Colaborativo Unidad 3

 

Auohecojs< Zob auoheøo js uob rjibheøo y jotrj ncs mbgoetunjs, nj mbojrb quj b hbnb vbicr nj ib

pre premjr mjrbb ij hcrrjs hcrrjspco pconj nj uo üoehc üoehc vbicr vbicr nj ib sjguon sjguonbb (c oeoguo oeoguoc), c), quj iibmbm iibmbmcs cs embgjo embgjo c trbosacrmbnc.  B ib auoheøo sj ij sujij njsegobr pcr a y b ib embgjo pcr a(x), sejonc x ib vbrebfij eonjpjonejotj.  

  Qbrebfij eonjpjonejot eonjpjonejotj< j< ib quj sj aelb prjvebmjotj. prjvebmjotj.   Qbrebfij njpjonejotj< Ib quj sj njnuhj nj ib vbrebfij eonjpjonejotj.

Vjhefjo rjspjhtevbmjotj ji ocmfrj nj ncmeoec y hconcmeoec. Ji rbogc  nj ib auoheøo js ji hcoluotc nj tcncs ics vbicrjs quj ib auoheøo  tcmb. Jljmpic< V= {(2,6) (6,>) (8,1) (>,0)} se js uob auoheøo. Ncmeoec y rbogc nj uob auoheøo< pbrb ji ncmeoec njspjlbmcs U, y bobiezbmcs nconj qujnb P

Xe P qujnb jo uo pcieocmec ji ncmeoec sco tcncs ics oümjrcs rjbijs Xe P qujnb jo uo njocmeobncr sj `bhj tcnc ji njocmeobncr nesteotc nj hjrc y sj njspjlb x, ji njocmeoc sjreb tcncs ics oümjrcs rjbijs mjocs ji oumjrc njspjlbnc Xe P qujnb jo uob rbíz sj `bhj ib hbotenbn suf rbnehbi mbycr c egubi quj hjrc y sj rjsujivj ib eojhubheøo, ji ncmeoec js ib sciuheøo nj ib eojhubheøo. Jo ji rbogc sj `bhj ji mesmc prchjsc scic quj sj njspjlb P y sj bobiezb nøonj qujnb U 2 Jljmpic< a(x) = ∜x-2 U= ∜x-2 P-2 ≪ 4 P≪ 2 {2, ∟) Ncmeoec

6 Jljmpic< :x + 0y= > 0y= > ‘ :x U= > -:x / 0 Nm< Vjbijs Vbogc< :x + 0y= > :x= >- 0y P= > ‘ 0y / : Vg< Vjbijs  

 

Auoheøo Auo heøo ieojbi< ieojbi) m= > ‘ 8 / 6 ‘ (-2) = 2/8 U ‘ 8 = 2/8 {x ‘ (-2)} 8y ‘ 5 = x + 2 Empiíhetb -x + 8y -5 -2 = 4 -x + 8y -24 = 4 8y-5 = x+2 8y= x +2 + 5 8y= x + 24 U= x + 24 / 8 U= 2/8x + 24/8 Jxpiehetb

Auoheøo hubnrâtehb< Zob auoheøo hubnrâtehb js bqujiib quj pujnj jshrefersj hcmc uob jhubheøo

nj ib acrmb< A(P) = BP6 + fx + h Nconj b, f y h (iibmbncs tårmeocs) sco oümjrcs rjbijs hubijsquejrb y b js nesteotc nj hjrc (pujnj sjr mbycr c mjocr quj hjrc, pjrc oc egubi quj hjrc). Ji vbicr nj f y nj h sí pujnj sjr hjrc. Jo ib jhubheøo hubnrâtehb hbnb uoc nj sus tårmeocs tejoj uo ocmfrj.  Bsí<   

  BP6 js ji tårmeoc hubnrâtehc   fx js ji tårmeoc ieojbi   H js ji tårmeoc eonjpjonejotj

^cnb auoheøo hubnrâtehb pcsjj uo mâxemc c uo míoemc, quj js ji vårtehj nj ib pbrâfcib. Xe ib pbrâfcib tejoj hcohbvenbn `bheb brrefb, ji vårtehj hcrrjspconj b uo míoemc nj ib auoheøo; mejotrbs quj se ib pbrâfcib tejoj hcohbvenbn `bheb bfblc, ji vårtehj sjrâ uo mâxemc. Jo ib auoheøo hubnrâtehb johcotrbmcs<    

       

ji vårtehj ji jlj nj semjtríb eotjrhjphecojs ncmeoec y rbogc

 

Ib grbaehb nj uob auoheøo hubnrâtehb js uob hurvb hco uob acrmb nj Z iibmbnb pbrâfcib. Tujnj sjr trbzbnb nefulbonc sciuhecojs nj ib jhubheøo, johcotrbonc ji vårtehj y usbonc ji jlj nj semjtríb pbrb grbaehbr puotcs sjijhhecobncs, c johcotrbonc ibs rbíhjs y ji vårtehj. Jljmpic<

Jljmpic< nbnb ib jhubheøo ‘x6 + 0x ‘ 26  B= -2, F= 0, H= -26 Hcmc B= -2, y -2 js mjocr quj hjrc jotcohjs ib pbrâfcib bfrj `bheb bfblc. @biijmcs ics eotjrhjptc< `bhjmcs x=c pbrb `biibr ics eotjrsjhtcs hco ji jlj U U= -x6 + 0x ‘ 26 U= - 46 + 0(4) ‘ 26 U= -26 Eotjrhjptc jo ji jlj U (4, -26)  Bvjregubmcs se `by eotjrsjhtcs hco ji jlj x pbrb jsc `biibmcs ji njshrjhemejotc f6 ‘ >bh= 06 ‘ > . (-2) . (-26) = 1> ‘ 21 Hcmc 21?4 tejoj ncs eotjrsjhhecojs hco ji jlj x @biijmcs jsj eotjrhjptc, sj `bhj y=4 U= -x6 + 0x ‘ 26 4= -x6 + 0x ‘ 26 Muitepiehbmcs pcr (-2) 4= x 6 ‘ 0x + 26 Fushbmcs uo hbsc nj abhtcrezbheøo. P6 ‘ 0x + 26= 4 P6 ‘ 0x + 26= (x ‘ 1) (x ‘ 6)= 4 P ‘ 1= 4

x ‘ 6= 4

P= 1

x= 6

Ics eotjrhjptc rjprjsjotbncs jo ib grâaehb nji nji jlj x (1,4) (6,4)

Qårtehj nj ib pbrâfcib<

(-f /6b, >bh ‘ f 6 / >b)= (-0 / 6 . (-2) , > .(-2) .(-26) ‘ 06 /   > .(-2) Q= (-0 / -6, >0 ‘ 1> /->) = Q (>, (>, >)

 

sco ibs auohecoj auohecojss quj tejojo tejojo ib vbrebfij vbrebfij eonjpjo eonjpjonejotj nejotj x jo ji Auoheøo Auo heøo jxpc jxpcojoh ojohebi< ebi< Jstbs sco jxpcojotj, js njher sco nj ib acrmb< acrmb< A(P)= Bx Xu jxprjseøo js< sejonc b uob rjbi pcsetevc, b ? 4, y neajrjotj nj 2, b = 2. Hubonc 4 9 b 9 2, jotcohjs ib auoheøo jxpcojohebi js uob auoheøo njhrjhejotj y hubonc b ? 2, js uob auoheøo hrjhejotj. Ji ncmeoec nj ib auoheøo jxpcojohebi js ji hcoluotc nj tcncs ics oümjrcs rjbijs. Ji rbogc js ji hcoluotc nj tcncs ics oümjrcs rjbijs pcsetevcs. Zob auoheøo jxpcojohebi, pcr ic tbotc, pjrmetj biuner b ajoømjocs quj hrjhjo hbnb vjz hco mbycr  rbpenjz. Js uob auoheøo eoyjhtevb Jljmpic< U= 6x P (-6) (-2) (4) (2) (6) U (2/>) (2/6) (2) (6) (>)

Xe x= -6, y= 6-6 = 2/ 66 = 2/> Xe x= -2, y= 6-2 = 2/ 62 = ½ Xe x= 4, y= 64 = 2 Xe x= 2, y= 62 = 6 Xe x= 6, y= 66 = > Xu grbaehb js hrjhejotj.

Auoheøo icgbrítmehb.

Zob auoheøo icgbrítmehb js bqujiib quj gjoårehbmjotj sj jxprjsb hcmc a (x) =   icgbx sejonc b ib fbsj nj jstb auoheøo, quj `b nj sjr pcsetevb y nesteotb nj 2. Ib auoheøo icgbrítmehb js ib eovjrsb nj ib auoheøo jxpcojohebi, nbnc quj< icgb x = f     

         

Ib auoheøo icgbrítmehb js eovjrsb b ib auoheøo jxpcojohebi Oc jxestj ji icgbretmc nj oümjrcs nj fbsj ojgbtevb. Oc jxestj ji icgbretmc icgbretmc nj oümjrcs ojgbtevcs. Ji icgbretmc nji oümjrc 2 js egubi b hjrc. Oc jxestj ji icgbretmc icgbretmc nji oümjrc hrjc. hrjc.

Ib auoheøo icgbrítmehb tejoj ibs seguejotjs prcpejnbnjs<     

         

Ji ncmeoec js ji hcoluotc nj tcncs ics oümjrcs oümjrcs rjbijs pcsetevcs. Ji rbogc js ji hcoluotc nj tcncs ics ics oümjrcs rjbijs. Ib auoheøo js hcoteoub y uoc b uoc. Ji jlj nj ibs U js ib bsíotctb nj ib grâaehb. Ib grbaehb eotjrs eotjrsjhtb jhtb bi jlj nj ibs P jo (2,4). (2,4). Jstc js, ib eotjrhjpheøo jo P js 2.

 

Jljmpic< Vjprjsjotb grâaehbmjotj y ne ibs prcpejnbnjs nj ib auoheøo A huyb jhubheøo js A(x) = icg6(x + 2) ‘ 2.

Tbrb rjprjsjotbr grâaehbmjotj jstb auoheøo premjrc njtjrmeobs su bsíotctb vjrtehbi nj x = ‘ 2. Ib bsíotctb vjrtehbi pbsb pcr ji vbicr quj `bhj hjrc bi brgumjotc, jo jstj hbsc ‘ 2. Hbihuibs su hjrc< Icg6(x + 2) ‘ 2 = 4 (Egubibs ib jhubheøo b hjrc) Icg6(x + 2) = 2 (^rbospcojs ji ‘ 2) P + 2 =62 (Bpiehbs ib njaeoeheøo nj icgbretmc) (ji 2 js ji jxpcojotj nji 6) x = 2 (Njspjlbs x) @biibmcs ji eotjrhjptc nj ib grâaehb hco ji jlj U y = icg6 (4 + 2) ‘ 2 (Xustetuyjs P pcr hjrc jo ib jhubheøo) y = icg62 ‘ 2 (Jajhtübs ics hâihuics eonehbncs) (ji 6 jstâ fblc nji icg) y=4‘2 y=‘2 ^rbzbmcs ib bsíotctb vjrtehbi, ufehbs ics puotcs `biibncs jo ji sestjmb nj hccrnjobnbs y trbzbs ib hurvb.   Bpiehbheøo nj ib auoheøo ieojbi. Bpiehbhecojs Bpiehbheco js nj ib auoheøo ieojbi< Hubonc ji prjhec nj uob hâmbrb actcgrâaehb js nj :4 nøibrjs,

sj carjhjo :4 nj jiibs jo ji mjrhbnc; se ji prjhec js nj 7: nøibrjs `by uob nespcoefeienbn nj 244 hâmbrbs. ³Huâi js ib jhubheøo nj ib cajrtb3 Xciuheøo Ji prcfijmb ocs nb ics puotcs (:4, :4) y (244, 7:) hbihuijmcs ib pjonejotj m= 7: ‘ :4 / 244 ‘ :4 = 6: / :4 = ½ Hcmc sj tejoj ib pjonejotj y puotc (:4,:4) bpiehbmcs ib aørmuib U ‘U2 = m(x- xe) U - :4= 2/6 (x ‘ :4) P ‘ 6y + :4 = 4

 

Bpiehbhec Bpie hbheco o jo ibs auoheco auohecojs js hubnrâteh hubnrâtehbs< bs< Mbouji Gcozbijs js prcpejtbrec nj uo ojgchec nj

tjiåacocs hjiuibrjs. Ji eogrjsc E nj vjotbs nj tjiåacocs hjiuibrjs sj njtjrmeob muitepiehbonc ji oümjrc nj hjiuibrjs pcr ji hcstc (E = o.p) supcogb supcogb quj ji eogrjsc pcr ib vjotb nj O hjiuibrjs hco O ≮ :4 js E(o)= o (:4 ‘ 4,6o) nconj :4 ‘ 4,6o js ji prjhec jo nøibrjs pcr tjiåacocs. b). Njtjrmeobr ji eogrjsc hubonc sj vjonjo 84 hjiuibrjs E(o) = o (:4 ‘ 4,6o) E (84) = 84 (:4 ‘ 4,6. 84) E (84)= 84 (:4 ‘ 1) E (84)= 84 x >> E (84)= 2864. f). Tbrb tjojr uo eogrjsc nj >04 nøibrjs hubotcs hjiuibrjs njfjo vjonjr E(o) = o (:4 ‘ 4,6o) >04 = o (:4 ‘ 4,6o) >04 = :4o ‘ 4,6o6 6o6 ‘ :4o + >04= 4  B= 4,6 F= -:4 H=>04 O= - (-:4) ² ∜(-:4)6 - > (4,6) (>04) / (>04) / 6 . (4,6) O= :4 ² ∜6:44 ‘ 80> / 80> / 4,> O= :4 ² >1 / >1 / 4,> O2= :4 + >1 / >1 / 4,> = 6>4 O6= :4 - >1 / >1 / 4,> = 24 Hcmc ji prcfijmb nehj quj O ≮ :4 jotcohjs ib rjspujstb js o= 24 Tcr ic tbotc sj njfjo vjonjr 24 hjiuibrjs

Bpiehbhecojs Bpiehbheco js jo ibs auohecojs jxpcojohebijs

Xe ji oümjrc nj brtíhuics y abfrehbncs pcr níb nurbotj x níbs njspuås nji eoehec nj uo pjrecnc nj prcnuhheøo, jstâ nbnc pcr  U= 644(2 ‘ j-4.2x) ³Huâotcs brtíhuics sco abfrehbncs nebrebmjotj 24 níbs njspuås nj eoehebnb ib Trcnuhheøo, y quå pcrhjotblj nj ib prcnuhheøo mâxemb prjsjotb3 U = 644 (2 - j-2) U = 644(2 - 4.810) = 644(4.186)

 

U =261.> c bprcxembnbmjotj 261 brtíhuics, brtíhuics, quj sco ji 18.6% nji m mâxemc âxemc nj 644. Bpiehbhecojs Bpiehbheco js jo ibs auohecojs icgbrítmehbs

Zob pjrscob njpcsetb jo hbpetbiezbheøo uob sumb nj $:444 pcr jljmpic, pjscs, jth. Bi >% nj eotjrås. ³Nj huâotc sjrâ su hbpetbi aeobi (preohepbi mâs eotjrjsjs) bi hbfc nj 24 bòcs< (b) se ji eotjrås js pbgbnjrc boubimjotj, y (f) se ji eotjrås js pbgbnjrc tremjstrbimjotj3  B) U=x (2 + e)o U=:.444 (2 +4.4>)24 Icg = icg :444 + 24 icg 2.4>  = 8.1554 + 24(4.4274)  = 8.0154 U = $ 7851.17 F) U = x (2 + e / k) ok >4

U= :44 (2+ 4.4> / >) Icg y = icg :444 + >4 icg 2.42  = 8.1554 + >4 (4.44>8) = 8.0724 U = $ 7>84.44 Nj mbojrb quj ji njpcsetbotj tejoj, njspuås nj '24 bòcs, uob sumb nj $ 7851.17 se pjrhefj eotjrjsjs boubimjotj, y nj $7>84.44 se ib pjrhjpheøo nj eotjrjsjs js hbnb trjs mjsjs.

Trcfijmbs nj bpiehbheøo nj Eogrjsc, hcstc y uteienbnjs<

Ib schejnbnnjjhciøgehb ib uoevjrsenbn hejotíaehb jstâ crgboezbonc boubi nj bnqueseheøo aconcs, jinjhcmínbic, sj hcfrbrb :4 hjotbvcs pcr pjrscobsu pcrhcmpbòb sjrverij uob crnjo nj pbstb. Ics üoehcs gbstcs nj ib schejnbn sco ji gbstc nj ib pbstb, quj sj jstemb jo 2: hjotbvcs pcr rbheøo y 8:4 $ pcr ib rjotb nj ibs eostbibhecojs. b) jshrefb ibs jhubhecojs hcrrjspconejotjs nj hcstc, eogrjsc y uteienbn. f) hubotbs rbhecojs nj pbstb njfj vjonjr ib schejnbn pbrb iijgbr bi jqueiefrec. h) quj uteienbn c pårnenb rjsuitbrb bi vjonjr 2:44 rbhecojs nj pbstb. Xciuheøo< h=hv + ha. E=prjhec * uoenbn Z= eogrjsc ‘ hcstc

 

h= 2:x+8:4 Eogrjsc= :4x Zteienbn= :4x-.2:x+8:4 u= 8:x-8:4=4 u= 8:x=8:4 x=8:4/8:x x=2444 u= 8:(2:44)+8:4 :6:-8:4=27: