgungunktrktrbb ncnc vbcvbcerer mnc cìlotn fbgonkmemnc cìlotn fbgonkme uussee mmnn ccbb bgterozbgo÷k.bgterozbgo÷k. (2 pukte)(2 pukte)

^sb cb ketbgo÷k mn^sb cb ketbgo÷k mn cìlotn y ukmblnktbcìlotn y ukmblnktb cbcb okmnokmntnrltnrlokbgokbgo÷ko÷k :/::/: (2 pukte)(2 pukte)

NkgNkgunkunktrbtrb ncnc vbvbcercer mnc cìlotn fbgonkmemnc cìlotn fbgonkme uussee mmnn ccbb rbgoekbcozbgo÷krbgoekbcozbgo÷k (2 pukte)(2 pukte) ììttnnll bb bb hh hh

. 8Zâ. 8

8.8. (= puktes) Wnb cb ukgo÷k ehdntove(= puktes) Wnb cb ukgo÷k ehdntove^ ZZ

<< == xx

++ yy, sudntb b cbs rnstroggoekns4, sudntb b cbs rnstroggoekns

8686 xx++ 7272 yy ≭≭22=622= μμ 8686 xx ++>7>7 yy ≭≭201=201= μμ >1>1 xx ++8>8> yy ≭≭20:=20:= xx ≮≮ :: yy ≮≮::

Mntnrlokn4Mntnrlokn
b)b) Cb rnCb rno÷k bo÷k bgtohcngtohcn fbgonkfbgonkme usme use mne mn cbs rncbs rnstroggstroggoekns.oekns.
h)h) Cbs geCbs geermnkbermnkbmbs mncmbs mnc vïrtogvïrtogn qun ln qun lbxolozb cbxolozb cb ukgb ukgo÷k eo÷k ehdntovehdntove..

. =Zâ. =

7.7. (= puktes) Gbcgucn ces souonktns cìlotns4(= puktes) Gbcgucn ces souonktns cìlotns

bb..

colcol x↘x↘ 77

xx^88 ∘∘??

xx^77 ∘∘ xx^88 ∘∘ 11 xx

,, (8puktes)(8puktes)

hh..

colcol xx ↘↘ ::

∔∔ xx^88 ++^ pp^88 ∘∘^ pp

∔∔xx^88 ++qq^88 ∘∘qq

(8puktes)(8puktes)

. >Zâ. >

>.>. (= puktes) Nk cb nlprnsb OLB@N W.B. cb(= puktes) Nk cb nlprnsb OLB@N W.B. cb ukgo÷k geste tetbc mn premuggo÷k (nk locns mn secns) ns G(x),ukgo÷k geste tetbc mn premuggo÷k (nk locns mn secns) ns G(x),

guyb rncb mn gerrnspekmnkgob nstâ mbmb per4guyb rncb mn gerrnspekmnkgob nstâ mbmb per4 GG ((^ xx ))<<^ ??^ xx

(^88) ∘∘?::?:: xx ∘∘2:2:

mekmn xmekmn x ns nc kôlns nc kôlnre mnnre mn brtìgucesbrtìguces

premugomes (nk gonktes) per cb nlprnsbpremugomes (nk gonktes) per cb nlprnsb..

b)b) μGuâc ns nc geste tetbc mn premuggo÷k pbrb >:: brtìguces5μGuâc ns nc geste tetbc mn premuggo÷k pbrb >:: brtìguces
h)h) Wo cb premuggo÷k sn bprexolb b ces 2::: brtìguces, μb quï vbcer sn bprexolb nc geste tetbc5Wo cb premuggo÷k sn bprexolb b ces 2::: brtìguces, μb quï vbcer sn bprexolb nc geste tetbc

. 0Zâ. 0

]XBHBDE]XBHBDE ZXÂG]OGEZXÂG]OGE ‗‗ ]7]

LB]NLÂ]OGBLB]NLÂ]OGB HÂWOGBHÂWOGB KN@EGOEWKN@EGOEW

1.1. (= puktes) Mbme(= puktes) Mbme nc Wostnlb mn Okngubgoekns Coknbcns (WOC)nc Wostnlb mn Okngubgoekns Coknbcns (WOC)

{{

88 xx++ yy ≭≭ 7:7: 77 xx

∘∘ 88 yy ≭≭

8=8= xx ≮≮:: yy ≮≮ :: Mntnrlokn4Mntnrlokn g)g) Cb rnCb rno÷k seck secugougo÷k ln÷k lnmomobktbktn uk rân ukrâageage.. m)m) Ces vCes vïrtoïrtogns qgns qun geun gekerkerlbk clbk cb rnb rno÷k so÷k secugecugo÷k.o÷k.

0.0. (= puktes) Wnb cb ukgo÷k ehdntove(= puktes) Wnb cb ukgo÷k ehdntove ZZ<< == xx ++ yy, sudntb b cbs rnstroggoekns4, sudntb b cbs rnstroggoekns

. 6Zâ. 6

bb..

colcol x↘x↘ 77

xx^88 ∘∘??

xx^77 ∘∘ xx^88 ∘∘ 11 xx

,, (8puktes)(8puktes)

hh..

colcol xx ↘↘ ::

∔∔ xx^88 ++^ pp^88 ∘∘^ pp

∔∔xx^88 ++qq^88 ∘∘qq

(8puktes)(8puktes)

. 2:Zâ. 2:

?.?. (= puktes) ^kb nlprnsb mnmogbmb b cb vnktb mn huzes nk Cb Qogterob, mospekn pbrb cb geknggo÷k mn(= puktes) ^kb nlprnsb mnmogbmb b cb vnktb mn huzes nk Cb Qogterob, mospekn pbrb cb geknggo÷k mn

:: l mn tncb bcem÷k y >8:>:: l mn tncb bcem÷k y >8: l mn tncb pecoïstnr. Gbmb pbktbc÷k mnc huze kngnsotb 2 l mn bcem÷k y 8l mn tncb pecoïstnr. Gbmb pbktbc÷k mnc huze kngnsotb 2 l mn bcem÷k y 8 l mn pecoïstnr9 pbrb gbmb gbsbl mn pecoïstnr9 pbrb gbmb gbsbgb sn kngnsotbk 8 l mn bcem÷k y 2 l mn pecoïstnr.gb sn kngnsotbk 8 l mn bcem÷k y 2 l mn pecoïstnr. Cb bkbkgob perCbbkbkgob per gbmb pbktbc÷k ns mn W/? y cb bkbkgob per gbmb gbsbgb ns mn W/ 28.gbmb pbktbc÷k ns mn W/? y cbbkbkgob per gbmb gbsbgb ns mn W/ 28. Nc bhrogbktn sbhn qun susNc bhrogbktn sbhn qun sus lâlâququokokbsbs nknk prprelelnmnmoeoe mnmnlelerbrbkk == lolokukutetess nknk gegekkngnggogoekekbrbr ukuk pbpbktktbcbc÷k÷k yy 2:2: lolokukutetess nknk geknggogeknggoekbr ukb gbsbgb9 ïcekbr ukb gbsbgb9 ïc prelntn nktrnbprelntn nktrnbr ukr uk pnmome nk ukb snlbkb trbhbdbkme sus lâquokbs mnpnmome nk ukb snlbkb trbhbdbkme sus lâquokbs mn

cukns b vonrkns 6 ferbs mobrobscukns b vonrkns 6 ferbs mobrobs..

g)g) Lemncn cb ukgo÷k ehdntove y ncbhern nc WOC pbrb cbs rnstroggoekns.Lemncn cb ukgo÷k ehdntove y ncbhern nc WOC pbrb cbs rnstroggoekns. m)m) μGuâkte snrâ nc pnmome lâxole mnμGuâkte snrâ nc pnmome lâxole mn huzes (gbsbgb y pbktbc÷k) qun pemrâ nktrnbr nkhuzes (gbsbgb y pbktbc÷k) qun pemrâ nktrnbr nk ukb snlbkbukb snlbkb mn trbhbde5 μGuâkte snrâ sumn trbhbde5 μGuâkte snrâ su bkbkgob lâxolb5bkbkgob lâxolb

. 22Zâ. 22

28.28. (= puktes) Wnb cb ukgo÷k ehdntove(= puktes) Wnb cb ukgo÷k ehdntove ZZ<< == xx ++ yy, sudntb b cbs rnstroggoekns4, sudntb b cbs rnstroggoekns

8686 xx++ 7272 yy ≭≭22=622= μμ 8686 xx ++>7>7 yy ≭≭201=201= μμ >1>1 xx ++8>8> yy ≭≭20:=20:= xx ≮≮ :: yy ≮≮::

Mntnrlokn4Mntnrlokn
n)n) Cb rnCb rno÷k bo÷k bgtohcngtohcn fbgonkfbgonkme usme use mne mn cbs rncbs rnstroggstroggoekns.oekns.
)) Cbs gCbs geermneermnkbmbskbmbs mnc vmnc vïrtognïrtogn qun lqun lbxolozbbxolozb cb cb ukgo÷ukgo÷k ehk ehdntovedntove..

. 27Zâ. 27

27.27. (= puktes) Gbcgucn ces souonktns cìlotns4(= puktes) Gbcgucn ces souonktns cìlotns

bb..

colcol x↘x↘ 77

xx^88 ∘∘??

xx^77 ∘∘ xx^88 ∘∘ 11 xx

,, (8puktes)(8puktes)

hh..

colcol xx ↘↘ ::

∔∔ xx^88 ++^ pp^88 ∘∘^ pp

∔∔xx^88 ++qq^88 ∘∘qq

(8puktes)(8puktes)

. 2=Zâ. 2=

2>.2>. (= puktes) Nk cb nlprnsb OLB@N W.B. cb(= puktes) Nk cb nlprnsb OLB@N W.B. cb ukgo÷k geste tetbc mn premuggo÷k (nk locns mn secns) ns G(x),ukgo÷k geste tetbc mn premuggo÷k (nk locns mn secns) ns G(x),

guyb rncb mn gerrnspekmnkgob nstâ mbmb per4guyb rncb mn gerrnspekmnkgob nstâ mbmb per4 GG (( xx ))<<^ ??^ xx

(^88) ∘∘?::?:: xx ∘∘2:2:

mekmn xmekmn x ns nc kôlns nc kôlnre mnnre mn brtìgucesbrtìguces

premugomes (nk gonktes) per cb nlprnsbpremugomes (nk gonktes) per cb nlprnsb..

n)n) μGuâc ns nc geste tetbc mn premuggo÷k pbrb >:: brtìguces5μGuâc ns nc geste tetbc mn premuggo÷k pbrb >:: brtìguces
)) Wo cb premuggo÷k sn bprexolb b ces 2::: brtìguces, μb quï vbcer sn bprexolb nc geste tetbc5Wo cb premuggo÷k sn bprexolb b ces 2::: brtìguces, μb quï vbcer sn bprexolb nc geste tetbc

]XBHBDE ZXÂG]OGE ‗ ]7]XBHBDE ZXÂG]OGE ‗ ]

LB]NLÂ]OGBLB]NLÂ]OGB HÂWOGBHÂWOGB KN@EGOEWKN@EGOEW

21.21. (= puktes) Mbme(= puktes) Mbme nc Wostnlb mn Okngubgoekns Coknbcns (WOC)nc Wostnlb mn Okngubgoekns Coknbcns (WOC)

{{

88 xx++ yy ≭≭ 7:7: 77 xx ∘∘ 88 yy ≭≭ 8=8= xx ≮≮:: yy ≮≮ (^) :: Mntnrlokn4Mntnrlokn

. 21Zâ. 21

)) Cb rCb rno÷no÷k sek secugocugo÷k l÷k lnmonmobktnbktn uk ukrâagrâage.e. f)f) Ces vCes vïrtoïrtogns qgns qun geun gekerkerlbk clbk cb rnb rno÷k so÷k secugecugo÷k.o÷k.

20.20. (= puktes) Wnb cb ukgo÷k ehdntove(= puktes) Wnb cb ukgo÷k ehdntove ZZ<< == xx ++ yy, sudntb b cbs rnstroggoekns4, sudntb b cbs rnstroggoekns

8686 xx++ 7272 yy ≭≭22=622= μμ 8686 xx ++>7>7 yy ≭≭201=201= μμ >1>1 xx ++8>8> yy ≭≭20:=20:= xx ≮≮ :: yy ≮≮::

Mntnrlokn4Mntnrlokn
)) Cb rnCb rno÷k bo÷k bgtohcngtohcn fbgonkfbgonkme usme use mne mn cbs rncbs rnstroggstroggoekns.oekns.
f)f) Cbs geCbs geermnkbermnkbmbs mncmbs mnc vïrtogvïrtogn qun ln qun lbxolozb cbxolozb cb ukgb ukgo÷k eo÷k ehdntovehdntove..

. 20Zâ. 20

2?.2?. (= puktes) ^kb nlprnsb mnmogbmb b cb vnktb mn huzes nk Cb Qogterob, mospekn pbrb cb geknggo÷k mn(= puktes) ^kb nlprnsb mnmogbmb b cb vnktb mn huzes nk Cb Qogterob, mospekn pbrb cb geknggo÷k mn

:: l mn tncb bcem÷k y >8:>:: l mn tncb bcem÷k y >8: l mn tncb pecoïstnr. Gbmb pbktbc÷k mnc huze kngnsotb 2 l mn bcem÷k y 8l mn tncb pecoïstnr. Gbmb pbktbc÷k mnc huze kngnsotb 2 l mn bcem÷k y 8 l mn pecoïstnr9 pbrb gbmb gbsbl mn pecoïstnr9 pbrb gbmb gbsbgb sn kngnsotbk 8 l mn bcem÷k y 2 l mn pecoïstnr.gb sn kngnsotbk 8 l mn bcem÷k y 2 l mn pecoïstnr. Cb bkbkgob perCbbkbkgob per gbmb pbktbc÷k ns mn W/? y cb bkbkgob per gbmb gbsbgb ns mn W/ 28.gbmb pbktbc÷k ns mn W/? y cbbkbkgob per gbmb gbsbgb ns mn W/ 28. Nc bhrogbktn sbhn qun susNc bhrogbktn sbhn qun sus lâlâququokokbsbs nknk prprelelnmnmoeoe mnmnlelerbrbkk == lolokukutetess nknk gegekkngnggogoekekbrbr ukuk pbpbktktbcbc÷k÷k yy 2:2: lolokukutetess nknk geknggogeknggoekbr ukb gbsbgb9 ïcekbr ukb gbsbgb9 ïc prelntn nktrnbprelntn nktrnbr ukr uk pnmome nk ukb snlbkb trbhbdbkme sus lâquokbs mnpnmome nk ukb snlbkb trbhbdbkme sus lâquokbs mn

cukns b vonrkns 6 ferbs mobrobscukns b vonrkns 6 ferbs mobrobs..

)) Lemncn cb ukgo÷k ehdntove y ncbhern nc WOC pbrb cbs rnstroggoekns.Lemncn cb ukgo÷k ehdntove y ncbhern nc WOC pbrb cbs rnstroggoekns.

. 2?Zâ. 2?

f)f) μGuâkte snrâ nc pnmome lâxole mnμGuâkte snrâ nc pnmome lâxole mn huzes (gbsbgb y pbktbc÷k) qun pemrâ nktrnbr nkhuzes (gbsbgb y pbktbc÷k) qun pemrâ nktrnbr nk ukb snlbkbukb snlbkb mn trbhbde5 μGuâkte snrâ sumn trbhbde5 μGuâkte snrâ su bkbkgob lâxolb5bkbkgob lâxolb

. 8:Zâ. 8: