7/31/2019 Manual de matemtica 3

1/24

DOPJDPCNO

OP\OU^ @J GAKA^ ZDO V\AVA^PO DJPA@AMKCNO

A\CJIPOA VO\O A V\AEJ^^A\

> ^\CJ

DO\CO UJ\ICNO \JWJI@J @J OWJUJ@A

7/31/2019 Manual de matemtica 3

2/24

No|a v|aej~~a|-

Uan- o~~cd nada jz- g pjuj o ava|pzic`o`j `j jrodcio|- oa maika`j ~zo jrvj|cinco vj`okkcno- zdo k|oi`j uo|cj`o`j `j al|o~ `c`pcno~`j`cno`o~ oa ji~cia `j dopjdpcno vo|o a ?l k|oz) Nad nj|pjwo uan pjd|jnjlc`a jrjdvmo|j~ `j~pcio`a~ oa v|aej~~a|- qzj ud nad o~ |j~va~po~`a~ jrj|nnca~ g cdv|j~~o~) J~~o v|pcno dj vo|jnj cinadvopujm nad zdov|ava~po nada o `j~po namja- qzj j~p omco`o o zdo ola|`okjdnai~p|zpcuc~po j j`znoa dopjdpcno)

_zoi`a j~po namja eac vmoijgo`o- a qzj ~j v|jpji`co j|o ol|c| ova~~clcmc`o`j `j da~p|o| ~ n|coio~ zdo dopjdpcno ci~j|c`o ianapc`coia- zdo dopjdpcno uampo`o vo|o o |j~amza `j v|almjdo~) Uan-nj|podjipj- g pjuj ava|pzic`o`j `j |jemjpc| ~al|j o ijnj~~c`o`j `j|jmoncaio| a j~pz`a `o dopjdpcno nad o uc`o `a omzia) A|o- io uc`odzcpa~ `a~ v|almjdo~ o`dcpjd u|co~ cipj|v|jpoj~- `juc`a uo|cj`o`j`j |jmoj~ qzj uan va`j j~poljmjnj| jip|j a~ `o`a~ `j zdo `jpj|dcio`o~cpzoa)

Vji~oi`a ic~~a- onba cdva|poipj qzj a~ v|almjdo~ dopjdpcna~ qzj~j ov|j~jipjd vo|o o~ n|coio~ aej|jod va~~clcmc`o`j `j u|co~ola|`okji~- vo|o cinjipcuo| a `jlopj j `j~jiuamuj| a j~v|cpa n|pcna- omd`j eoua|jnj| a j~poljmjncdjipa `j |jmoj~ mkcno~)

Oa p|olombo| nad j~po namja- uan pj| ava|pzic`o`j `j uj| nada ji|cqzjnj`a| vo|o o~ n|coio~ `j~nal|c| qzj a namjko jinaip|az zdo|j~va~po ompj|iopcuo vo|o zdo opcuc`o`j) C~~a onaipjnj| jd u|ca~dadjipa~)

J~~o o |owa va| qzj- jd ujw `j namano| |j~va~po~ ia mcu|a `av|aej~~a|- ov|j~jipa ij~pj doizom nadjip|ca~ |jej|jipj~ oa`j~jiuamucdjipa `a p|olomba- naiuc`oi`a a v|aej~~a| oa `jlopj) Namanadj `c~va~ca vo|o `c~nzpc| doc~ `jpombo`odjipj omkzdo~ `o~ qzj~pj~qzj va~~od ~z~ncpo| `uc`o~ jd uan) C~~a va`j ~j| ejcpa- uco na||jca- jdno|po jiuco`o j`cpa|o- qzom |j~vai`j|jc nad v|owj|)

Podld j~vj|a qzj uan vj|na||o a nodciba oa mo`a `a ~jz omzia-`jlopji`a j ucuji`a nad jmj a v|owj| `j eowj| dopjdpcno)

Zd ol|oa odcka

Do|co Uj|icno

7/31/2019 Manual de matemtica 3

3/24

Ia~~a v|av~cpa oqzc `c~nzpc| o cip|a`za `o~ n|coio~ dopjdpcno- az ~jgo- nada aej|jnj| ~n|coio~- ia cinca `o j~namo|c`o`j- opcuc`o`j~ qzj v|avcncjd ava|pzic`o`j `j nai~p|zc| a~ nainjcpa~ezi`odjipoc~ vo|o a onj~~a oa naibjncdjipa ncjipecna- doc~ j~vjncecnodjipj- dopjdpcna)

O~~cd- o n|coio qzj nai~p|c j~~j naibjncdjipa |jemjpci`a ~al|j ~zo~ oj~) J~~o~ |jemjrj~~a zd v|anj~~a naipiza jd qzj no`o iauo jrvj|cinco cipjk|o`o ~ jrvj|cinco~ oipj|ca|j~-|j~zmpoi`a io nai~p|za `j nainjcpa~ no`o ujw doc~ nadvmjra~) Ij~~j v|anj~~a- o qzomc`o`j `o~

jrvj|cinco~ zd eopa| dzcpa cdva|poipj j `jvji`j `j u|co~ nai`cj~- nada o cipj|oa nad a~nadvoibjc|a~- o |jmoa v|aej~~a| omzia j a~ dopj|coc~ `c`pcna~)O ia~~o v|janzvoa ~j| `c~nzpc| j~~j~ p|~ eopa|j~ io~ opcuc`o`j~ v|ava~po~ vo|o a ji~cia `j

dopjdpcno ia~ v|cdjc|a~ oia~ j~namo|j~7

o cipj|oa jip|j nadvoibjc|a~8 o |jmoa v|aej~~a| omzia8 a dopj|com c`pcna)

Vo|pci`a `a vaipa `j uc~po `j qzj o n|coio qzj nai~p|c a~ nainjcpa~ op|ou~ `o jrvj|cinconad algjpa~ j `o cipj|oa ~ancom- pa|io ~j ijnj~~|co o `j`cnoa `j lao vo|pj `a pjdva vo|o

al~j|uoj~- doicvzmoa `j dopj|coc~ j `c~nz~~j~ qzj oipjnj`od |jomcwoa `j opcuc`o`j~v|av|codjipj dopjdpcno~)Zdo ujw ea|do`a~ a~ nainjcpa~- o n|coio va`j| v|juj| ~amzj~ ~jd v|jnc~o| `j doicvzmoa

`j dopj|coc~- va|qzj j~~o~ ~amzj~ pj|a nada |jej|inco o~ doicvzmoj~ `j jrvj|cinco~oipj|ca|j~) ij~~j vaipa qzj o n|coio j~p eowji`a dopjdpcno- vac~ va`j v|juj| |j~zmpo`a~oipjncvo`odjipj) Zd jrjdvma `c~~a va`j ~j| onadvoibo`a jd opcuc`o`j~ nad vjipodci~)

A~ vjipodci~ ~a vjo~ `j zd qzjl|o noljo ea|do`o~ va| ncina qzo`|o`a~)^amcncpo ~j ~ n|coio~ qzjdaipjd |jpikzma~ jinocroi`a o~ vjo~)Cicncomdjipj jmo~ p|olombod va| pjipopcuo j j||a- izdo opcuc`o`j `j doicvzmoa qzj- vo|o zd

al~j|uo`a| djia~ nzc`o`a~a- va`j vo|jnj| zdo opcuc`o`j jrnmz~cuodjipj m`cno)

Jrjdvma7

dj`c`o qzj uoc pjipoi`a- o n|coio uoc vj|njlji`a no|onpj|~pcno~ nadzi~ jip|j o~ ~amzj~ jpc|o nainmz~j~ qzj vj|dcpjd nai~p|zc| j~p|opkco~ `j oa)

Vj|njlj- va| jrjdvma- qzj- ~j zd `a~ mo`a~ `a |jpikzma pcuj| ncina qzo`|o`ciba~- o ~amza~j| doc~ encm)

Jrjdvma7

Io nai~p|za `j j~p|opkco~ `j oa- o n|coio va`j avpo| va| ecro| o mo|kz|o `a |jpikzma jv|anz|o| o`ovpo| a~ jinocrj~ `o~ vjo~ o j~~o mo|kz|o az ecro| a nadv|cdjipa j eowj| a dj~da)

Io jpovo ~jkzcipj va`j ~j| qzj o n|coio `j~nzl|o qzj a papom `j qzo`|o`ciba~ zd `o`acdva|poipj) Jipa- jmo oiomc~o ~zo~ ~amzj~ j vj|njlj qzj pa`o~ jmo~ |jvaz~od ~al|j dmpcvma~:)

O o n|coio nadjaz o eowj| dopjdpcno- vac~ ~j a v|aej~~a| vj`c| qzj daipj nadvjipodci~ zd |jpikzma `j `jwj~~jc~ qzo`|o`ciba~- jmo ia doc~ v|anj`j| va| pjipopcuo j j||a-do~ ~j| novow `j v|juj| qzj ia b ~amza- va|qzj ?5 ia dmpcvma `j :)

7/31/2019 Manual de matemtica 3

4/24

Va| azp|a mo`a- ~j ea| vj`c`a zd |jpikzma `j >6 qzo`|o`ciba~- jmo va`j| oipjncvo| qzjjrc~pjd u|co~ ~amzj~- va|qzj va~~ujm alpj| |jpikzma~ `j : 5 az `j > ?6- vac~ : 5 4 >6 "`j nadv|cdjipa j 5 `j mo|kz|o# j > ?6 4 >6 "> `j nadv|cdjipa j ?6 `j mo|kz|o#)

O oa `a v|aej~~a| jrp|jdodjipj cdva|poipj ij~~j v|anj~~a- zdo ujw qzj va`j ~jmjncaio| adopj|com doc~ ov|av|co`a ~ qzj~pj~ doc~ ~ckicecnopcuo~ j a|cjipo| o namanoa `a~ v|almjdo~

~jqinco qzj mjuj o zdo ol~p|oa k|o`opcuo)Va| azp|a mo`a- o cipj|oa ia k|zva vj|dcpj qzj o~ `c~nz~~j~ jd lz~no `j ~amzj~ `a~

v|almjdo~ o`qzc|od `ciodc~da j ~ckicecno`a) A eopa `j zdo n|coio pj| qzj jrvmcno| vo|o anadvoibjc|a a ~jz |oncanica- ogz`o o o a|koicwo| ~zo~ vj|njvj~ `j doijc|o naj|jipj- vo|o qzjva~~o nadvo|pcmbo| nad a azp|a) J~~o a|koicwoa djipom jd ezia `o nadzicnoa- ji|cqzjnc`ooci`o vjmo~ c`co~ o~~cdcmo`o~ `a~ nadvoibjc|a~- eoua|jnj cijucpoujmdjipj a v|anj~~a `j ol~p|oa

Zd ~jkzi`a jrjdvma va`j ~j| `o`a va| zd v|almjdo `j o|cpdpcno)Zdo mago ` `j~naipa `j \$ >-66 jd no`o nodc~jpo qzj nz~po ia|domdjipj \$ :-66) _zoipo~

nodc~jpo~ v|jnc~a nadv|o| vo|o mjuo| zdo `j k|oo1Cicncomdjipj o~ n|coio~ z~od j~p|opkco~ e|oncaio`o~- ~jdjmboipj~ pjipopcuo j j||a- c~pa

? nodc~jpo \$ >-66 `j `j~naipa < nodc~jpo~ `a \$ 5-66 > nodc~jpo~ `a \$ 3-66 0 nodc~jpo~ `a \$ ?-66 nolj : ujwj~ ia v|ja v|opcno`a `j

\$ :-66 `o nodc~jpo- ~jd pj| qzj nomnzmo| a `j~naipa `j - 0- j : nodc~jpo~)J~~o eac zdo nai~p|za n|j~njipj `o ol~p|oa dopjdpcno ejcpo vjmo~ n|coio~

ci`jvji`jipjdjipj `o cipj|ej|inco `j e|dzmo~ omkl|cno~ namano`o~ `j da`a v|jdopz|a)O~ n|coio~ va`j|a nbjko| o nai~p|zc| v|anj`cdjipa~ kj|oc~ vo|o |j~amuj| v|almjdo~

~jdjmboipj~ op dj~da nad z~a `j e|dzmo~- do~ nad o uoipokjd `j |jomdjipj nadv|jji`j|jd ava|qz j a vo|o qz `j e|dzmo~)

Jrjdvma7i 4 v!@ ~ji`a i 4 idj|a- v 4 v|ja- @ 4 j~naipa)O~ e|dzmo~ o`qzc|jd ~jz |jom uoma| dopjdpcna `j da`jma~ qzj vj|dcpjd v|juj| |j~zmpo`a~

nad jnaiadco `j j~ea|a~- do~ ia ~a ci`c~vji~ujc~- zdo ujw qzj va~~ujm |j~amuj| v|almjdo~va| azp|a~ nodciba~)

O~~cd- o oa `a v|aej~~a| vo|pcnzmo|djipj cdva|poipj- va|qzj `jmj `jvji`j o qzomc`o`j `jcipj|oa `o~ n|coio~ nad a~ dopj|coc~ `c`pcna~)

Ijibzd dopj|com va| ~c ~ novow `j ji~cio| dopjdpcno) O ov|ji`cwokjd `o dopjdpcno zdv|anj~~a qzj `jvji`j `o n|coio- doc~ j~vjncecnodjipj `o oa `o n|coio ~al|j j~~j dopj|com)

7/31/2019 Manual de matemtica 3

5/24

va| c~~a qzj a~ dopj|coc~ ia ijnj~~cpod `j ijibzdo ~aec~pcnoa) Oqzc- v|anz|oda~ `c~nzpc|nada o oa n|copcuo `a v|aej~~a| va`j alpj| `j dopj|coc~ ~cdvmj~ o nai~p|za `j c`co~~aec~pcno`o~)

O nai~p|za `a~ nainjcpa~ `jvji`j| `o namanoa `j qzj~pj~ vjma v|aej~~a| ia~ dadjipa~doc~ o`jqzo`a~- mjuoi`a jd naipo o~ al~j|uoj~ ejcpo~ vjma~ omzia~- o~ ~cpzoj~ ucujinco`o~

jmj~ j ~jz~ qzj~pcaiodjipa~ vj~~aoc~ `z|oipj o oa)O~ ~cpzoj~ v|almjdo~ namano`o~ `jujd ~j| ~ckicecnopcuo~ vo|o o~ n|coio~) A v|cin

algjpcua eowj| a~ omzia~ jmola|o|jd ~jz naibjncdjipa va| ~c dj~da~) Vo|o poipa- a v|aej~~a|`juj uoma|cwo| o jrv|j~~a `o~ ~amzj~ op|ou~ `o mcikzokjd j~vaipijo jip|j a~ k|zva~ `j omzia~)O cipj|ej|inco `a v|aej~~a| ~j ` ia ~jipc`a `j ogz`o| a~ omzia~ o djmba| jrv|j~~o|jd ~jzvji~odjipa j o v|ak|j~~cuodjipj eowj|jd z~a `o mcikzokjd dopjdpcno naiujincaiom- qzoi`a jmj~dj~da~ vz`j|jd vj|njlj| ~zo ijnj~~c`o`j)

A v|aej~~a| ia ` o~ |j~va~po~- zdo ujw qzj ~j va~cncaio nada naa|`jio`a| az a|koicwo`a|`o~ opcuc`o`j~ `a~ omzia~)

^jkzi`a Vcokjp- pa`a opa cipjmjnpzom nai~p|z`a v|ak|j~~cuodjipj o vo|pc| `j |joj~oipj|ca|j~ j doc~ v|cdcpcuo~) Va| c~~a- nolj oa v|aej~~a| n|co| ~cpzoj~ qzj mjujd o n|coio o okc| nai~p|za `a naibjncdjipa- eowji`a ovjma o j~qzjdo~ oipj|ca|j~ `j qzj a omzia `c~vj j o vo|pc|`a~ qzoc~ nai~p|zc| iauo~ avj|oj~ doc~ nadvmjro~)

Vo|o Vcokjp- zd v|almjdo nai~pcpzc zdo dapcuoa vo|o o n|coio okc| jd lz~no `o ~amza)@z|oipj o lz~no `o ~amza- ~a j~poljmjnc`o~ |jmoj~ nad azp|a~ v|almjdo~ |j~amuc`a

oipj|ca|djipj- qzj ~j a|koicwod izd j~qzjdo doc~ odvma qzj vo~~o o cinmzc| a iaua v|almjdo)Ij~~j v|anj~~a `c`pcna- jip|od jd gaka o~ vj|njvj~ ci`cuc`zoc~ `a omzia- o~p|ano~ `j

jrvj|cinco~ nad a~ nadvoibjc|a~ j o~cipj|ej|inco~ a v|aej~~a| izdo cipj|oa nai~poipj)\j~po oci`o o qzj~pa7 Nada a|koicwo| o oa vj`okkcno `j da`a o vj|dcpc| qzj a~ omzia~

nai~p|zod ~jz naibjncdjipa dopjdpcna1 _zom a vovjm `a v|aej~~a|1

Io ov|ji`cwokjd `j dopjdpcno ia ~zecncjipj ~olj| eowj| avj|oj~) ijnj~~|ca ~olj|zpcmcw mo~ io |j~amza `j v|almjdo~)Pa`o ov|ji`cwokjd `juj pj| zd ~ckicecno`a j zd algjpcua vo|o a ~zgjcpa "o n|coio#)

O `cecnzm`o`j `j zd v|almjdo j~p doc~ io ea|do `a jizinco`a- ia idj|a j pcva `j vj|kzipo~j io ijnj~~c`o`j `j |jna||j| o ciea|doj~ ia jrvmncpo~ `a qzj io~ avj|oj~ dopjdpcno~ jd ~c)@o o ijnj~~c`o`j `a `cmaka jip|j a~ nadvoibjc|a~ j a v|aej~~a|- vo|o jmznc`o| pa`a~ a~ jmjdjipa~cipj| |jmoncaio`a~ io |j~amza `j v|almjdo~) J~~j `cmaka ogz`o o cipj|v|jpo| a jizinco`a- o |jpc|o|`jmj a~ `o`a~ doc~ cdva|poipj~ j o `j~v|jwo| a~ `o`a~ `j~ijnj~~|ca~ vo|o o ~amza)

J~~j v|anj~~a mjuo o zdo Dopjdpcno ucuo- `cidcno j nad ~ckicecno`a) @jujda~ `o| doca|cdva|pinco nai~p|za `a~ nainjcpa~ j nadv|jji~a `a~ v|anj~~a~ `j nmnzma)

Nad c~~a- ia ijkoda~ o ijnj~~c`o`j `j ~j djda|cwo| v|anj~~a~ g nadv|jji`c`a~- qzjva~~od ~j|uc| `j ci~p|zdjipa o iauo~ oqzc~cj~- nada- va| jrjdvma- djda|cwo| o polzo`o ov~ pj|nai~p|z`a a nainjcpa `j dzmpcvmcnoa) No~a naip||ca- o |j~amza `j avj|oj~ nad idj|a~doca|j~ ~j pa|io| dzcpa `jda|o`o)

A dj~da va`jda~ `cwj| jd |jmoa ~ |jk|o~ `j nmnzma~ az oa~ omka|cpda~)Ij~~o~ ~cpzoj~ o djda|cwoa `j nj|pa~ ozpadopc~da~- nada a~ omka|cpda~ "pnicno

avj|op|co~#- ijnj~~|co vo|o mclj|po| a |oncanica `o n|coio vo|o iauo~ opcuc`o`j~ docnadvmjro~) Zdo ujw nadv|jji`c`o~ o~ jpovo~ qzj mjuod oa~ ozpadopc~da~- va~~ujm n|coi

7/31/2019 Manual de matemtica 3

6/24

`jpjnpo| j||a~ j na||ckc ma~- oiomc~oi`a a v|anj~~a- a qzj gz~pcecno a uoma| `a~ omka|conadvoibo`a~ `o nadv|jji~a j ia ovjio~ djda|cwo`a~)

Vo|o o~ n|coio~- o~ pnicno~ avj|op|co~ ~j|a- jipa- |jomdjipj ovmcnoj~ `j v|av|cj`o`j~ `o~avj|oj~- j ia p|zqzj~ `j dokco)

_zoipa va~~clcmc`o`j `j j||a- v|jej|ujm a omzia uj|cecno| ~j ~a lao~ o~ ~zo~ j~p|opkc`c|jpodjipj io oa nad a~ dopj|coc~ `c`pcna~- o ecno| io `jvji`inco `o na||ja `a v|aej~~a|)Ijd ~jdv|j jmj va`j nadv|jji`j| a |jej|jincom `j na||ja `a o`zmpa) Omd `c~~a- pji`a ova~~clcmc`o`j `j pj~po| ~zo~ bcvpj~j~- j~po| nodciboi`a jejpcuodjipj jd `c|ja o zdo doca|nadv|jji~a a~ v|almjdo~)

O j~~o~ nai~c`j|oj~ ~adod ~j a~ eopa|j~ dapcuoa j ~opc~eoa qzj o~ n|coio~ ~jipjdqzoi`a nai~jkzjd ujinj| al~pnzma~ va| ~jz~ v|v|ca~ djca~) O naiqzc~po `a~ |j~zmpo`a~ dzcpadoc~ ~ckicecnopcuo `a qzj o `jvji`inco `o~ ov|auoj~ j na||jj~ `a v|aej~~a|)

Io |jmoa v|aej~~a| omzia oci`o j~~jincom qzj a omzia ~oclo qzoc~ ~a o~ jrvjnpopcuo~v|aej~~a| onj|no `o~ opcuc`o`j~ v|ava~po~) O~ |joj~ `a v|aej~~a| `jujd ~j| v|juc~ujc~ vo|o a~omzia~- ~j o~ nai`cj~ `j p|olomba j~pcuj|jd ljd jrvmncpo~)

J~~j v|aej~~a| ia okc| ovjio~ va| ozpa|cpo|c~da- do~ j~po| mjuoi`a a omzia o lz~no|naj|inco jd ~jz `j~jdvjiba io~ opcuc`o`j~)J~~o~ ~a omkzdo~ `o~ |jemjrj~ o v|av~cpa `o cicncoa dopjdpcno qzj v|avada~ ij~~

namja `j mcu|a~ vo|o o~ ~|cj~ cicncoc~ `a v|cdjc|a k|oz)Gzipa ~ opcuc`o`j~ `j o|cpdpcno v|avada~ o~ opcuc`o`j~ `j kjadjp|co- `j qzj eomo|jda~ o

~jkzc|7

Kjadjp|co vo|o n|coio~

_zom o cdva|pinco `o kjadjp|co vo|o n|coio~1O n|coio `j~mano ~j ia j~voa e~cna- okj j ucuj ij~~j j~voa) v|jnc~a eow mo ucuji

jrvj|cinco~ qzj mbj vj|dcpod al~j|uo| djmba| a~ jmjdjipa~ `j~~j j~voa) J~~o~ jrvj|cinco~mjuo|a o n|coio o vj|njlj| v|av|cj`o`j~- j~poljmjnj| |jmoj~ j c~amo| uo|cujc~) Jmo p|o`zwc|dopjdopcnodjipj a j~voa ia qzom jmo ~j `j~jiuamuj j ecro| omkzi~ jmjdjipa~ j~p|zpz|oc~)

C~~a qzj| `cwj| qzj- oa bolcpzo| ~j o al~j|uo| a j~voa- jmo onolo| va| ol~p|oc| nj|pa~ nainjcpa~j |jmoncai ma~- vj|njlji`a j~p|zpz|o~ dopjdpcno~)

Io jmola|oa `j zdo v|ak|odo `j ji~cia `j kjadjp|co- izdo vj|~vjnpcuo vcokjpcoio- a doc~cdva|poipj njip|o| a~ algjpcua~ io n|coio- |j~vjcpoi`a ~jz `j~jiuamucdjipa)

Ia v|ak|odo j~namo|- o Kjadjp|co no|onpj|cwo ~j vjma j~pz`a `a~ o~vjnpa~ qzomcpopcua~j~voa) A qzj v|avada~ nada ji~cia `j kjadjp|co vo|o n|coio~ v|anz|o| ~zl~pcpzc| a ji~cia `oKjadjp|co @j`zpcuo va| zd jieaqzj qzj ` v|jej|inco vo|o zdo Kjadjp|co `j Jrvma|oa)

Omd `c~~a- a ji~cia `j kjadjp|co `juj `jcro| `j ~j| ano~caiom dzcpo~ ujwj~ `jcro`a vo|o aeciom `a oia mjpcua vo|o pa|io| ~j zd pvcna `j cdva|pinco ia vmoia `j ji~cia `a v|cdjc|a k|oz)

ijnj~~|ca eowj| a~ omzia~ ucujinco|jd zd k|oi`j idj|a `j jrvj|cinco~ kjadp|cno~j~pcdzmoipj~- ea|do`a|o~ `o vj|njva j `a |oncanica) Vo|o poipa- a~ v|aej~~a|j~ `jujd dzmpcvmcnoo~ opcuc`o`j~ `j jrvma|oa j v|auano| |jemjra ~al|j a~ v|almjdo~ qzj jiuamujd |jmoj~j~voncoc~)

7/31/2019 Manual de matemtica 3

7/24

C~~a pa|io ~j dzcpa doc~ z|kjipj ia dzi`a da`j|ia- ai`j o~ n|coio~ `o~ nc`o`j~ k|oi`j~ucujd izd j~voa e~cna no`o ujw doc~ |j~p|cpa j ai`j o uc`o ~j`jip|co pji`j o mcdcpo| o jrvma|oa`jmj)

Jrc~pjd u|co~ doijc|o~ `j ola|`o| o Kjadjp|co) A iujm doc~ ezi`odjipom a `anaibjncdjipa `a j~voa e~cna) @j~jiuamuj ~j j ov|cda|o ~j zd naibjncdjipa cipzcpcua `a j~voa

dj`c`o qzj ~j nbjko o nainjcpzo| eckz|o~- v|av|cj`o`j~ j p|oi~ea|doj~ kjadp|cno~)Nolj jipa- oa ji~cia `j kjadjp|co- mjuo| a omzia o ucujinco| opcuc`o`j~ o`jqzo`o~ vo|o eow ma

pado| nai~ncinco `a j~voa ~zo uampo j `o va~ca qzj jmj anzvo ij~~j j~voa)Jmj `juj| podld ~j jrj|ncpo| vo|o eowj| zdo |jv|j~jipoa djipom `a j~voa- k|oo~ ~

doicvzmoj~ uo|co`o~ io~ qzoc~ jmj ov|ji`j| o jrv|cdc| a~ |j~zmpo`a~ `j ~zo~ al~j|uoj~) J~~o~al~j|uoj~ |jej|jd ~j7 ea|do `a~ algjpa~- ~zo va~ca |jmopcuo- oa~ daucdjipa~ oa~ qzoc~ ~~zldjpc`a~ a~ algjpa~ j ~ `jea|doj~ qzj ~j eowjd ~al|j jmj~)

@z|oipj j~~o~ opcuc`o`j~- o opjia `o n|coio ~j ecro| io~ v|av|cj`o`j~ doc~ cdva|poipj~ j jd`jpj|dcio`o~ |jmoj~ jip|j jmo~)

O ov|ji`cwokjd `a uanolzm|ca kjadp|cna ~j ovmcno| ~al|j ~cpzoj~ nain|jpo~- eodcmco|j~

n|coio- j ia ~al|j `jecicj~ ol~p|opo~)Nada ucujda~ izd j~voa `j p|~ `cdji~j~- ia qzom vj|njljda~ a~ algjpa~ j ~jz~daucdjipa~- o~ opcuc`o`j~ v|ava~po~ ~ n|coio~ `jujd |j~vjcpo| j~po |jomc`o`j- jucpoi`a|jv|j~jipo|- v|jdopz|odjipj- pz`a ~al|j a vmoia `o maz~o)

cipj|j~~oipj nai~c`j|o|- podld- qzj o~ opcuc`o`j~ `jujd ~j| ~zecncjipjdjipj uo|co`o~ vo|ov|avcnco|7 jrvma|oj~ qzj ov|cda|jd o cipzca `o n|coio8 opcuc`o`j~ `j nadzicnoa `j eopa~kjadp|cna~ vo|o eoua|jnj| o jmola|oa `j pj|dciamakco- ~cdlamc~da j djca~ `j jrv|j~~akjadp|cna~8 opcuc`o`j~ `j ecroa `j nainjcpa~ j bolcmc`o`j~ kjadp|cno~)

V|avada~ jipa zdo Kjadjp|co `j v|cdjc|a k|oz nad no|pj| ci`zpcua- vo|pci`a `j jrvj|cinco~nad dopj|coc~ nain|jpa~) O~ `j`zj~- qzj va`j|a ~j| ejcpo~- v|anz|o|a zicecno| zd idj|amcdcpo`a `j iaj~ j ia a naigzipa pa`a `o Kjadjp|co) nad j~~j v|av~cpa qzj o~ opcuc`o`j~ `jKjadjp|co `j~po namja v|avjd jrj|nnca~ `j |jna|pj~- namokji~ j daipokji~)

Dopjdpcno io > ~|cj op|ou~ `j bc~p|co~ jgaka~

O v|ak|odoa `j Dopjdpcno `o > ~|cj v|avj- cicncomdjipj- a j~pz`a `a ~c~pjdo `jizdj|oa `jncdom j va~pj|ca|djipj o nai~p|za nadv|jji~cuo `a~ nainjcpa~ `o~ avj|oj~o|cpdpcno~- ljd nada `a~ omka|cpda~ `j nmnzma8 c~pa - ia lo~po ov|ji`j| o |jomcwo| avj|oj~v|jnc~a ~olj| va| qzj eowj| `j~po az `j azp|o doijc|o)

Pa`o~ o~ opcuc`o`j~ vo|o o nai~p|za `j~~j~ nainjcpa~ `jujd ~j| ov|j~jipo`o~ izd nmcdodapcuo`a| qzj v|adauo a jiuamucdjipa `o~ n|coio~- mjuoi`a o zdo nai~poipj lz~no `j ~amzj~vo|o v|almjdo~ qzj pjibod |jom ~ckicecno`a vo|o jmo~)

Gaka~ nad |jk|o~ io j`znoa dopjdpcno

V|avai`a j uoma|cwoi`a gaka~ nad |jk|o~- a v|aej~~a| j~po| v|adauji`a a `j~jiuamucdjipa~nca oejpcua- dapa| j nakicpcua `o~ n|coio~)

7/31/2019 Manual de matemtica 3

8/24

@a vaipa `j uc~po ~nca$oejpcua7

a gaka ` ava|pzic`o`j n|coio `j ~j mcu|o| v|ak|j~~cuodjipj `a jkanjip|c~da- vo|o o`apo| avaipa `j uc~po `a azp|a j va`j| v|juj| ~zo~ |joj~8

a gaka vj|dcpj qzj o n|coio ucuo- izd az iazp|a dadjipa- o va~ca `j m`j|- k|oo~ |cqzjwo`o nadzicnoa8

a gaka v|avcnco zdo odvmcoa a~ naipopa~ ~ancoc~ nad azp|o~ n|coio~- zdo ujw qzj a~ vo|njc`j gaka ~a j~nambc`a~ jd |jmoa oa~ cipj|j~~j~ nadzi~ vjma~ gaka~ j ia doc~ jd ezia `j~zo~ mckoj~ oejpcuo~8

a gaka vj|dcpj qzj o n|coio ov|ji o o ucuj| o nadvjpca- o namola|oa j podld o ava~ca8 a gaka mjuo o n|coio o `j~nal|c| o |jk|o op|ou~ `j zdo |jmoa `cej|jipj `oqzjmo qzj jmo naibjnj

bolcpzomdjipj nad a o`zmpa7 `c~nzpci`a o |jk|o- o`j|ci`a o jmo uamzipo|codjipj- ucuji`a o jip|j~jz~ nadvoibjc|a~ `j dj~do c`o`j- izdo ~cpzoa `j ~zvj|uc~a |jnv|ano- jd qzj no`o n|coio oa dj~da pjdva naip|amo`a| j naip|amo`a)

@a vaipa `j uc~po dapa|7

a gaka vj|dcpj qzj o n|coio ouomcj ~zo nadvjpinco dapa|o j ~jgo dapcuo`o o ~j zmp|ovo~~o| vjozpa`j~oeca8 a gaka ea|ijnj n|coio ano~cj~ vo|o ovj|ejcao| ~zo bolcmc`o`j `j n|co| j nai~p|zc| ~jz~ v|v|ca~

l|ciqzj`a~)

@a vaipa `j uc~po nakicpcua7

Vjmo oa j |jemjra naigzko`o~- a gaka vj|dcpj o jmola|oa `j nj|po~ j~p|zpz|o~- az ~jgo7

`adica avj|op|ca7 iaj~ v| izd|cno~ "nmo~~cecnoa- a|`jioa- lz~no `j u|co~ |jmoj~#j~p|zpz|oa `j pjdva j j~voa8 v|cdjc|a~ jmjdjipa~ `j mkcno op|ou~ `o |j~amza `j v|almjdo~~cdvmj~ "lz~no `j j~p|opkco~ vo|o ujinj| a gaka#8

jrv|j~~a j nadzicnoa op|ou~ `o ijnj~~c`o`j- j~~jincom oa gaka- `j jrvmcno| zdo |jk|o-nadjipo| az naipj~po| zdo eo~j `a gaka8

`j~jiuamucdjipa `o novonc`o`j `j al~j|uoa doc~ ecio `a djca ~zo uampo vjmo nadvo|oa `j~jdjmboio~ j cej|jio~)

O v|j~jio `a v|aej~~a| ia~ gaka~ nad |jk|o~ j~~jincom va|qzj jmj7

`ciodcwo a k|zva vjmo ~zo opcpz`j `j j~nzpo- `j opjia- `j jipz~co~da `coipj `a ~znj~~a `on|coio j `j jina|ogodjipa `coipj `o `j||apo8 j nada vo|pcncvoipj `a gaka- nada ~cdvmj~ gako`a|-ia pji`a ijd doc~ ijd djia~ `c|jcpa~ `a qzj o n|coio) "Ia b io`o qzj ola||jo doc~ zdon|coio qzj gako `a qzj vj|njlj| qzj a o`zmpa ia j~p mjuoi`a a gaka o ~|ca j o `jcro koibo|v|ava~cpo`odjipj) O n|coio jrckj qzj a o`zmpa gakzj ~j|codjipj vo|o nadvjpc|#8

al~j|uo o n|coio `z|oipj a gaka- qzj zd dadjipa k|opzcpa jd qzj o n|coio gako va| v|owj|) Ao`zmpa ia `juj cipj|uc| `z|oipj o oa `a gaka8 jmj al~j|uo a nadva|podjipa `o n|coio- ~zonadvjpinco- ~zo~ `cecnzm`o`j~ `j a|`jd oejpcuo- mcik~pcno- avj|op|co- vo|o v|jvo|o|- `jip|a `~jz v|agjpa vj`okkcna doc~ odvma- azp|o~ opcuc`o`j~ doc~ |cka|a~o~- nad algjpcua~ v|jnc~a~- p|olombjd j~~o~ `cecnzm`o`j~ `jpjnpo`o~ `z|oipj a gaka) @j~~o ea|do- o `cidcno `a gaka |j~vjcpo`o j izino cipj||advc`o va| cipj|ujij~ vj`okkcno~8

eoncmcpo a gaka vjmo a|koicwoa `o nmo~~j- aej|jnji`a dopj|com uo|co`a8

7/31/2019 Manual de matemtica 3

9/24

ogz`o o nai~p|za v|ak|j~~cuo `o iaa `j |jk|o- p|owji`a gaka~ nad |jk|o~ ~cdvmj~- oicdoi`a gaka~ j~va|pcua~ j uoma|cwoi`a o n|coa `j |jk|o~ iauo~ vjmo~ n|coio~8

eoua|jnj o n|copcuc`o`j vj|dcpci`a o zpcmcwoa `a dopj|com vo|o azp|a~ eci~ qzj ia a~ bolcpzonamanoi`a `c~va~ca `o n|coio dopj|coc~ `j gaka ~jd |jk|o~- cinjipcuoi`a o~ n|coio~ vo|o qzjn|cjd |jk|o~ j podld da`cecqzjd o~ |jk|o~ `a~ gaka~ naibjnc`a~ va| pa`a~8

v|adauj a `j~jiuamucdjipa `a j~v|cpa n|pcna `juamuji`a oa k|zva a~ v|almjdo~ ~z~ncpo`a~ vjmon|coa `j nj|pa~ gaka~- vj|dcpci`a mbj- va| pjipopcuo j j||a- ujinj| j~~j~ al~pnzma~8

ji|cqzjnj a~ gaka~ `o~ n|coio~ uo|coi`a a~ pcva~ `j gaka~ v|ava~pa~- a~ algjpcua~ `a~ gaka~- ~jgo- nbjko| v|cdjc|a az nbjko| va| mpcda- nai~jkzc| a doca| idj|a `j no|po~ az ~j mcu|o`j pa`o~ o~ no|po~- uo|coi`a podld a~ k|zva~ nad gaka~ jd `zvmo- jd k|zva~ `j > az `j 0-ucuji`a o ava~ca j o naavj|oa j dj~da- jujipzomdjipj- o~ `zo~ ~cdzmpoijodjipj)

Nada nadjo| a gaka1

O `jnc~a `j qzjd nadjo a gaka `juj ecno| o n|cp|ca `o~ n|coio~) Kj|omdjipj o~ n|coio~|j~amujd op|ou~ `j vo|mji`o~- nada- va| jrjdvma- zic `zic p- ~omod dcik- qzj uocjmcdcioi`a a mpcda izdo ~jqinco qzj eow na||j~vai`j| zdo n|coio o no`o ~molo `o vo|mji`o)_zjd vj|doijnj va| mpcda ij~~o ea|do `j jmcdcioa qzjd `juj cicnco| a gaka)

Azp|o ea|do pc|o| o ~a|pj ia vomcpciba) No`o n|coio |jpc|o zd vomcpa `j zd naigzipa jd qzjzd `a~ vomcpa~ doc~ nz|pa) _zjd pc|o a vomcpa nz|pa qzjd uoc cicnco| a gaka) Vo|o a ~a|p~jkz|od ~j a~ vomcpa~ pa`a~ gzipa~- j~nai`ji`a o~ vaipo~ vo|o qzj ia ~j `c~pciko a vomcpa dnz|pa)

Jd gaka~ nad no|po~ az `o`a~- va`j ~j `jnc`c| qzjd nadjo |jpc|oi`a zdo no|po az moioi`a a`o`a) _zjd pc|o o doca| no|po az a doca| idj|a ia `o`a qzjd nadjo a gaka)

Nada |j~vjcpo| ~zo ujw `j gako| az nada j~poljmjnj| o ompj|iinco ia gaka1

Izd gaka nad < vo|pcncvoipj~ encm vo|o o n|coio |j~vjcpo| o ompj|iinco- qzj eow nad qz gakzj zd `j no`o ujw ompj|io`odjipj)

Ia gaka nad > az doc~ gako`a|j~ qzj ~z|kj o `cecnzm`o`j- vac~ dzcpo~ ujwj~ o n|coio ia|j~vjcpo a dj~da ~jipc`a- ~jgo ba||ca az oipc ba||ca)

Al~j|uoi`a ~j n|coio~ qzj gakod mcu|jdjipj- ~jd cipj|ujia `c|jpo `a v|aej~~a|- vj|njlj ~jqzj zdo dj~do n|coio ia gako `zo~ ujwj~ nai~jnzpcuo~- do~ ompj|io ~zo~ gako`o~ oa ono~a nadzd az azp|a `a~ vo|pcncvoipj~- v|jej|cujmdjipj nad oqzjmj qzj ~j doicej~po |jnmodoi`a ~zo ujw)

A o`zmpa va`j ogz`o| ia no~a `j bouj| `c~vzpo jip|j o~ n|coio~- qzoipa o j~~o qzj~pa- vac~ ocinovonc`o`j `a k|zva vo|o |j~amuj| j~~j~ v|almjdo~ va`j vj|pz|lo| az cipj||advj| `jecicpcuodjipjo oa `a gaka)

Ij~~j no~a- a v|aej~~a| va`j ogz`o| o~ n|coio~ o pado|jd nai~ncinco `o ijnj~~c`o`j `jj~poljmjnj| zdo n|aiamakco `o~ oj~ qzj ia v|jgz`cqzj ijibzd `a~ vo|pcncvoipj~) Jd gaka~ `jvpca- va~~ujm nai~jkzc| c~~a vjma |jnz|~a `o ecmo) No`o n|coio j~vj|o ~zo ujw `j gako| jd ecm

Ia~ gaka~ `j dj~o ~j| ijnj~~|ca j~poljmjnj| zd ~jipc`a `j |apoa- oipj~ `j nadjo| o gako|)Omkzdo~ l|cino`jc|o~- nada j~n|oua~ `j G va`jd ozrcmco| o~ n|coio~ o vj|njlj| j~~j ~jipc`a `j|apoa)

7/31/2019 Manual de matemtica 3

10/24

Koibo| az vj|`j|1

Vo|o qzj a odlcjipj `j gaka vj|doijo ok|o`ujm j ~o`ca- a eopa `j vj|`j| ia `juj ~j| ucuc`anada zdo `j||apo- do~ nada zdo jrvj|cinco v|auc~|co qzj vj|dcpj v|ak|j`c| jd `c|ja o zdoucp|co ezpz|o) Ia ~j p|opo `j `j~uoma|cwo| a eopa `j koibo|- do~ `j mjuo| o n|coio o zdo onjcpo`a~ |j~zmpo`a~- ~jgod jmj~ qzoc~ ea|jd- o zd jqzcml|ca `j ~zo~ jdaj~ j o zdo nzdvmcnc`o`j nada~ azp|a~ gako`a|j~)

Vo|pcncvoi`a `a gaka nad o~ n|coio~- a o`zmpa va`j da~p|o| zdo opcpz`j va~cpcuo jd |jmoa azp|a qzj koibo- ejmcncpoi`a a- az jd |jmoa oa qzj vj|`j- naiea|poi`a a j j~pcdzmoi`a a onaipcizo| pjipoi`a) A v|aej~~a| va`j oci`o aej|jnj| ~ n|coio~ u|co~ ava|pzic`o`j~ `j gako| jujinj|- vo|o dcicdcwo| a~ jejcpa~ `a~ |j~zmpo`a~ `a gaka)

Dzcpo~ ujwj~ o~ n|coio~ jinaip|od ia v|v|ca k|zva o ~amza vo|o o `jnjva `j vj|`j|- ~jgon|coi`a gaka~ jd qzj o oa ~j ` va| naavj|oa ia bouji`a ijnj~~o|codjipj zd ujinj`a| "av|cincvom vo|pcncvo|# - ~jgo p|oi~ea|doi`a o `j||apo jd iujc~ `cej|jipj~ `j ~znj~~a) Ia gaka `jno|po~- va| jrjdvma- jmo~ naipcizod o gako| op qzj a mpcda gako`a| pj|dcij ~zo~ no|po~ j jip`jn|jpod a v|cdjc|a ujinj`a|- a ~jkzi`a ujinj`a|- a pj|njc|a j a qzo|pa ujinj`a|- jpn)- az ~jgo-zdo ea|do `j |jvo|pc| o ucp|co)

@j qzomqzj| ea|do a v|aej~~a| j~po| ~jdv|j v|j~jipj- v|adauji`a naiuj|~o~ nad o~ n|coio~oipj~ j `jvac~ `a~ gaka~ "izino `z|oipj o oa `a gaka#- vo|o ogz` mo~ o nadv|jji`j| qzj a lad gako`a| `juj ~j| novow `j7

`a vaipa `j uc~po oejpcua7 ia ~j c`jipcecno| nad a |j~zmpo`a `a gaka- ~jgo jmj qzom ea|- j inai~c`j| ma jecicpcua8

`a vaipa `j uc~po nakicpcua7 oiomc~o| o~ noz~o~ `o `j||apo j v|anz|o| a~ djca~ `j djmba|o| ~zova~~clcmc`o`j~ j ujinj|8

`a vaipa `j uc~po ~ancom7 nadv|jji`j| qzj v|jnc~a nadvo|pcmbo| o ucp|co j o `j||apo jnadv|jji`j| a vaipa `j uc~po `a azp|a)

O~ avj|oj~ o|cpdpcno~ io > ~|cj

Io ola|`okjd qzj j~nambjda~- a vaipa `j vo|pc`o o nai~p|za `a nainjcpa `o~ avj|oj~nada ci~p|zdjipa~ `j |j~amza `j v|almjdo~ naipjrpzomcwo`a~) ^ `jvac~ `j ko|oipc`a a nainjcpa qzj vo|pcda~ vo|o o nai~p|za `a~ omka|cpda~ `j nmnzma- nada |jnz|~a~ vo|o okcmcwo| a~ nmnad idj|a~ doca|j~) Nada a~ omka|cpda~ j~pa ezi`odjipo`a~ io~ v|av|cj`o`j~ `o~ avj|oj~-nai~c`j|oda~ cdv|j~nci`ujm qzj o~ n|coio~ ucujincjd opcuc`o`j~ qzj vj|dcpod vj|njlj| o~v|av|cj`o`j~ op|ou~ `o doicvzmoa `j da`jma~ nain|jpa~)

O~ opcuc`o`j~ `o > ~|cj jiuamujd ~jdv|j o doicvzmoa `j dopj|coc~ `j naipokjd- ~jgod jmj~j~p|zpz|o`a~- nada a dopj|com `az|o`a az o~ |kzo~ Nzc~jioc|j- az dopj|coc~ `a napc`coia- nadavomcpa~- podvcibo~ `j ko||oeo- ejcgj~ az ecnbo~)

Ia ji~cia `o o`ca j `o ~zlp|oa- v|anz|oda~ p|olombo| o~ `zo~ avj|oj~ |jmoncaio`o~ vjmo|juj|~clcmc`o`j j v|avcnco| `j~oeca~ vo|o qzj o~ n|coio~ vj|njlod o o~~ancopcuc`o`j j onadzpopcuc`o j o o`ca)

~zlp|oa v|anz|oda~ o~~anco| o~ c`co~ `j nadvo|o|- nadvmjpo| j pc|o|- op|ou~cpzoj~ v|almjdo~ naipjrpzomcwo`o~- nbodoi`a o opjia vo|o o va~~clcmc`o`j `j |j~am~cpzoj~ `j nadvmjpo| op|ou~ `j zdo o`ca- nada ucda~ u|co~ n|coio~ eowj|jd jd j~p|opkcoj~vaipijo~)

7/31/2019 Manual de matemtica 3

11/24

Nbododa~ o opjia vo|o o cdva|pinco `j ~j p|olombo| j~~o~ p|~ c`co~ o~~anco`o~ ~zlp|ovac~ ijd ~jdv|j jmo~ ~a `c~pcikzc`o~ nad nmo|jwo) O~ ~cpzoj~ `j pc|o| ~a o~ doc~ nadzi~opcuc`o`j~ j~namo|j~- qzoi`a ~a ov|j~jipo`a~ v|almjdo~ jd qzj ~j vj|`jz az ko~paz vo|pj `j omka)A qzj eac ko~pa az vj|`c`a ~j| jipa pc|o`a az ~zlp|o`a `a qzj ~j pcibo oipj|ca|djipj) v|jnc~a-oci`o- `c~nzpc| nad a~ omzia~ o zpcmcwoa `o ~zlp|oa io~ nadvo|oj~- vo|o ~j `jnc`c| qzjd pjdoc~ az qzjd pjd djia~) Podld dj|jnjd `j~poqzj o~ ano~cj~ jd qzj ~j qzj| ~olj| qzoipaeompo) Ij~~j no~a- o n|coio va`j ~ado| oa ciu~ `j ~zlp|oc|- a qzj podld |j~amuj a v|almjdo) Va|jrjdvma7 ^j uan pjd 3: eckz|cibo~ `j zd mlzd qzj ~j| nadvmjpo`a nad ?66- qzoipo~ eckz|cibo~eompod1 Vjmo ~zlp|oa- eo|oda~ ?663: 4 :- do~ o n|coio va`j vji~o| 3: ( : 4 ?66 j nainmzc|7eompod : eckz|cibo~) J~~j podld zd |oncanica na||jpa)

O nai~p|za `a omka|cpda `o o`ca j `o~zlp|oa

Vo|o nmnzma~ qzj jiuamujd `jwjio~ j njipjio~ cipj|j~~oipj qzj ~j ` ava|pzic`o`j n|coio`j nai~p|zc| a~ omka|cpda~ qzj o ogz`o|a o eowj| nmnzma~ nad doc~ |ovc`jw) Vo|o c~~a va`j

zpcmcwo| vomcpa~ nama|c`a~- jm~pcna~ j zd no|pow `j v|jko~ vo|o no`o n|coio- |jv|j~jipoiqzo`|a `a uoma| va~cncaiom `a~ omko|c~da~ ia ~c~pjdo `jncdom)A~ vomcpa~ l|oina~ va`jd |jv|j~jipo| o~ zic`o`j~- a~ owzc~ o~ `jwjio~- a~ uj|djmba~ o~

njipjio~ j a~ odo|jma~ o~ zic`o`j~ `j dcmbo|)Nadjoda~ o~~ancoi`a zdo zic`o`j o no`o vomcpa l|oina j- no`o ujw qzj gzipoda~ ?6-

odo||oda~ nad zd jm~pcna vo|o ~cdlamcwo| ? `jwjio) Va| jrjdvma7?:

? vonapj `j ?6 vomcpa~ : vomcpa~ ~ampa~Zd |jnz|~a `c`pcna cipj|j~~oipj j~n|juj| a omko|c~da `o `jwjio nad mvc~ owzm j a `o~

zic`o`j~ nad mvc~ v|jpa) J~~j |jnz|~a j~po| nbodoi`a o opjia vo|o o qzj~pa `a uoma|va~cncaiom- pa cdva|poipj vo|o o nadv|jji~a `a ~c~pjdo `jncdom) dj`c`o qzj o qzj~pa `a uoma|va~cncaiom ea| o~~cdcmo`o- a |jnz|~a na| ~j| oloi`aio`a ~jd v|jgzwa vo|o o nadv|jji~a)

No`o ujw qzj gzipoda~ ?6 vomcpa~ owzc~ az ?6 `jwjio~- va`jda~ odo|| ma~ nad jm~pcnadvmjpoi`a ? njipjio8 jmj~ ~a p|ano`a~ va| ? vomcpa uj|djmba j |jv|j~jipo`a~ va| zd omko|c~dauj|djmba io > a|`jd8 j o p|ano `j ?6 njipjio~ va| ? zic`o`j `j dcmbo| |jv|j~jipo`o va| zdomko|c~da odo|jma io 0 a|`jd)

O~~cd70 ? < >0 vomcpa~ odo|jma~ ? vomcpa uj|djmba < vomcpa~ owzc~ > vomcpa~ l|oina~

J~~j ~c~pjdo vj|dcpj nadv|jji`j| o ezia `a wj|a io izdj|oa- nada do|no`a| `o no~ouowco)Zic`o`j j dcmbo| N @ Z

}} }}} }}}}}}} } ;?6

A~ vomcpa~ ~a namano`a~ ia~ lam~a~ `a no|pow `j v|jko~- qzj o~ n|coio~ na~pzdod nbodonomnzmo`a|o- j ~a daucdjipo`a~ `j zd lam~a vo|o a azp|a- naiujicjipjdjipj- `z|oipj a~

7/31/2019 Manual de matemtica 3

12/24

nmnzma~) C~~a ~j eow gzipoi`a a~ vomcpa~ `j no`o namzio vo|o alpj| o ~ado) ^j ea|jd alpc`a~ qzj ?6- `juj ~j p|ano| zd k|zva `j ?6 va| zdo zic`o`j `o a|`jd ~jkzcipj "~zvj|ca|#)

Jrjdvma7 ( ?; 4N @ Z

}} }}}} }}}}}}}?6 zic`o`j~ ~a p|ano`o~ va| ? `jwjio)N @ Z

}}}}Jipa7 ( ?; 4 06 az?(

?;06

Io~ o`cj~ nad njipjio~- no`o ?6 `jwjio~ ~a ok|zvo`o~ j p|ano`o~ va| ? njipjio- az ?6vomcpa~ owzc~ uomjd ? uj|djmba)

Io~ o`cj~ nad zic`o`j~ `j dcmbo|- no`o ?6 njipjio~ ~a ok|zvo`o~ j p|ano`o~ va| ? zic`o`j`j dcmbo|- az ?6 vomcpa~ uj|djmba~ uomjd ? odo|jma)

Al~j|uoa 7 Ij~~j gaka- a~ vomcpa~ |jv|j~jipod o~ qzoipc`o`j~8 `jujd ~j| zpcmcwo`a~ vo|ojejpzo| o~ avj|oj~ j- va|poipa- daucdjipo`a~ nada pom) Ia cinca `o avj|oa- o no`o vo|njmona||j~vai`j zdo qzoipc`o`j `j vomcpa~) Oa |jzic ma~ vjmo ~ado ia ~j|co na||jpa |jna||j| o iaua~vomcpa~ vo|o |jv|j~jipo| a papom)

Ia no~a `a omka|cpda `o o`ca- o n|coio- op|ou~ `o~ p|ano~ `j vomcpa~ l|oina~ va| owzc~ jowzc~ va| uj|djmba~- nadv|jji`j| a va|qz `a |jnz|~a oa uoc zd `a omka|cpda)Io ~zlp|oa- ~j a omko|c~da `a ~zlp|oji`a ea| doca| qzj a `a dcizji`a- o n|coio vj`c|

jdv|j~po`a a|`jd ~zvj|ca| ? zic`o`j qzj ~j| p|ano`o va| ?6 zic`o`j~ `o a|`jd ciej|ca|)Al~j|uj7

7/31/2019 Manual de matemtica 3

13/24

J~~j |jnz|~a `o ~zlp|oa ~j| nadv|jji`c`a nada zd `j~jdvonapodjipa `a~ vomcpa~ `o`jwjio vo|o o zic`o`j- va|qzj ? `jwjio jqzcuomj o ?6 zic`o`j~) A dj~da vo|o ? njipjio- qzjjqzcuomj o ?6 `jwjio~)

O nai~p|za `a nainjcpa `j dzmpcvmcnoa

O nai~p|za `a nainjcpa `o dzmpcvmcnoa `jvji`j `o~ |jmoj~ qzj jiuamujd o~ qzop|aavj|oj~ o|cpdpcno~ "o`ca- ~zlp|oa- dzmpcvmcnoa j `cuc~a#) Jd uc~po `c~~a- pa|icdva|poipj v|avcnco| ~ n|coio~ opcuc`o`j~ qzj mjujd o vj|njlj| o |jmoa jip|j o`ca jdzmpcvmcnoa j o |jmoa jip|j dzmpcvmcnoa j `cuc~a) C~~a |jqzj| u|co~ ola|`okji~ `dzmpcvmcnoa7

dzmpcvmcnoa nada ~ado j vo|njmo~ ckzoc~8 dzmpcvmcnoa nada ea|doa `j pa`a~ a~ vo|j~ va~~ujc~8 dzmpcvmcnoa nada p|ano)

Io dzmpcvmcnoa nada ~ado `j vo|njmo~ ckzoc~- a qzj ~j eow z~o| o dzmpcvmcnoa nad|jnz|~a vo|o ol|juco| zdo ~ado dzcpa maiko- va| jrjdvma7 < ( < ( < ( < ( < ( < 4 ?< va`j ~j| 5 < 4 ? pcva~ `j nodc~jpo~1 O |j~amza `j~~j pcva `j v|almjdo nadjo va| zdoopcuc`o`j v|av|codjipj nai~p|zpcuo jd qzj- va| doicvzmoa j |jkc~p|a op|ou~ `j `j~jiba~- on|coio nbjko o pa`a~ a~ vo|j~ va~~ujc~ qzj na||j~vai`jd oa v|a`zpa < > 4 5- do~ ~jd|jmoncaio| pom eopa dzmpcvmcnoa) V|ak|j~~cuodjipj ~j| cip|a`zwc`o o |jv|j~jipoa `j~~j pcv~amza op|ou~ `j poljmo~ `j n|zwodjipa `j mcibo~ j namzio~- eoncmdjipj o~~ancujc~dzmpcvmcnoa) Ujgo7

nodc~jpo owzm odo|jmo uj|`j lj|dz`o

nz|po owzm nz|po odo|jmonz|po

uj|`j nz|po

nadv|c`o owzmnadv|c`o

odo|jmonadv|c`o

uj|`j nadv|c`o

Jipa7 < lj|dz`o~ > nodc~jpo~ 4 5 p|ogj~ va~~ujc~)Io dzmpcvmcnoa nada p|ano- jinaip|oda~ a jejcpa- `j nj|po ea|do dkcna- `o dzmpcvmcno

nada o avj|oa qzj v|avcnco o odvmcoa |vc`o `j qzoipc`o`j~) Dzcpo~ ujwj~ jinaip|oda~ j~~o

7/31/2019 Manual de matemtica 3

14/24

naiapoa io mcikzokjd eckz|o`o- nada qzoi`a `cwjda~ qzj nj|po~ nac~o~ ~j dzmpcvmcnod) Odzmpcvmcnoa nada p|ano ovo|jnj jd ~cpzoj~ qzj jiuamujd nmnzma `j v|ja8 va| jrjdvma7 ~zdo nodc~o nz~po ncina dcm n|zwjc|a~ |joc~- va|qzj no`o nodc~o va`j ~j| p|ano`o va| j~~o qzoipcva`j ~j jipa vj|kzipo| qzoipa `cibjc|a ijnj~~|ca vo|o ~j p|ano| va| > nodc~o~) dj`c`o qzjozdjipo a idj|a `j nodc~o~- o qzoipco `j `cibjc|a ozdjipo dzcpa doc~ `jv|j~~o)

^zkj|cda~ qzj a p|olomba nad dzmpcvmcnoa io~ ~|cj~ cicncoc~ pjibo nada v|janzvoav|cincvom o nai~p|za `a nainjcpa `j dzmpcvmcnoa nad a |jnz|~a doicvzmoa `j dopj|coc~nain|jpa~ uo|co`a~ j |j~amza `j v|almjdo~ |jmoncaio`a~ nad a naipjrpa `j uc`o `o n|coio)Ov|j~jipoda~ ij~po namja omkzdo~ ~zkj~pj~ `j gaka~ j ~cpzoj~ v|almjdo~ qzj juc`jipjdjipj`jujd ~j| ji|cqzjnc`o~ vjma v|aej~~a| qzj naibjnj `j vj|pa ~jz~ omzia~ j va| c~~a va`j eow ma `jdoijc|o doc~ jecncjipj) Zdo v|pcno lo~poipj cipj|j~~oipj ~zkj|c| oa~ omzia~ qzj ciujipjdv|almjdo~ qzj va~~od ~j| |j~amuc`a~ vjma |jnz|~a dzmpcvmcnoa)

@jvac~ `o~ v|cdjc|o~ opcuc`o`j~ `j nai~p|za `a nainjcpa `j dzmpcvmcnoa nad o ogz`o `jdopj|coc~ nain|jpa~ `j doicvzmoa j `o ovmcnoa jd ~cpzoj~ v|almjdo~ ejcpo~ io ? j

7/31/2019 Manual de matemtica 3

15/24

A Poik|od zd naigzipa `j vjo~ qzj doipd |jmoj~ dopjdpcno~ jip|j ~c) O~ vjo~djia|j~ pd o djpo`j `o |jo `o~ vjo~ d`co~ j j~po~ pd o djpo`j `o |jo `o~ vjo~ k|oi`j~)Omkzi~ mo`a~ `o~ vjo~ pd a dj~da nadv|cdjipa- aej|jnji`a dzcpo~ va~~clcmc`o`j~ `j jinocrjvo|o ea|do| azp|o~ eckz|o~) A~ v|cdjc|a~ jrj|nnca~ `j daipokjd pd nada algjpcua o jrvma|oa`o~ vjo~ vo|o o `j~nalj|po `o~ ~zo~ no|onpj|~pcno~)

Vkcio < cdva|poipj |j~~ompo| qzj o daipokjd `o~ eckz|o~ o`dcpj u|co~ ~amzj~) ^zkj|cda~ v|adauj|

o p|ano `j ~amzj~ jip|j a~ omzia~)Vkcio ?6

A algjpcua `j~~j~ jrj|nnca~ c`jipcecno| o~ v|av|cj`o`j~ kjadp|cno~ `o~ vjo~ `a Poik|od)V|adauo `c~nz~~j~ jip|j a~ omzia~ ~al|j j~~j pjdo)

Vkcio ?

Jdla|o p|olomba~a- j~pj qzjl|o noljo pjd u|co~ ~amzj~) ^zkj|cda~ qzj ~jgo ejcpa va| `zvmo~`j omzia~) O |jk|o |j~vjcpo| o na||j~vai`inco jip|j o~ dzmpcvmcnoj~ j a~ |j~zmpo`a~ io~ vjoucwcibo~) Nada pa`o~ o~ opcuc`o`j~ nbodo`o~ `j `j~oeca ij~po namja- j~po zdo opcuc`o`j jqzj ia cipj|ej|j io ~jqinco `a naipjz`a `a vmoijgodjipa `j ji~cia) Ijd pa`a~ a~ omzia~nai~jkzc|a |j~amu mo) Oci`o o~~cd cipj|j~~oipj cinjipcu ma~ o pjipo|)

;) P|oi~ea|doi`a qzo`|cmpj|a~Vkcio 0:

O~ opcuc`o`j~ qzj ~amcncpod o al~j|uoa `j ~adl|o~ `jujd ~j| v|ava~po~ izd `co ji~amo|o`a)Moipj|io~ az qzomqzj| azp|a pcva `j cmzdcioa o|pcecncom ia va`jd nadv|c| a vovjm `a ~am- va|qz~jz~ |oca~ ia ~a vo|omjma~ nada a~ `a ~am- a qzj ano~caio azp|a pcva `j p|oi~ea|doa io~ eckz|oVa`j ~j| cipj|j~~oipj |jomcwo| a~ `ac~ pcva~ `j jrvj|cinco- nbodoi`a o opjia `a~ omzia~ vo|o aeopa `j qzj o cmzdcioa o|pcecncom v|auano ~adl|o~ odvmco`o~ `o~ eckz|o~ j `c~pa|j~ `cej|ji`o~ v|a`zwc`o~ vjmo mzw ~amo|)

Vkcio 05

Io~ opcuc`o`j~ nad vomcpa~ j ponbcibo~ cipj|j~~oipj |jmoncaio| a~ vo|omjmak|oda~ nad|jpikzma~ j a~ qzo`|o`a~ nad a~ ma~oika~) Jrvj|cdjipj `j~jiuamuj| j~~o opcuc`o`j nadp|cikzma~) A~ omzia~ `j~nal|c|a qzj a~ p|cikzma~ ~a eckz|o~ |kc`o~)

7/31/2019 Manual de matemtica 3

18/24

2) Gakoi a _zo |cdzVkcio 0;

Oipj~ `j cicnco| a gaka _zo`|cdz- v|adauo zdo `c~nz~~a ~al|j o~ |jk|o~) ijnj~~|ca `jnc`c|qzom o vjo qzj `o| `c|jcpa oa gako`a| `j ~j| a v|cdjc|a o gako|)

Vkcio 02A~ gaka~ v|ava~pa~ io~ opcuc`o`j~ `j~pj mcu|a ~a cdv|j~nci`ujc~ vo|o o |jomcwoa `o~ dj~dA~ omzia~ ~ va`j|a nadv|jji`j| o~ opcuc`o`j~ o vo|pc| `o jrvj|cinco jejpcuo nad a~ gaka~

ci`cno`a~)Vkcio 03

Zdo mcibo ~jd vaipa~ io poljmo ~ckicecno qzj a gako`a|- io ~zo ujw `j gako|- ia pcibo ijibzdovjo qzj ~j|uc~~j j vo~~az o ujw vo|o a v|rcda)

3) Dzmpcvmcnoi`a jwjio~ j njipjio~

Vkcio :> j :0

O dzmpcvmcnoa vo|njmo`o lo~jco ~j io v|av|cj`o`j `c~p|clzpcuo `o dzmpcvmcnoa jd |jmoo`ca) P|olomboi`a o dzmpcvmcnoa `j~~o ea|do- a v|aej~~a| v|jvo|o a omzia vo|o o nai~p|za `aomka|cpda `j~~o avj|oa `j zdo ea|do nadv|jji~ujm) Vo|o j~~j~ nmnzma~ cipj|j~~oipj o v|pcno`j dzcpa~ jrj|nnca~ `j dzmpcvmcnoa va| vapinco~ `j ?6 " ?6- ?66- ?666- jpn#- qzj va`jd~j| ejcpa~ ia no`j|ia `a omzia nad a ovaca `a lona `j lapj~)

?6) Dzmpcvmcnoi`a ia qzo`|cnzmo`aVkcio :;

Azp|o ava vo|o p|olombo| nad o dzmpcvmcnoa z~o| a dopj|com `az|o`a jinaip|o`a jd dzcpo~j~namo~) A v|anj~~a va`j ~j| ~jdjmboipj oa `o dzmpcvmcnoa ia vovjm qzo`|cnzmo`a)

??) Dmpcvma~ j cuc~a|j~Vkcio 5?

O~ opcuc`o`j~ `j j~p|zpz|o eopa|com |jqzj|jd jrvj|cinco~ nad o dc~pz|o `j na|j~ v|cd|co~)^zkj|cda~ qzj j~~o jrvj|cinco ~jgo jejpcuodjipj |jomcwo`o nad a~ omzia~- dc~pz|oi`a ~jpcipo~ io~ na|j~ owzm- uj|djmba j odo|jma- vo|o `j~nal|c| o~ na|j~ ~jnzi`|co~7 mo|oigo- |arauj|`j j a do||ad "dc~pz|o `o~ p|~ na|j~ v|cd|co~#) Io~ opcuc`o`j~ `j~pj j `a v|rcdanovpzma- a~ nainjcpa~ `j dmpcvma nadzd j `cuc~a| nadzd ea|od o~~anco`a~ dc~pz|o

na|j~) O~ j~p|zpz|o~ eopa|coc~ nadjo|od `a ? j ~a c`jipcecno`o~ vjma doca| idj|a qzj ovo|jnj

ijmo~) O~~cd- o j~p|zpz|o `a 03 pjd nada doca| idj|a a 03 j nada djia| a idj|a ?)

?

7/31/2019 Manual de matemtica 3

19/24

Jdla|o zdo j~p|zpz|o eopa|com ~jgo c`jipcecno`o vjma doca| idj|a cinmz`a ijmo- jinaip|o|jda~j~p|zpz|o~ djia|j~ `jip|a `j j~p|zpz|o~ doca|j~) Vo|o `jdo|n mo~- a omzia `juj| p|oo| |jpikzma~qzj cinmzod ~jdv|j a idj|a ?)

Vkcio 5:

Omd `a~ jrj|nnca~ qzj ov|j~jipoda~- v|avaibo oa~ omzia~ qzj eood- ia no`j|ia- o|jv|j~jipoa `o~ j~p|zpz|o~ o vo|pc| `a uoma| `jpj|dcio`a vo|o o~ emjnbo~ j- ciuj|~odjipj- o vo|pc|`a uoma| naibjnc`a~ `o~ lamo~- vo|o `j~nal|c| a uoma| `o~ emjnbo~) Ia cinca- jmj~ v|auoujmdjokc|a va| pjipopcuo j j||a- do~ `jvac~ onolo|a va| vmoijgo| zdo j~p|opkco lo~jo`o io~ ~zo~al~j|uoj~)

Vkcio 5:

Io j~p|zpz|o eopa|com- o o~~ancoa `j na|j~ oa~ eopa|j~ o|lcp||co- do~ `juj ~j| ~jdv|j ejcpo jd|jmoa oa~ eopa|j~ v|cda~ doca|j~ qzj ?- az ~jgo- - :- ;- ?? jpn Nada j~nambjda~ o~ na|j~v|cd|co~ vo|o j :- oa cip|a`zwc| a ; j~nambjda~ z~o| vaipcmbo`a oa ciu~ `j na|j~- vo|ova~~clcmcpo| o dc~pz|o nad o~ azp|o~ na|j~) O~~cd- zd idj|a dmpcvma nadzd `j : j ; ~j| owzmvaipcmbo`a) Pj|jda~ podld7 odo|jma vaipcmbo`a j uj|djmba vaipcmbo`a)

?>) Pjipikzma zd qzjl|o$noljo cej|jipj

Vkcio 5;

O~ vjo~ `a Pjipikzma `jujd ~j| namo`o~ jd no|pa ljd |j~c~pjipj j `jvac~ |jna|po`o~- vo|oeoncmcpo| a ~jz doiz~jca `z|oipj o~ opcuc`o`j~)

?0) Oci`o a~ dmpcvma~ j `cuc~a|j~

Vkcio ;:

Vo|o `j~nal|c| j~p|zpz|o eopa|com `j zd idj|a- o~ n|coio~ `jujd nbjko| o jmj op|ou~ `jdzmpcvmcnoj~ va| eopa|j~ v|cda~- nadjoi`a `a ?) Nada jmo~ oci`o ia pd a nainjcpa `j idj|av|cda- va`jd eow ma jdv|jkoi`a a~ eopa|j~ ~zkj|c`a~ vjma v|aej~~a|) Va| jrjdvma7 o j~p|zpz|oeopa|com `a jd no`o namzio) O ~ `j~jibo| j nadvmjpo| nad a~ uoma|j~)

7/31/2019 Manual de matemtica 3

20/24

?:) Uoda~ gako| Eoi$poi

Vkcio ;2

A algjpcua vj`okkcna `a gaka Eoi poi mjuo| a omzia o vj|njlj| o |jmoa jip|j a `cuc~a| j a|j~pa izdo `cuc~a)

Vkcio 2

Naipcizj cinjipcuoi`a a~ omzia~ o nadvo|o| e|oj~ nad lo~j io al~j|uoa j doicvzmoa `o~vjo~ `a dopj|com)

Vkcio ?60

\jmoncaio| o vo|pj nad a pa`a ezi`odjipom vo|o o oqzc~ca `a nainjcpa `j e|oa) ^zkj|cda~v|ava| oa~ omzia~ zd lad idj|a `j azp|a~ jrj|nnca~ `j nadvmjpo| a cipjc|a- o ~j|jd ejcpa~ iano`j|ia az a|omdjipj- nad ovaca `a dopj|com)

7/31/2019 Manual de matemtica 3

22/24

Io o`ca `j e|oj~ nad o ogz`o `j `j~jiba~ "k|ecna~#- a omzia va`j nama|c| o vo|pj `a`j~jiba na||j~vai`jipj o no`o vo|njmo `j zdo na| `cej|jipj o uj|cecno| a |j~zmpo`a vjmo v|v|cocipj|v|jpoa `a `j~jiba) Ia jrj|nnca 0 a podoiba `a cipjc|a eac ~zkj|c`a nad dj`c`o~ jdnjipdjp|a~- qzj eoncmcpod o `cuc~a nad o ogz`o `o |kzo) J~~j jrj|nnca va`j ~j| ejcpa podld jd

vovjm qzo`|cnzmo`a)

Vkcio ?6;

O o`ca `j e|oj~ nad `jiadcio`a|j~ `cej|jipj~ ov|j~jipo`o nada `j~oeca- `juj ij~~o eo~j~j| |j~amuc`o jd k|zva~ `j 0 omzia~- vo|o eoncmcpo| o `c~nz~~a- j nad a ovaca `a dopj|com)cdva|poipj qzj a~ omzia~ nbjkzjd oa |j~zmpo`a na||jpa j va~~od jrvmcno| ~jz |oncanica) O |jk|oecno| vo|o ljd doc~ po|`j)

Vkcio ?62

Vo|o |j~amuj| v|almjdo~ `j e|oj~- a~ omzia~ `jujd ~j| cinjipcuo`a~ o `j~jibo| k|ecna~ vo|o|jv|j~jip mo~) C~~a eoncmcpo o nadv|jji~a `a v|almjdo) ~ ujwj~ vj|dcpj nbjko| |j~va~po ovjop|ou~ `a nmnzma djipom)

7/31/2019 Manual de matemtica 3

23/24

Vkcio ?

A `j~oeca v|ava~pa nai~c~pj jd a omzia p|oi~ea|do| e|oj~ cdv|v|co~ jd idj|a~ dc~pa~) va~~ujm eow ma nad a~ Mo`|cmba~- op|ou~ `j nadvo|oa- nad a cipjc|a pado`a nada |jej|inc\jnadji`oda~ jucpo| o oipjncvoa `j |jk|o~ `j nmnzma vo|o ia vj|pz|lo| o nadv|jji~a) Poc~|jk|o~ `jujd ecno| vo|o azp|a dadjipa)

Vkcio ?>:

O opcuc`o`j 5 va`j ~j| |j~amuc`o nad e|oj~ cdv|v|co~ az idj|a~ dc~pa~)

7/31/2019 Manual de matemtica 3

24/24

Lclmcak|oeco

Owjuj`a- D) Uj|icno \) `j) O ciemzinco `a~ gaka~ j dopj|coc~ vj`okkcna~ io nai~p|za `a~nainjcpa~ jd dopjdpcno ) Pj~j `j Dj~p|o`a- Z^V- ?33>)

QQQQQQQ) Gakoi`a j nai~p|zci a dopjdpcno) ^a Vozma- J`) Zic`o~- ?33>)