aplicaciones de matemática y cálculo a situaciones reales

7/23/2019 Aplicaciones de matemtica y clculo a situaciones reales

1/21


2/21

Ihon) g}bhk fm 40>4


3/21

]bhqmtuhfnf Pmt}nbn fm Ehmbehnu Npihenfnu *]PE(Pthomtn p}dihenehb7 g}bhk fm 40>4Hoptmuk mb mi Pmt / Pthbvmf hb Pmt}

]bhqmtuhfnf Pmt}nbn fm Ehmbehnu Npihenfnu *]PE(Embvtk fm Hbaktonehb

Lhtb U}nwk) Onthm Ekumvvm" Npihenehkbmu fm onvmovhen x eie}ik n uhv}nehkbmu tmnimu

Ihon7 ]bhqmtuhfnf Pmt}nbn fm Ehmbehnu Npihenfnu *]PE() 40>4HUDB7 84/606>/:5/8

ONVMOVHENU) EIE]IK) PTKDIMONU ONVMOVHEKU) L]NU FM MUV]FHK

=>0"1>>) Ihon 55 *Pmt(Vmia7 5>5/5555|||"}pe"mf}"pmPthomtn mfhehb7 g}bhk fm 40>4Vhtngm7 >000 mgmopintmu

Muvm ihdtk um vmtohb fm hopthoht mb mi omu fm g}bhk fm 40>4) mb iku vniimtmu ltheku fm LthenDhdiku U" N" }dhenfku mb Gt" Oktkekejn >=4 U}t~}hiik / Ihon 56 / Pmt"

Jmejk mi Fmpuhvk Imlni mb in Dhdihkvmen Bnehkbni fmi Pmt b" 40>4/01>0>

Ptkdimonu pntn umt tmu}mivku fm onbmtn hbfhqhf}ni >5

Ptkdimonu }unbfk me}nehkbmu >5

Ptkdimonu pntn umt tmu}mivku omfhnbvm kpmtnehkbmu ekb onvthemu 55

Ptkdimonu pntn umt tmu}mivku pkt uhuvmonu fm me}nehkbmu ihbmnimu *UMI( 6=

Ptkdimonu pntn umt tmu}mivku }unbfk in me}nehb fm in tmevn =1

Ptkdimonu pntn umt tmu}mivku }unbfk ptkltnonehb ihbmni 15

Ptkdimonu pntn umt tmu}mivku }unbfk in vmktn fm a}behkbmu fm qnthndim tmni 5>

Ptkdimonu }unbfk vthlkbkomvtn >6=

Ptkdimonu pntn umt tmu}mivku mb lt}pk >6" Muvtnvmlhnu pntn tme}pmtnt in pntvhehpnehb mb mi omtenfk >6==

Ptkdimon 6" Mi omtenfk hgnt iku ptmehku fmi lnu fmi ikvm =< >=8

Ptkdimon =" Fkm T}b Pmt >15

Ptkdimon 1" Nbihuhu fm }b ptkxmevk fm hbqmtuhb pdihen mb mi umevkt v}thuok7

p}muvn mb qnikt v}tuvhek fm iku tmuvku nt~}mkilheku fmi fhuvthvk fm Qhienuj}nob >11


6/21

Ptkdimon :5

Ptkdimon 8" Upn mekilhek fm Ekien Ikflm >:0" Ptkxmevk Fmmp Hopnev >8>

Nbmzk 5" Qnikt nev}ni bmvk *QNB( 4>4

Nbmzk 6" U}on fm bhqmimu fm ptmuhb neuvhen mb fD 4>5

Nbmzk =" Nemtebfkbku n in tmnihfnf 4>6

Dhdihkltnan 4>


7/21

bfhem pkt npihenehkbmu nmupmehnihfnfmu

Ptkdimonunpihenfkuninjkvmimtnxv}thuok

>) 4) 5) 6) =) 1) ) 44) 45) 46) 4=) 4:) 50) 55) 56) 5=) 51) 5) =4) =5) =6) =5) >6) >=) >1) 48) 6:) 68) =1)


8/21

]bhqmtuhfnf Pmt}nbn fm Ehmbehnu Npihenfnu 8

Ptiklk

Npihenehkbmu fm onvmovhen x eie}ik n uhv}nehkbmu tmnimu mu }b ihdtk ~}m um mbaken

mb tminehkbnt ekbempvku onvmovheku ekb enu}uvhen dnunfn mb uhv}nehkbmu tmnimu

x ekvhfhnbnu" Nu) muvn kdtn vhmbm ekok hb anehihvnt mi nptmbfhwngm fm iku muv}fhnbvmu"

Pntn ikltnt muvm ptkpuhvk) Ekumvvm pkbm mb ptevhen u} ltnb anehihfnf pntn pinuont

onvmovhenombvm ik ~}m ke}ttm n b}muvtk nitmfmfkt"

In ltnb fhhe}ivnf fm iku ni}obku mb mi nptmbfhwngm fm in onvmovhen mb mi odhvk

}bhqmtuhvnthk jn uhfk uhmoptm mi bhqmi fm nduvtneehb ~}m o}ejku vmonu tm~}hmtmb"

Iku eiuheku ptkdimonu ~}m mi ni}obk bk iimln n ekoptmbfmt x ~}m tmu}miqm ukdtm in

dnum fm }b ovkfk k tmemvn ptmmuvndimehfn ik bhek ~}m ikltnb mu ~}m mi muv}fhnbvm

omokthem iku pnuku ~}m fmdm uml}ht uhb iimlnt n mbvmbfmt tmniombvm mb ~} uhv}nehkbmu

fmdm }unt vni k e}ni ekbempvk onvmovhek"

Mb mum umbvhfk) in ptkamuktn Onthm Ekumvvm Lhtb jn fmokuvtnfk n ik intlk fm u}

vtnxmevkthn ekok fkembvm ~}m bk ukik muv hbvmtmunfn mb mbumnt onvmovhen) uhbk ~}m

um jn ptmke}pnfk) ukdtm vkfk) pkt mbekbvtnt in omgkt akton pntn ~}m iku muv}fhnbvmu

in nptmbfnb" Mb u} mzpmthmbehn) Ekumvvm jn fmue}dhmtvk ~}m mi muv}fhnbvm um hbvmtmun

mb in onvmovhen e}nbfk qhmbm nekopnnfn fm }b mbvktbk ekbkehfk fm in tmnihfnf

~}m ik tkfmn" Mi mbumnt vmonu onvmovheku n vtnqu fm mgmtehehku x mgmopiku ihlnfku n

jmejku tmnimu x) ukdtm vkfk) ekvhfhnbku ptko}mqm }b nemtenohmbvk mbvtm mi ni}obk x

in onvmovhen x) pkt ekbuhl}hmbvm) }b omgkt nptmbfhwngm"

Ni tmupmevk) mi ptmumbvm ihdtk mu }bn pt}mdn fm muvm mua}mtwk fm ihlnt iku

nptmbfhwngmu fm vmonu onvmovheku ekb jmejku tmnimu ~}m ke}ttmb mb b}muvtk ptkphkmbvktbk" Pntn Ekumvvm mu a}mbvm hbnlkvndim fm hbuphtnehb vkfk u}emuk ~}m ke}ttm mb

b}muvtn tmnihfnf x ~}m iku muv}fhnbvmu p}mfnb tminehkbnt ekb nilb vmon fm u}u e}tuku

fm onvmovhen" Pkt miik) mb iku :8 ptkdimonu pntn tmuki}ehb hbfhqhf}ni ~}m ekbvhmbm)

vkon jmejku umbehiiku mzvtnfku fm b}muvtn tmnihfnf pntn hbfhentimu n iku muv}fhnbvmu


9/21

>0

Onthm Ekumvvm Lhtb U}nwk y Npihenehkbmu fm onvmovhen x eie}ik n uhv}nehkbmu tmnimu

]bhqmtuhfnf Pmt}nbn fm Ehmbehnu Npihenfnu

~}m mu nii fkbfm um mopimn in onvmovhen" Fmufm }b pktembvngm) pnunbfk pkt mi }uk fm

me}nehkbmu jnuvn in npihenehb fm inu e}tqnu fm inu a}behkbmu ukb vmonu ihlnfku pkt iku

ptkdimonu ekb u}emuku ~}m ke}ttmb mb b}muvtnu qhfnu ekvhfhnbnu"

Npihenehkbmu fm onvmovhen x eie}ik n uhv}nehkbmu tmnimu umt }b ihdtk ~}mfm uml}tk ekbvthd}ht ni omgkt mbvmbfhohmbvk fm iku ekbempvku fm in onvmovhen

x npktvnt o}ejk mb iku tmu}ivnfku kdvmbhfku pkt iku muv}fhnbvmu mb u}u e}tuku fm

onvmovhen x eie}ik"

G}nb I}hu Amtbnbfk Ukvmik Tnaak

Fhtmevkt fmi tmn fm Ehmbehnu



10/21

]bhqmtuhfnf Pmt}nbn fm Ehmbehnu Npihenfnu >>

Hbvtkf}eehb

Muvm ihdtk ekbvhmbm ptkdimonu onvmovheku dnunfku mb uhv}nehkbmu tmnimu)

ekx}bv}tnimu k ~}m jn qhqhfk mi Pmt" Mb mum umbvhfk) hbei}xm enuku u}tlhfku fm

ekbihevku ukehknodhmbvnimu3 enopnnu ~}m ptkokehkbnb mi v}thuok ekok E}uek

pkbm) Umiqn pkbm) mvevmtn3 ptkxmevku fm mzpiktnehb x mzpikvnehb fm xnehohmbvku

fmi lnu ekok iku fmi ekbuktehk Enohumn k enuku fm ekbvnohbnehb nodhmbvni fm in

eh}fnf In Ktkxn3 mbvtm kvtku"

Muv fhthlhfk pthbehpniombvm n iku ni}obku ~}m muv}fhnb iku e}tuku fm ptm/Eie}ik"

N~} mbekbvtntb }bn umthm fm ptkdimonu tmu}mivku pnuk n pnuk ~}m jntb ou umbehiin

u} ekoptmbuhb" Mi kdgmvhqk hbni mu hbembvhqnt n iku ni}obku n nbnihwnt ptkdimonu

k uhv}nehkbmu tmnimu x tmukiqmtiku }unbfk jmttnohmbvnu onvmovhenu"

In okvhqnehb pntn muethdht Npihenehkbmu fm onvmovhen x eie}ik n uhv}nehkbmu

tmnimu a}m in anivn fm omvkfkikln x nev}nihwnehb fm iku enuku pinbvmnfku ~}m um

ptmumbvnb mb iku ihdtku bnehkbnimu" Nuhohuok) mu hopktvnbvm ombehkbnt ~}m iku ihdtku

fm mfhvkthnimu mzvtnbgmtnu ~}m um }vhihwnb mb b}muvtk pnu ekbvhmbmb npihenehkbmu

onvmovhenu fmuekbvmzv}nihwnfnu fm b}muvtn tmnihfnf" Pkt ik ombehkbnfk ibmnu

nbvmthktmu muvm ihdtk p}mfm ekbuhfmtntum ekok bhek mb mi Pmt x }bk fm iku pkeku mb

Invhbknothen fmunttkiinfk n pntvht fm ptkdimovhenu tmnimu"

Enfn ptkdimon bneh fm }b enuk tmni) i}mlk um ptkemfh n in hbqmuvhlnehb fm in

hbaktonehb mzhuvmbvm mb a}mbvmu hndimu x) pkt ivhok) n in etmnehb fmi ptkdimon" Enfn

ptkdimon mb mi ihdtk um ptmumbvn ekb }bn dtmqm fmuethpehb fmi u}emuk) n ekbvhb}nehb

um mzpkbmb iku fnvku b}otheku) ~}m p}mfmb muvnt mb ltheku k onvthemu) x hbniombvminu ptml}bvnu" In uki}ehb um mbe}mbvtn n ekbvhb}nehb fm muvnu"

Muvm ihdtk muv fhqhfhfk mb fku pntvmu7 in pthomtn ekbvhmbm ptkdimonu ektvku

~}m u}mimb umt fmunttkiinfku mb }bn einum k mb mqni}nehkbmu) x in uml}bfn pntvm)

ptkdimonu pntn umt tmu}mivku mb lt}pku) ~}m u}pkbmb }b onxkt bhqmi fm fhhe}ivnf"

Vkfku iku ptkdimonu muvb ktfmbnfku pkt vmonu7 fmufm me}nehkbmu) pnunbfk pkt

onvthemu x a}behkbmu fm qnthndim tmni) jnuvn eie}ik hbvmltni" Nfmou) mb in pntvm hbni

um mbe}mbvtnb iku ptkdimonu nlt}pnfku pkt mupmehnihfnf fm enfn enttmtn"


11/21

>4



Ahbniombvm) ~}huhmtn mzptmunt oh ptka}bfk nltnfmehohmbvk n iku ohmodtku

fmi Akbfk Mfhvkthni fm in ]bhqmtuhfnf Pmt}nbn fm Ehmbehnu Npihenfnu) ~}hmbmu jnb

vtndngnfk o}ejk vhmopk tmqhunbfk mi vmzvk x ltheku ekb }b e}hfnfk omvhe}ikuk)

n I}hu Jkxku x n iku ptkamuktmu Kuent Jhfnilk) Pn}i Iin~}m) Amtbnbfk Ukvmik) GkuE}mqnu x Mbth~}m Phmxtk pkt u}u mt}fhvku ekbumgku"


12/21

]bhqmtuhfnf Pmt}nbn fm Ehmbehnu Npihenfnu >5

P

Ptkdimon>"HbltmukfmnmtkibmnupktinenopnnE}uekpkbm

Pkt in enopnn E}uek pkbm) INB Pmt qmbfh >5: 056 nuhmbvku n }bn vnthan fm & 68)

vnthan ekb in e}ni vthpihentkb mi hbltmuk muvhonfk" Ohmbvtnu ~}m UvntPmt qmbfh 4 0>0

nuhmbvku n in vnthan fm &


13/21

>6



n" Fmvmtohbm mi hbltmuk hbhehni muvhonfk pkt INB Pmt

d" Fmvmtohbm mi hbltmuk fm fhqhunu lmbmtnfku pkt enfn nmtkibmn kdvmbhfk pkt iku

v}thuvnu mzvtnbgmtku f}tnbvm in enopnn"

Uki}ehb7

n" Mi hbltmuk muvhonfk pkt INB Pmt a}m * ( 4 4=6 ===)55&68>5: 0565

>? "

d" Mi hbltmuk fm fhqhunu lmbmtnfk pkt enfn nmtkibmn um enie}in fm in uhl}hmbvm onbmtn7

INB Pmt7 * ( 6 5: 056>> >=5&0"

V}thuvnu fm iku ehbek ekbvhbmbvmu

Uh dhmb jnx }bn ontenfn pntvhehpnehb fm v}thuvnu bktvmnomthenbku x m}tkpmku)

mzhuvm }b etmehmbvm hbvmtu fm pntvm fm v}thuvnu invhbknomthenbku) ~}m pkt u} emtenbn

fhupkbmb fm onxktmu anehihfnfmu pntn qhngnt m) hbei}uk) tmpmvht in qhuhvn"

> Eat" OHBEMV]T 40>>"


14/21


Ptkdimonu pntn umt tmu}mivku fm onbmtn hbfhqhf}ni

>=

Ntthdkfmv}thuvnuumlbtmlhbfmptkemfmbehn

Nothen fmi U}t

Nothen fmi Bktvm

Nothen Embvtni

Kemnbn

athen

Bk mupmehhenfk

M}tkpn

Nuhn

=4)>6 -

0)>< -

0)04 -

>)=5 -

5)6= -

>:):6 -

44)>> -

>)>"

Pkuhehkbnohmbvk mb mi umlombvk Ptmoh}o

Mi pmthi fmi v}thuvn mzvtnbgmtk ekttmupkbfm) mb u} onxktn) n pmtukbnu ~}m iimlnb n

qnenehkbnt ni Pmt"

OkvhqkufmiqhngmniPmt

0 -

Kvtku

Ekbamtmbehnu k ekbltmuku

Uni}f

Qhuhvn n anohihntmu x nohlku

Bmlkehku

Qnenehkbmu

>0 - 40 - 50 - 60 - =0 - -

>= -

>5 -

5 -

5 -

= -

10 -

A}mbvm7 OHBEMV]T 40>>"


15/21

>1



OkfnihfnfmufmqhngmniPmt

Qhngn pkt e}mbvn ptkphn

Ekbvtnvn nlmbehn fm qhngmu

mb u} pnu

Ekbvtnvn nlmbehn fm qhngmu

mb mi Pmt

1: -

50 -

4 -

A}mbvm7 OHBEMV]T 40>>"

Fmvmtohbm mi bomtk fm v}thuvnu ptkqmbhmbvmu fm M}tkpn ~}m qhuhvntkb mi Pmt

mb mi nk 4008 pkt okvhqku fm qnenehkbmu x ~}m qhngntkb pkt e}mbvn ptkphn"

Uki}ehb

Mi bomtk fm v}thuvnu pmfhfk mu7

>>>>161:)01>)06::>)0>0>)41

?

Ptkdimon5"Ptkxmevkfm}bnidmtl}mv}tuvhekpntnpmtukbnufminvmtemtnmfnf

]dhenehb

Ikenihfnf fm Unbvk Vkou) fhuvthvk fm Unb G}nb Dn}vhuvn) ptkqhbehn fm Onxbnu)

fmpntvnombvk fm Iktmvk"

Fmuethpehb

PtkoPmt hbfhen ~}m Muvnfku ]bhfku lnuvn > 600 ohiikbmu fm fintmu mb v}thuok)

kthmbvnfku n nukehnehkbmu fm g}dhinfku" Pkt miik) mi ptmumbvm muv}fhk ptmvmbfm


16/21



>>"


17/21

>:



Ptkdimon6"Ptkxmevkehte}hvkv}tuvhekH~}hvku/Nonwkbnu/Bn}vn

]dhenehb

Inu wkbnu thdmtmnu hbflmbnu fm Vnoujhxne}) enumtk fm xnenpnbn) eh}fnf fm Bn}vn x inwkbn fm hbi}mbehn fm in enttmvmtn H~}hvku / Bn}vn) }dhenfku mb mi fmpntvnombvk fm Iktmvk"

Fmuethpehb

In nonwkbn) pkt u} ekbhl}tnehb lmklthen x th~}mwn bnv}tni) katmem }bn qnthmfnf

fm ptkf}evku v}tuvheku pntn unvhuanemt n iku qhuhvnbvmu aktbmku ~}m iiml}mb n u}u

vhmttnu" Mi ptkpuhvk fm muvm ptkxmevk mu in etmnehb fm ekbfhehkbmu bmemunthnu pntn

omgktnt in kamtvn v}tuvhen mb in tmlhb Iktmvk) ukdtm in dnum fm u}u tme}tuku v}tuvheku"

Pkt muvn tnwb um jn ekbuhfmtnfk hopimombvnt }b ltnb ehte}hvk v}tuvhek fm ne}mtfk

ni Pinb Onmuvtk5"

Qnthndimu mekbohenu muvhonfnu

Ekuvk hgk ombu}ni7 & = =00

Ekuvku fm kpmtnehb x onbvmbhohmbvk *ekuvku qnthndimu(7 4 fintmu+v}thuvn

Ptmehk fm mbvtnfn ni ehte}hvk7 5 fintmu+v}thuvn

n" Uh ~ tmptmumbvn ni bomtk fm v}thuvnu ~}m qhuhvntkb mi ehte}hvk ni omu) fmvmtohbm inu

me}nehkbmu ekuvk vkvni) hbltmuk x }vhihfnf fmi ptkxmevk) mb vtohbku fm ~"

d" Fmvmtohbm mi QOP" Muethdn u}u }bhfnfmu"

e" Fmvmtohbm in }vhihfnf ombu}ni fm ptkxmevk uh um tmlhuvt }b hbltmuk fm > =00

v}thuvnu mb }b omu"

Uki}ehb

n" Inu me}nehkbmu ukb7

E? = =00 # 4~

H ? 5~

]? ~ = =00

5 Eat" Modngnfn fm iku Muvnfku ]bhfku fm Bktvmnothen 40>>"


18/21



>8

d" Mi QOP um enie}in7

]? 0 ? ~ = =003 QOP ? = =00 v}thuvnu ni omu

e" In }vhihfnf pkt mi hbltmuk fm > =00 v}thuvnu ni omu mu7

]? > =00 = =00 ? 6 000 fintmu *ptfhfn(

Ptkdimon="Ptkxmevkekopimgkv}tuvheke}iv}tnUhpb

]dhenehb

Fmpntvnombvk fm Inodnxm~}m) ptkqhbehn fm Ejheinxk) fhuvthvk fm Uhpb" Mi ekopimgk

v}tuvhek umtn mfhhenfk mb }b vmttmbk fm in ekkpmtnvhqn Pkonien"

Fmuethpehb

Mi ekopimgk v}tuvhek katmemt u}u umtqhehku) vnbvk n v}thuvnu bnehkbnimu ekok

mzvtnbgmtku) f}tnbvm vkfk mi nk) dthbfnbfk }b pn~}mvm fm umtqhehku ~}m um

ekopimombvmb mbvtm u" Mi ptkxmevk jkvmimtk vmbft }bn envmlktn fm 5 muvtmiinu)

ekb =4 jndhvnehkbmu x 54 d}blnik|u) tmn pntn tmuvn}tnbvm x dnt) unin fm tm}bhkbmu)

phuehbn) enbejnu fmpktvhqnu) uninu fm g}mlku) fhuekvmenu) nbhvmnvtk) mvevmtn6"


Ekuvk hgk ombu}ni7 & >0 000

Ekuvku fm kpmtnehb x onbvmbhohmbvk *ekuvku qnthndimu(7 =0 fintmu+v}thuvn

Ptmehk fmi pn~}mvm ptkokehkbni *5 fnu fm jkupmfngm(7 >00 fintmu+v}thuvn

n" Uh ~ tmptmumbvn ni bomtk fm pn~}mvmu ptkokehkbnimu ni omu) fmvmtohbm inu

me}nehkbmu ekuvk vkvni) hbltmuk x }vhihfnf fmi ptkxmevk) mb vtohbku fm ~"

d" Fmvmtohbm mi QOP ni omu" Muethdn u}u }bhfnfmu"

e" Fmvmtohbm in }vhihfnf fmi ptkxmevk uh um tmlhuvt in qmbvn fm =00 pn~}mvmu

ptkokehkbnimu ni omu"

6 Eat" Modngnfn fm iku Muvnfku ]bhfku fm Bktvmnothen 40>>"


19/21

40



Uki}ehb


E? >0 000 # =0~

H ? >00~

]? =0~ >0 000


]? 0 ? =0~ >0 0003 QOP ? 400 pn~}mvmu v}tuvheku ni omu

e" In }vhihfnf pkt mi hbltmuk fm =00 pn~}mvmu v}tuvheku ni omu mu7

]? =0*=00( >0 000 ? >= 000 fintmu ombu}nimu

Ptkdimon1"Ptkxmevkekopimgkv}tuvhektmetmnehkbniDnkufmiHben

]dhenehb

Fmpntvnombvk fm Engnonten) ptkqhbehn fm Engnonten) fhuvthvk fm Dnku fmi Hben"

Fmuethpehb

Mi ekopimgk v}tuvhek nev}niombvm e}mbvn ekb : d}blnik|u x vhmbm }bn envmlktn

nuhlbnfn fm 4 muvtmiinu" Mu bmemunthn in hopimombvnehb fm hbatnmuvt}ev}tn ~}m

pkuhdhihvm mi kvktlnohmbvk fm nfme}nfku umtqhehku n iku v}thuvnu3 pntn miik um

pinbhhen in ekbuvt}eehb fm }b pndmiib fm d}blnik|u) ekb u}u tmupmevhqnu tmnu fm

tmetmnehb" Nuhohuok) um ptkxmevn in envmlktn fm 5 muvtmiinu="


Ekuvk hgk ombu}ni7 & 5 =00

Ekuvku fm kpmtnehb x onbvmbhohmbvk *ekuvku qnthndimu(7 40 fintmu+v}thuvn

Ptmehk fmi pn~}mvm v}tuvhek7 =0 fintmu+v}thuvn

= Eat" Modngnfn fm iku Muvnfku ]bhfku fm Bktvmnothen 40>>"


20/21



4>

n" Uh ~ tmptmumbvn mi bomtk fm pn~}mvmu v}tuvheku ~}m um ekbvtnvntkb ni omu) fmvmtohbm

inu me}nehkbmu ekuvk vkvni) hbltmuk x }vhihfnf fmi ptkxmevk) mb vtohbku fm ~"

d" Fmvmtohbm mi QOP ni omu" Muethdn u}u }bhfnfmu"

e" Fmvmtohbm in }vhihfnf fmi ptkxmevk uh um tmlhuvt in qmbvn fm 500 pn~}mvmu v}tuvhekuni omu"

Uki}ehb


E? 5 =00 # 40~

H? =0~

]? 50~ 5 =00


]? 0 ? 50~ 5 =003 QOP ? >>< pn~}mvmu v}tuvheku

e" In }vhihfnf pkt in qmbvn fm 500 pn~}mvmu v}tuvheku mu7

]? 50*500( 5 =00 ? = =00 fintmu

Ptkdimon10 fintmu pkt v}thuvn"

n" Uh ~ tmptmumbvn mi bomtk fm v}thuvnu ~}m vkontkb fhejk vk}t) fmvmtohbm in me}nehb

fmi hbltmuk"

1 Eat" ]ivhonjktn"eko 40>>"


21/21

Uh fmumn ekbvhb}nt imxmbfk) p}mfm nf~}htht mi ihdtk mb aktonvk auhek n

vtnqu fm b}muvtn vhmbfn qhtv}ni"
http://www3.upc.edu.pe/0/modulos/TIE/TIE_DetallarProducto.aspx?CAT=859&PRO=860http://www3.upc.edu.pe/0/modulos/TIE/TIE_DetallarProducto.aspx?CAT=859&PRO=860http://www3.upc.edu.pe/0/modulos/TIE/TIE_DetallarProducto.aspx?CAT=859&PRO=860

aplicaciones de matemática y cálculo a situaciones reales

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