Practica #10

 

]ZGOX@OG # 7= CG_K_. ―ZKGOO@ßA HK [A NKXGE OFA BOe.„

NGZ@G ]G[EG GZGCFA ][KZXG_ GAG DGZ@AG OGEHKZ@A _@NGAOG GE[NAG_

@AFOKAO@F NF_OFXK ]ZFIK_FZ 

W[@N@OG @ACKA@KZ@G @AH[_XZ@GE [A@YKZ_@HGH HK OFZHFJG

 

7. @AXZFH[OO@FA Ka ke s`cu`katk `aifrnk hk egjfrgtfr`f kstgrknfs prkskatgahf eg rkego`ÿa quk kx`stk ka uag rkgoo`ÿa quk sk prfhuok katrk ua nktge y ke ço`hf BOe ka hfahk tkahrknfs ka oukatg eg ofaofrhgao`g quk bgy katrk eg prks`ÿa, eg hkas`hgh y ke vfeunka, ofa ke i`a hk hktkrn`agr egs ouge`hghks, ogrgotkrîst`ogs y h`st`atgs prks`faks quk sk prfhuoka ougahf sk ks ogpgz hk kaofatrgr ke vfeunka quk foupg uag nfe hk b`hrfckaf cgskfsf g ofah`o`faks afrngeks. G ofat`augo`ÿa, sk hksgrrfeegrç ua oukst`fagr`f gokrog hk ef quk ka eg gatkr`fr rkgoo`ÿa sk  prfhuok y sk pukhk hkhuo`r, ofnf< eg prks`ÿa b`hrfstçt`og, prks`ÿa hke vgpfr hk gcug,  prks`ÿa hke b`hrÿckaf cgskfsf, rkgot`vf r kgot`vf eîn`tk y ftrfs, fjskrvgahf gsî eg `npfrtgao`g hk kstg rkgoo`ÿa.

:. I[AHGNKAXF_ XKFZ@OF_ Hk eg rkgoo`ÿa hk Ngcaks`f nktçe`of ofa ço`hf oefrbîhr`of ka sfeuo`ÿa goufsg sk fjt`kak b`hrÿckaf cgskfsf, gsî ]7 + ]:+ … + ]a

Ke b`hrfckaf rkofc`hf ka eg prfjktg kstgrç sgturghf ofa vgpfr hk gcug, hk tge ngakrg quk eg  prks`ÿa hke cgs ofrrkspfahkrç ofrrkspfah krç g eg sung hk eg prks`ÿa pgro`ge hke b`hrfckaf y eg prks`ÿa quk klkrok ke vgpfr hk gcug.  ]cgs> ] B  + ] Ygp ( gcug) :

Kstg üet`ng pukhk ofafokrsk ofasuetgahf uag tgjeg quk prfpfro`fak eg prks`ÿa hk vgpfr hke gcug ofa eg tknpkrgturg. Ghknçs, kx`stk uag ofeunag hk eîqu`hf quk klkrok uag prks`ÿa hkafn`aghg prks`ÿa b`hrfstçt`og (]b), quk sk fjt`kak hk eg s`cu`katk kxprks`ÿa<  ]

 E Ó

>

b

 χ B  = :

 χ Bc

 

Hfahk< E > Geturg ka nn hk eg ofeunag hk eîqu`hf. χB:F > Hkas`hgh hke gcug. χBc> Hkas`hgh hke nkrour`f. Ge tkrn`agr eg rkgoo`ÿa sk fjt`kak klkro`hg hkatrf hk eg prfjktg, gsî< ua kqu`e`jr`f katrk eg prks`ÿa gtnfsiàr`og y eg prks`ÿa  ]gtn> ] cgs + ]b

_k sgjk quk<  ]cgs> ] B  + ] Ygp ( gcug) :

Bgo`kahf usf hk egs kougo`faks gatkr`frks sk hkhuok<  ] B  > ]gtn√ ]Ygp ( gcug )√ ]b :

]grg ogeouegr eg ogat`hgh hk b`hrfckaf prfhuo`hf ka eg rkgoo`ÿa (nfeks hk b`hrfckaf) sk knpekg eg kougo`ÿa hk kstghf hk efs cgsks `hkgeks.  ] B  Y > aZX  :

Hfahk< Y > Ks ke vfeunka foupghf pfr eg nkzoeg hk cgsks. a >Nfeks hk b`hrÿckaf cgskfsf. Z > Ofastgatk hk efs cgsks `hkgeks, =.=2: gtn x E / nfe x D. X > Xknpkrgturg dkev`a > X ½O + :3?. ?. NGXKZ@GEK_. _fpfrtk ua`vkrsge. Xkrnÿnktrf. 7 ]rfjktg crghughg hk ;= nE. 7 Jkgdkr hk 7=== nE 7 ]`pktg hk 7= nE 7]`azg pgrg jurktg. 7 B`ef (X.K) 7 Ofrobf f `ofpfr (X.K) Zkceg crghughg (X.K). BOe 9 I O`atg hk ngcaks`f. @ah`oghfr ―agrgalg hk nkt`ef„

 

8. ]ZFOKH@N@KAXF. 

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]ksgr ua pkhgo`tf hk ngcaks`f (7 on gprfx`nghgnkatk). Çtkef g ua pkhgzf hk b`ef  luatf g ua pkhgzf hk ofrobf. Gh`o`fak 7= nE hk BOe 9 I g eg prfjktg y tkrn`ak hk eekagr ofnpektgnkatk ofa gcug. Eekak egs ?/8 pgrtks hke jkgdkr ofa gcug. @av`krtg rçp`hgnkatk eg prfjktg y ofeÿqukeg hkatrf hke gcug. Af hkjk katrgr g`rk g eg  prfjktg. _ulktk eg prfjktg ofa uag p`azg ge sfpfrtk. sfpfr tk.

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@atrfhuzog ke ofrobf ofa ke Nc hkatrf hk eg prfjktg. _k pukhk ofefogr pr`nkrf ke ofrobf gatks hk `avkrt`r eg prfjktg.

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Fjtkacg eg tknpkrgturg ge tkrn`agr eg rkgoo`ÿa. gh`o`fak ge jkgdkr : cftgs hk agrgalg hk nkt`ef.

;. O[K_X@FAGZ@F. ;.7. Hktkrn`agr eg prks`ÿa b`hrfstçt`og ka nn Bc. ]grg hktkrn`agr eg prks`ÿa b`hrfstçt`og hke B : pfhknfs bgokref ofa eg kougo`ÿa s`cu`katk< ]b (]rks`ÿa b`hrfstçt`og) > E (geturg hke eîqu`hf (nn)) * h B :F / h Bc. Hfahk ka ke ogsf gatkr`fr< E > 77: nn. h B:F > 7 c / nE h Bc > 7?.9 c / nE Katfaoks< ]b > 77:nn * (7c / ne / 7?.9 c/ ne) > 2.:?;: nnBc. Eg prks`ÿa b`hrfstçt`og hke B : ka nnBc ks hk 2.:?;: nnBc. ;.:. Hktkrn`agr eg prks`ÿa hk vgpfr hk gcug g eg tknpkrgturg hk eg rkgoo`ÿa. ]grg hktkrn`agr eg prks`ÿa hke vgpfr hk gcug y eg prks`ÿa hke b`hrfckaf cgskfsf tkaknfs ka oukatg egs s`cu`katks kougo`faks< ]gtn > ]cgs + ]b ] B: (cgs) > ]gtn + ] B :F (vgpfr) ] B: (cgs) * Y (cgs) > aB : (cgs) * Z * X Gbfrg tkaknfs quk< ]gtn > 7 gtn. ]b > 2.:?;: nnBc. > 7=.2?; * 7= -? gtn. X > ?=¼O > ?=?.7; D. Y (cgs) > 77nE > =.=77E. aB :  (cgs) > =.8:3 * 7= -? nfe. Z > =.=2: gtn * E / nfe * D.

 

]r`nkrf bgeegnfs eg prks`ÿa hke cgs< ]cgs > ]gtn ‟ ]b ]cgs > 7gtn ‟ 7=.2?; * 7= -? gtn ]cgs > =.020 gtn. Gbfrg bgeegnfs eg prks`ÿa hke b`hrfckaf cgskfsf< ] B: (cgs) > (aB: (cgs) *-?Z * X) / Y (cgs) ] B: (cgs) > (=.8:3 * 7= nfe * =.=2: gtn * E / nfe * D * ?=?.7; D) / =.=77E ] B: (cgs) > =.098gtn. Xka`kahf eg prks`ÿa hke b`hrfckaf cgskfsf pfhrîgnfs bgeegr eg prks`ÿa hke vgpfr hke gcug< ]cgs > ] B: (cgs) + ] B:F (vgpfr) ] B:F (vgpfr) > ]cgs ‟ ] B : (cgs) ] B:F (vgpfr) > =.020 gtn ‟ =.098 gtn ] B:F (vgpfr) > =.=:; gtn

Eukcf hk bgjkr rkge`zghf efs oçeouefs gatkr`frks pfhknfs aftgr quk eg prks`ÿa hke vgpfr hke gcug g eg tknpkrgturg hk eg rkgoo`ÿa iuk hk =.=:; gtn. ;.?. Ogeouegr eg prks`ÿa hke b`hrÿckaf cgskfsf ka nn Bc. Hk efs oçeouefs gatkr`frks pfhknfs aftgr quk eg prks`ÿa hke b`hrfckaf cgskfsf ks hk =.098gtn, pkrf ofnf afs ef p`hka ka nnBc skrîg eg s`cu`katk<

]rks`ÿa hke b`hrfckaf cgskfsf ka nnBc  > =.098gtn * (39=nnBc / 7gtn) ]rks`ÿa hke b`hrfckaf cgskfsf ka nnBc > 3?:.98 nnBc. Eukcf tkaknfs quk eg prks`ÿa hke b`hrfckaf cgskfsf ks hk 3?:.98 nnBc. ;.8. Ogeouegr egs Nfeks hk ngcaks`f cgstghf. :BOe + Nc ↚ NcOe : + B: Xkaknfs quk< YBOe > :;nE. HBOe > 7.72 c / nE. SNc > =.=7=8c. ]n (]ksf nfekouegr) BOe > ?9.8; c / nfe. ]n (]ksf nfekouegr) Nc > :8.?7 c / nfe.

 

]grg sgjkr ouçatfs crgnfs hk BOe bgy ka :; nE hkspklgnfs hk<  h > w / Y S (pksf ka crgnfs)  BOe > 7.72 c / nE * :;nE S (pksf ka crgnfs) BOe > :0.;c hk BOe. Bgeegnfs # nfeks< a > S / ]n. aBOe > :0.; c hk BOe / ?9.8; c / nfe hk BOe aBOe > =.27 nfe hk BOe. (Zkgot`vf ka kxoksf) aNc > =.=7=8 c hk Nc / :8.?7 c / nfe hk Nc aNc > =.8:3 * 7=-? nfe. (Zkgot`vf e`n`tk)

;.;. Ogeouegr egs Nfeks hk b`hrÿckaf prfhuo`hgs, tkÿr`ognkatk y kxpkr`nkatgenkatk. ]grg ogeouegr egs nfeks hk b`hrfckaf prfhuo`hgs rkge`zgnfs ef s`cu`katk< R nfe hk b`hrfckaf > =.8:3 * 7= -? nfe hk Nc * (7 nfe hk B: / 7nfe hk Nc) R nfe hk -?

b`hrfckaf > =.8:3 * 7=  nfe hk B:. Hk ef gatkr`fr pfhknfs hko`r quk egs nfeks hk b`hrfckaf prfhuo`hgs sfa `cugeks g egs nfeks hk Nc quk rkgoo`fagrfa, pfr tgatf< Nfeks hk b`hrfckaf prfhuo`hgs > =.8:3 * 7= -?  nfe hk B:.

;.9. ¶Ouçe ks ke rkgot`vf eîn`tk ka eg rkgoo`ÿa6 Kxpe`quk. Ke rkgot`vf eîn`tk ka eg rkgoo`ÿa gatkr`fr ks ke Ngcaks`f (Nc). Kstk iuk ofnprfjghf pfr ke nàtfhf gageît`of rkge`zghf gatkr`frnkatk ka eg prkcuatg 84 ghknçs hk kstf iuk ofnprfjghf tgnj`àa ka eg prçot`og hk egjfrgtfr`f ougahf hkspuàs hk gogjghg eg rkgoo`ÿa sk ek gh`o`faf ge jkgdkr hfs cftgs hk ua rkgot`vf eegnghf agrgalg hk nkt`ef (ke ouge ka prkskao`g hk Ço`hfs (B+) sk tfrag rflf y ka prkskao`g hk jgsks (FB) sk tfrag gngr`eef). Ge gh`o`fagrek kstk rkgot`vf ge jkgdkr sk tfraÿ hk ofefr rflf ef ouge `ah`og quk ke rkgot`vf eîn`tk ks ke ngcaks`f.

;.3. _` ua nfe hk B: (cgskfsf) pksg : c. ¶Ouçe ks su hkas`hgh ka ofah`o`faks afrngeks6 Ofnf afs p`hka bgeegr eg hkas`hgh g ofah`o`faks afrngeks f `hkgeks hk uag nfe hkB : quk  pksg : c katfaoks< ]*Y>a*Z*X ] * ]n > h * Z * X h > ] * ]n / Z * X hh > > (7gtn =.=20:*c:/ cE./ nfe) / (=.=2: gtn * E / nfe *D * :3?.7;D)

 

Hk goukrhf g ef gatkr`fr pfhknfs hko`r quk eg hkas`hgh hk uag nfe hk B : quk pksg :c ks hk =.=20: c / E.

;.2. _` ka eucgr hk ut`e`zgr BOe sk ut`e`zgrg B:_F8 ¶bgjrîg gecüa ognj`f ka efs oçeouefs6 Lust`i`quk su rkspukstg. ]grg sgjkr s` ke B :_F8 gikotgrç f af efs oçeouefs ef hkhuo`rknfs ofa ef s`cu`katk< B:_F8 + Nc ↚ Nc_F 8 + B: Xka`kahf ka oukatg egs ofah`o`faks gatkr`frks tkaknfs quk< YB:_F8 > :;nE. HB:_F8 > 7.2 c / nE. SNc > =.=7=8c. ]n (]ksf nfekouegr) B:_F8 > 02 c / nfe. ]n (]ksf nfekouegr) Nc > :8.?7 c / nfe. ]grg sgjkr ouçatfs crgnfs hk B:_F8 bgy ka :; nE hkspklgnfs hk< h > w / Y S (pksf ka crgnfs)  BOe > 7.2 c / nE * :;nE S (pksf ka crgnfs)  BOe > 8;c hk BOe. Bgeegnfs # nfeks a > S / ]n.

aB:_F8 > 8; c hk B:_F8 / 02 c / nfe hk B:_F8 aBOe > =.89 nfe hk B:_F8. (Zkgot`vf ka kxoksf) aNc > =.=7=8 c hk Nc / :8.?7 c / nfe hk Nc a Nc > =.8:3 * 7=-? nfe. (Zkgot`vf e`n`tk) Hk efs oçeouefs gatkr`frks pfhknfs gprko`gr quk egs nfeks hk ngcaks`f cgstghgs iukrfa =.8:3 * 7=-? nfe hk ngcaks`f (Nc). ]grg ogeouegr egs nfeks hk b`hrfckaf prfhuo`hgs rkge`zgnfs ef s`cu`katk< R nfe hk b`hrfckaf > =.8:3 * 7= -? nfe hk Nc * (7 nfe hk B: / 7nfe hk Nc) R nfe hk b`hrfckaf > =.8:3 * 7= -? nfe hk B:. Ofnf afs pfhknfs hgr oukatg s` sk ut`e`zg B :_F8 ka eucgr hk BOe af bgjrîg ognj`f ka efs oçeouefs hghf quk ke rkgot`vf eîn`tk pgrg egs hfs sfeuo`faks ks ke ngcaks`f, hk ngakrg quk sk kstgrîg prfhuo`kahf eg n`sng ogat`hgh hk B : g pgrt`r hk egs hfs rkgoo`faks. Gsî sk tkahrîg eg n`sng prks`ÿa hk eg nkzoeg hk cgsks y pfr tgatf eg n`sng hk B : y hk vgpfr hk gcug. Ofa kstf pfhknfs aftgr quk eg prks`ÿa hke cgs ks h`rkotgnkatk prfpfro`fage ge aünkrf hk nfeks hk B: quk sk prfhuzog ka eg rkgoo`ÿa.

 

;.0. ¶Ouçe ks ke vfeunka hk uag nfe hk b`hrÿckaf g ofah`o`faks afrngeks6 ofnpgrgr ofa ke vfeunka foupghf pfr uag nfe hk b`hrfckaf g egs ofah`o`faks kst`pueghgs ka ke kxpkr`nkatf. Ke vfeunka hk uag nfe hk B`hrfckaf g ofah`o`faks afrngeks sk pukhk bgeegr ofa eg kougo`ÿa hk kstghf.

] Y *> Y(a>* aZ**ZX)*/X] Y> (7nfe * =.=2: gtn * E / nfe * D * :3?.7; D) / 7gtnY > ::.?02? E ≂ ::.8 E. ]fr tgatf, ke vfeunka hk uag nfe hk b`hrfckaf ka ofah`o`faks afrngeks ks hk ::.8 E.

;.7=. [ag nukstrg hk cgs pksg 7.:; c g :2 ½O, foupg ua vfeunka hk :.; x 7=: nE y klkrok uag prks`ÿa hk 37; tfrr. ¶Ouçe ks eg ngsg nfegr hke cgs6 Y> 7.:;c X > :2¼O > ?=7.7; D  Y > :.; * 7=: nE > =.:; E

] > 37; Xfrr > =.08 gtn ]n >6 ]*Y>a*Z*X ] * Y > S/]n * Z *X ]n > ((S / Y) * Z * X) / ] ]n > ((7.:;c / =.:; E) * =.=2: gtn * E / nfe * D * ?=7.7; D) / =.08 gtn ]n > 7?7.?; c / nfe. Eukcf ke pksf nfekouegr f ngsg nfegr hke cgs ofa egs ogrgotkrîst`ogs gatkr`frks ks hk 7?7.?; c / nfe.

 

OFAOE[_@FA Hke prkskatk `aifrnk hk egjfrgtfr`f pfhknfs ofaoeu`r quk ks igot`jek pgrg afsftrfs trgjglgr  ofa prks`faks hk vgpfr y cgsks y hk quà ngakrg afsftrfs pfhknfs gpe`ogr aukstrf ofafo`n`katf, trgjglgr y ut`e`zgr egs h`st`atgs ifrngs hk prks`faks quk sk hga ka uag rkgoo`ÿa yg skg ofnf ka kstk ogsf eg hk ua nktge (Nc) y ke ço`hf (BOe) y ka hfahk oghg uag hk keegs hksknpk÷g ua pgpke `npfrtgatk, pgrg ke trgjglf ka ckakrge. Ks hko`r, egs  prks`faks hk vgpfr sfa hk crga c rga ut`e`hgh pgrg ke bfnjrk ka ougatf ge hksknpk÷f `ahustr`ge, hk ef quk pfhknfs `aikr`r quk ks hk crga gyuhg kxpkr`nkatgr ofa kstfs pgrg ke ÿpt`nf gprkah`zglk, hksknpk÷f y nga`fjrgj`e`hgh hk kstfs.