គណ ក ម ករនពន ឋ នង រ ប រ ង ក លម ផលគន ក សន ពសដ ឋ គណ ក ម ករ រត តពនតយប ចចក ទស

Átírás

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3 គណ ក ម ករនពន ឋ នង រ ប រ ង ក លម ផលគន ក សន ពសដ ឋ គណ ក ម ករ រត តពនតយប ចចក ទស ក លម ឆ ន ក អ ង ស ង ក រស ទ យ រ ក ទតយ ម ង ក នន ស ខ ក រពម ស នតយ គណ ក ម ករ រត តពនយអកខ វរ ទ ឋ ក លម មគកសរ ករយក ពយទ រ រចនទ ព រ នង រកប ក អ ង ស ង ក រព ម ក ញ ល គ ណ ក - i -

4 រមភកថ sysþimitþgñksiksa CaTIemRtI!! esovepa ក ណល ត គណតវ ទយ ថន ក ទ kmritmuldæanedleyigjmú)an eroberogenh rymmanbyncmbukfm² Edl)andkRsg;ecjBIesovePAKNitviTüa fñak;tikmritmuldæan rbs;rksyggb;rm yuvcn nigkila. kñúgcmbuknimyy²eyigjmú)ansegçbemeron nig dkrsg;lmhat;kñúgemeronéncmbuk nimyy²mkefivdmenahrsayy:agek,ahk,ay EdlGacCaCMnYysμartIdl;GñksikSa kñúgrkb;mcædæan. etahcay:agnak¾eday karbkrsaynuvral;lmhat;kñúgesovepa enhvaminl hysekhység TaMgRsugenaHeT. kmhusqágnana TaMgbec kets nig GkçraviruTæR)akdCaekItmanedayGectna BMuaneLIy GaRs½yehtuenH eyigjmúrg;camcanic nuvmtirihkn;ebbsßabnabismnak; GñksikSakñúgRKb;mCÆdæan edim,iektrmuvesovepaenh[kan;etmansurkitüpab EfmeTot. CaTIbBa b; eyigmjúgñkeroberog sumcunbrdl;gñksiksatamggs; TTYl)aneCaKC½ykñúgCIvit nig mansupabl. - ii - )at;dmbgéf TI 5 E emsa qñam Gñkeroberog lwm pláún

5 matikaerog TMB½r CMBUkTI emeronti limiténgnukmn_ emeronti PaBCab;énGnuKmn_ CMBUkTI emeronti edrievéngnukmn_ 58 emeronti Gnuvtþn_énGnuKmn_ 8 CMBUkTI emeronti GnuKmn_ snitan 6 emeronti GnuKmn_Gius,:ÚNg;Esül 5 emeronti GnuKmn_elakarIt 5 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt; 59 emeronti GaMgetRkalkMNt; 8 lmhat;benßm 89 - iii -

6 CMBUkTI emeronti emeronsegçb limiténgnukmn_ limiténgnukmn_ -limiténgnukmn_rtg cmnynkmnt niymn½y ³ GnuKmn_ f manlimitesµi L kalna itcit aebirkb; cmnyn mancmnyn Edl a nam[ f() L. eksresr ³ niymn½y ³ ekfagnukmn_ f iteta lim f() L. a b kalna itetacit Edl a ebicmebahrkb;cmnyn M man a nam[ f() M b f() M. eksresr lim f() b lim f() a a. - -

7 CMBUkTI emeronti limiténgnukmn_ -limiténgnukmn_rtgǵnnþ ekfagnukmn_ f manlimitesµi L kalna etacit b ebicmebahrkb;cmnyn ekgacrk b N nam[ f() L. eksresr lim f() L lim f() b L. ekfagnukmn_ f manlimit kalna etacit ebicmebahrkb;cmnyn M ekman N Edl N nam[ f() M.eKsresr lim f(). - - N Edl N ekfagnukmn_ f manlimit kalna etacit ebicmebahrkb;cmnyn M ekman N Edl N nam[ f() M.eKsresr lim f(). -RbmaNviFIelIlImIt ebi lim f() L ; lim g() M nig lim h() N a a a Edl L ; M ; N CacMnYnBitenaHeK)an ³ / g() L M lim f() a a CacMnYnkMNt; b Gnnþ.

8 CMBUkTI emeronti limiténgnukmn_ / / / 5 / lim g() h() k f() k limf() a lim f() a a lim f().g().h() a f() lim a g() L M ; M L.M.N L M N a n n L Edl n CacMnYnKt;FmµCatiminsUnü. 6 / lim f() -limiténgnukmn_gsnitan n n a n n a n f() n limf() n / lim cmebah a / lim cmebah a / lim a a ebi L nig ncacmnynkt;ku b 5-lImIténGnuKmn_bNþak a nig n IN a nig ncacmnynkt;ess L L nig ncacmnynkt;ess. ebi f nig g CaGnuKmn_Edlman lim[g()] L a nig lim f() f(l) enah lim f g() f (L). L a 6-lImIttamkareRbobeFob ebiekmangnukmn_ f ; g nigcmnynbit A Edl lim g() - -

9 CMBUkTI emeronti limiténgnukmn_ nig f() g() cmebahrkb; A enah lim f(). ebiekmangnukmn_ f ; g nigcmnynbit A Edl lim g() nig f() g() cmebahrkb; A enah lim f(). ebiekmangnukmn_ f ; g; h nigcmnynbit A Edl lim g() lim h( ) nig g() f() h() cmebahrkb; A enah lim f() ebiekmangnukmn_. lim g() ' nig () g() enah '. 7-lImItragminkMnt limitedlmanragminkmnt; f ; g nigcmnynbit A Edl lim f() f cmebahrkb; A vifan edim,iknnalimitedlmanragminkmnt; ekrtuvbmebkpak yk nig PaKEbgCaplKuNktþa ehiysrmylktþarym rycknna limiténkensamfµi. - -

10 CMBUkTI emeronti limiténgnukmn_ limitedlmanragminkmnt; vifan edim,iknnalimitedlmanragminkmnt; ekrtuvdak;ty EdlmandWeRkFMCageKenAPaKyk nig PaKEbgCaplKuNktþa ehiysrmylktþarym rycknnalimiténkensamfµi. limitedlmanragminkmnt; vifan edim,iknnalimitedlmanragminkmnt; ekrtuv dak;tyedlmandwerkfmcagekenapakyk nig PaKEbgCaplKuN énktþa ehiysrmylktþarym rycknnalimiténkensamfµi. 8-lImIténGnuKmn_RtIekaNmaRt -ebi a CacMnYnBitsßienAkñúgEdnkMnt;énGnuKmn_RtIekaNmaRtEdl [enahek)an lim sin sina ; lim cos cosa a a nig lim tan tana. a -vifan ebi Cargjvas;mMu b FñÚKitCar:adüg;enaHeK)an sin cos lim nig lim

11 CMBUkTI emeronti limiténgnukmn_ 9-lImIténGnuKmn_Giucs,:ÚNgÉsül / / / / 5 / lim e lim e e lim e lim n ( n ) n lim e (n ) -limiténgnukmn_elakariteneb / / lim ln lim ln ln / lim / lim ln ln 5 / lim (n n n 6 / lim ln ) - 6 -

12 CMBUkTI emeronti limiténgnukmn_ lmhat nig dmen¼rsay!-rsaybba ak;fa limitagerkamenhbitedayerbiniymn½y k> lim(5 ) > lim ( )( ) K> (a b) a b lim dmenahrsay k> lim(5 ) tag f() 5 ekman f() (5 ) f() 5 nam[ ebiektag 5 nam[ f() ( enahrkb; man Edl ekfa () ducenh lim(5 ) 5 f manlimitesµi kalna itcit.. 5 ) - 7 -

13 CMBUkTI emeronti limiténgnukmn_ > lim ( )( ) tag f() ( )( ) eday enaheyigsnµtfa. ek)an ( ) nig ( ). cmebahrkb; yk Edl nam[ f() ( )( ) ( )( ) f(). ducenhcmebahrkb; man Edl nam[ ( )( ). ducenh lim ( )( ). K> lim(a b) a b tag f() a b -cmebah a enahrkb; ekman a b b kñúgkrnienh (a b) a b b lim

14 CMBUkTI emeronti limiténgnukmn_ -snµtfa ekman () a. f (a b) a b a b a( ) f() (a b) a., a. a enahrkb; a b ebiektag a man Edl nam[ f() (a b) ducenh (a b) a b lim f() M nig lim g() L Edl M nig L Ca a a cmnynefrenah curbgðajfa k> lim[k.f()] k. L a > lim[f() g()] L M a dmenahrsay k> lim[k.f()] k. L a -ebi k enah kf() kl RKb;

15 CMBUkTI emeronti limiténgnukmn_ -snµtfa k ekman lim f() L man a cmebahrkb; Edl f() L cmebahrkb; Edl a. k ducenhrkb; Edl a enaheyig)an kf() kl k f() l k.. k edayrkb; enah a man Edl kf() kl RKb;. ducenh limkf() kl. a > lim[f() g()] L M a eday limf() L nig lim g() MenaHcMeBaHRKb; a a man nig Edl f() g() L M a cmebah a cmebah ekman f() g() L M (f() L) (g() M) f() L g() M - -

16 CMBUkTI emeronti limiténgnukmn_ yk nig enahrkb; Edl a ek)an f() g() L M f() L g() M edayrkb; man Edl a nam[ f() g() L M. ducenh lim[f() g()] L M. a. #-KNnalImItageRkam k> lim > ( )( K> lim ( lim )( ) )( ) X> lim( 8 ) dmenahrsay KNnalImItageRkam k> lim - -

17 CMBUkTI emeronti limiténgnukmn_ - - B A B) A B)( A ( ) ( ) )( ( lim ) ( ) )( ( lim ) )( ( ) )( ( lim ducenh lim. > ) )( ( ) )( )( ( lim ) )( ( ) )( )( ( lim ) )( ( ) )( )( ( lim ducenh ) )( ( ) )( )( ( lim.

18 CMBUkTI emeronti limiténgnukmn_ - - K> lim ) lim( ) ( lim ducenh lim. X> ) 8 lim( lim ) 8 ( ) (8 lim 8 8 lim 8 8 lim ducenh ) 8 lim(.

19 CMBUkTI emeronti limiténgnukmn_ $-kmnt;cmnynefr a edim,i[limitagerkamcalimiténcmnynefr ehiy kmnt;limitenhpg. a a k> lim > lim a a K> lim X> lim - - dmenahrsay kmnt;cmnynefr a edim,i[limitagerkamcalimiténcmnynefr ehiy kmnt;limitenhpg a k> lim edim,i[limitenhcacmnynefrluhrtaet Cab ssmikar a ek)an a a. cmebah lim a ek)an lim ( )( lim )

20 CMBUkTI emeronti limiténgnukmn_ a > lim edim,i[limitenhcacmnynefrluhrtaet Cab ssmikar a ek)an () a a. cmebah a lim ek)an lim ( ) lim a K> lim edim,i[limitenhcacmnynefrluhrtaet Cab ssmikar a ek)an a a. cmebah a lim ek)an lim lim ( )( ) - 5 -

21 CMBUkTI emeronti limiténgnukmn_ a X> lim edim,i[limitenhcacmnynefrluhrtaet Cab ssmikar a ek)an a a. cmebah lim a ek)an %-KNnalImItageRkam sin k> K> g> lim ( )( ) lim lim sin 5 cos lim sin > ( cos ) lim tan sin sin lim lim cos X> lim sin c> - 6 -

22 CMBUkTI emeronti limiténgnukmn_ dmenahrsay KNnalImItageRkam sin k> > lim sin 5 sin.. lim sin sin ducenh lim sin. 5 5 ( cos ) lim tan sin ( cos ) lim tan cos tan ( cos ) cos lim lim ( cos )tan ( cos cos )tan sin lim ( cos cos )tan sin lim ( ) tan ( cos cos ) - 7 -

23 CMBUkTI emeronti limiténgnukmn_ K> cos lim sin cos lim sin ( cos ) lim sin lim sin cos ( cos ) lim cos ducenh ( cos sin ( cos ) cos ) cos lim. sin 8 sin X> lim tag kalna t t enah t 8-8 -

24 CMBUkTI emeronti limiténgnukmn_ sin ek)an lim lim ducenh g> tag sin lim lim sin t sin( t) t cost lim t t sin lim t t sin lim t ). t ( t kalna enah t lim sin t lim sin t t ducenh lim sin. t t - 9 -

25 CMBUkTI emeronti limiténgnukmn_ c> lim cos tag lim cos t kalna enah t ek)an lim ducenh lim cos t sin lim t t cost t t sin lim t ). t ( t - -

26 CMBUkTI emeronti limiténgnukmn_ ^-KNnalImItageRkam k> lim( e ) K> lim( )e X> lim( )e > g> lim ln e lim e c> lim ln( ) q> lim ln C> lim{[ln( ) ln ]} dmenahrsay KNnalImItageRkam k> lim( e ) ekman lim( e ) lim( ) lim(e ) erbah ; lim e ehiy lim( e ) lim( ) lim(e ) erbah lim ; lim e ducenh lim( e ) lim. - -

27 CMBUkTI emeronti limiténgnukmn_ > lim( )e eday lim( ) nig lim e ducenh lim ( )e K> lim( )e lim e lim e erbah lim e ducenh lim( )e e X> lim lim e e e. ( ) e ( ) e e lim e erbah lim e, lim e g> lim ln lim ln ln( ) ducenh lim ln lim ln( ) lim ln. - -

28 CMBUkTI emeronti limiténgnukmn_ c> lim ln( ) lim lim ln( ) erbah lim nig lim ln( ) ducenh lim ln( ). q> lim ln tag t kalna enah t lim(t )ln t lim(t ) ln(5t t t lim(t )[ t (t ) (t ) ) ln t ln(5 t)] erbah lim(t ) ; lim[ln t t ducenh ln ln(5 t)] t lim. C> lim{[ln( ) ln ]} lim ln( ) lim ln( erbah lim( ) e ) ducenh lim{[ln( ) ln ]}. lne - -

29 CMBUkTI emeronti limiténgnukmn_ &-KNnalImItageRkam tan sin sin k> lim > lim sin cos cos K> lim X> lim cos cos cos cos dmenahrsay KNnalImItageRkam tan sin k> lim tan tan cos lim tan sin tan ( cos ) lim lim sin lim tan lim tan sin ducenh lim. - -

30 CMBUkTI emeronti limiténgnukmn_ > sin cos lim sin cos cos sin lim cos sin sin cos lim ; lim cos cos lim ( cos ) cos ( cos ) lim cos cos ( cos ) lim lim ( cos ) erbah K> lim sin lim cos sin ( cos ) ducenh sin sin cos lm lim cos cos cos lim

31 CMBUkTI emeronti limiténgnukmn_ X> lim cos cos ( cos ) cos ( cos ) lim cos cos ( cos ) lim lim sin lim ducenh cos sin lim sin lim lim cos 5 cos cos 5 lim. sin - 6 -

32 CMBUkTI emeronti limiténgnukmn_ *-KNnalImItageRkam sin k> sin lim sin > sin cos lim sin (a b) K> ( a, b, a b lim a sina sinb sin lim sin cos X> g> lim( ) c> sin lim dmenahrsay KNnalImItageRkam sin k> sin lim sin 6 (sin )(sin ) lim (sin )(sin ) 6 sin lim sin 6

33 CMBUkTI emeronti limiténgnukmn_ ducenh > nig sin sin lim sin 6 sin cos lim sin sin cos sin lim erbah lim sin sin sin cos lim lim lim. sin cos lim. sin (a b) lim a ( a, b, a sina sinb (a b) (a b) sina sinb sin cos (a b) lim (a b) (a b) sin cos (a b) lim (a b) (a b) sin cos ducenh K> b ekman

34 CMBUkTI emeronti limiténgnukmn_ X> sin lim sin cos sin ( sin cos ) lim sin cos sin lim ( sin sin sin cos ) sin ( sin cos ) lim sin sin lim lim( sin cos ) sin sin lim. sin cos ducenh g> lim( ) lim ducenh lim lim( )

35 CMBUkTI emeronti limiténgnukmn_ c> sin lim sin ( ) lim sin lim lim( 6 sin lim ducenh 6. (-kmnt;gnukmn_dwerktibiredlbmebjlkçnðlimittamgbiragerkam - - ) f() f() lim (i) nig lim (ii) dmenahrsay kmnt;gnukmn_dwerktibir tag f() a b c a b c ekman lim a lim smmul a

36 CMBUkTI emeronti limiténgnukmn_ - - GnuKmn_Gacsresr c b f(). ekman f() lim ektaj lim f() c) b ( lim b c c b GnuKmn_Gacsresr b b f() b) )( ( ) b( ) )( ( f() ehiy b) )( ( lim f() lim b ) )( ( b) )( ( lim b b b lim ducenh f().

37 CMBUkTI emeronti limiténgnukmn_!-ekmanbhuekanniyt½carwkkñúgrgvg;edlman nrcugnigkamesµi a. tag S ncaépþrklaénbhuekanenh. KNna S n ryc lim S n dmenahrsay KNna S n ryc lim S n -mmup itrbs;bhuekan n n -RkLaépÞRtIekaNOAB S OAB a sin n -RkLaépÞrbs;BhuekaNniy½t nrcug S n n S OAB n a nsin n. n eyig)an lim S lim a nsin a lim n a n n n ducenh lim S a. n n n C O n a sin n B H A - -

38 CMBUkTI emeronti emeronti PaBCab;énGnuKmn_ PaBCab éngnukmn_ -PaBCab RtgḿYycMnuc niymn½y ³ GnuKmn_ y f() Cab;Rtg;cMnuc c kalna f bmebj lkçnðtamgbiducagerkam!- f kmnt;cmebah f manlimitkalna c #- lim f() f(c) c -lkçn³éngnukmn_cab ebi f nig g CaGnuKmn_Cab;Rtg; c enahek)an > f() g() CaGnuKmn_Cab;Rtg; c > f ().g() CaGnuKmn_Cab;Rtg; c f () > CaGnuKmn_Cab;Rtg; c Edl (c) g() g. - -

39 CMBUkTI emeronti PaBCab éngnukmn_ -PaBCabélIcenøa¼ niymn½y ³ -GnuKmn_ f Cab;elIcenøaHebIk ( a,b ) luhrtaet f Cab;cMeBaH RKb;témø éncenøahebienah. -GnuKmn_ f Cab;elIcenøaHbiT a,b ] ebik a,b ) [ luhrtaet f Cab; elicenøah ( nigmanlimit lim f() f(a) ; lim f() f(b) a b GnuKmn_ f Cab;Rtg; a agsþam Cab;Rtg; bageqvg -PaBCab éngnukmn_bnþak ebignukmn_ g Cab;Rtg; c niggnukmn_ f Cab;Rtg; g (c) enahgnukmn_bnþak; ( f g)() f g() Cab;Rtg; c. 5-Gnuvtþn_ GnuKmn_bnøaytamPaBCab ebi f CaGnuKmn_minkMnt;Rtg; a nigmanlimit lim f() L a enahgnukmn_bnøayén f tampabcab;rtg; a kmnt;eday f() ebi a g() L ebi a - -

40 CMBUkTI emeronti PaBCab éngnukmn_ 6-RTwsþIbTtémøkNþal RTwsþIbT ³ ebignukmn_ f Cab;elIcenøaHbiT [ a, b ] nig k CacMnYnmYyenA cenøah f (a) nig f (b) enahmancmnynbit c myyya:gtickñúg cenøahbit [ a, b ] Edl f(c) k. y ( c) : y f() f (b) k f (a) O a c b - 5 -

41 CMBUkTI emeronti PaBCab éngnukmn_ lmhat nig dmen¼rsay!-bba ak;fa etignukmn_agerkamcab;rtg;témø Edl[b et? k> f() 5 6, > f(), f(), f(), ebi f(), ebi ebi f() ebi, K> X> g> c> dmenahrsay siksapabcab;éngnukmn_ k> f() 5 6, man f() 9 kmnt; - 6 -

42 CMBUkTI emeronti PaBCab éngnukmn_ lim f() lim(5 6 ) 9 eday lim f() f() 9 nam[ f CaGnuKmn_Cab;Rtg; > f(), man () lim f() lim f kmnt;. eday lim f() f() nam[ f CaGnuKmn_Cab;Rtg; K> f(), man f() minkmnt; ducenh f CaGnuKmn_dac;Rtg; X> f(),. man f( ) minkmnt; ducenh f CaGnuKmn_dac;Rtg;

43 CMBUkTI emeronti PaBCab éngnukmn_ g> ebi f(), ebi man f() lim f() lim( ) nig lim f() lim eday lim f() lim f() enah f () KµanlImIt ducenh f () CaGnuKmn_dac;Rtg;. c> ebi f(), ebi man f( ) kmnt; lim f() lim f() lim lim ( ) lim ( ) eday lim f() lim f() ducenh f () CaGnuKmn_Cab;Rtg;

44 CMBUkTI emeronti PaBCab EdleFVI[GnuKmn_ageRkamCaGnuKmn_dac;. k> f() > K> X> g> c> 6 f() 5 f() f() 8 f() f() ebi ebi ebi ebi ebi dmenahrsay rktémø EdleFVI[GnuKmn_ageRkamCaGnuKmn_dac;. k> f() 6 GnuKmn_ f CaGnuKmn_dac;luHRtaEt

45 CMBUkTI emeronti PaBCab éngnukmn_ > f() 5 GnuKmn_ f CaGnuKmn_dac;luHRtaEt 5 ektajb s ; 5. K> f() GnuKmn_ f CaGnuKmn_dac;luHRtaEt ektajb s ;. X> f() 8 GnuKmn_ f CaGnuKmn_dac;luHRtaEt ektajb s 6 ; g> f(). ebi ebi ( d ) : y, tag nig ( d ) : y, - -

46 CMBUkTI emeronti PaBCab éngnukmn_ y ( d ) ( d ) - tamrkahvikeyigexijfarkb; IR, f CaGnuKmn_dac;. c> f() ebi ebi ebi edahrsayduclmhat;ti g ek)anrkb; IR GnuKmn_ f CaGnuKmn_dac;. - -

47 CMBUkTI emeronti PaBCab éngnukmn_ #-siksapabcab;éngnukmn_agerkamelicenøahedl[. k> f() elicenøah (,) nig [,] > f() ( ) elicenøah (,) nig [,] ( ) ebi K> f() 9 ebi elicenøah (, ) nig [, ] dmenahrsay siksapabcab;éngnukmn_ k> f() elicenøah (,) nig [,] -elicenøah (,) eyigexijfagnukmn_ f kmnt;)ancanic ducenhgnukmn_ f CaGnuKmn_Cab;elIcenøaH (, ). -elicenøah [, ] GnuKmn_ f mingackmnt;)ancanic et erbah 7 ebi enah f( ) minkmnt;. ducenh f CaGnuKmn_dac;Rtg; nigdac;kñúgcenøah, ] - - [.

48 CMBUkTI emeronti PaBCab éngnukmn_ > f() ( ) elicenøah (,) nig [,] -elicenøah (, ) eyigexijfa f() ( ) kmnt; ducenh f CaGnuKmn_Cab;elIcenøaH (,). -elicenøah [, ] eyigexijfa f CaGnuKmn_dac;Rtg; erbahgnukmn_ f minkmnt;rtg;. ducenh f CaGnuKmn_dac;elIcenøaH [, ]. ( ) ebi K> f() 9 ebi elicenøah (, ) nig [, ] ebi enah f() ( ) 6 kmnt; man lim f() lim ( ) 6 9 nig lim f() lim lim( ) 6 eday lim f() lim f() 6 lim f() 6 f() ducenh f CaGnuKmn_Cab;elI IR. ehtuenhvacagnukmn_cab; elicenøah (, ) nig [, ]. - -

49 CMBUkTI emeronti PaBCab éngnukmn_ $-rktémø A EdleFVI[ f () Cab;RKb;témø A ebi k> > f() f() A ebi ebi ebi dmenahrsay rktémø A EdleFVI[ f () Cab;RKb;témø A k> ebi f() ebi edim,i[ f () Cab;RKb;témø luhrtaet lim f() lim f() f() ek)an lim(a ) lim( ) A A. ducenh 6 A 8 9 A 6 - -

50 CMBUkTI emeronti PaBCab éngnukmn_ > f() edim,i[ () lim f() ebi A ebi f Cab;RKb;témø luhrtaet lim f() f() ek)an lim( ) lim(a ) ducenh A 6A 8 6A 5. A - 5 -

51 CMBUkTI emeronti PaBCab éngnukmn_ %-rktémø A nig BEdleFVI[GnuKmn_kMNt;eday A 5 9 ebi f() B ebi Cab;RKb;témø. ( )(A ) ebi dmenahrsay rktémø A nig B edim,i[ f () Cab;RKb;témø luhrtaet lim f() ek)an lim f() lim(a A 5 9 f() 5 9) (A ) lim( B )(A ) B A A B A, B. ektaj - 6 -

52 CMBUkTI emeronti PaBCab éngnukmn_ ^-kñúgbnþagnukmn_agerkam eti f GacmanbnøaytamPaBCab; Rtg; a b et? 9 k> f(), a. f(), a. f(), a. 6 f(), a > K> X> dmenahrsay rkgnukmn_bnøaytampabcab; 9 k> f(), a. 9 lim f() lim ekman lim( ) 6 ducenhekgacbnøay f tampabcab;rtg; a. ebi g CaGnuKmn_bnøaytamPaBCab;enaHGnuKmn_ g kmnt;eday - 7 -

53 CMBUkTI emeronti PaBCab éngnukmn_ f() 9 ebi ebi 6 f(), a. lim f() lim lim( 5) a. > ekman 7 ducenhekgacbnøay f tampabcab;rtg; ebi g CaGnuKmn_bnøaytamPaBCab;enaHGnuKmn_ g kmnt;eday ebi f() 7 ebi K> f(), a. ekman lim f() lim lim( ) ducenhekgacbnøay f tampabcab;rtg; a. ebi g CaGnuKmn_bnøaytamPaBCab;enaHGnuKmn_ g kmnt;eday ebi f() ebi - 8 -

54 CMBUkTI emeronti PaBCab éngnukmn_ 6 X> (), a ekman f 6 lim f() lim ( )( ) lim ( )( ) lim ducenhekgacbnøay f tampabcab;rtg; a. ebi g CaGnuKmn_bnøaytamPaBCab;enaHGnuKmn_ g kmnt;eday 6 ebi f() ebi - 9 -

55 CMBUkTI emeronti PaBCab éngnukmn_ &-erbirtwsþibttémøknþal bgðajfa GnuKmn_ageRkammancMnYn c kñúgcenøahedl[. k> f(), [,5], f(c). > f() 6 8, [, ], f(c). K> f(), [, ], f(c). 5 X> f(), [, ], f(c) 6 dmenahrsay karbgðaj k> f(), [,5], f(c). ekman f() nig f(5) eday f() f(c) f(5) 9 ducenhtamrtwsþibttémøknþaly:agehacnas;man c myyén [,5] Edl (c) f

56 CMBUkTI emeronti PaBCab éngnukmn_ > f() 6 8, [, ], f(c). ekman f() 8 nig f() eday f() f(c) f() 8 ducenhtamrtwsþibttémøknþaly:agehacnas;man c myyén [,] Edl (c) f. K> f(), [, ], f(c). ekman f() nig f() eday f() f(c) f() 9 ducenhtamrtwsþibttémøknþaly:agehacnas;man c myyén [,] Edl f(c). f() 5 X>, [, ], f(c) 6 ekman eday f( f( ) nig 5 6 f() ) f(c) 6 f() ducenhtamrtwsþibt 6 5 [, Edl f(c) 6. témøknþaly:agehacnas;man c myyén ] - 5 -

57 CMBUkTI emeronti PaBCab éngnukmn_ *-ek[gnukmn_ nig cenøahbitducagerkam.erbirtwsþibttémøknþal rktémø c ebieksáal;témø k. k> f(), [,], k. > f() 5, [ 5, ], k. dmenahrsay rktémø c k> f(), [,], k. edahrsaysmikar f(c) k Edl c [, ] ek)an c c c c, 5 5 ektajb s, c b 5 c 5 c. f() 5, [ 5, ], k. ducenh > cemøiy c sissedahrsayøünég

58 CMBUkTI emeronti PaBCab éngnukmn_ (-k>rsaybba ak;fasmikar tan cos y:agehacnas;man b sbitmyycenøah [, ]. >RsaybBa ak;fasmikar ( n )cos sin y:agehacnas;manb sbitmyycenøah (, ). dmenahrsay k>rsaybba ak;fasmikar tan cos y:agehacnas;man b sbitmyycenøah [, ]. taggnukmn_ f() tan cos ekman f() nig f( ) ) eday f() f( ) ( enahtamrtwsþibttémø knþaly:agehacnas;mancmnynbit c myyén [, ] Edlf (c). ducenhsmikar tan cos y:agehacnas; manb sbitmyycenøah [, ]

59 CMBUkTI emeronti PaBCab éngnukmn_ >RsaybBa ak;fasmikar ( n )cos sin y:agehacnas;manb sbitmyycenøah (, ). n taggnukmn_ f() ( )cos sin ekman f() nig f() sin sin eday f() f() enahtamrtwsþibttémø knþaly:agehacnas;mancmnynbit c myyén [, ] Edl (c) f. ducenhsmikar ( n )cos sin y:agehacnas;manb sbitmyycenøah [, ]

60 CMBUkTI emeronti PaBCab éngnukmn_ lmhatćmbukti nig dmen¼rsay!-knnalimitagerkam k> lim( ) > lim( ) sin K> X> lim sin sin lim cos dmenahrsay KNnalImItageRkam k> lim( ) lim lim lim erbah.

61 CMBUkTI emeronti PaBCab éngnukmn_ > ) lim( lim ) ( ) ( lim lim lim ducenh ) lim( K> sin sin lim.sin sin sin lim ) sin ( sin ) ( lim ducenh sin sin lim.

62 CMBUkTI emeronti PaBCab éngnukmn_ X> sin lim cos ( sin ) cos lim cos ( sin ) cos lim sin sin lim sin sin cos -ebi enah sin sin ek)an lim sin cos lim sin ebi enah sin sin ek)an lim sin cos lim sin cos cos

63 CMBUkTI emeronti edrievéngnukmn_ emeronti edrievéngnukmn_ -edrievéngnukmn_rtg niymn½y ³ edrievéngnukmn_ y f() CalImIt ebiman énplefob y kmenin kalna itetacit. ekkmnt;sresr y' y lim f() lim -edrievéngnukmn_bnþak ebi y f(u) nig g() dy dy du y' u' d du d d f[u()] f'(u) u'( d f( u enahek)an b ) ) f( lim h h) f( h )

64 CMBUkTI emeronti edrievéngnukmn_ -edrievéngnukmn_rtiekanmart!> y sin y' y cos y' sin #> y tan y' tan $> y cot y' %> y sinu y' u'cosu ^> y cosu y' sin u u' &> tanu y' y cos ( cot sin cos u u' y y' sin u *> cotu -edrievlmdab <s ebignukmn_ y f() manedrievbnþbnþab;dl;lmdab; n (n) (n) enah y f () ehafaedrievti n éngnukmn_ y f() ( n ) d (n) ehiy f () f (). d )

65 CMBUkTI emeronti edrievéngnukmn_ lmhat nig dmen¼rsay!-knnaedrievéngnukmn_agerkam k> f() ( ) > f() K> f() ( 8) X> f() ( 7 9) g> () c> f() f 5 6 dmenahrsay KNnaedrIevénGnuKmn_ageRkam k> f() ( ) n n tamrubmnþ (u )' n.u'.u eyig)an f'() ( )'( ) 8( ) ducenh f'() 8( ) > f() 5 6 tamrubmnþ ( u)' u' u (6 5 )

66 CMBUkTI emeronti edrievéngnukmn_ eyig)an 5 )' (5 '() f K> 8 5 8) ( () f eyig)an ) 8)'( 8( '() f ) 8)( 8(5 8) 8)( 8(5 X> 5 9) 7 ( () f eyig)an 9) 7 9)'( 7 5( '() f 9) 7 7)( (6 9) 7 )( 5( g> 6 5 () f eyig)an ) 6 (5 )' 6 (5 f'() ) 6 (5 6

67 CMBUkTI emeronti edrievéngnukmn_ c> f() (6 5 ) ekgacsresr f() (6 5 ) ek)an f'() (6 5 )'(6 5 ) ( 5) (6 5 k> y > K> y X> y 5 y 5 g> y ( ) ( ) c> y ( ) (5 ) dmenahrsay k> y ekgacsresr y ( ) ek)an y' ( )'( ) ( ) - 6 -

68 CMBUkTI emeronti edrievéngnukmn_ > 5 y ekgacsresr ) (5 y ek)an ) )'(5 (5 y' ) (5 5 K> y ek)an )'. ( y' ) )( ( ) (. ) ( ) ( ) ( X> y ek)an )' ( ' y 5 ) ( ) (. ) ( ) ( ) (

69 CMBUkTI emeronti edrievéngnukmn_ 5 g> y ( ) ( ) 5 ek)an y' ( ) ( ) ( ) ( ) ( ) ( ) (6 7)( ) (6 ( ) ) c> y ( ) (5 ) eyig)an y' ( ) (5 ) (5 )( ) (5 )( ) ( ) (5 )(5 )( ) #-KNnaedrIevénGnuKmn_ f () tamvifibiry:agkw tamrubmnþedriev énplkun nigedrievénsv½ykun rycbgðajfavifitamgbirman ltæplduckña k> f() ( 5) > f() (7 ) dmenahrsay k> f() ( 5) ekgacsresr f() ( 5)( 5) - 6 -

70 CMBUkTI emeronti edrievéngnukmn_ tamrubmnþ ek)an f'() ( uv)' u' v v' u ( 5)'( 5) ( ( 5) ( 5) ( 5) n n müa:getotedayerbirubmnþ (u )' nu'.u ek)an f'() ( 5)'( 5) 5)'( 5) 6( 5) > f() (7 ) ekgacsresr f() (7 )(7 ) ek)an f'() (7 )'(7 ) (7 )'(7 ) (7 ) (7 ) 8(7 ) n n müa:getotedayerbirubmnþ (u )' nu'.u ek)an f'() (7 )'(7 ) 8(7 )

71 CMBUkTI emeronti edrievéngnukmn_ $-KNnaedrIevénGnuKmn_ageRkam k> y sin cos K> y sin g> y tan > y 5 sin cos X> y sin cos c> y cot dmenahrsay KNnaedrIevénGnuKmn_ageRkam k> y sin cos eday sin cos ek)an y y' > y 5 sin cos ekman ek)an sin y y' 5 sin cos sin 5 cos K> y sin y' ( sin)'( sin) ( cos)( sin)

72 CMBUkTI emeronti edrievéngnukmn_ X> y sin cos y' (sin )' ( )'cos cos cos ( )cos g> y tan y' (tan )'tan ( tan ( tan tan c> y cot y' ()'cot cot sin sin )tan ) (cot 6( cot (cos)' )' )cot cot (cot cot )

73 CMBUkTI emeronti edrievéngnukmn_ %-KNnaedrIevénGnuKmn_ageRkam k> sin y > y cos 5 tan K> y X> y sin( 5) cos g> y sin cos( ) c> y tan( 5) dmenahrsay KNnaedrIevénGnuKmn_ageRkam k> y sin ek)an y' ( )'cos sin cos sin > y cos 5 ek)an y' sin sin 5cos5 K> 5sin tan y cos ( tan )( cos ) sin tan y' ( cos )

74 CMBUkTI emeronti edrievéngnukmn_ X> y sin( 5) y' ( 5)'cos( ( 5)cos( g> y sin cos( ) y' cos (cos c> y tan( 5) ( 5)' y' cos ( 5) 6 5 cos ( 5) sin( sin( 5) 5) ) ))

75 CMBUkTI emeronti edrievéngnukmn_ éngnukmn_agerkam k> y sin > y K> y X> y g> y ( ) c> y ( ) dmenahrsay éngnukmn_agerkam k> y sin ek)an y' y' 6 cos ( cos ) ehiy y'' ( sin ) > y ek)an y' ehiy y'' 6-7 -

76 CMBUkTI emeronti K> y ek)an y' ehiy y'' X> y ekgacsresr y ( ek)an y' ( ) y''. ( ) ehiy ) g> y ( ) ek)an y' 6( ) ehiy y'' 6( ) ( ) 6( 6( c> y ( ) y'' )( )(5 ) ( )( ). ) edrievéngnukmn_ - 7 -

77 CMBUkTI emeronti edrievéngnukmn_ &-rk y ' CaGnuKmn_én nig y k> tan y > sin y K> y sin y X> sin y y g> tan(y) c> y sin cos dmenahrsay rk y ' CaGnuKmn_én nig y k> tan y efviedrieliggátamgbirek)an ()' (tan y)' y' y' cos y > sin y efviedrieliggátamgbirek)an ()' (sin y)' y'cos y y' cos cos y y - 7 -

78 CMBUkTI emeronti edrievéngnukmn_ K> y sin y efviedrieliggátamgbirek)an (y sin y)' y y' y'cos y y'( cos y) X> sin y y efviedrieliggátamgbirek)an ( sin y)' (y)' y'cos y y y' y y' y'(cosy ) y y' g> tan(y) efviedrieliggátamgbirek)an ( tan(y))' (y)' cos (y) y y' cos (y) y' y cos y cos y y cos y (y) - 7 -

79 CMBUkTI emeronti edrievéngnukmn_ c> y sin cos ekman sin cos ek)an y sin efviedrieliggátamgbirek)an (sin cos sin yy' sin cos ektaj sin y'. y sin ) sin cos - 7 -

80 CMBUkTI emeronti edrievéngnukmn_ *-rk y ' nig y '' CaGnuKmn_én nig y k> y K> y y > y 8 X> y y g> y y c> y y dmenahrsay rk y ' nig y '' CaGnuKmn_én nig y k> y efviedrieliggátamgbirek)an yy' y' y ehiy y'' ducenh y' y y' y y ( ) y y y y y'' y y nig y

81 CMBUkTI emeronti edrievéngnukmn_ > 8 y efviedrieliggátamgbirek)an y y' y'y 6 ehiy y yy' y y'' 5 y 8 y y ) y ( y y y' y ducenh y ' y nig 5 y 8 y y'' K> y y efviedrieliggátamgbirek)an y y y' y) ( y)y' ( yy' y' y

82 CMBUkTI emeronti edrievéngnukmn_ ehiy y) ( y) y')( ( y) y')( y'' y) ( ) y y 6( y) ( y 6 6y y y) ( ) y y ( y y) ( y' y y) ( yy' y' y yy' y' y eday y y ducenh y) ( 6 y''.

83 CMBUkTI emeronti edrievéngnukmn_ (-KNna k> f ) () f() > f 6) () f() 7 sin 7 K> f 8) () 8 f() 5 cos ( ) X> f () ebi f() 6 dmenahrsay ( ) k> f () man f() ek)an f'() 6 f''() f f () () ( 6) > f () man f() ek)an () () f'() 7 6 sin cos

84 CMBUkTI emeronti edrievéngnukmn_ 5 f''() 6 sin () f () 8cos () f () 6sin (5) f () 6 cos f (6) ( 8) K> f () man f() ek)an () 7 6sin f'() f f f f f f () () 8 f''() () () (5) (6) (7) (8) cos sin 6 cos sin cos () sin () cos () sin () 8! cos ( ) X> f () ekman f() 6 ek)an f() sissedahrsayøünég 6 5!

85 CMBUkTI emeronti edrievéngnukmn_ emeronti Gnuvtþn_énedrIev -el,ónénclna niymn½y ³ el,ónénclnamyyenan³ t KW V(t) S'(t) Edl S (t) Cacm ayenan³ t. -smtu¼énclna dv(t) smtuhénclnamyyenanh t KW a(t) V'(t) dt Edl V (t) Cael,ÓnénclnaenAN³ t. ds dt - 8 -

86 CMBUkTI emeronti edrievéngnukmn_ lmhat nig dmen¼rsay!-bira:mitctumuniy½t SABCD manépþrklaag S. t KNnapleFobrgVas;km<s; SO nigrcugmyyén)atedim,i[mad BIra:mItctumuniy½tmantémøGtibrma. dmenahrsay KNnapleFobrgVas;km<s; SO nigrcugmyyén)at S tag AB, CaRbEvg RCug)at ehiyyk I CacMnuckNþal én [ AB] nam[ SICaGab:UEtm. ek)anrklaépþagrbs;bir:amit S L.SI SI RtIekaN SOI EkgRtg;O ek)ansi SO OI S L b SO S L D C O A I B - 8 -

87 CMBUkTI emeronti edrievéngnukmn_ S S L L SO SO Edl S L b S Et L ehtuenh S. L tag V CamaDrbs;BIr:amIt SABCD ek)an V() S SO ABCD V() S L 6 S L V'() 6 S eyig)an ebi V'() L S L S L V '() + Gtibrma V () S L S L _ - 8 -

88 CMBUkTI emeronti edrievéngnukmn_ tamtaragagelieyigexijfa V () mantémøgtibrmaluhrta Et S L. S L SO S L eday SO b SO S L S L. S L ducenhplefobrgvas;km<s; nigrcug)aténbira:mitkw

89 CMBUkTI emeronti nig rgvg;myymanrbevg L. KNnapleFobrvagrgVas;kaMrgVg; nig RCugkaeredIm,I[plbUk épþrklakaernigrgvg;mantémøgb,brma? dmenahrsay KNnapleFobrvagrgVas;kaMrgVg; nig RCugkaer a R tag a CargVas;RCugénkaer nig R CargVas;kaMénrgVg; eday plbukbrimartkaermyy nig rgvg;myymanrbevg L enah L R ek)an a R L a () tag CaplbUképÞRkLakaernigrgVg;. ek)an a R () - 8 -

90 CMBUkTI emeronti edrievéngnukmn_ yk ) ( CMnYskñúg ( ) ek)an L R ( R) R L LR R 6 L R ek)an '(R) 6 ebi '(R) enah L R R L 5 mü:agetoteday ''(R) nam[ (R) man témøgb,brmartg; tam ) ek)an ( ek)an R a 6 L R. L L a L 5. L 5 L 5 ducenhplefobrvag rgvas;kamrgvg; nig RCugkaerKW

91 CMBUkTI emeronti edrievéngnukmn_ #-RbGb;Rtg;mYymanKRmbelI nig )aterkamcakaerehiymanmad 5 cm. smöarhsrmab;efvikrmbeli nig)aterkamtémø erolkñúgmyysgþiem:rtkaer ehiysmöarhsrmab;efvi épþagmantémø erolkñúgmyysgþiem:rtkaer. kmnt;rgvas;rtnugénrbgb;edim,i[r)ak;cmnayelismöarhman témøgb,rma rycknnar)ak;cmnaygb,brmaenah? dmenahrsay kmnt;rgvas;rtnugénrbgb; nig R)ak;cMNayGb,brma tag CaRCug)at nig y CaRTnugag madrbs;rbgb;kw V y 5 5 ektaj y () R)ak;cMNaysmÖarHeFVIKRmb nig )at rbs;rbgb;kw R)ak;cMNaysmÖarHeFVIépÞagrbs;RbGb;KW y y. y

92 CMBUkTI emeronti edrievéngnukmn_ tag T CaGnuKmn_R)ak;cMNaysrubelIsmÖarHsRmab;eFVIRbGb; ek)an T y ( y ) () yk ( ) CMnYskñúg ( ) ek)an 65 T() ( ) 5 T'() ( 5 5 ebi '() 5 ek)an ) T enah 6 ektaj ekman T''() ( ) 6 cmebah 5 T''(5 ) nam[ T () mantgb,brma 5 5 Rtg; 5 ehiy y 5. 5 ducenhrtnugrbs;rbgb;kw 5 cm. KNnaR)ak;cMNayGb,brma cmebah 5, y 5 ek)an T (5 5), erol CaR)ak;cMNayGb,brma

93 CMBUkTI emeronti edrievéngnukmn_ $-TUkmYycab;epþImecjdMeNIrBIcMnucRtYtBinitü EdlryHeBl t natierkaymktukenahmancm aybicmnucrtytbinitüedltageday GnuKmn_ S(t) t 6t KitCaEm:Rt. k> rkel,ónéntukrtg;cmnuccab;epþim? > kmnt;el,ónéntuknh t mn? dmenahrsay k> rkel,ónéntukrtg;cmnuccab;epþim tag V (t) CaGnuKmn_el,ÓnénTUkenAN³ t enahek)an V(t) ds(t) dt S'(t) t 6 ebit V(t ) 6m / mn ducenhel,ónéntukrtg;cmnuccab;epþimkw > kmnt;el,ónéntuknh t mn ebi mn V() () V 6m / mn. t enah 6 87m / mn ducenh el,ónéntuknh t mn KW 87 m / mn

94 CMBUkTI emeronti edrievéngnukmn_ %-rfynþmyycab;epþimecjdmeniredayel,ónedltageday t GnuKmn_ V(t) (m / s) t 5 kmnt;smtuhénrfynþnhebl k> 5s t K> t s > t s dmenahrsay k> t 5s tag (t) dv(t) 5 a(t) V'(t) dt (t 5) 5 ebi t 5s a(5).75m / s 5 > ebi t s a().m / s 65 5 t s a().m / s 5 a CaGnuKmn_sMTuHénrfynþenAN³ t enahek)an K> ebi

95 CMBUkTI emeronti edrievéngnukmn_ ^-rfynþmyyepþimecjdmeniredayel,inedl t vinatitageday t GnuKmn_ V(t) Em:RtkñúgmYyvinaTI..t. kmnt;smtuhrbs;rfynþenan³ebl ³ k- 5vinaTI -vinati K- vinati X- vinati. dmenahrsay kmnt;smtuhrbs;rfynþenan³ebl t man V(t) t t.t. t t 5 a(t) V'(t) (t 5) 5 t 5s a(5).m / s 5 5 t s a().8m / s 65 5 t s a().5m / s 5 5 t s a().m / s 5 ek)an k- - K- X- t t 5-9 -

96 CMBUkTI emeronti edrievéngnukmn_ lmhatćmbukti nig dmen¼rsay!-knnaedrievéngnukmn_agerkam k> f() (7 ) > g() ( 8) K> y ( 9 ) X> f() 5 5 g> g() ( ) c> y dmenahrsay KNnaedrIevénGnuKmn_ k> f() (7 ) f'() (7 )'(7 ) (7 > g() ( 8) g'() ( ( 8)'( 8) K> y ( 9 ) y' ( 8) 9 )'( (8 9)( 9 ) 9 ) ) - 9 -

97 CMBUkTI emeronti edrievéngnukmn_ X> ) (5 5 () f ) (5 5 ) )'(5 (5 f'() g> ) ( () g ) 5 )( ( ) ( ) ( )' ) (( ) ()'( g'() c> 5 y 6 5 )' ( )' 5 ( y' ) 6(

98 CMBUkTI emeronti k> y sin y sin K> y cos( ) X> f() g> > cos f() tan c> f() cos(sin ) dmenahrsay KNnaedrIevénGnuKmn_ k> y sin y' sin (sin > y cos y' 9 cos K> y cos( ) y' cos( sin f() X> 6 f'() cos cos ) sin ) 6 sin( sin cos sin ) sin 6 sin

99 CMBUkTI emeronti edrievéngnukmn_ g> tan f() ( ) tan f'() cos ( ) ( ( c> f() cos(sin ) f'() (sin ) sin ) cos )'sin(sin sin 6 sin(sin ) ) 6sin cos sin(sin ) - 9 -

100 CMBUkTI emeronti edrievéngnukmn_ k> f() 8 > f() 6 9 K> f() ( 7) X> f() 5 g> 7 f() c> f() dmenahrsay KNnaedrIevTI@énGnuKmn_ k> f() 8 ek)an f'() 8 nig f''() 6 8 ( ) > f() 6 9 ek)an f'() 9 9 nig f''()

101 CMBUkTI emeronti edrievéngnukmn_ K> f() ( 7) ek)an f'() ( 7)'( 7) ( 8 f''() ( 8 7) 8 ehiy 7) ( 7)'( 7) 8 ( 7) f() 5 f'() 5 ( ) f''() ( ) f() f'() ( ) 8 f''() ( ) 7 f() 7 X> ek)an nig g> ek)an nig ( 7) c> ek)an f'() 5 6 nig f''().

102 CMBUkTI emeronti edrievéngnukmn_ $-rk y ' CaGnuKmn_én nig y k> y 5 > y K> y X> y y 5 g> (y ) 9 c> y y dmenahrsay rk y ' CaGnuKmn_én nig y k> y 5 ek)an ( y )' (5)' y'y y'. y ektaj > y ek)an (y )' ( )' ektaj y y'y y y'. y

103 CMBUkTI emeronti edrievéngnukmn_ K> y ek)an ( y)' ()' y' y y' y. ektaj X> y y 5 ek)an (y y)' ektaj y' (5 (y)' y )' y y' y' y y y y'. y g> (y ) 9 ek)an 8y'(y ) y'. (y ) c> y y ek)an y y' y y'y 9 9 y y ektaj y'. y

104 CMBUkTI emeronti edrievéngnukmn_ %-rk y ' nig y '' CaGnuKmn_én nig y k> y 5 > y K> y 6 y g> X> y y c> y dmenahrsay rk y ' nig y '' CaGnuKmn_én nig y k> y 5 ek)an y y' ehiy y' y' y'' ( y') ektaj y'' eday y y' y' ( y ) ( ek)an y'' ( y) ducenh y''. y y)

105 CMBUkTI emeronti edrievéngnukmn_ - - > y ek)an y'y y b y y y y y' yy' y ehiy y y' y y' y y'' y y y y y y y y y y y y y. y y y y y' y y y y y y' y y' y ducenh y y y y''. K> 6 y ek)an y y' y'y

106 CMBUkTI emeronti edrievéngnukmn_ y y y' y y ehiy y'' y y y X> y y ek)an y y' y' y'. y'( ) y ehiy y'' ( ) ( y) y''. ( ) ducenh g> y ek)an ehiy y'y y''y y' 9 6. y' y y'' y ( ) ( ) 6 y' y 6 y y b y'' 9. y 8y c> y nam[ yy' y' y y' ehiy y''. y y y y y - -

107 CMBUkTI emeronti edrievéngnukmn_ ^-RtIekaNsm)at ABC myymanbrimart p vilcmuvijkm<s; EdlKUsecjBIkMBUl A kmnt;)anlulitmyymanmadfmbmput. KNnargVas;RCug nig km<s;énrtiekansm)atenah? dmenahrsay KNnaRCug nig km<s;énrtiekansm)at A tag AB AC nig AH h Edl ; h ek)an p BC b BC (p ). h RtIekaNsm)at ABC vilcmuvij km<s; AH begáit)ancasulid B H brivtþn_myykwcaekanedlman km<s; AH CenRt AB AC nigfas)atcargvg;mankam BC BH HC p. yk V CamaDrbs;ekaN nig SCaépÞRkLafas)at. C - -

108 CMBUkTI emeronti edrievéngnukmn_ eyig)an V S h eday S (p ) kñúgrtiekanekg AHBeKman AB AH HB ektaj h (p ) p p nam[ h p p Edl p p b ek)an V (p ) p p p p p p ' p ;. 5 p 5 8p eyig)an V'. ebi V p. p V '() p p 5 + _ Gtibrma V () - -

109 CMBUkTI emeronti edrievéngnukmn_ tamtaraggefrpabageli CamYynwglkçNÐkMNt;eK)an p CatémøEdleFVI[ V mantémøgtibrma. cmebah ehiy ducenh 5 p p p BC (p ) p p 5 AH h p p p 5 5 p p p 5 AB AC, BC, AH &-rfynþmyycab;epþimecjdmenirbicmnucrtytbinitümyy. enanh t vinatirfynþenahxøatbicmnucrtytbinitüedltagedaygnukmn_ cm aycr S(t).t KitCaEm:Rt ehiy t. erbignukmn_agelirycbmebjtaragagerkam $ ^ *! S (t) V (t) a (t) - -

110 CMBUkTI emeronti edrievéngnukmn_ dmenahrsay bmebjtarag ekman S(t).t KitCaEm:Rt ehiy t ek)an V(t) S'(t) 8.8t nig a(t) S''(t) 8. 8 efr. $ ^ *! S (t)!&>^ $$ V (t)!&>^ #%>@ %@>* &>$ ** a (t) *>* *>* *>* *>* *>* *>* - 5 -

111 CMBUkTI emeronti GnuKmn_sniTan emeronti GnuKmn_sniTan -sikßagnukmn_ y a b c p q Edl p, a nig b c EdnkMnt; D IR { } a q p TisedAGefrPaB ap aq -edriev y' -ebi ' nig Gb,brmaeFobmYy. bq cp (p q) cmebah y manb sbiruskñaenahgnukmn_mangtibrmaefobmyy -ebi y' manb sdúbenahgnukmn_køaycadwerktimyy -ebi y' Kµanb senahgnukmn_kµanbrmaet GasIumtUt q -bnþat; CaGasIumtUtQr p q p

112 CMBUkTI emeronti GnuKmn_sniTan -GnuKmn_Gacsresr y p q Edl lim enahbnþat; y CaGasIumtUteRTt p q manrkabcagiuebbuledlmanp itqøúhcacmnucrbsbvrvaggasiumtu TaMgBIr. a b c -sikßagnukmn_ Edl y p q r p, a, b, c, p, q, r IR. mangasiumtutedkmyycanic cmnyngasiumtutqrkwgars½ynwgb ssmikar q r -ebi q pr KµanGasIumtUtQreT nigrkabmanemk EtmYy. -ebi q pr mangasiumtutqrmyy p nigrkabmanemkbir. -ebi q pr mangasiututqrbirkw nig q ehiyrkabmanemkbi. p q p q p - 7 -

113 CMBUkTI emeronti GnuKmn_sniTan lmhat nig dmen¼rsay!-rkgasiumtuténgnukmn_agerkam k> y > K> y y dmenahrsay rkgasiumtuténgnukmn_agerkam k> y ek)an ( ) ( ) y ( ) ( ) lim y lim( ) lim eday nam[bnþat; CaGasIumtUtQr ehiy nam[bnþat; y CaGasIumtUteRTt.

114 CMBUkTI emeronti GnuKmn_sniTan > y ekman y ( )( ) lim ( )( ) eday nig lim ( )( ) ducenhbnþat;, CaGasIumtUtQr. ehiy lim nam[bnþat; y CaGasIumtUtedk K> y ekman lim nam[bnþat; CaGasIum tutqr nig lim nam[bnþat; y CaGasIumtUteRTt

115 CMBUkTI emeronti nig sg;rkabéngnukmn_agerkam 6 k> y > 9 y K> y dmenahrsay siksagefrpab nig sg;rkabéngnukmn_agerkam 6 k> y > EdnkMNt; D IR { } 9 > sresrcaragkanunic y > TisedAGefrPaB 9 ( ) 9 ( )( -edriev y' ( ) ( ) ( ) ebi y', -brmaéngnukmn_ y ) - -

116 CMBUkTI emeronti GnuKmn_sniTan cmebah GnuKmn_manGtibrmaeFob y f( ) 5 cmebah GnuKmn_manGb,brmaeFob y f(). -KNnalImIt 9 lim 9 lim 9 lim -GasIumtUt eday enahbnþat; ehiy lim 9 9 lim CaGasIumtUtQr. enahbnþat; y CaGasIumtUteRTt. - -

117 CMBUkTI emeronti GnuKmn_sniTan -taraggefrpab y ' y 5 >smng;rkab -cmnucrbsbvrvagrkabcamyygks½kugredaen ebi enah y nam[rkabkat;gks½ ( oy) Rtg; (, ) ebi y 6 ( ) 5 Kµanb s nam[rkabminkat;gks½ ( o) et. -p itqøúh GasIumtUtQr RbsBVCamYyGasIumtUteRTt y Rtg;cMnuc I(, ) KWCap itqøúhénrkabtag y 6 - -

118 CMBUkTI emeronti GnuKmn_sniTan y (c) : y 6 y > y >EdnkMnt; D IR { ; } ( )( ) > ekman y >TisedAGefrPaB ( )( ) - -

119 CMBUkTI emeronti GnuKmn_sniTan -edriev y' nam[ y CaGnuKmn_ekIn. -KNnalImIt ( ) lim nig lim -GasIumtUt eday lim nam[bnþat; y CaGsIumtUtedk ehiy lim nam[bnþat; CaGsIumtUtQr. -taraggefrpab y ' y >smng;rkab -ebi o enah (. y nam[rkabkat; ( oy) Rtg;, ) - -

120 CMBUkTI emeronti GnuKmn_sniTan -ebi y nam[rkabkat; ( o) Rtg; ( ; ). -p itqøúh GasIumtUtQr kat;gasiumtutedk y Rtg; I (, ) Cap itqøúhénrkab. y y y - 5 -

121 CMBUkTI emeronti GnuKmn_sniTan 9 K> y >EdnkMNt; D IR { ; } >TisedAGefrPaB -edriev y' ( ) -ebi y' 9 -brmaén y cmebah GnuKmn_manGtibrma -KNnalImIt lim 9 lim 9 9 lim lim ( )( ) 9 9 lim lim ( )( ) 9 9 lim lim ( )( ) 9 9 lim lim ( )( ) y 9-6 -

122 CMBUkTI emeronti GnuKmn_sniTan -GasIumtUt 9 9 eday lim nig lim nam[bnþat; nig CaGasIumtUtQr. 9 ehiy lim nam[bnþat; y CaGasIumtUtedk -taraggefrpab y ' y 9 9 >smng;rkab y -kugredaencmnucrbsbvrvagrkabcamyygks½kugredaen 9 ebi enah y nam[rkab ( c) kat;gks½ ( oy) Rtg;cMnuc 9 mankugredaen ( ; )

123 CMBUkTI emeronti GnuKmn_sniTan ebi y ek)an 9, ducenhrkabkat; ( o) Rtg; (,) nig (, ). -GkS½qøúH 9 GnuKmn_ y CaGnuKmn_KU. ducenhgks½ ( oy) CaGkS½qøúHrbs;Rkab. y y y

124 CMBUkTI emeronti GnuKmn_sniTan #-k> siksagefrpab nig sg;rkabéngnukmn_ y > siksaetatamtémø m GtßiPaB nig sbaøaénb ssmikar (m ) m 8 edayerbirkabtag y. 8 dmenahrsay k> siksagefrpab nig sg;rkabéngnukmn_ y 8 > EdnkMNt; D IR { } > sresrcaragkanunic y > TisedAGefrPaB ( ) -edriev y' ( ) ( ) ebi y', -brmaéngnukmn_ y cmebah GnuKmn_manGtibrmaeFob y f() cmebah GnuKmn_manGb,brmaeFob y f()

125 CMBUkTI emeronti GnuKmn_sniTan -KNnalImIt lim lim lim -GasIumtUt eday lim enahbnþat; CaGasIumtUtQr. ehiy lim enahbnþat; y CaGasIumtUteRTt. -taraggefrpab y ' y - -

126 CMBUkTI emeronti GnuKmn_sniTan >smng;rkab -cmnucrbsbvrvagrkabcamyygks½kugredaen ebi enah y nam[rkabkat;gks½ ( oy) Rtg; (, ) ebi y 8 ( ) Kµanb s nam[rkabminkat;gks½ ( o) et. -p itqøúh GasIumtUtQr RbsBVCamYyGasIumtUteRTt y 8 Rtg;cMnuc I (, ) KWCap itqøúhénrkabtag y. y y = - = - -

127 CMBUkTI emeronti GnuKmn_sniTan $-kmnt;témø a edim,i[gnukmn_ ekinelicenøah (, ). y a a a dmenahrsay kmnt;témø a y a a a EdnkMNt; D IR {a} ( ekman y f() a) a a a ( a) a a a a a ( a) a ( a) ek)an y' f'() edim,i[gnukmn_ y ekinelicenøah (, ) luhrtaet f'() cmebahrkb; elicenøah (, ) ebalkw ekrtuv[ ( a) a,. - -

128 CMBUkTI emeronti GnuKmn_sniTan m m m %-ek[gnukmn_ y k> RsaybBa ak;fagasiumtuterttkat;tamcmnucnwgmyycmebah RKb;témø)a:ra:Em:Rt m EdlRtUvkMnt;kUGredaen. > rktémø m edim,i[bnþat; y m b:hnwgrkab. K> siksagefrpab nigsg;rkabcmebah m. dmenahrsay k> RsaybBa ak;fagasiumtuterttkat;tamcmnucnwgmyy m m m ekman y EdnkMnt; D IR { } GnuKmn_Gacsresr (m m) (m m) y m m eday lim y m nam[ m CaGasIumtUteRTt - -

129 CMBUkTI emeronti GnuKmn_sniTan ehiyeday y m( ) smikarenhepþógpþat;rkb; m IR luhrtaet b nig y. ducenhgasiumtutertténrkabkat;tamcmnucnwg I(, ) cmebah RKb;témø)a:ra:Em:Rt m. > rktémø m edim,i[bnþat; y m b:hnwgrkab smikargab;sius m m m ' m m m m m m m m edim,i[bnþat; m m m(m ) y m b:hnwgrkabluhrtaet m ' m(m ) m. ektaj)an K> siksagefrpab nigsg;rkabcmebah m cmebah m ek)an y EdnkMNt; D IR { }. - -

130 CMBUkTI emeronti GnuKmn_sniTan > TisedAGefrPaB ( ) -edriev ' y ( ) ( ) nam[gnukmn_ f cuhcanic eliednkmnt; nam[vakµanbrmaet. -KNnalImIt lim lim lim -GasIumtUt eday enahbnþat; ehiy lim lim CaGasIumtUtQr. enahbnþat; y CaGasIumtUteRTt

131 CMBUkTI emeronti GnuKmn_sniTan -taraggefrpab y ' y >smng;rkab -cmnucrbsbvrvagrkabcamyygks½kugredaen ebi enah y nam[rkabkat;gks½ ( oy) Rtg; (, ) ebi y , -p itqøúh GasIumtUtQr RbsBVCamYyGasIumtUteRTt y Rtg;cMnuc I(, ) KWCap itqøúhénrkabtag y

132 CMBUkTI emeronti GnuKmn_sniTan y y (c) : y ^-rktémøbrmaefobrbs;gnukmn_ dmenahrsay rktémøbrmaefob y y - 7 -

133 CMBUkTI emeronti GnuKmn_sniTan EdnkMNt; D IR GnuKmn_Gacsresr y( ) b y y y (y ) (y 5) y ( E) ' (y 5) (y )(y ) y y 5 y y 9y 6 y 9y 5 ebi ' y 9y 5 5 manb s 7 ; y. y edim,i[gnukmn_ y mantémøbrmaefobluhrta EtsmIkar ( E) manb s ebalkwekrtuv[ ' y 9y 5 5 ektaj y 7. 5 ducenhtémøgtibrmaefobkw 7 nigtémøgb,rmaefobkw

134 CMBUkTI emeronti GnuKmn_sniTan a &- rktémøbrmaefobrbs;gnukmn_ y ( ) Edl a Ca)a:ra:Em:RtusBIsUnü. dmenahrsay rktémøbrmaefob a a ekman y ( ) ( ) ( ) tamvismpab mfümnbvnþ-mfümfrnimartek)an a a y ( ) ( ) a ( ) ( ) ducenhgnukmn_mantémøgb,brmaefobesµinwg a

135 CMBUkTI emeronti GnuKmn_sniTan *-siksagefrpab nig sg;rkabéngnukmn_ y dmenahrsay siksagefrpab nig sg;rkabéngnukmn_ y > EdnkMNt; D IR { } > TisedAGefrPaB ( ) -edriev y' ( ) ( ) ebi y', -brmaéngnukmn_ y cmebah GnuKmn_manGtibrmaeFob y f( ) cmebah GnuKmn_manGb,brmaeFob y f(). -KNnalImIt - -

136 CMBUkTI emeronti GnuKmn_sniTan lim lim lim -GasIumtUt eday enahbnþat; ehiy lim lim CaGasIumtUtQr. enahbnþat; y -taraggefrpab CaGasIumtUteRTt. y ' y - -

137 CMBUkTI emeronti GnuKmn_sniTan >smng;rkab -cmnucrbsbvrvagrkabcamyygks½kugredaen ebi enah y nam[rkabkat;gks½ ( oy) Rtg; (, ) ( ) ebi y Kµanb s nam[rkabminkat;gks½ ( o) et. -p itqøúh GasIumtUtQr RbsBVCamYyGasIumtUteRTt y Rtg;cMnuc I(, ) KWCap itqøúhénrkabtag y y

138 CMBUkTI emeronti GnuKmn_sniTan (-ek[ C m CaRkabtagGnuKmn_ y ( m) k>etimanrkab C m cmnynb:unµanedlkat;tamcmnuc A (, ) > RsaybBa ak;fa bnþat;b:hrkab C m Rtg;cMnucmanGab;sIus m RsbnwgGkS½ o. dmenahrsay k>etimanrkab C m cmnynb:unµanedlkat;tamcmnuc A (, ) ykkugredaenéncmnuc A epþógpþat;kñúgsmikar C m ek)an (m ) (m ), m. ektaj)an m ducenhmanrkab C m BIrEdlkat;tam A(, ) mansmikarerogkña y ( ) ( ) nig y > RsaybBa ak;fa bnþat;b:hrkab C m Rtg;cMnucmanGab;sIus m RsbnwgGkS½ o - -

139 CMBUkTI emeronti GnuKmn_sniTan ekman y ( m) EdnkMnt; D IR { m} 8 edriev y' ( m) emkunr)ab;tisénbnþat;b:h C m Rtg; m KWCacMnYnedrIev Rtg; m EdlkMNt;eday a f'( m). 8 ek)an a f( m) ( m m) ltæplenhbba ak;fa bnþat;b:hrkab C m Rtg;cMnucmanGab;sIus m RsbnwgGkS½ o. - -

140 CMBUkTI emeronti emeronti GnuKmn_Gics,: ÚNg;Esül GnuKmn_Gics,ÚëNgÉsül cmnyn e b ekal e EdleKniymeRbImantémø e. 788 Edl e n lim n. n GnuKmn_ f() e ehafagnukmn_gius,:úng;esülekal e. edriev -ebi y e enah y' e u() u() -ebi y e enah y' u'().e RkabénGnuKmn_ y e y y e

141 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül lmhat nig dmen¼rsay!-knnalimitagerkam k> lim e 5 > lim e K> lim 6e dmenahrsay k> lim e > lim e 5 K> lim 6e 6 lim e - 6 -

142 CMBUkTI emeronti k> lim e > lim e n K> lim ( ) dmenahrsay KNnalImItageRkam k> lim e e lim () lim e erbah lim nig lim e > lim e erbah lim e. n n n K> lim ( ) lim e erbah n n lim e. n n - 7 -

143 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül #-KNnalImItageRkam k> e lim > lim K> e e lim dmenahrsay KNnalImItageRkam k> e lim e e lim > lim lim ) ( e lim erbah e lim nig ) ( lim.

144 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül e e e K> lim lim e lim e e ducenh lim e.. e $-KNnaedrIevageRkam k> y e > f() e e e K> g() e e dmenahrsay KNnaedrIevageRkam k> y e ek)an y' e e ducenh y' ( )e (. )e - 9 -

145 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül - - > e () f ek)an )e ( e e '() f ducenh )e ( '() f. K> e e e e () g ek)an ) e (e ) e (e ) e (e '() g ) e (e ) e (e e e e e %-KNnaedrIevageRkam k> e e y > cos cos ) ( e f() K> e () g

146 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül dmenahrsay KNnaedrIevageRkam e e k> y e e y' e ( cos ) f() cos ek)an > ek)an f'() e e ( cos ) e sin ( cos) sin e ( cos) ( cos) (sin sin sin cos sin sin cos ) ( cos ) (sin cos )sin e ( cos ) K> g() e ek)an g'() ( )'e e ( ) g'() e. ( ) ducenh - -

147 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül ^-siksagefrpab nig sg;rkabéngnukmn_agerkam k> y e > y 5 e K> g() e dmenahrsay siksagefrpab nig sg;rkabéngnukmn_agerkam k> y e >EdnkMNt; D IR >TisedAGefrPaB -edriev y' e e ( )e -ebi y' ; cmebah GnuKmn_manGb,brmaeFob y f() cmebah GnuKmn_manGtibrmaeFob y e -KNnalImIt lim e nig lim e - -

148 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül -GasIumtUt eday lim e nam[bnþat; y CaGasIumtUtedk. -taraggefrpab y ' y e >smng;rkab (C) : y e y

149 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül > y 5 e >EdnkMNt; D IR >TisedAGefrPaB -edriev y' e, IR nam[gnukmn_ f cuhcanic. nam[gnukmn_kµancmnucbrmaefobet. -KNnalImIt lim (5 e ) nig lim (5 e ) 5 -GasIumtUt eday lim (5 e ) 5 nam[bnþat; y 5CaGasIumtUtedk. -taraggefrpab y ' y 5 - -

150 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül >smng;rkab (C) : y 5 e y K> g() e y C : y e - 5 -

151 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül &-siksagefrpab nig sg;rkabéngnukmn_agerkam k> f() e e e > g() K> g() dmenahrsay siksagefrpab nig sg;rkabéngnukmn_agerkam k> f() e >EdnkMNt; D IR { ln } >TisedAGefrPaB e -edriev ' y ( e ) nam[gnukmn_ f cuhcanic. nam[gnukmn_kµancmnucbrmaefobet. -KNnalImIt lim nig lim e e - 6 -

152 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül -GasIumtUt eday lim nig lim e nam[ y nig y CaGasIumtUtedkénRkab. -taraggefrpab e ln y ' y >smng;rkab e ( C) : y y

153 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül e > g() y K> g() e y

154 CMBUkTI emeronti GnuKmn_Gics,ÚëNgÉsül *-]bmafaekykr)ak; 5 duløa etaepjienafnakarmyyeday TTYl)anGRtakarR)ak;smas % kñúgmyyqñam. ebiekepijryhebl! qñam etiekttyl)anr)ak;srubtamggs;cmnyn b:unµanebikartutat;ercindgkñúgmyyqñam? dmenahrsay KNnaR)ak;srubEdleKTTYl)an edaykartutat;manercindgkñúgmyyqñamenahtamrubmnþ r.t A p e Edl A : CaR)ak;srub p : CaR)ak;edIm r : CaGRtakarR)ak;smas t : CaryHeBldak;R)a;snSM (.)() ek)an A 5 (.78) 59 duløa

155 CMBUkTI emeronti GnuKmn_elakarIt emeronti GnuKmn_elakarIt e k elakarienebéncmnynvic man k KWCaniTsSnþ én e Edl ekkmnt;sresr ln k ehiy ln k e k. GnuKmn_ f() ln ehafagnukmn_elakaritenebén limit!> lim lim n ln, n #> lim ln ln $> lim n edriev -ebi y ln enah -ebi ln u() y enah y' y' u'() u()

156 CMBUkTI emeronti GnuKmn_elakarIt RkabénGnuKmn_ y ln y y ln

157 CMBUkTI emeronti GnuKmn_elakarIt lmhat nig dmen¼rsay!-knnatémøénkensamagerkam ln 7 k> e > ln e 7 K> ln e dmenahrsay KNnatémøénkenSamageRkam k> e ln 7 7 > ln e K> ln e

158 CMBUkTI emeronti k> y ln > y ln( ) K> y ln( e ) dmenahrsay sg;rkabéngnukmn_agerkam k> y ln >EdnkMNt; D ( ; ) >TisedAGefrPaB -edriev y' (, ) nam[vacagnukmn_cuhcanic eliednkmnt; nigkµancmnucbrmaefobet. -KNnalImIt lim ( ln ) nig lim ( ln ) -GasIumtUt eday lim ( ln ) nam[gks½ oy) ( CaGasIumtUtQr

159 CMBUkTI emeronti GnuKmn_elakarIt -taraggefrpab y ' y >smng;rkab y ln y

160 CMBUkTI emeronti GnuKmn_elakarIt > y ln( ) y K> y ln( e ) y

161 CMBUkTI emeronti GnuKmn_elakarIt #-KNnalImIténGnuKmn_ageRkam k> lim 5 ln ln > lim K> lim ln dmenahrsay KNnalImIténGnuKmn_ageRkam 5 ln k> lim ln lim 5 ln ln > lim lim K> lim ln $-KNnaedrIevénGnuKmn_ageRkam k> y.ln > y ln K> y ln( )

162 CMBUkTI emeronti GnuKmn_elakarIt dmenahrsay KNnaedrIevénGnuKmn_ageRkam k> y.ln ek)an y' ln > y ln (ln ) ek)an y' (ln ) (ln ) K> y ln( ) ek)an y' ( ) y'. ducenh

163 CMBUkTI emeronti GnuKmn_elakarIt %-KNnaedrIevénGnuKmn_ageRkam e k> y ln e > y ln K> y ln dmenahrsay KNnaedrIevénGnuKmn_ageRkam e k> y ln ln e ln(e e ) ln(e e ek)an y' e e > y ln ek)an y' ln. ln K> y ln ln ek)an ' ln. (ln ) y )

164 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt emeronti RBImITIv nig GaMgetRkalminkMnt; niymn&y F () CaRBImITIvén () kñúgednkmnt;éngnukn_ () RTwsþIbT ebi () f kalna F '() cmebahrkb;témø f. F CaRBImITIvmYyén f () enahrbimitivtamggs;én f () mantrmg;tueta F() C Edl C CacMnYnBitefr. niymn&ygamgetrkalminkmnt ebi F () CaRBImITIvmYyén f () enahgamgetrkalminkmnt; én () f kmnt;eday f ().d F() Edl C CacMnYnBitefr. lkçn GaMgetRkalminkMnt!> f().d k. f(). d k Edl k f () g().d f().d g(). d C

165 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt rubmnþgamgetrkalminkmnt!> k.d k C n C, C IR, n n #>.d ln C $>.d C %>.d C ^> e.d e C &> sin.d cos C *> cos.d sin C (>.d tan C cos!>.d cot C sin!!> k.f'().d k.f() C n n!@> f ().f'().d f () C n f () f ()!#> f'().e.d e C - 6 -

166 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt f'()!$>.d ln f() C f() f'().d n n f () (n )f () f'().d C f () f() f'().d f() C f()!%> C!^>!&>!*> sin[ f()].f'().d cos[f()] C!(> cos[ f()].f'().d sin[f()] tan[f()] f'() cos [f()] f'().d sin f() rubmnþgamgetrkaledayepñk cot[f()] C f ().g'().d f().g() g().f'(). d - 6 -

167 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt lmhat nig dmen¼rsay!-bgðajfa F () CaRBImITIvén f(), IR Edl k> F() 7 nig f() 7 > F() 7 nig f() 9 7 K> F() e 7 nig f() 6e X> F() ln(e ) nig f() e e dmenahrsay bgðajfa F () CaRBImITIvén f(), IR k> F() 7 nig f() 7 ekman F'() 7 f(), IR ducenh F() 7 CaRBImITIvén f() 7. > F() 7 nig f() 9 7 ekman F'() 9 7 f(), IR ducenh F() 7 CaRBImITIvén f()

168 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt K> F() e 7 nig f() 6e ekman F'() 6e f(), IR ducenh F() e 7 CaRBImITIvén f() 6e e X> F() ln(e ) nig f() ekman F'() (e e )' e e ducenh F() ln(e () f() e,. CaRBImITIvén IR f() e e F én () k> f() 5, I IR > f(), I IR K> f() 5, I IR X> f() ; I (, ) g> f(), I (, ) c> f() q> f() e 5 C> f() e, I IR. f kmnt;nigcab;elicenøah I eday - 6 -

169 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt dmenahrsay rkrbimitiv () F én f () k> f() 5, I IR tamniymn½yek)an F'() f() 5 ektaj F () 5d 5 C, C > f(), I IR IR tamniymn½yek)an F'() f() ektaj F() ( )d C, C K> f() 5, I IR tamniymn½yek)an F'() f() 5 ektaj F() ( 5).d 5 C f() ; I (, X> ) tamniymn½yek)an F'(), C IR f() IR - 6 -

170 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt ektaj F () d ln C, C IR g> f(), I (, ) tamniymn½yek)an F'() f() 5 ektaj F () d ln( ) C, C c> f() 5 tamniymn½yek)an F'() f() 5 ektaj F() d ln( 5) C, C 5 q> f() e tamniymn½yek)an F'() f() e ektaj F() e d e C, C IR C> f() e, I IR tamniymn½yek)an F'() f() e ektaj F() e d e C, C IR. IR IR

171 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt #-rkrbimitiv F () éngnukmn_ f EdlkMnt;eday k> f() nig F( ) cos > f() nig F(o) K> f() e nig F() X> f() e nig F() dmenahrsay rkrbimitiv F () éngnukmn_ f EdlkMnt;eday k> f() nig F( ) cos ek)an F'() f() cos ektaj F() d tan C cos ebi enah F( ) tan C C ducenh F() tan

172 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt > f() nig F(o) ek)an F'() f() ektaj F() ( )d C ebi enah F() C ducenh F(). K> f() e nig F() ek)an F'() f() e ektaj F() ( e )d e C ebi enah F() C C ducenh F() e. X> f() e nig F() ek)an F'() f() e ektaj F() ( e )d e ebi enah F() C C ducenh F() e C

173 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt $-rkrbimitiv F () éngnukmn_ f EdlkMNt;eday k> f() sin nig F() > f() e nig F() e K> f() nig F(5) ( ) X> f() sin cos nig F( ) 6 dmenahrsay rkrbimitiv F () éngnukmn_ f EdlkMNt;eday k> f() sin nig F() ek)an F'() f() sin ektaj F () sin d cos C ebi enah F() C C ducenh F() cos

174 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt > f() e nig F() e ek)an F'() f() e ektaj F() ( e )d e ebi enah F() e C e ducenh F() e. K> f() nig F(5) ( ) ek)an F'() f() ektaj ( ) F(). d ( ) tag u du d ek)an F() u du u b (). C F ebi 5 enah ducenh () ( ) C F(5) C F ( ) 76 C C C

175 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt X> f() sin cos nig F( ) 6 ek)an F'() f() sin cos ektaj F() sin cos d cos ebi enah F( ) C 6 C 6 57 ducenh F() cos. 6 C 57 6 %-KNnaGaMgetRkalageRkam k> ( 5 ) d > ( 5 ) d K>. d X> ( ) d 5 g> ( 6 ). d c> (e ln 5) d 5 q> d C>. d Q> (sin 5cos ) d ( 5) sin cos e d>. d z> e ( e ) d j> d - 7 -

176 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt dmenahrsay KNnaGaMgetRkalageRkam k> ( 5 d 5 5 ) > ( 5 d 5 d d 5 C K>. d 8 d 5 ( ) 5 X> d d g> ( ). d 5 5 ) d d C C 5 d d d d 7 7 d C C - 7 -

177 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt 6 c> (e ln 5) d d e 6 ln 5 d e 6ln ln 5 C 5 q> d d d C>. d 5 d C 6 5 Q> (sin 5cos ) d sin d 5 d C cos 5sin C j> ( 5) d sin cos cos d d d Cot tan 5 C - 7 -

178 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt e d>. d e (e )(e e (e e e e z> ( e ) d ( e d e cos sin cos d sin cos e )d C e e e ).d d C ).d e d cos D>.d d C sin sin d F> cot C sin - 7 -

179 CMBUkTI emeronti RBImITIv nig GaMgetRkalminkMNt ^-KNnaGaMgetRkalminkMNt;ageRkam k> e d > e d 5 7 K> 5e d X> (6 7)e d sin g> ( )e d c> cos e d q> e d C> 6 e d Q> ( )e d j> d e dmenahrsay k> e d e.()'d e C > e d e.( )'d e C K> 5e d e ( 5)'d e C 7 7 X> (6 7)e d e ( 7)' d e 7 C - 7 -