DIAPOSITIVA #15 - Final-2021-2

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ZKLRAGOKLPN HÒIOBN RKROÒL 1;

GN@KHAGOKLPN KL ROP[ABONLKR BNLPKVP[AHKR

ZTNITAGA ABA@ÏGOBN @K CNTGABOÒL F[GALÇRPOBA

ZTNITAGA ABA@ÏGOBN @K CNTGABOÒL F[GALÇRPOBA

 

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Zartobopa rkspnl`okl`n kh soiuokltk quozozz=

ftps=//quozozz.bng/a`gol/quoz/310dc:4:`:k::1k20:;3 

ZTNITAGA ABA@ÏGOBN @K CNTGABOÒL F[GALÇRPOBA

   

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….. ….

 Tkbnr`kgns bnlbkptns quk lns skrál ýtohks ¾Q[K KR [L GN@KHN HOLKAH> Rk `obk quk ul gn`khn ks holkah bual`n su irába ks ula hçlka rkba5 y kl bnlskbuklboa klk ha + d nrga= y 7 c(x) 7 gx + d

G[_ OGZNTPALPK

Kl vosa `kbuáhks quk luksrn gn`khn ks `k haa g nrga 7 gx + d, khsouaboòl prolbopahbnlkxuah. ndmkvn skrá - o`klbar snl hns `ans qukholkah rkprksklal  y a d(x) kl `kkrgola`a - Ro ln ukra pnsodhk o`klbar o`klbar a ïsns `k galkra kxphçboa, kxphçboa, klnlbks sk kl`rá kl`rá quk udobar `ns `ans rkhabonla`ns bnl ha souaboòl bnlkxuah (pulns Z1= (x (x1 , y1 ) y Z4= (x (x4 , y4 ) ) para fahhar g a ravïs `k ha òrguha y hukin d.

 

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C[LBOÒL HOLKAH

Bual`n sk bnlnbkl hns pultns Z1= ( x1 , y1 ) y Z4= ( x4 , y4 ) quk pkrtklkbkl a ha irácoba `k ula culboòl holkah, ha pkl`okltk ‗g‗ voklk `a`n pnr ha soiuokltk còrguha=  

[la culboòl holkah ks ula rkiha `k bnrrkspnl`klboa buya kxprksoòl alahçtoba ks=

y7 c(x) 7 gx + d @nl`k= ‘g‗= ks hhaga`a pkl`okltk ‘d‗ ks kh pultn `k bnrtk bnl kh kmk _ ‘x‗ ks ha varoadhk ol`kpkl`okltk ‘y7 c(x)‗ ks ha varoadhk `kpkl`okltk

Bual`n sk bnlnbkl kh pultn Z1= ( x1 , y1 ) y ha pkl`okltk ‗g‗ `k ula rkbta, ïsta voklk `a`a pnr= y-y1 7 g(x ‒ x1)

 

 

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Sosuahozar y alahozar kh vç`kn ‘ Aláhosos Kxpnrtabonlks Bacï Zkrualn‗ fps=//www.ynutudk.bng/watbf>v7ekYLCPZECD6&ad^bfallkh7BOPKZKTF[RA   Bngpartgns= 1)

4)

2)

Kxpr Kxprks ksa a ttu u npo npolo loòl òl sn sndr drk k kh kh vo` vo`kn kn ndskrva`n y gklbonla ha varoadhk ol`kpkl`okltk y ha varoadhk `kpkl`okltk  Muscba tu pultn pultn `k vosta. ¾Bu ¾Buáh áh ks kh bngp bngpnr nrta tagok goklt ltn n `k `k has has varoadhks kl ha irácba= ‘Bngpnrtagokltn gklsuah `k has kxpnrtabonlks‗> Rk p puk uk`k `k gn`k gn`kha harr gat gatkg kgá ába bagk gklt ltk k ul  klògkln `k ha vo`a rkah a travïs `k has  ulbonlks>

ZTNITAGA ABA@ÏGOBN @K CNTGABOÒL F[GALÇRPOBA  

 

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RKROÒL Lº 1; GN@KHAGOKLPN KL ROP[ABONLKR BNLPKVP[AHKR TKR[HPA@N @K AZTKL@O]AMK Cnrguha gn`khns `k bngpnrtagokltn holkah quk rkprksklal souabonlks bnlkxuahks quk pkrgoal olkror k olkrprkar bnlskbuklboas `k ha gosga, fabokl`n usn `k hns bnlbkpns `k prnpnrbonlaho`a` prnpnrbonlaho` a` y ulboòl holkah.

 

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¾Q[K KR [L GN@KHN GAPKGÁPOBN>

- [l gn`khn gatkgátobn bnlstotuyk ula rkprkskltaboò rkprkskltaboòl l n adstrabboòl `k ha rkaho`a` - Ks ul oltkltn `k `ksbrodor ahiula partk `kh gul`n rkah kl tïrgolns gatkgátobns

KMKGZHNR @K GN@KHNR GAPKGÁPOBNR



Kh tagañn `k ula pndhaboòl.  Ha `kgal`a pnr ul prn`ubtn. 

Ha rapo`kz `k ha baç`a `k ul ndmktn. Ha kxpkbtatova `k vo`a `k ula pkrsnla bual`n labk.  Ha varoaboòl `kh árka `k ul tkrrkln `k abukr`n a sus `ogklsonlks. 

 

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¾B[ÁH KR KH KH ZTNZÒROPN @K [L GN@KHN GA GAPKGÁPOBN> PKGÁPOBN> Klkl`kr ul klògkln y fabkr prk`obbonlks bnl rkspkbn ah bngpnragokln uurn. NDRKTSABONLKR= - [l gn`khn gakgábn lulba ks ula rkprksklaboòl nahgklk prkbosa `k ula souaboòl rkah, ks ula o`kahozaboòl. - [l dukl gn`khn gakgábn sogphoba ha rkaho`a`  rkaho`a`  hn suboklk bngn para pkrgor báhbuhns gakgábns, hn suboklkgklk prkbosn bngn para prnvkkr bnlbhusonlks y ksgabonlks vahonsas.

¾Q[Ï POZN @K C[LBONLKR RK Z[K@KL [RAT BNGN GN@KHN GAPKGÁPOBN> Kxoskl `okrklks pns `k ulbonlks quk puk`kl usarsk para gn`khar bnrrkspnl`klboas quk sk ndskrval kl kh gul`n rkah, sokl`n ula `k khhas ha ulboòl holkah.

 

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GN@KHNR HOLKAHKR

• Hhagagns gn`khns holkahks a aqukhhas souabonlks quk `kspuïs `k fadkr

so`n alahoza`as gakgábagklk, sk rkprksklal pnr gk`on `k ula ulboòl holkah. Kl nrns ahiulns basns luksrn bnl ula rkba5 kl basns, a pksar `k gn`khn quk has bnolbo`k varoadhks prkbosagklk quk lns olkrksal ln pkrklkbkl n`as a ha gosga hçlka, ks pnsodhk klbnlrar ula ulboòl holkah quk gkmnr sk aprnxogk a luksrn prndhkga, ayu`ál`nlns a ndklkr olnrgaboòl vahonsa. • Luksrn gn`khn holkah sk puk`k `kkrgolar `k galkra irába n dokl, pnr

gk`on `k ula kbuaboòl.

 

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Bnlcrgagns luksrns bnlnbogoklns sndrk kh kga …

I[ÇA ZTÁBPOBA Lº 1; BARNR @O@ÁBPOBNR

ZTNITAGA ABA@ÏGOBN @K CNTGABOÒL F[GALÇRPOBA

ZTNITAGA ABA@ÏGOBN @K CNTGABOÒL F[GALÇRPOBA      

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BARN @O@ÁBPOBN Lº :1 Kl kh 4:4: sk brkarnl gás `k 42; ::: kgprksas kl kh Zkrý Ah bokrrk `kh añn pasa`n, kh lýgkrn `k bngpañças asbkl`oò a 4,99 gohhnlks. @uralk ha pal`kgoa bkrrarnl pnbn gás `k 0; ::: kgprksas, skñahò OK@KZ. OK@KZ. Kl rkhaboòl a ha irácba, `ksarrnhhar=  

fps=//habagara.pk/kl-kh-4:4:-sk-brkarnl-gas-`k-42;-:::-kgprksas-kl-kh-pkru/ 

A. O`klba has varoadhks y sus bbnrrkspnl`okl nrrkspnl`oklks ks ulo`a`ks quk sskk boal y rkhabo rkhabonlal, nlal, para huki hukin n asoilarhks ula lnaboòl a has gosgas (vahnr hokrah).   D. Musba hhaa rkhaboòl `k `kpkl`klboa kl klrk rk has vvaroadhks aroadhks boa` boa`as as k ol`oba ha varoadhk ool`kpkl`okl l`kpkl`oklkk y varoadhk `kpkl`oklk.

 

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B. Bnl Bnlso` so`kr kral` al`n n qu qukk ha k kl`k l`klbo lboaa `k `khh lý lýgkr gkrn n `k kgprksas kl kh Zkrý ks brkbokltk y holkah holkah,, uhozal`n hns `ans `k 4:12 4:12   y 4:4:  4:4:  `kkrgola ha rkiha `k bnrrkspnl`klboa quk hns rkhabonla ulbonlahgklk y prnykbtk kh lýgkrn `k kgprksas para kh 4:44. @. @k ha gosga gal galkra kra bn bnlso` lso`kral kral`n `n bngn kkxrk xrkgns gns `k ha kl`klboa holkah hns añns 4:16 y 4:4: 4:4:   `kkrgola ha rkiha `k bnrrkspnl`klboa y prnykbk kh lýgkrn `k kgprksas kspkra`ns para kh 4:44. K. ¾Buáh bbrkks rkks q quk uk ks ha g gkmnr kmnr pr prnykb nykbboòl boòl>> ¾Zn ¾Znrr quï>

   

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A. Saroadhks

RNH[BO ÒL Saroadhk

Lntaboòl asoila`a

Lýgkrn `k kgprksas kl kh Zkrý

K

kgprksas

Añns tralsburro`ns a parr `k 4:12



añns

ulo`a`

D.  kh lýgkrn `k kgprksas sk olbrkgklal bnl kh ralsbursn `k hns añns, pnr hn aln sus bnrrkspnl`oklks bal`a`ks `kpkl`král `kh kgpn, hn quk `ará huiar a quk ksas varoadhks sk `klal bngn=  )= Lýgkrn `k kgprksas ( K  )=

varoadhk `kpkl`okl `kpkl`oklk k

 Añns tralsburro`ns tralsburro`ns (  )  )==

varoadhk ol`kpkl`okl ol`kpkl`oklk k

 

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B. Tkiha `k bnrrkspnl`klboa Zara ksadhkbkr ha rkiha `k bnrrkspnl`klboa klrk has varoadhks quk `ksbrodorál ha prnykbboòl `kh lýgkrn `k kgprksas, bnlso`krarkgns ha soiuoklk skbuklboa= Zuksn quk a parr `k ha rkprksklaboòl irába, bnlso`krarkgns quk ha kl`klboa `kh lýgkrn `k kgprksas kl kh Zkrý ks brkboklk brkboklk y holkah, uhozarkgns ha rkiha rkiha `k bnrrkspnl`klboa `k ula ulboòl holkah, `klo`a `klo`a bngn=

 

…(1)

 

…(4)

Ha kbuaboòl (1) kl su nrga `k ulboòl holkah ho lkah skrça kquovahklk a ha kxprksoòl=

• @nl`k= =   K y   =

snl has varoadhks `kpkl`oklk k ol`kpkl`oklk, rkspkbvagklk. rkspkbvagklk.   g =  bnlsalk nlkxn `k ha ssouaboòl ouaboòl prndhkgátba, bnlsalk `klo`a bngn pkl`oklk `k ha rkba (kbuaboòl 1), ha buah kl kh bbnlkxn   sk olkrprka bngn kh olbrkgkln n ha varoaboòl `kh lýgkrn `k kgprksas pnr añn (kbuaboòl 4) .   d =  bnlsal bnlsalkk `klo`a bngn olkrbkpn olkrbkpn bnl kh kmk `k nr`kla`as (kbuaboòl 1), ha buah sk olkrprka olkrprka bngn kh lýgkrn `k  kgprksas kl `kkrgola`n añn (kbuaboòl 4) y quk skrá puln oloboah `k rkkrklboa para rkahozar has ksgabonlks `k hns pròxogns añns, asnboál`nhn bnl kh kquovahklk `kh añn bkrn.

    

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bnrrkspnl`oklks ks a hns añns 4:12 y 4:4: uthozal`n hns `ans bnrrkspnl`okl

K() 7 g + d

Zara `kkrgolar kh vahnr `k has bnlsalks g y d, rkahozarkgns hns soiuoklks báhbuhns= Báhbuhn `k g=

Rkiýl kbuaboòl `k ha rkba, quk para luksra souaboòl prndhkgába skrá=

Tkkgphazagns ‘g‗ kl ha culboòl=

K(t) 7 104 941t + d Báhbuhn `k d= rkkgphazagns kh puln (: 5 1 996 299)  

K() 7 104 941 + d

1 996 299 7 104 941(:) + d d 7 1 996 299 

Añns brnlnhò iobns

Añns tralsburro `ns

P (añns)

t (añns)

K (kgprksas)

4:12

:

1 996 299

4:4:

9

4 999 040

Kgprksas

(: 5 1 996 299) (9 5 4 999 040)

Rkiýl `an `k irábn pn`kgns hkkr quk para 4:12 kh lýgkrn `k kgprksas bnrrkspnl`k a 1 996 299, pnr hn aln=    

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uthozal`n hns `ans bnrrkspnl`oklks bnrrkspnl`oklks a hns añns 4:16 y 4:4:

K() 7 g + d

Zara `kkrgolar kh vahnr `k has bnlsalks g y d, rkahozarkgns hns soiuoklks báhbuhns= Báhbuhn `k g=

Rkiýl kbuaboòl `k ha rkba, quk para luksra souaboòl prndhkgába skrá=

Tkkgphazagns Tkkgphaz agns kh vahnr `k ha pkl`oklk kl ha ulboòl=

K(t) 7 gt + d K(t) 7 1?4 1?;,;t 1?;,;t + d

Báhbuhn `k d= Tkkgphazagns Tkkgphazag ns kh puln (: 5 4 2?2 :22) kl go ulboòl=

Añns brnlnhòiobns

Añns tralsburro`ns

Kgprksas

P (añns)

t (añns)

K (kgprksas)

4:16

:

4 2?2 :22

4:4:

4

4 999 040

K(t) 7 1?4 1?;,; 1?;,;tt + d   

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K. ¾Buáh ks ha gkmnr prnykbboòl> Ha gkmnr prnykbboòl ks ha asnboa`a ah pkron`n `kh 4:16 ah 4:4:   , tn`a vkz quk ah kvahuar kh lýgkrn `k kgprksas a travïs travïs `kh kgpn, sk bnlso`krará bnlso`krará kh bngpnrtagokl bngpnrtagokltn tn n tkl`klboa tkl`klboa rkiuhar `kh `kh  klògkln kgprksaroah kl hns ýhtgns añns , hn quk lns pkrgorá tklkr olnrgaboòl n barabtkrçsbas  pkrlkltks `k ha ha prkskltk rkaho`a`. rkaho`a`.

   

F KSNH[BOÒL @KH ZDO ZKT BÁZOPA KL KH ZKTÝ

BARN @O@ÁBPOBN Lº :4 fttps=//www.bngkrbonkxtkronr.ud.k`u/cp aos/pkru/kvnhubonl.ftg

A) Kl tu parkbkr bògn sk ksgarça kh ZOD pkr bápota para ul `ktkrgola`n añn. Kh ZOD pkr bápota sk pn`rça ksgar skiýl=   

  Ha `aa, fkrragoklas olnrgábas y ksa`çsbas quk `ospnlk ha kl`a` `kh Pfk Ynrh` Dale. gn`khn gakgábn quk `ksbroda ha prnykbboòl `kh ZOD pkr bápoa, klokl`n bngn rkkrklboa ha olnrgaboòl prnpnrbonla`a pnr kh Pfk Ynrh` Dale y asugokl`n quk kh brkbogokln `kh ZOD pkr bápoa a ravïs `k hns añns prnykba ula kl`klboa prnpnrbonlah, prnpnrbonlah, ha buah sk kxprksarça kl pahadras `k ha galkra soiuoklk=

[l

ZOD ZKT BÁZOPA KRPOGA@A A [L @KPKTGOLA@N @KPK TGOLA@N AÑN  

7

OLBTKGKLPN N SATOABOÒL ZTNGK@ON @KH ZOD ZKT BÁZOPA ZNT AÑN

LÝGKTN @K AÑNR PTALRB[TTO@NR A ZATPOT @KH 4::0

+

 

ZOD ZKT BÁZOPA KL KH AÑN 4::0

  

F KSNH[BOÒL @KH ZOD ZKT BÁZOPA KL KH ZKTÝ

D)  D)  O`klba has varoadhks y sus bnrrkspnl`oklks ulo`a`ks quk sk boal y rkhabonlal para ksgar kh ZOD pkr bápoa. Hukin asçilahks ula ul a lnaboòl.

Rnhuboòl

Saroadhk

Lnaboòl asoila`a

[lo`a`

Kh ZOD pkr bápoa

Z

@òharks

Pokgpn



Añns

 

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B) B)   Musba ha rkhaboòl `k `kpkl`klboa klrk has varoadhks boa`as k ol`oba ha varoadhk ol`kpkl`oklk y varoadhk `kpkl`oklk. Rnhuboòl Kh ZOD pkr bápoa sk olbrkgkla bnl kh ralsbursn `k hns añns, pnr hn aln kh ZOD pkr bápoa `kpkl`krá `kh kgpn hn quk `ará huiar quk ksas varoadhks sk `klal bngn= ZOD pkr pkr bá bápo poaa (Z) = varoadhk `kpkl`okltk Pokgpn ()

=

varoadhk ol`kpkl`oklt ol`kpkl`okltk k

   

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@) @kkrgola ha rkiha `k bnrrkspnl`klboa (gn`khn gatkgábn gatkgábn)) quk rkhabonla hns `ans varoadhks para ksgar kh ZOD pkr bápoa para ul kgpn `kkrgola`n. Kh ZOD pkr bápoa `kh Zkrý kl ulboòl `kh tkgpn ksarça `kclo`a pnr=

  7

Buyas bnlstaltks g y d sk bahbuharçal=

* Kl rkhaboòl a hns añns añns brnlnhòiobns= brnlnhòiobns= (4::05 4022,6) y a`kg a`kgás ás (4:105 3;?0,0) Zara kh añn 4::0 quk ks go añn oloboah kh t 7 : Znr hn taltn kh añn 4:10, kh t 7 1: * Ro asnboagns kh añn : y kh añn 1: bngn kquovahklks= (:5 4022,6) y a`kgás (1:5 3;?0,0)

  4 ∔   1 4∔ 1

  

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K) [sk kh gn`khn gakgábn `kkrgola`n para ksgar ha pndhaboòl para kh añn 4:11, 4:12 y 4:41. Bngpara us rksuha`ns bnl hns boa`ns kl ha irába y rkahoza us bngklarons.

Rnhuboòl

Z(t) 7 013,:3 013,:3tt + 4022,6

Zndhaboòl `kh Zkrý ksga`a para=  Añn 4:11,  7 9  Añn 4:12, 7 ?

Z(9) 7 013,:3(9) + 4022,6 7 ; 203,44 `òharks pnr pkrualn Z(?) 7 013.:3 (?) + 4022,6 7 3 196,20 `òharks pnr pkrualn

 Añn 4:41,  7 19 Z(19) 7 013.:3 (19) + 4022,6 7 ? ;:3,64 `òharks pnr pkrualn

 

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Zarbopa rkspnl`okl`n ha soiuokltk abvo`a`=

ftps=//ks.k`ubaphay.bng/rkbursns-k`ubav ftps=//ks.k`ubaphay .bng/rkbursns-k`ubavns/1:20?294-gn`khagokln^holkah^4.fgh ns/1:20?294-gn`khagokln^holkah^4.fgh

   

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@KRATTNHHN @K HA I[ÇA ZTÁBPOBA L½ 1;

ZKLRAGOKLPN KL ABBOÒL

Ba`a kquopn `ksarrnhhará kmkrbobons `k ha Iuça Zrábba 1; , skiýl ha asoilaboòl `kh `nbkltk. Ba`a kquopn prkskltará ul olcnrgk kh buah skrá sustklta`n kl ul phklaron.

 

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