Avaliação Final - Introdução ao Cálculo - UNIASSELVI

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11//;;==//==33==33 RRIIHHKKWWWWBB@@ZZH H - - GGbbiittrrf f RRiihhvvbbrrsshhttëërrhhf f @@bbffiikkrraaf f AAk k ZZhhiiggh h - - SSffrrttkk` ` aaf f KK`̀uuiif f - - SSffrrttkk` ` aaf f KK`̀uuiif f - - OOrruuppf f RRIIHHKKWWWWBB@@ZZHH
llttttppss55////ppffrrttkk`̀aaffkk`̀uuiiffbbkkaa..uuiihhkkssssbb`̀vvhh..ggffnn..mmrr//kkvvkk//kkvvkk`̀hhkkggkkff//kkvvkk`̀hhkkggkkffVV`̀hhssttkk..ppllpp ;;//11
Kgkaånhgf5Kgkaånhgf5 Nkrhk Grhsthkib ak Guilk Nkglkaf (=22;<7?)Nkrhk Grhsthkib ak Guilk Nkglkaf (=22;<7?)
Ahsghp`hik5Ahsghp`hik5 Hitrfauïçf kf Gë`gu`f (NKA37)Hitrfauïçf kf Gë`gu`f (NKA37)
Kvk`hkïçf5Kvk`hkïçf5 Kvk`hkïçf Jhik` (Fmebthvk) - Hiahvhauk` J@B\ ( Gfa.52;=711) ( pbsf.57,33)Kvk`hkïçf Jhik` (Fmebthvk) - Hiahvhauk` J@B\ ( Gfa.52;=711) ( pbsf.57,33)
Srfvk5Srfvk5 ;2882233;2882233
IIffttk k aak k SSrrffvvkk55 ;;33,,3333
@bobiak5 @bobiak5 ]bspfstk ]bspfstk GbrtkGbrtk   Wuk ]bspfstk BrrkakWuk ]bspfstk Brrkak  
;.;. [uk` f gfieuitf iunèrhgf qub gf[uk` f gfieuitf iunèrhgf qub gfitèn fs iõnbrfs qub sçf aîzhnks itèn fs iõnbrfs qub sçf aîzhnks pbrh÷ahgks, hstf è, un iõnbrf abghnk`, gfn k pkrtb abpbrh÷ahgks, hstf è, un iõnbrf abghnk`, gfn k pkrtb abghnk` jfrnkak pfr hi﬋ihtfsghnk` jfrnkak pfr hi﬋ihtfs
k`okrhsnfs qub sb rbpbtbn pbrhfahgknbitb, gfnf pfr k`okrhsnfs qub sb rbpbtbn pbrhfahgknbitb, gfnf pfr bxbnp`f5 1,2222... b ;3,<?<?<?...6bxbnp`f5 1,2222... b ;3,<?<?<?...6
 k) k) GfieuGfieuitf afitf afs Iõnbs Iõnbrfs ]krfs ]kghfikghfikhs.hs.
 m) m) GfieuGfieuitf afitf afs Iõnbs Iõnbrfs Sbrfs Sbrh÷ahgrh÷ahgfs.fs.
 g) g) GfieuGfieuitf aitf afs Ifs Iõnbrfõnbrfs His Hitbhrfstbhrfs..
 a) a) GfieuGfieuitf afitf afs Iõns Iõnbrfs brfs IkturkIkturkhs.hs.
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=.=. Skrk gk`gu`kr k ërbk ab un qukarkafSkrk gk`gu`kr k ërbk ab un qukarkaf, mkstk qub sb nu`thp`hqubn afhs afs sb, mkstk qub sb nu`thp`hqubn afhs afs sbus `kafs bitrb sh. Skrk f gë`gu`f ab ërbk ab uus `kafs bitrb sh. Skrk f gë`gu`f ab ërbk ab un rbtíiou`f, tbnfs k nu`thp`hgkïçfn rbtíiou`f, tbnfs k nu`thp`hgkïçf
ak mksb pb`k k`turk. Wbiaf ak mksb pb`k k`turk. Wbiaf ksshn, gk`gu`b k ërbk ak ksshn, gk`gu`b k ërbk ak ﬋ourk k sbouhr, rbprbsbitkiaf unk nu`thp`hgkïçf ab nfiñnhfs. Wfmrb f ﬋ourk k sbouhr, rbprbsbitkiaf unk nu`thp`hgkïçf ab nfiñnhfs. Wfmrb f bxpfstf, ksshik`b k k`tbrikthvkbxpfstf, ksshik`b k k`tbrikthvk
GF]]BQK5GF]]BQK5
 k) k) K ërbk bsK ërbk bstë rbprbtë rbprbsbitksbitkak pfr =xak pfr =x½ + ;1x.½ + ;1x.
 m) m) K ërbk K ërbk bstë rbbstë rbprbsbprbsbitkak itkak pfr 1xpfr 1x½ + :.½ + :.
 g) g) K ërbk bsK ërbk bstë rbprbtë rbprbsbitksbitkak pfr (=ak pfr (=x + ;)(7 + =x).x + ;)(7 + =x).
 a) a) K ërbk bsK ërbk bstë rbprbtë rbprbsbitksbitkak pfr =xak pfr =x½ + =x + :.½ + =x + :.
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7.7. F bstuaf aks bqukïòbs F bstuaf aks bqukïòbs è nuhtf uth`hzkaf if ahk k ahk. [ukiaf bstuakè nuhtf uth`hzkaf if ahk k ahk. [ukiaf bstuaknfs pb`k prhnbhrk vbz ks `btrks bn nktbnëthgk, gfinfs pb`k prhnbhrk vbz ks `btrks bn nktbnëthgk, gfishabrknfs un nuiaf ifvf. Skrkshabrknfs un nuiaf ifvf. Skrk
kuxh`hkr ik gfnprbbisçf abssb gfitbõaf, sçf uth`hzkaks ahjbrbitbs nbtfaf`fohks. Ksshn, vknfs hkuxh`hkr ik gfnprbbisçf abssb gfitbõaf, sçf uth`hzkaks ahjbrbitbs nbtfaf`fohks. Ksshn, vknfs hnkohikr un vk`fr absgfilbghaf b abtbrnhikr qubn nkohikr un vk`fr absgfilbghaf b abtbrnhikr qubn sbrë fsbrë f
iõnbrf absgfilbghaf. F quîitup`f ab un iõnbrf iõnbrf absgfilbghaf. F quîitup`f ab un iõnbrf nkhs ;2 è houk` nkhs ;2 è houk` kf afmrf abssb iõnbrf kahghfikaf ab 12. kf afmrf abssb iõnbrf kahghfikaf ab 12. [uk` è bssb iõnbrf6[uk` è bssb iõnbrf6
 k) k) F iõnF iõnbrf abrf absgfbsgfilbghilbghaf è af è =2.=2.
 m) m) F iõnF iõnbrf abrf absgfbsgfilbghilbghaf è af è 73.73.
 g) g) F iõF iõnbrf nbrf absgfabsgfilbghilbghaf è af è ;2.;2.
 a) a) F iõnF iõnbrf abrf absgfbsgfilbghilbghaf è af è ;3.;3.
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1.1. F orë﬋gf ab unk juiïçf af prhnbhrf orku F orë﬋gf ab unk juiïçf af prhnbhrf orku è unk rbtk b skmbnfs, pb`f quhitf pfstuè unk rbtk b skmbnfs, pb`f quhitf pfstùkaf ab Bug`habs, qub tbiaf afhs p`kaf ab Bug`habs, qub tbiaf afhs pfitfs gfisbouhnfs abtbrnhikfitfs gfisbouhnfs abtbrnhikr unk rbtk.r unk rbtk.
Kik`hsb f orë﬋gf k sbouhr Kik`hsb f orë﬋gf k sbouhr b ksshik`b k k`tbrikthvk GF]]BQK qub kprbsbitk k juiïçf ak b ksshik`b k k`tbrikthvk GF]]BQK qub kprbsbitk k juiïçf ak rbtk5rbtk5
 k) k) K juK juiïçiïçf è y f è y > x - > x - ;.;.

  
11//;;==//==33==33 RRIIHHKKWWWWBB@@ZZH H - - GGbbiittrrf f RRiihhvvbbrrsshhttëërrhhf f @@bbffiikkrraaf f AAk k ZZhhiiggh h - - SSffrrttkk` ` aaf f KK`̀uuiif f - - SSffrrttkk` ` aaf f KK`̀uuiif f - - OOrruuppf f RRIIHHKKWWWWBB@@ZZHH
llttttppss55////ppffrrttkk`̀aaffkk`̀uuiiffbbkkaa..uuiihhkkssssbb`̀vvhh..ggffnn..mmrr//kkvvkk//kkvvkk`̀hhkkggkkff//kkvvkk`̀hhkkggkkffVV`̀hhssttkk..ppllpp ==//11
 m) m) K juK juiïçiïçf è y f è y > - x > - x - ;.- ;.
 g) g) K juK juiïçiïçf è y f è y > - x > - x + ;.+ ;.
 a) a) K juK juiïçiïçf è y f è y > x + > x + ;.;.
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2.2. Kik`hsb k shtukïçfKik`hsb k shtukïçf5 Efçf gfnbu ;/7 5 Efçf gfnbu ;/7 ab unk tfrtk. Gknh`f gab unk tfrtk. Gknh`f gfnbu 2/;= ak nbfnbu 2/;= ak nbsnk tfrtk. [uk` k jrkïçf snk tfrtk. [uk` k jrkïçf tftk` ak tfrtk qub jfh tftk` ak tfrtk qub jfh gfnhak6gfnhak6
 k) k) F tftkF tftk̀ gfnha` gfnhaf è ab 7 f è ab 7 pkrtbpkrtbs ab 1.s ab 1.
 m) m) F tftkF tftk̀ gfnha` gfnhaf è ab ; f è ab ; pkrtbpkrtbs ab =.s ab =.
 g) g) F tftkF tftk̀ gfnh` gfnhaf è af è ab = pab = pkrtbs krtbs ab 2.ab 2.
 a) a) F tftkF tftk̀ gfnha` gfnhaf è ab ; f è ab ; pkrtbpkrtbs ab 1.s ab 1.
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:.:. K jrkïçf k`oèmrhgk è f qufghbitb pfK jrkïçf k`oèmrhgk è f qufghbitb pf̀ hifnhk` kprbsbitkaf sfm k jfrnk ab jrkïçf`hifnhk` kprbsbitkaf sfm k jfrnk ab jrkïçf, if quk` f abifnhikafr kprbsbitk un, if quk` f abifnhikafr kprbsbitk unk fu nkhs vkrhëvbhs. Ksshn, içf bxhstbk fu nkhs vkrhëvbhs. Ksshn, içf bxhstb
ahvhsçf pfr zbrf if gfieuitf afs ahvhsçf pfr zbrf if gfieuitf afs Iõnbrfs ]bkhs. Wfmrb abtbrnhikr f Iõnbrfs ]bkhs. Wfmrb abtbrnhikr f vk`fr ab "y" ak bxprbssçf k`oèmrhgk k vk`fr ab "y" ak bxprbssçf k`oèmrhgk k sbouhr, ksshik`b k k`tbrikthvk GF]]BQK5sbouhr, ksshik`b k k`tbrikthvk GF]]BQK5
 k k)) - - ==..
 m m)) - - ;;//==..
 g g)) ;;//==..
 a a)) ==..
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?.?. Fs shstbnks ab bqukïòbFs shstbnks ab bqukïòbs sçf jbrrknbitks mkstkitb gfnuis sçf jbrrknbitks mkstkitb gfnuis ik rbsf`uïçf ab prfm`bnks iks ik rbsf`uïçf ab prfm`bnks iks ahvbrsks ërbks af gfilbs ahvbrsks ërbks af gfilbghnbitf (Nktbnëthgk, Jîshgk, [uînhgk,ghnbitf (Nktbnëthgk, Jîshgk, [uînhgk,
Biobilkrhk btg). Ab nkibhrk obrk`, k rbsf`uïçf ab un shstbnk è mbn shnp`bs. F qub abvbnfs fmsbrvkr b vbrh﬋gkr è quk` f nètfaf nkhs prëthgf Biobilkrhk btg). Ab nkibhrk obrk`, k rbsf`uïçf ab un shstbnk è mbn shnp`bs. F qub abvbnfs fmsbrvkr b vbrh﬋gkr è quk` f nètfaf nkhs prëthgf pkrk gkakpkrk gkak
shtukïçf. Fs nètfafs ab rbsf`uïçf af shstbnk shtukïçf. Fs nètfafs ab rbsf`uïçf af shstbnk sçf f ak kahïçf b sçf f ak kahïçf b f ak sumsthtuhïçf. ]bsf`vk f f ak sumsthtuhïçf. ]bsf`vk f shstbnk b k sbouhr ksshik`b shstbnk b k sbouhr ksshik`b k k`tbrikthvk GF]]BQK qub hiahgk kk k`tbrikthvk GF]]BQK qub hiahgk k
sf`uïçf5sf`uïçf5
 k) k) K sfK sf`uï`uïçf è Wçf è W>{=>{==, ;3=, ;3} } 
 m) m) K sfK sf`uï`uïçf è Wçf è W>{=>{=3, ;33, ;3} } 
 g) g) K sfK sf̀ uï`uïçf è çf è W>{=W>{=, ;} , ;} 
 a) a) K sf`K sf ùïçuïçf è W>{f è W>{=3, ;=3, ;;} ;} 
<.<. [ukiaf tbnfs unk juiïçf, pf[ukiaf tbnfs unkjuiïçf, pfabnfs gfnpkrkr f kunbitf fu k ahnhiabnfs gfnpkrkr f kunbitf fu k ahnhiuhïçf afs vk`frbs ab x bn rb`kïçf uhïçf afs vk`frbs ab x bn rb`kïçf kf kunbitf fu ahnhiuhïçf afs vkkf kunbitf fu ahnhiuhïçf afs vk̀frbs y ak hnkobn.`frbs y ak hnkobn.
Fmsbrvb k juiïçf ak ﬋ourk k Fmsbrvb k juiïçf ak ﬋ourk k sbouhr b ksshik`b k k`tbrikthvk sbouhr b ksshik`b k k`tbrikthvk GF]]BQGF]]BQK5K5
 k) k) K juiK juiïçf è gïçf è grbsrbsgbigbitb.tb.
 m) m) K juiK juiïçf è gïçf è gfisfistkitkitb.tb.
 g) g) K juiK juiïçf è aïçf è abgrbgrbsgbsgbitbitb.b.
 a) a) K juK juiïçiïçf è `f è h̀ibhibkrkr..
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8.8. F gfisunf ab bibrohk b`ètrhgk pkrk uF gfisunf ab bibrohk b`ètrhgk pkrk unk rbshaåighk if abgfrrbr afs nbsbnk rbshaåighk if abgfrrbr afs nbsbs è akaf pfr5 B(t) > t½ - <t + =;3, ik quk` B rbprbsbitk f gfisunf ab bis è akaf pfr5 B(t) > t½ - <t + =;3, ik quk` B rbprbsbitk f gfisunf ab bibrohk akafbrohk akaf
bn Dwl b t è f tbnpf nbahaf bn bn Dwl b t è f tbnpf nbahaf bn nbsbs, sbiaf t > 3 (ekibhrf), t > nbsbs, sbiaf t > 3 (ekibhrf), t > ; (jbvbrbhrf) b ksshn pfr ahkitb. Ksshik`b k k`tbrikthvk GF]]BQK qub rbprbsbitk f(s); (jbvbrbhrf) b ksshn pfr ahkitb. Ksshik`b k k`tbrikthvk GF]]BQK qub rbprbsbitk f(s)
nås(bs) bn qub f gfisunf è houk` k ;82dwl.nås(bs) bn qub f gfisunf è houk` k ;82dwl.
 k) k) WfWfnbnbititb nkb nkrïrïf.f.
 m) m) KmKmrhrh̀ b e` b euiuilflf..
 g) g) WfWfnbnbititb nkb nkhfhf..
 a) a) NkNkrïrïf b f b nknkhfhf..
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;3.;3. F bstuaf afs rkahgkhs è un afs gfitbõafs af ifif kif af Bishif Juiaknbitk`. Skrk hssf, prbghsknfs `bnmrkr ab suks prfprhbakabs nktbnëthgks. KF bstuaf afs rkahgkhs è un afs gfitbõafs af ifif kif af Bishif Juiaknbitk`. Skrk hssf, prbghsknfs `bnmrkr ab suks prfprhbakabs nktbnëthgks. K
jktfrkïçf ab rkahgkhs ifs kuxh`hk if jktfrkïçf ab rkahgkhs ifs kuxh`hk if absbivf`vhnbitf ab k`ounks bxprbssòbs k`oèmrhgks. Wbiaf ksshn, quk` prfprhbakab vfgå pfabrë uth`hzkr pkrk absbivf`vhnbitf ab k`ounks bxprbssòbs k`oèmrhgks. Wbiaf ksshn, quk` prfprhbakab vfgå pfabrë uth`hzkr pkrk abtbrnhikr fabtbrnhikr f
vk`fr ak bxprbssçf k sbouhr6vk`fr ak bxprbssçf k sbouhr6
 k) k) F vk`F vk f̀r ab fr ab x bstx bstë bitë bitrb = b rb = b 7.7.
  
11//;;==//==33==33 RRIIHHKKWWWWBB@@ZZH H - - GGbbiittrrf f RRiihhvvbbrrsshhttëërrhhf f @@bbffiikkrraaf f AAk k ZZhhiiggh h - - SSffrrttkk` ` aaf f KK`̀uuiif f - - SSffrrttkk` ` aaf f KK`̀uuiif f - - OOrruuppf f RRIIHHKKWWWWBB@@ZZHH
llttttppss55////ppffrrttkk`̀aaffkk`̀uuiiffbbkkaa..uuiihhkkssssbb`̀vvhh..ggffnn..mmrr//kkvvkk//kkvvkk`̀hhkkggkkff//kkvvkk`̀hhkkggkkffVV`̀hhssttkk..ppllpp 77//11
 m) m) F vk`fF vk`fr ab x br ab x bstë bistë bitrb zbtrb zbrf b =.rf b =.
 g) g) F vk`F vk f̀r ab fr ab x è nx è nkhfr fkhfr fu houu houk` k :k` k :..
 a) a) F vkF vk`fr a`fr ab x è ib x è ibokbokthvthvf.f.
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;;.;;. (BIKAB, =33<) Ks pftbighk`hakabs pbako÷ohgks ak lhst÷rhk if bishif ab nktbnëthgk tån shaf mkstkitb ahsguthaks. Bitrb ks eusth﬋gkthvks pkrk f usf ak lhst÷rhk(BIKAB, =33<) Ks pftbighk`hakabs pbako÷ohgks ak lhst÷rhk if bishif ab nktbnëthgk tån shaf mkstkitb ahsguthaks. Bitrb ks eusth﬋gkthvks pkrk f usf ak lhst÷rhk
if bishif ab nktbnëthgk, hig`uh-sb f if bishif ab nktbnëthgk, hig`uh-sb f jktf ab b`k susghtkr jktf ab b`k susghtkr fpfrtuihakabs pkrk k hivbsthokïçf. Gfishabrkiaf bssk eusth﬋gkthvk, un prfjbssfr prfpñs unkfpfrtuihakabs pkrk k hivbsthokïçf. Gfishabrkiaf bssk eusth﬋gkthvk, un prfjbssfr prfpñs unk
kthvhakab k pkrthr ak hijfrnkïçf lhst÷rhgk ab qub f jknfsf nktbnëthgf Shbrrb Jbrnkt T;:3;-;::2^, qub sb hitbrbsskvk pfr iõnbrfs prhnfs, pbrgbmbu k`ounkskthvhakab k pkrthr ak hijfrnkïçf lhst÷rhgk ab qub f jknfsf nktbnëthgf Shbrrb Jbrnkt T;:3;-;::2^, qub sb hitbrbsskvk pfr iõnbrfs prhnfs, pbrgbmbu k`ounks
rb`kïòbs bitrb iõnbrfs prhnfs înpkrbs b rb`kïòbs bitrb iõnbrfs prhnfs înpkrbs b qukarkafs pbrjbhtfs. Skrk qub fs k`uifs qukarkafs pbrjbhtfs. Skrk qub fs k`uifs tknmèn absgfmrhssbn bssk rb`kïçf, pbahu qub b`bs gfnp`btkssbn k tkmb`ktknmèn absgfmrhssbn bssk rb`kïçf, pbahu qub b`bs gfnp`btkssbn k tkmb`k
k sbouhr, vbrh﬋gkiaf qukhs iõnbrfs prhnfs înpkrbs pfabn sbr bsgrhtfs k sbouhr, vbrh﬋gkiaf qukhs iõnbrfs prhnfs înpkrbs pfabn sbr bsgrhtfs gfnf sfnk ab afhs qukarkafs pbrjbhtfs. K`èn gfnf sfnk ab afhs qukarkafs pbrjbhtfs. K`èn ahssf, sf`hghtfu qub fmsbrvkssbnahssf, sf`hghtfu qub fmsbrvkssbn
k`ounk prfprhbakab gfnun k bssbs iõnbrfs.k`ounk prfprhbakab gfnun k bssbs iõnbrfs.
 k) k) HHHH, k, kpbpbikiks.s.
 m) m) H, H, kpkpbibiksks..
 g) g) HH b HHH b HHHHH, kp, kpbibiksks..
 a) a) H b HHH b HHH, kH, kpbpbikiks.s.
;=.;=. (BIKAB, =33<) K prfjbssfrk Gkr`k prfpñs k sbus k`uifs qub bigfitrkssbn k sf`uïçf ak sbouhitb bqukïçf af sbouia(BIKAB, =33<) K prfjbssfrk Gkr`k prfpñs k sbus k`uifs qub bigfitrkssbn k sf`uïçf ak sbouhitb bqukïçf af sbouiaf orku5f orku5
x½ - ; > (=x + 7)(x - ;)x½ - ; > (=x + 7)(x - ;)
Sbarf b Efçf rbsf`vbrkn ak sbouhitb Sbarf b Efçf rbsf`vbrkn ak sbouhitb nkibhrk.nkibhrk.
]bsf`uïçf ab Sbarf5]bsf`uïçf ab Sbarf5
x½ - ; > (=x + 7)(x - ;)x½ - ; > (=x + 7)(x - ;)
x½ - ; > =x½ + x - 7x½ - ; > =x½ + x - 7
= - x > x½= - x > x½
Gfnf ; è sf`uïçf abssk bqukïçf, bitçf W > {;} Gfnf ; è sf`uïçf abssk bqukïçf, bitçf W > {;} 
]bsf`uïçf ab Efçf5]bsf`uïçf ab Efçf5
x½ - ; > (=x + 7)(x - ;)x½ - ; > (=x + 7)(x - ;)
(x - ;)(x + ;) > (=x + 7)(x - ;)(x - ;)(x + ;) > (=x + 7)(x - ;)
x + ; > =x + 7x + ; > =x + 7
x > -=x > -=
Sfrtkitf, W > {-=} Sfrtkitf, W > {-=} 
Sbarf b Efçf pbrouitkrkn ã Sbarf b Efçf pbrouitkrkn ã prfjbssfrk pfr qub bigfitrkrkn sf`uïòbs ahjbrbitbs. K prfjbssfrk fmsbrvfu qub prfjbssfrk pfr qub bigfitrkrkn sf`uïòbs ahjbrbitbs. K prfjbssfrk fmsbrvfu qub futrfs k`uifs lkvhkn kprbsbitkaf sf`uïòbsfutrfs k`uifs lkvhkn kprbsbitkaf sf`uïòbs
pkrbghaks gfn ks ab`bs. Bitrb ks pkrbghaks gfn ks ab`bs. Bitrb ks bstrktèohks kprbsbitkaks iks fpïòbs k sbouhr, bsgf`lk k nkhs kabqukak k sbr kaftkak pfr bstrktèohks kprbsbitkaks iks fpïòbs k sbouhr, bsgf`lk k nkhs kabqukak k sbr kaftkak pfr G`krk vhskiaf ã kprbiahzkobnG`krk vhskiaf ã kprbiahzkobn
shoih﬋gkthvk pfr pkrtb afs k`uifs5shoih﬋gkthvk pfr pkrtb afs k`uifs5
 k) k) Bsgrbvbr k sf`uïçf af bxBsgrbvbr k sf`uïçf af bxbrgîghf if qukarf, uskiaf k j÷rnu`k ak rbsbrgîghf if qukarf, uskiaf k j÷rnu`k ak rbsf`uïçf ak bqukïçf af =f`uïçf ak bqukïçf af =µ orku, pkrk qub fs k`uifs pbrgbmkn qµ orku, pkrk qub fs k`uifs pbrgbmkn qub bssb è f nètfaf qubub bssb è f nètfaf qub
jfribgb k rbspfstk gfrrbtk.jfribgb k rbspfstk gfrrbtk.
 m) m) Sbahr k Sbarf b Efçf qub kprbsSbahr k Sbarf b Efçf qub kprbsbitbn ã g`kssb suks sf`uïòbitbn ã g`kssb suks sf`uïòbs pkrk ahsgussçf b bsthnu`kr fs kbs pkrk ahsgussçf b bsthnu`kr fs k̀uifs k tbitkrbn gfnprbbiabr fia`uifs k tbitkrbn gfnprbbiabr fiab bstë k jk`lk iksb bstë k jk`lk iks
sf`uïòbs kprbsbitkaks b gfnf abvbn jkzbr pkrk gfrrhoh-`ks.sf`uïòbs kprbsbitkaks b gfnf abvbn jkzbr pkrk gfrrhoh-`ks.
 g) g) ]bsf`vbr hiahvhauk`nbitb f bx]bsf`vbr hiahvhauk`nbitb f bxbrgîghf pkrk gkak k`uif, uskiaf k j÷brgîghf pkrk gkak k`uif, uskiaf k j÷rnu`k ak rbsf`uïçf ak bqukïçrnu`k ak rbsf`uïçf ak bqukïçf af =µ orku, nfstrkiaf qub bssb è f nèf af =µ orku, nfstrkiaf qub bssb è f nètfaf qub jfribgbtfaf qub jfribgb
k rbspfstk gfrrbtk.k rbspfstk gfrrbtk.
  
11//;;==//==33==33 RRIIHHKKWWWWBB@@ZZH H - - GGbbiittrrf f RRiihhvvbbrrsshhttëërrhhf f @@bbffiikkrraaf f AAk k ZZhhiiggh h - - SSffrrttkk` ` aaf f KK`̀uuiif f - - SSffrrttkk` ` aaf f KK`̀uuiif f - - OOrruuppf f RRIIHHKKWWWWBB@@ZZHH
llttttppss55////ppffrrttkk`̀aaffkk`̀uuiiffbbkkaa..uuiihhkkssssbb`̀vvhh..ggffnn..mmrr//kkvvkk//kkvvkk`̀hhkkggkkff//kkvvkk`̀hhkkggkkffVV`̀hhssttkk..ppllpp 11//11
 a) a) Hiahgkr hiahvhauk`nbitb, pkrk gkak kHiahgkr hiahvhauk`nbitb, pkrk gkak k̀uif qub kprbsbitfuunk rbsf`u`uif qub kprbsbitfu unk rbsf`uïçf higfrrbtk, fiab bstë f brrf b gfnf gfrrhoh-`f, k pkrthr ak bstrktèohk hihgïçf higfrrbtk, fiab bstë f brrf b gfnf gfrrhoh-`f, k pkrthr ak bstrktèohk hihghk` bsgf`lhakhk` bsgf`lhak
pb`f k`uif.pb`f k`uif.
Srfvk ﬋ik`hzkak gfnSrfvk ﬋ik`hzkak gfn ;= kgbrtfs;= kgbrtfs b b 3 qubstòbs brrkaks3 qubstòbs brrkaks..