GñkcUlrYmRtYtBinitübec kets elak lwm qun elak Esn Bisidæ elakrsi Tuy rina elak Titü em g elak RBwm sunitü elak pl b unqay GñkRtYtBinitüGkçraviruTæ

Átírás

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2 sñaédeá Bum<pßay esovepaknitvitüa ½ -dmena RsaylMhat KNitviTüa eá Bum< qñam ( srmabértomrblgculsaklvitüal&y ig GaharUbkrN_ ) -BiPBsVIútcMYBit ( srmab fñak TI ig sisßbuekknitvitüa ) - GuKm_RtIekaNmaRt ( srmab fñak TI ig sisßbuekknitvitüa ) - dmena RsayKMrU cmykmupøic limit edriev ( srmab fñak TI) 5- segçbrubmþknitvitüa ( srmab fñak TI- ) 6- KMrUsikßaGuKm_ ( srmab fñak TI- ) 7- kmelmhat KNitviTüafñak TIkmμviFIsikßafμI ( PaK qñam8 ) 8-5 KNalImIt ( srmab fñak TI- ) 9- lmhatḿadmena Rsay ( srmab fñak TI )

3 GñkcUlrYmRtYtBiitübec kets elak lwm qu elak Es Bisidæ elakrsi Tuy rina elak Titü em g elak RBwm suitü elak pl b uqay GñkRtYtBiitüGkçraviruTæ elak lwm mikþsir karikmubüút&r kbaøa li KuNÑaka GñkiBæ ig eroberog elak lwm plþú ig elakes Bisidæ

4 GarmÖkfa esovepa lmhat madmena Rsay EdlGñksikßakMBugka eakñúgéde MJúáTáeroberogeLIgkñúgeKalbMgTukCaÉksar srmab CaCMYydlǴñksikßaykeTAsikßaRsavRCavedayøÜÉg ig müa getot kñúgekalbmgculrymeliksþüyvis&yknitvitüaearbetskm<úcaeyig [kaétrikcermiefmetotedim,ibeg;iffamusß[makaéterci edim,icyygpivdä_rbetscatirbséyig eakñúgesove eyigmjúáitmrsavrcaverciserisyklmhatýa g srmamgbmputykmkefivdmena Rsayya gek,a k,ayedlgac[elakgñk gayyl qabćgcamgmbisil, ékareda RsayTaMgGsé b ueþeta Ca ya gnak¾eday kgv at ig kmhusqþgedaygectarákdcamatamg bec kets ig GkçraviruTæ GaRs&yehtue eyigmjúcagñkeroberogrgćam edayrikraycaic Uvmtiri KÉbbsSabaBIsMNakǴñksikßakñúgRKbḿCÄdæa edim,icyyeklmg esovepae [ákaétsurkirtpabefmetot CaTIbBa bé eyigmjúgñkeroberogsumekarbcubrdlǵñksikßatamggs [masupabmammy ig TTYlC&yCM RKbṔarkic át dmbgéf TI7 kumö¼ 9 GñkiBæ lwm plþú

5 -ek[smikar (E) : z ia z a ib Edl a, b IR k-kmt a ig b edim,i[ z i Ca smyyrbs smikar ( E) ryckna smyyetot z z -cursresr z, z ig CaTRmgŔtIekaNmaRt 5 K-TajbBa ak témørákdé cos ig -ek[gukm_ f () kmtéli IReday ½ f() l( z 5 si ) l( ) f( ), f( ig f ( ) k-curknatémø ) bghajfa f() CaGuKm_ess -KNaedrIev f '() ig f ''() -ek[cmykmupøic z i y Edl ig y CaBIrcMYBit curkmt témø ig y ebiekdwgfa½ ( i)z ( i) z i ( z CacMYkMupøicqøas é z ) - -

6 -ekegaycmykmupøic z cos isi 7 7 cursresr ( z) CaragRtIekaNmaRt 6 i 5-eKeGaycMYkMupøic ³ z ig z i z k>cursresr z, z ig Z CaragRtIekaNmaRt >cursresr z Z CaragBiCKNit z 6 K>TajeGay)afa cos ig 6-KNalImItageRkam ½ k e lim si 7-cUrKNalImIt ½ e k cos lim 8-cUrKNalImItageRkam ½ k 5 lim z 6 si lim e e lim si lim 7 y i 9-kMt;cMYBit ig y edim,iegay ( ) ( ) 9i i - -

7 -ekegay f ( z) z ( i) z ( i ) z 8i k>curbgðajfa z f ( z) ( z i)( z z ) >edahrsaysmikar f ( z) kñúgsmnmukmupøic -curknalimit ³ cos si 9 k> lim a ( ) > lim a -ekegaygukm_ f ( ) 6 9 k>bgðajfamatmél Edl < < ehiy ( ) >KNaedrIev f' ( ) ehiysiksasbaøaé ( ) sg;taraggefrpabé f ( ) -ekmagukm_ f () f f', f'() kmt;eli k> cmebahrkb; [,5] curbgðajfa 8 > edayerbivismpabkmenimakmt;etawggukm_ f cmebahrkb; [,5] curbgðajfa

8 -ek[gukm_ f () a bl kmt;eliceøah ], [ curkmt;cmybit a ig b edim,i[esekag ( c) taggukm_ y f() b:hwgbþat; ( T) : y Rtg;cMuc A(, ) 5-eK[GuKm_ si cos ( ) f() cursiksapabcab;égukm_ f Rtg;cMuc ebi ebi l 6-eK[GuKm_ f ( ) a b Edl > ehiy a ig b CacMYBit k-bgðajfacmebahrkb;cmybit a ig b Edl a ESekag ( C ) taggukm_ f ( ) magasiumtuterttmyyedlekwgbba ak;smikar -kmt;cmybit a ig b edim,i[esekag ( C ) taggukm_ f ( ) b:hetawgbþat; ( T ): y Rtg;cMuc A (,5) si( ) 7-eK[GuKm_ f() kmt;rkb; etiekgacbøaygukm_ f [Cab;Rtg;cMuc )ab et? ebigackmt;rkgukm_bøaytampabcab;égukm_ f() Rtg; - -

9 8-eK[GuKm_BIr F ( ) ( a b c d) e ig f ( ) e kmt;eli IR kmt;cmybit a,b, c ig d edim,i[ F( ) CaRBImITIvéGuKm_ f ( ) 9-eK[GuKm_ f ( ) a b e markabtmag ( c ) kmt;cmybit a ig b edim,i[esekag ( c ) b:hwgbþat;( d ): y Rtg;cMuc A (,) -ek[gukm_ f ( ) a b l markabtmag ( c ) kmt;cmybit a ig b edim,i[esekag ( c ) b:hwgbþat; ( d ): y Rtg;cMuc (,) A m -ek[gukm_ f() Edl CacMYBit ig m Ca)a:ra:Em:Rt k> curkmt;témø m edim,i[gukm_ f() matémøbrmartg;cmuc > curkmt;témø m edim,i[gukm_ f() matémøbrmaetmyykt; 7 6 -ekmagukm_ f ( ) k-sresr f ( ) CaTMrg; rycknatmél A A, B ig C ( ) ( ) B C ( )

10 -KNa f ( ) dedaysresrcmeliycatmrg; a lbedl aigb CacMYsiTa -ek[gukm_ g() 5 Edl, A B C k kmtćmybita, B ig C edim,i[ g() curkna I g() d -ekmagukm_ f() Edl ig A B C k-kmtćmybit A, B, C edim,i[ f() -KNaGaMgetRkal I f() d eKmaGuKm_ f() ( ) ( ) Edl ig k-kmt bicmybit a, b, c edim,i[ a b c f() -KNaGaMgetRkal I f()d 6-eK[GuKm_ f() kmt; ig maedrievrtg;cmuc c f (c h) f (c h) currsaybba ak;fa lim f'(c)f(c) h ( ) h

11 e 7-eK[GuKm_ f() Edl a,a, b IR a b k-curknaedriev f '( ) ig f ''( ) -kmt;cmybit a ig b edim,i[gukm_ f () matémøgb,brmaesμi ecmebah 8-eK[GuKm_ f kmt;eli IR eday f () si ( ) curbgðajfaedrievti égukm_ f kmt;eday f () si( ) 9-eKmaGuKm_ f() Edl a b c k-kmt;bicmybit a, b, c edim,i[ f() -KNaGaMgetRkal I f() d 5 -ekmagukm_ f() Edl ig k-kmt;bicmybit -KNaGaMgetRkal -ek[gukm_ k-curkmt;bicmybit ( )( ) a b c a, b, c edim,i[ f () ( ) I f() d f() Edl ( ) CacMYBitusBIsUü A B C A, B ig C edim,i[ f()

12 -KNaGaMgetRkal K-TajrkGaMgetRkal -ek[gukm_ f() I f() d J e ( l d ) Edl CacMYBit k-curkmt;bicmybit A ig B edim,i[ -KNaGaMgetRkal f() d e K-TajrkGaMgetRkal J -ek[gamgetrkal k-curkna -Tajrk I ig J -ek[gamgetrkal ³ I J ig I J cos d si cos I (e ) d I e cos d I ig J k-kna -Tajrk I ig J I J ig I J f() Be A e ig J si d si cos e si d - 8 -

13 5- ek[gamgetrkal I t ig I dt, ( dt t t IN ) t t k-curknatémøé I ryc Rsayfa ( I ) CasIVútcu¼ -RsaybBa ak fa I I I, ( ) K-Taj[áfa I ( ) TajrklImIt lim ( I ) 6-eK[GaMgetRkal I cos d ig J si d k-curkna I J ig I J -Tajrk I ig J eKmaGuKm_ f() Edl {,, } k-kmt;bicmybit a, edim,i[ -KNaGaMgetRkal ( )( )( ) a b c b, c f() I f() d - 9 -

14 8-eKmaGaMgetRkal I cos d ig J si k-curkna I J ig I J -Tajrktémøé I ig J 9-eKmaGuKm_ f() Edl ig ( ) A B k-kmt;bicmybit Aig B edim,i[ f() -KNaGaMgetRkal f()d -ekmagamgetrkal I I ig J d si - - d si d asi k-kmt;bircmybit a, b edim,i[ si cos -KNaGaMgetRkal I ryctajrktémø J e -ekmagukm_ f() Edl CacMYBit (e ) Be k-kmt;bicmybit A, B edim,i[ f() A -KNaGaMgetRkal I f()d b si cos (e )

15 -ek[gukm_ f ( ) k-kmt;bicmybit f ( ) ( )( ) A, B ig C edim,i[gukm_ f ( ) A B C ( ) I f d -KNaGaMgetRkal ( ) e -ek[gukm_ f ( ) kmt;eli IR k-cursresr ( ) e Be f Carag f ( ) A -KNaGaMgetRkal I f ( ) d e GacsresrCarag edaysresrltæplcarag a l bedl a ig b CaBIrcMYBitRtUvrk -ek[gukm_ f ( ) ( 7) e kmt;eli IR k-kmt;cmybit a, big c edim,i[gukm_ F ( ) ( a b c) e CaRBImITIvéGuKm_ f ( ) -KNaGaMgetRkal I f ( ) d - -

16 5-eK[GuKm_ f ( ) k-kmt;cmybit A ig BedIm,I[ f ( ) 5 -KNaGaMgetRkal f ( ) d A B I 6-eKdwgfa 6 f (t )dt currkgukm_ () f 7>edaHRsaysmIkar g ''( ) 5g' ( ) 6g( ) ( E) > kmt;cmeliy g ( ) myyésmikar ( E ) Edl g ( ) ig g '( ) 8-edaHRsaysmIkarDIepr:g;Esül ( E ) : y'' y' y edaydwgfa y ( ), y' ( ) 9-eK[smIkarDIepr:g;Esül ( E) : y'' y' y k-kmt;cmybit P a, big c edim,i[gukm_ y ( ) a b c CacMelIyedayELkmYyrbs;smIkar ( ) E -bgðajfagukm_ y y ( ) y ( ) CacMelIyTUeTArbs; ( E ) P h - -

17 luhrtaetgukm_ y h ( ) CacMelIyrbs;smIkarGUmU:Es ( E '): y'' y' y K-edaHRsaysmIkar ( E' ) ryctajrkcmeliytuetarbs;smikar ( E ) 5-k-edaHRsaysmIkarDIepr:g;Esül ( E ): f'' ( ) f' ( ) 6f ( ) -kmt;gukm_ y f ( ) CacMelIymYyrbs;smIkar ( E ) ebiekdwgfaesekag ( C ) tag f ( ) b:hetawgbþat; ( T ): y Rtg;cMuc M (,) 5-eK[smIkarDIepr:g;Esül ( E ): y'' 9y k-edahrsaysmikar ( E ) -kmt;gukm_ f ( ) CacMelIymYyrbs;smIkar ( E) ebiekdwgfa ³ f ( ),f'( ) 5-eK[smIkarDIepr:g;Esül : y'' y 8 ( E) k-kmt;gukm_ ϕ ( ) a b CacMelIyedayELkmYyrbs;( ) -rkcmeliytuetarbs;smikar ( ) E E - -

18 5-eK[RbEvgERbRbYlmYy MN Edl f ( ) GuKm_ f ( ) CacMelIysmIkarDIepr:g;Esül ³ ( E ): f'' ( ) f' ( ) f ( ) k-knarbevg MN ebiekdwgfa ( ) -kmt;rbevggtibrmaé MN - - MN f ig '( ) 5-eK[smIkarDIepr:g;Esül ( E ): y'' y k-edahrsaysmikar ( E ) -kmt;gukm_ f ( ) CacMelIymYyésmIkar ( ) ( ) f ig '( ) K-cUrsresrGuKm_ ( ) Edl, ω f f E ebiekdwgfa f Carag f ( ) kcos( ω ϕ) k ig ϕcabicmybit X-KNaGaMgetRkal d I f ( ) e 55-eK[GaMgetRkal d, I IN e I k-kna I I, I rytajrk -KNa I CaGuKm_é I

19 56-eK[GuKm_ f kmt;eli IR { } ehiyepþogpþat;tmak;tmg³ f ( ) f ( ) 5 ( ) curknagamgetrkal³ f ( ) I d 57-eKsμt;fa f CaGuKm_mYykMt;elI IR ehiyepþogpþat;tmak;tmg³ f ( ) f ( ) cos curkna f ( ) I d b b 58-cUrbgðajfa f ()d f(a b a Guvtþ_ ³ curkna I log ( 59-eK[ f CaGuKm_Cab;elI [,] curbgðajfa f(si )d Guvtþ_³ curkna a )d ta ) d f(si )d? si d I cos - 5 -

20 6-eK[ f CaGuKm_KUelI [ a,a] a a ()d k> curbgðajfa a > Guvtþ_ ³ KNa I f f()d, q >,q q cos d 6-k-KNaGaMgetRkalkMt; ( ) I d, IN -Tajbgðajfa C C C 6-eKmasIVút ( I ) kmt;cmebahrkb; eday I! ( ) e d C k-curknaty -curbba ak; I CaGuKm_é I ryctaj[)afa K-cUrrklImIt I lim I I e Tajfa lim e 788!!!! p P! - 6 -

21 EpñkdM MeNa¼Rsay - 7 -

22 lmhat TI ek[smikar (E) : z ia z a ib Edl a, b IR k-kmt a ig b edim,i[ z i Ca smyyrbs smikar ( E) ryckna smyyetot z z -cursresr z, z ig CaTRmgŔtIekaNmaRt z 5 K-TajbBa ak témørákdé cos 5 ig si dmena¼rsay k-kmt a ig b ½ edim,i[ z i Ca smyyrbs ( E) lu¼naet vaepþógpþatńwgsmikar eká ( i ) ia( i ) a ib ektaj i ia a ( a a ) i( a a a b a a ib a b) eda¼rsayrbb&æeká ½ duce¼ ig b - 8 -

23 KNa smyyetot z ½ eday z ig z Ca srbsśmikar ( E) ea¼tamrtwsþibtevüteyigá z ia am[ z z ia z i ( ) ( i ) i z i duce¼ -sresr z, z ig eyigá ig z z z CaTRmgŔtIekaNmaRt ½ z z i ( i ) (cos isi ) z i ( i ) (cos isi ) 5 5 cos( ) isi( ) (cos isi ) K-TajbBa ak témørákdé 5 cos 5 ig si z 5 5 (cos isi ) z tamsrmayagelieyigma () müa getot tam ( ) ig ) 5 cos z i ( i )( i) i z i ( ektajá ½ 6 ig 5 si 6 () - 9 -

24 lmhat TI ek[gukm_ f () kmtéli IReday ½ f() l( ) l( ) f ( k-curknatémø f( ), f() ig ) bghajfa f() CaGuKm_ess -KNaedrIev '() dmena¼rsay f ig ''() f k KNatémø f( ), f() ig f ( ) ma f() l( ) l( ) eyigá f ( ) l( ) l( ) l 5 f() f( l( ) l( ) ) l( ) l( ) l 5 duce¼ f ( ) l5, f(), f( ) l5 bghajfa f () CaGuKm_ess ½ eyigma IR ig IR eyigá f( ) l( ) l( ) f() - -

25 - - duce¼ ) ( f CaGuKm_ess -KNaedrIev ) '( f ig ) ''( f eyigá ) ( )' ( ) ( )' ( '() f ) ( ) )( ( ) )( ( ) )( ( duce¼ '() f müa getot ) ( ) )'( ( ) )'( ( f''() 5 5 ) ( ) ( ) ( ) ( ) ( ) ( duce¼ ) ( ) ( f''()

26 lmhat TI ek[cmykmupøic z i y Edl ig y CaBIrcMYBit curkmt témø ig y ebiekdwgfa½ ( i)z ( i) z i ( z CacMYkMupøicqøas é z ) dmena¼rsay kmt témø ig y ekma ( i) z ( i) z eday eká i i y am[ z i y z ( i)( iy) ( i)( iy) i ( i) iy i y iy i y 5 ( y) i (5 y) i y ektajá 5 y ekma D 8 5, D

27 ig 8 D y 5 D 6 D D D, y y eká, y duce¼ lmhat TI ekegaycmykmupøic z cos isi 7 7 cursresr ( z) CaragRtIekaNmaRt dmena¼rsay sresr ( z) CaragRtIekaNmaRt½ eká z cos isi eday ektaj z cos 7 cos cos 7 7 si si cos isi cos cos (cos isi tamrubmþdwmr&ekgacsresr½ 7 )

28 ( z) duce¼ cos (cos isi ) cos cos isi ( z) 6cos cos isi lmhat TI5 6 i ekegaycmykmupøic ³ z ig z i z k>cursresr z, z ig Z CaragRtIekaNmaRt >cursresr K>TajeGay)afa dmena¼rsay k>sresr z, z ig ekma duceh z Z CaragBiCKNit z z 6 cos ig z Z CaragRtIekaNmaRt³ z 6 si 6 i i cos isi 6 6 z cos( ) isi( ) 6 6 z - -

29 ekma duceh ekma duceh i i cos z cos( ) isi( ) z > sresr ek)a Z z Z cos( ) isi( ) z 6 6 Z cos isi z Z CaragBiCKNit z 6 i ( 6 i )( i) ( i) ( i)( i) 6 6 Z i duceh 6 K> TajeGay)afa cos ig tamsrmayageliekma ³ 6 i si isi 6 i 6 Z cos isi () ig 6 6 Z i () pþwmtmak;tmg! ek)a ³ cos isi 6 6 i 6 6 duceh cos ig si - 5 -

30 - 6 - lmhat TI6 KNalImItageRkam ½ k si e lim lim dmena¼rsay KNalImItageRkam ½ k si e lim si lim e lim ) si e lim ( duce¼ si e lim lim e e ) ( lim lim duce¼ e lim

31 lmhat TI7 curknalimit ½ e k cos lim dmena¼rsay KNalImIt e cos k lim duce¼ (e lim e lim e lim e e lim si (e lim ) ( cos) si lim cos ) (e si e lim e lim si si e duce¼ e lim e lim e e lim si si si ) e e lim lim si si - 7 -

32 lmhat TI8 curknalimitagerkam ½ k 5 lim dmena¼rsay KNalImIt ½ k lim 5 lim tag kalna ea¼ 5 eká lim lim( ) 5 5 lim( ) ( ) e duce¼ 5 lim e lim tag am[ kalna ea¼ ig eká lim lim( ) lim( ) - 8 -

33 lim e ( ) ( ) e duce¼ lim e lmhat TI9 kmt;cmybit ig y edim,iegay ( ) ( y) dmena¼rsay kmt;cmybit ig y 7 9i ekma ( ) ( y) i ek)a i 7 9i i ( )( i) ( )( i) i 7i 8i 9 i 9 i i amegay y y,y duceh i 7 9i i - 9 -

34 lmhat TI ekegay f ( z) z ( i) z ( i ) z 8i k>curbgðajfa z f ( z) ( z i)( z z ) >edahrsaysmikar f ( z) kñúgsmnmukmupøic dmena¼rsay k> bgðajfa z : f ( z) ( z i)( z z ) eyigma f ( z) ( z i)( z z ) edaybøatgukm_eheyig)a ³ duceh f(z) z z z z z i(z z z iz z ) iz 8i ( i) z ( i ) z 8i Bit z f ( z) ( z i)( z z ) > edahrsaysmikar ebi f () z amegay ( z i)( z z ) ' ektajb s z i ehiy, z z, Δ i amegay i,z i z - -

35 lmhat TI curknalimit ³ cos k> lim si 9 a a ( ) > lim dmena¼rsay cos si 9 k> lim eday cos si si si 9 lim si si 9 lim a a lim a a lim > ( ) lim lim ( ) ( ) ( ) ( ) ( ) ( ) ( a )( ) ( ) ( ) a 9 a lim a 8 ( )( a ) ( )( ) - -

36 lmhat TI ekegaygukm_ f ( ) 6 9 k>bgðajfamatmél Edl < < ehiy ( ) >KNaedrIev f' ( ) ehiysiksasbaøaé ( ) sg;taraggefrpabé ( ) dmena¼rsay k-bgðajfamatmél f f f' Edl < < ehiy f ( ) f ( ) CaGuKm_kMt;Cab;elI IR ekma f () 6 9 ig f ( ) 6 9 eday f () f () < tamrtwsþibttmélknþalmatmél Edl < < ehiy f ( ) -KNaedrIev f' ( ) ehiysiksasbaøaé f' ( ) eyig)a f' ( ) 9 smikar f' ( ) 9 mab s,! # f '() - -

37 tamtaragageliek)a f '( ) > cmebah ], [ U ], [ f '( ) < cmebah ], [ sg;taraggefrpabé f ( ) ekma () f ig f ( ) f' ( ) f ( ) - -

38 lmhat TI ekmagukm_ f () kmt;eli, k> cmebahrkb; [,5] curbgðajfa f'() 8 > edayerbivismpabkmenimakmt;etawggukm_ f cmebahrkb; [,5] curbgðajfa dmena¼rsay 8 k> cmebahrkb; [,5] bgðajfa ekma f () am[ 8 8 f'() f'() cmebahrkb; [,5] ekma 5 duceh 8 5 b 6 8 f '() cmebahrkb; [,5] - -

39 > bgðajfa 8 8 f'() cmebahrkb; [,5] ekma 8 tamrtwsiþbtvismpabkmenimakmt; cmebah ekma ( ) f() f() ( ) 8 eday f () ek)a am[ duceh lmhat TI ek[gukm_ f () a bl kmt;eliceøah ], [ curkmt;cmybit a ig b edim,i[esekag ( c) taggukm_ y f() b:hwgbþat; ( T) : y Rtg;cMuc A(, ) dmena¼rsay kmt;cmybit a ig b edim,i[esekag ( c) taggukm_ GuKm_ y f() b:hwgbþat; - 5-5

40 ( T) : y ekma ek)a ek)a f'() Rtg;cMuc A(, ) luhrtaet f() () a b l cmebahrkb; ], [ f f '() (a bl )' a f'() a b f() a bl a, b duceh 6 b am[ a b a 6 lmhat TI5 ek[gukm_ si cos ( ) f() cursiksapabcab;égukm_ f Rtg;cMuc dmena¼rsay siksapabcab;égukm_ f Rtg;cMuc si cos ekma lim f() lim ( ) ebi ebi

41 tag t am[ t kalna eah t ek)a si( t) lim f() lim eday am[ () t lim lim cost cost t si lim t ) t ( t cos( t) t si cost sitcos cos cost si sit lim t t t t lim f() f( ) f CaGuKm_Cab;Rtg; sit lim t f( ) cost t (cos t ) t sit si t t - 7 -

42 lmhat TI6 l ek[gukm_ f ( ) a b Edl > ehiy a ig b CacMYBit k-bgðajfacmebahrkb;cmybit a ig b Edl a ESekag ( C ) taggukm_ f ( ) magasiumtuterttmyyedlekwgbba ak;smikar -kmt;cmybit a ig b edim,i[esekag ( C ) taggukm_ f ( ) b:hetawgbþat; ( T ): y Rtg;cMuc A (,5) dmena¼rsay k>bgðajfaesekag ( C ) taggukm_ f ( ) magasiumtuterttmyy l ekma f ( ) a b Edl > l eday lim am[bþat; y a b CaGasIumtUteRTté( ) duceh ESekag ( C ) taggukm_ f ( ) magasiumtutertt y a b >kmt;cmybit a ig b ekma f ( ) a b l C (l )' ()'l l ek)a f' ( ) (a b)' a edim,i[esekag ( C ) taggukm_ f ( ) b:hetawgbþat; ( T ): y - 8 -

43 Rtg;cMuc A (,5) luhrtaet f' f am[ a a b a b 5 b a,b duceh ( A ) ( ) A a lmhat TI7 si( ) ek[gukm_ f() kmt;rkb; etiekgacbøaygukm_ f [Cab;Rtg;cMuc )ab et? ebigac curkmt;rkgukm_bøaytampabcab;égukm_ f () Rtg;cMuc dmena¼rsay kmt;rkgukm_bøaytampabcab; si( ) ekma limf() lim tag t am[ t kalna eah t si( t) ek)a limf() lim t ( t ) y T A - 9 -

44 si( t) lim t t t si( t) lim t t( t t ) t si( t) lim t t t t limf() eday kmt; eahekgacbøaygukm_ f ()[Cab; Rtg;cMuc ebieyigtag g() CaGuKm_bøaytamPaBCab;éGuKm_ f () Rtg;cMuc duceh si( ) f() g() f() ebi ebi lmhat TI8 ek[gukm_bir F ( ) ( a b c d) e ig f ( ) e kmt;eli IR kmt;cmybit a,b, c ig d edim,i[ F( ) CaRBImITIvéGuKm_ f ( ) - -

45 dmena¼rsay kmt;cmybit a,b, c ig d ekma F ( ) ( a b c d) e ig f ( ) e edim,i[ F ( ) CaRBImITIvéGuKm_ f ( ) eli IR luhrtaet F' ( ) f ( ) IR : F' ( ) (a b c d)'e (e )'( a b c d) ( a b c) e e ( a b c d) [ a ( ) ( ) ( )] a b b c c d e [ a a b b c c d ]e e ek)a ( ) ( ) ( ) ektaj a a a b b am[ b c c 6 c d d 6 a,b,c 6,d duceh 6 - -

46 lmhat TI9 ek[gukm_ f ( ) a b e markabtmag ( c ) kmt;cmybit a ig b edim,i[esekag ( c ) b:hwgbþat;( d ): y Rtg;cMuc A (,) dmena¼rsay kmt;cmybit a ig b ekma f ( ) a b e amegay f' ( ) a e edim,i[esekag ( c ) b:hwgbþat; ( d ): y Rtg;cMuc A (,) luhrtaet ³ f' ( ) a amegay smmul a f ( ) b b duceh a,b lmhat TI ek[gukm_ f ( ) a b l markabtmag () kmt;cmybit a ig b edim,i[esekag ( c ) b:hwgbþat; ( d ): y Rtg;cMuc (,) A c - -

47 dmena¼rsay kmt;cmybit a ig b ekma f ( ) a b l ek)a f' ( ) (a b l )' a l edim,i[esekag ( c ) b:hwgbþat; ( d ) : y Rtg;cMuc A (,) luhrtaet ³ f' () a amegay smmul a f () a b b duceh a,b lmhat TI m ek[gukm_ f() Edl CacMYBit ig m Ca)a:ra:Em:Rt k> curkmt;témø m edim,i[gukm_ f() matémøbrmartg;cmuc > curkmt;témø m edim,i[gukm_ f() matémøbrmaetmyykt; - -

48 dmena¼rsay k> kmt;témø m edim,i[gukm_ f() matémøbrmartg;cmuc luhrtaet f '() m ekma f() / u u'v v'u v v ( m )'( f'() tamrubmþ ) ( ek)a ( m)( f'() ( m m ( ) ( ( ) ( ) 6 m ) ) m )'( m ) m m ) 8 m m m cmebah ek)a f '( ) am[ m ( ) 5 - -

49 > kmt;témø m edim,i[gukm_ f() matémøbrmaetmyykt;luhrtaetsmikar f '() smmul m 6 m mab setmyykt; ebalkwrtuv[ m lmhat TI ekmagukm_ f ( ) k-sresr f ( ) CaTMrg; 7 6 ( ) ( ) A B C ( ) rycknatmél A, B ig C -KNa f ( ) dedaysresrcmeliycatmrg; a lbedl a igb CacMYsiTa dmena¼rsay k-sresr f ( ) CaTMrg; A B C ( ) 7 6 edayekmagukm_ f ( ) eahek)a ³ ( ) ( ) - 5 -

50 A A A B C 7 6 ( ) ( ) ( ) ( ) B( )( ) C( ) 7 6 ( )( ) ( ) ( ) ( ) B( )( ) C( ) 7 6 ( ) cmebah tam! ek)a ³ 6 A am[ cmebah tam! ek)a ³ A A ( ) B( )( ) C( ) ( ) 7( ) 6 C 7 6 am[ cmebah tam! ek)a ³ 9 A B C 6 am[ duceh f ( ) B C ( ) ehiy A,B ig C -KNa f ( ) dedaysresrcmeliycatmrg; a lb ek)a f ( ) d ( ) d - 6 -

51 lmhat TI ek[gukm_ duceh ( ) ( ) ' ( ) ( ) ' ( ) ( ) ( ) l l ' d [ l l ] l l l l 6 l l l 8 f d l 8 g() 5 Edl, k kmtćmybita, B ig C edim,i[ curkna I g() d dmena¼rsay k kmtćmybit A, B ig C edim,i[ g() A B C g() A B C - 7 -

52 eyigá A B C 5 A( )( ) B( ) C( ) 5 ebi ea¼ A ebi ea¼ C 6 ebi ea¼ B B duce¼ A, B, C A C KNa I g() d cmeba¼ A, B, C ekma g() eká ½ d d d I l l l C duce¼ I g()d l l l C - 8 -

53 lmhat TI ekmagukm_ f() Edl A k-kmtćmybit A, B, C edim,i[ f() -KNaGaMgetRkal I f() d dmena¼rsay k kmtćmybit A, B, C B C eká ektajá ( ) A A( ) B( ) C ( ) ig B C (A C) (A B) B A C A B am[ A, B, C B A, B, C duce¼ -KNaGaMgetRkal I f() d tamsrmayagelicmeba¼ A, B, C - 9 -

54 eká ½ f() d d I d I l l C eyigá duce¼ lmhat TI5 ekmagukm_ 7 6 f() ( ) ( ) d Edl ig k-kmt bicmybit a, b, c edim,i[ a b c f() -KNaGaMgetRkal dmena¼rsay kmt bicmybit a, b, c I f()d 7 6 a b c eká ( ) ( ) ( ) 7 6 a( ) b( )( cmeba¼ eká 6 6a am[ a ( ) ) c( )

55 cmeba¼ eká c am[ c 9a c 6 cmeba¼ eká 6 9a b cam[ b duce¼ a, b,c -KNaGaMgetRkal f() eká duce¼ I f()d 7 6 ( ) ( ) ( ) I d ( ) l l l l 6l l [ l l ] 8 I f()d l 8 l - 5 -

56 lmhat TI6 ek[gukm_ f() kmt; ig maedrievrtg;cmuc c f (c h) f (c h) currsaybba ak;fa lim f'(c)f(c) h h dmena¼rsay f (c h) f (c h) RsaybBa ak;fa lim f'(c)f(c) h h f (c h) f (c h) tag L lim h h [ f(c h) f(c h) ][ f(c h) f(c h) ] lim h h f(c h) f(c h) lim lim[ f(c h) f(c h) ] h h h f(c h) f(c h) [ f(c h) f(c) ] [ f(c h) f(c) ] lim lim h h h h f(c h) f(c) f(c h) f(c) lim lim h h h h f(c h) f(c) f ( c ( h) ) f(c) lim lim h h h ( h) f'(c) f'(c) f'(c) ig lim[ f(c h) f(c h) ] f(c) f(c) f(c) h am[ L f'(c) f(c) f'(c)f(c) f (c h) f (c h) duceh lim f'(c)f(c) h h - 5 -

57 lmhat TI7 e ek[gukm_ f() Edl a,a, b IR a b k-curknaedriev f '( ) ig f ''( ) -kmt;cmybit a ig b edim,i[gukm_ f () matémøgb,brmaesμi ecmebah dmena¼rsay k-knaedriev f '( ) ig f ''( ) (e )'(a b) (a b)'e ek)a f'() (a b) e (a b) ae (a b a)e (a b) (a b) (a b a)e f'() (a b) duceh [(a b a)e ]'(a b) ig f'' ( ) [ae e (a b) e (a b) (a b a)](a b) a(a b a)e (a b) (a b) duceh ( ) [(a b) a(a b a) ] f'' [(a b) a(a b)(a b a)e ]'(a b a)e [(a b) a(a b a) ] (a b) e (a b) e

58 -kmt;cmybit a ig b edim,i[gukm_ f () matémøgb,brmaesμi ecmebah luhrtaet f'() f() e f''() > bþab;biedahrsayek)a a,b lmhat TI8 ek[gukm_ f kmt;eli IR eday f () si ( ) curbgðajfaedrievti égukm_ f kmt;eday f () si( ) dmena¼rsay ( ) bgðajfaedrievti égukm_ f kmt;eday f () si( ) ekma f () si ek)a f'() cos si( ) erbah θ) si θ f''() ( )'cos( ) si( ) f'''() ( )'cos( ) si( ) >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> si(

59 ( ) ]bmafavabitdl;edrievlmdab;ti KW f () si( ) Bit eyigwgrsayfavabitdl;edrievlmdab;ti ( ) KW ( ) ( ) f () si Bit ( ) () eyigma f () (f ( ) eday f () si( ) f ( ) ())' '() ( )'cos( ) ( ) si( ) si ( ) f () si( duceh ) lmhat TI9 ekmagukm_ f() Edl a k-kmt;bicmybit a, b, c edim,i[ f() -KNaGaMgetRkal I f() d dmena¼rsay k-kmt;cmybit a, b, c b c

60 ek)a c b a c) (a c) b a ( b) (a c b c b a a a ) )( ( c) )(b ( ) a( ) )( ( ektaj)a c a c b a b a am[,c b, a duceh,c b, a -KNaGaMgetRkal d f() I tamsrmayagelicmebah,c b, a ekma ³ f() ek)a d ) ( I C l l d )' ( d ) ( )' ( )d ( d

61 lmhat TI 5 ekmagukm_ f() Edl ig k-kmt;bicmybit ( )( ) a b c a, b, c edim,i[ f() ( ) I f() d -KNaGaMgetRkal dmena¼rsay k-kmt;cmybit a, b, c 5 a b c ek)a ( )( ) ( ) 5 a( ) b( )( ) c( ) ( )( ) ( )( ) 5 a 6a 9a b b b c c 5 (a b) ( 6a b c) (9a b c) a b 5 6a b c 9a b c a, b, c a, b, c ektaj)a am[ duceh

62 -KNaGaMgetRkal I f() d tamsrmayagelicmebah a, b,c ekma ³ f ( ) ek)a 5 ( lmhat TI ek[gukm_ )( ) ( I ( ) d ( ) d d d ( ) l l f() Edl ( ) CacMYBitusBIsUü A B k-curkmt;bicmybit A, B ig C edim,i[ f() -KNaGaMgetRkal I f() d l d K-TajrkGaMgetRkal J dmena¼rsay k-kmt;bicmybit A, B, C A B ek)a ( ) ( C ) c ) C

63 ektaj ( ) A A A( ) (B ( ) B (A B) C A A B C am[ A, B, C A A, B, C duceh C C) -KNaGaMgetRkal I f() d tamsrmayagelicmebah A, B, C ekma f() ( ) ek)a d I f()d d duceh I l l( ) C K-TajrkGaMgetRkal tag u l d dv ( ) J am[ ( l d ) du d d v ( ) d

64 ek)a eday duceh lmhat TI ek[gukm_ l J ( ) d l d l I ( ) I l l( ) C l J l l( ) C f() Edl e CacMYBit Be f() A e I k-curkmt;bicmybit A ig B edim,i[ -KNaGaMgetRkal f() d e K-TajrkGaMgetRkal J dmena¼rsay k-kmt;bicmybit ek)a e e A, B (e Be A e A(e ) Be e d ) (A B)e A e - 6 -

65 ektaj A B am[ A, B A duceh A, B -KNaGaMgetRkal I f() d tamsrmayagelicmebah A, B e ekma f() e ek)a e e d I ( )d d l(e ) duceh C e e I l(e ) C K-TajrkGaMgetRkal tag ek)a eday duceh e d J (e ) u du d e d am[ e d dv v (e ) (e ) e d J I e e e I l(e ) C J l(e ) C e - 6 -

66 lmhat TI ek[gamgetrkal k-curkna -Tajrk I ig J dmena¼rsay k-kna ig J I e cos d I J ig I J I J ig I J ek)a I J e cos d e si d (e cos e si )d e d I J e C duceh ek)a I J e cos d tag ek)a (e e cos e cos d u e dv cos d I J e si si am[ e e si )d e d (cos du e d v si si d e si d si )d - 6 -

67 tag ek)a u e du e d am[ dv si d v cos I J e si e cos e cos d I J e si e cos e cos d I J (si cos )e (I J) 5 (I J) (si cos )e I J (si cos )e C 5 duceh -Tajrk I ig J tamsrmayageliekma I J e C I J (si 5 bþab;biedahrsayrbb½æsmikarageliehek)a ³ I ( si cos )e K 5 5 J ( si cos )e K 5 5 ig cos )e C - 6 -

68 lmhat TI ek[gamgetrkal ³ cos si cos I d ig J k-kna -Tajrk I ig J dmena¼rsay k-kna ek)a I J ig I J I J ig I J si d si cos cos I J d si cos cos si ( si cos si cos si d d si cos I J C duceh ek)a cos I J d si cos cos ( si cos cos si d si cos l si cos C si d si cos )d cos C si d si cos si )d si cos ( si cos)' si cos d - 6 -

69 duceh I J l si cos C -KNaGaMgetRkal I ig J ekma I J C I J l si cos buksmikar! ek)a ³ C () () I l si cos C C I l si cos K C duceh Edl dksmikar! ek)a ³ J l si cos C C I l si cos K C K C duceh Edl lmhat TI5 dt ek[gamgetrkal I t t t ig I dt, ( IN ) t t k-curknatémøé I ryc Rsayfa ( I ) CasIVútcu¼ -RsaybBa ak fa I I I C K

70 K-Taj[áfa I, ( ) ( ) TajrklImIt lim ( I ) dmena¼rsay k> KNatémøé I ryc Rsayfa ( I ) CasIVútcuH dt dt eyig)a I t t ( t) tag U t am[ du dt ehiycmebah t [,] eah U, du U ek)a I arcta duceh I ( ) U arcta arcta 6 dt t t

71 mü:agetotekma dt t t t I ig dt t t t I cmebahrkb; [ ], t ekma t t am[ t t t t t t ektaj dt t t t dt t t t b IN, I I duceh ) I ( CasIVútcuH > RsaybBa ak;fa I I I eyig)a t t dt t t t dt t t t dt t I I I t dt t t t )dt t t ( t t t )dt t t (t duceh I I I K> Taj[)afa, ) ( I ) ( eyigma ) I ( CasIVútcuH tamlkçn³ésivútcuheyigma ³ I I I I I I I eday I I I am[ I I I

72 ektaj I am[ I, ( ) ( ) TajrklImIt lim ( I ) ma I, am[ ( ) ( ) I ( ) ( ) lim I duceh ( )

73 lmhat TI6 ek[gamgetrkal k-curkna -Tajrk I ig J dmena¼rsay k-kna I cos d I J ig I J I J ig I J ek)a I J cos d ig J si d si d ( cos si )d (cos si )d d I J C duceh ek)a I J cos d si d ( cos si )d (cos si )d tag dv cos d ek)a cos d u am[ I J si du d v si si d

74 duceh J -Tajrk I ig J I si cos C tamsrmayageliekma I J I J C si bþab;biedahrsayrbb½æsmikarageliehek)a ³ I ( si cos ) K J ( si cos ) K ig lmhat TI7 6 8 ekmagukm_ f() k-kmt;bicmybit a, b, c edim,i[ -KNaGaMgetRkal f() dmena¼rsay k- kmt;bicmybit a, b, c 8 ek)a cos C Edl {,, } ( )( )( ) a b c f() I d 6 a b c ( )( )( ) - 7 -

75 6 8 ( )( )( ) 6 a( )( ) b( )( ) c( )( ) ( )( )( ) 8 a( )( ) b( )( ) c( )( ) -cmebah ek)a a am[ a -cmebah ek)a b am[ b -cmebah ek)a 6 c am[ c duceh a, b, c -KNaGaMgetRkal I cmebah a, b, c ekma f() ek)a I ( duceh f() d ) d d d d l l l C I f()d l l l C - 7 -

76 lmhat TI8 ekmagamgetrkal k-curkna -Tajrktémøé I ig J dmena¼rsay k-kna I J ig I J I cos d ig J si I J ig I J eyig)a I J cos d duceh I J (cos si eyig)a I J cos d (cos si si si si si d )d )d d si d cos d d

77 duceh I J -Tajrktémøé I ig J ekma I J I J lmhat TI9 ekmagukm_ f() am[ I 8 k-kmt;bicmybit Aig B edim,i[ -KNaGaMgetRkal ig J 8 ( ) A B f() I f()d Edl ig dmena¼rsay k- kmt;bicmybit Aig B ek)a A B am[ A( ) ( ) B -cmebah am[ A b A -cmebah am[ B b B duceh A, B - 7 -

78 -KNaGaMgetRkal I f ()d cmebah A, B ekma f () ek)a I ( )d [ l l ) ] ( l l ) ( l l ) duceh I l l l 6 lmhat TI ekmagamgetrkal I ig J d si k-kmt;bircmybit a, b edim,i[ si -KNaGaMgetRkal I ryctajrktémø J dmena¼rsay k- kmt;bicmybit a, b eyig)a si asi b si cos cos d si asi b si cos cos - 7 -

79 a si ( cos ) b si ( cos ) si cos si (a a cos b b cos ) si si (a b) (a b)cos si si b a b ektaj)a a am[ a b duceh a, b -KNaGaMgetRkal I ryctajrktémø J cmebah a, b si si ekma ek)a I si cos si cos si d cos cos si d cos si d cos

80 [ ] [ ] ) l( ) l( ) l( ) l( ) l( l ) l( l cos l cos l d cos ) ( cos)' ( d cos) ( cos)' ( I duceh ) l( I müa:getot si d si si d J tag si d dv si u am[ si cos cot si d v si cos du

81 ek)a J J cos si cos si si d si am[ d J J I J J l( duceh ) I d si d si eday I l( ) lmhat TI e ekmagukm_ f() Edl CacMYBit k-kmt;bicmybit -KNaGaMgetRkal (e ) Be A, B edim,i[ f() A (e ) I f()d dmena¼rsay k- kmt;bicmybit A, B e (e e ) e (e ) ekma f() (e ) (e ) (e ) e

82 ) (e Be A ) (e e f() duceh B, A -KNaGaMgetRkal f ()d I ekma ) (e e f() ek)a ]d ) (e e [ I e e e d ) (e )' (e d duceh e I

83 lmhat TI ek[gukm_ f ( ) k-kmt;bicmybit f ( ) ( )( ) A, B ig C edim,i[gukm_ f ( ) A B C ( ) I f d -KNaGaMgetRkal ( ) dmena¼rsay k-kmt;bicmybit A, B ig C ekma f ( ) ig f ( ) ( )( ) A A B C ek)a ( )( ) ( ) ( )( ) A A GacsresrCarag B ( ) B ( )( ) C ( ) ( )( ) C ( ) ( ) B ( )( ) C ( ) ( ) cmebah tam! ek)a A am[ A cmebah tam! ek)a 8 8 C am[ C cmebah tam! ek)a A B C

84 am[ B A C duceh A,B,C -KNaGaMgetRkal I f ( ) d cmebah A,B,C ek)a f ( ) ek)a I f ( ) I I I ( ) ( ) ' ( ) ' ( ) d d ( ) ( ) ( ) [ l ] [ l ] I [ l l] [ l l ] I l l l l I l l I f d d d ( ) d d d duceh ( ) ' d ( ) - 8 -

85 - 8 - lmhat TI ek[gukm_ ( ) e e f kmt;eli IR k-cursresr ( ) f Carag ( ) e Be A f -KNaGaMgetRkal ( ) d f I edaysresrltæplcarag b l a Edl a ig bcabircmybitrtuvrk dmena¼rsay k- sresr ( ) f Carag ( ) e Be A f ( ) ( ) ( ) e e e e e e e e e e f duceh ( ) e e f ehiy A ig B -KNaGaMgetRkal ( ) d f I ek)a d e e I erbah ( ) e e f

86 ( e )' d ( ) [ l( e )] e [ l( e )] [ l( ) ] l duceh I f ( ) ehiy a ig lmhat TI ( e ) l e l e [ l( e ) l ] l e d l e b e l l e ek[gukm_ f ( ) ( 7) e kmt;eli IR k-kmt;cmybit a, big c edim,i[gukm_ F ( ) ( a b c) e CaRBImITIvéGuKm_ f ( ) -KNaGaMgetRkal I f ( ) d dmena¼rsay k-kmt;cmybit edim,i[gukm_ ( ) a, big c F CaRBImITIvéGuKm_ f ( ) luhrtaet ³ - 8 -

87 ( ) f ( ) IR : F' ek)a F' ( ) ( a b c) 'e ( e )'( a b c) ( a b) e e ( a b c) [ a ( ) ( )] a b b c e eday F '( ) f ( ) am[ a ( a b) ( b c) [ ] e ( 7) e ektaj)a a a a b am[ b b c 7 c 6,b,c duceh a 6 -KNaGaMgetRkal I f ( ) d eday F ( ) CaRBImITIvéGuKm_ f ( ) eahek)a ³ I f ( ) d [ F( ) ] F( ) F( ) cmebah a,b,c 6 ekma F( ) ( 6) e ek)a F( ) ( 9 6) e ig F( ) ( 6) e 6 duceh I ( 6) 6-8 -

88 lmhat TI5 ek[gukm_ f ( ) k-kmt;cmybit A ig BedIm,I[ f ( ) 5 -KNaGaMgetRkal f ( ) d dmena¼rsay k-kmt;cmybit a ig b A ek)a I B A B ( ) ( ) ( )( ) ( )( ) A ( ) B ( ) A A B B ( A B) ( A B) A B A,B ektaj)a A B am[ A B duceh A B - 8 -

89 KNaGaMgetRkal ( ) 5 d f I cmebah,b A ek)a ( ) ( ) ( ) f ek)a ( ) 5 d f I ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] [ ] [ ] l l l l 6 l l l l6 l l l l d ' d ' d d d duceh ( ) l d f I 5

90 lmhat TI6 ekdwgfa (t )dt currkgukm_ () dmena¼rsay 6 f f rkgukm_ f () ekma f (t )dt 6 tag g(t) f(t ) ig G (t) CaRBImITIvé (t) ek)a 6 g (t)dt [ G (t )] G ( ) G () 6 efviedrieveliggátamgbirétmak;tmgehek)a ³ 5 G'( ) am[ '( ) 6 g G eday G '(t) g(t) ektaj g ( ) Et g(t) f(t ) ek)a f ( ) tag y am[ y

91 y am[ f (y) (y ) f () ( ) duceh lmhat TI7 >edahrsaysmikar g ''( ) 5g' ( ) 6g( ) ( E) > kmt;cmeliy g ( ) myyésmikar ( E ) Edl g ( ) ig g '( ) dmena¼rsay > edahrsaysmikar g ''( ) 5g' ( ) 6g( ) ( E) masmikarsmkal; r 5r eday Δ 5 Ma[mab s,r tamrubmþ g( ) r r Ae Be, A, B IR ducehcmeliysmikarcagukm_ g( ) r Ae Be, A, B IR > kmt;cmeliy g ( ) myyésmikar ( E ) Edl g ( ) ig g '( )

92 ekma g ( ) Ae Be am[ g '( ) Ae Be ( ) tambmrab;ekma g g' ( ) smmul B A B duceh g ( ) e e A am[ A B lmhat TI8 edahrsaysmikardiepr:g;esül ( E ): y'' y' y edaydwgfa y ( ), y' ( ) dmena¼rsay edahrsaysmikardiepr:g;esül³ ( E ): y'' y' y masmikarsmkal; r r c eday a b c am[,r tamrubmþ r r y Ae Be r a

93 ek)a y Ae Be ig ( ) edaytambmrab;ekma y' Ae duceh y e e CacMelIysmIkar lmhat TI9 Be, A, B IR y b A B am[ A y' ( ) A B B ek[smikardiepr:g;esül ( E) : y'' y' y k-kmt;cmybit P a, big c edim,i[gukm_ y ( ) a b c CacMelIyedayELkmYyrbs;smIkar ( ) E -bgðajfagukm_ y y ( ) y ( ) CacMelIyTUeTArbs; ( E ) P luhrtaetgukm_ y h ( ) CacMelIyrbs;smIkarGUmU:Es ( E '): y'' y' y K-edaHRsaysmIkar ( E' ) ryctajrkcmeliytuetarbs;smikar ( ) h E

94 dmena¼rsay k- kmt;cmybit a, big c ( E) : y'' y' y edim,i[gukm_ y ( ) a b c CacMelIyedayELkmYy P rbs;smikar ( E) luhrtetgukm_ yp ( ), y' p ( ) ig y'' p ( ) epþógpþat;wgsmikar ( ) E ek)a( E) : y'' ( ) y' ( ) y ( ) p P p eday yp y' p y'' ( ) ( ) ( ) p a a b a b c ek)a ( a) ( a b) ( a b c) am[ a ( b 8a) ( a b c) - 9 -

95 ektaj)a a b 8a a b c am[ a b c duceh a,b,c ig y ( ) ( ) P -karbgðaj GuKm_ y y P ( ) yh ( ) CacMelIyrbs; ( E ) luhrtagukm_ y, y', y'' epþógpþat;smikar edayekma y' y' p ( ) y' h ( ) ig y'' y'' p ( ) y'' h ( ) eahek)a ³ [ y'' p ( ) y'' h ( ) ] [ y' p ( ) y' h ( ) ] [ yp( ) yh( ) ] [ y'' ( ) y' ( ) y ( ) ] [ y'' ( ) y' ( ) y ( ) ] () p tamsrmayageliekma y'' p p ( ) y' ( ) y ( ) ( ) p P p erbah y p ( ) CacMelIyrbs;smIkar ( E ) h h h - 9 -

96 tamtmak;tmg! ektaj)a ³ [ y'' ( ) y' ( ) y ( ) ] h h h y'' ( ) y' ( ) y ( ) TMak;TMgeHbBa ak;fagukm_ ( ) h h h CacMelIyrbs;smIkar ( E ') : y'' y' y K-edaHRsaysmIkar ( ') smikarsmkal; r r E ³ y '' y' y, Δ ' am[smikarmab sdúb r r r ducehcmeliysmikar ( E ') CaGuKm_ y h ( ) ( A B) e, A,B IR TajrkcMelIyTUeTArbs;smIkar ( ) E tamsmrayagelicmeliysmikar ( E ) KWCaGuKm_TMrg; y y ( ) y ( ) p h y h - 9 -

97 edayekma ( ) ( ) yp ig y h ( ) ( A B) e duceh y ( ) ( A B) e, A,B IR CacMelIyrbs;smIkar lmhat TI5 k-edahrsaysmikardiepr:g;esül ( E ): f'' ( ) f' ( ) 6f ( ) -kmt;gukm_ y f ( ) CacMelIymYyrbs;smIkar ( E ) ebiekdwgfaesekag ( C ) tag f ( ) b:hetawgbþat; ( T ): y Rtg;cMuc M (,) dmena¼rsay k-edahrsaysmikardiepr:g;esül³ ( E ): f'' ( ) f' ( ) 6f ( ) masmikarsmkal; r r 6 Δ 5 am[ r r 5 5 smikarmacmeliycagukm_ f ( ) Ae Be, A, B IR - 9 -

98 -kmt;gukm_ y f ( ) ebi y f ( ) CacMelIyrbs;smIkar ( E) eahek)a ³ ( C ) ³ y f ( ) Ae Be ig y ' f' ( ) Ae Be edim,i[esekag ( C ) tag f ( ) b:hetawgbþat; ( T ): y '( ) A B Rtg;cMuc M (,) luhrtaet am[ A, B duceh y f ( ) e e f b f ( ) A B lmhat TI5 ek[smikardiepr:g;esül ( E ) : y'' 9y k-edahrsaysmikar ( E ) -kmt;gukm_ f ( ) CacMelIymYyrbs;smIkar ( E) ebiekdwgfa ³ f ( ),f'( ) - 9 -

99 dmena¼rsay k-edahrsaysmikar ( E ) ( E ): y'' 9y masmikarsmkal; r 9 r 9 am[ r i b r i ektaj)a α ig β α cmeliysmikarcagukm_tmrg; y ( A cosβ B siβ) e duceh y A cos B si -kmt;gukm_ f ( ) ekma ( ) f A cos Bsi am[ f '( ) A si Bcos eday f ( ) Acos Bsi am[ A ig f '( ) A si Bcos am[ duceh f ( ) cos si B

100 lmhat TI5 ek[smikardiepr:g;esül : y'' y 8 ( E) k-kmt;gukm_ ϕ ( ) a b CacMelIyedayELkmYyrbs;( ) -rkcmeliytuetarbs;smikar ( E ) dmena¼rsay k-kmt;gukm_ ϕ ( ) a b ekma : y'' y 8 ( E) ebi ϕ ( ) CacMelIysmIkar ( E ) eahvartuvepþógpþat;wgsmikar ( ) ek)a ϕ ''( ) ϕ( ) 8 ( ) E eday ϕ ( ) a b am[ ϕ '( ) a ig ϕ ''( ) smikar ( E ) Gacsresr³ ( a b) 8 am[ a b 8 a 8 b a b b ϕ duceh ( ) E E

101 -rkcmeliytuetarbs;smikar ( E ) dksmikar ( E ) ig ( E ) ek)a '' ϕ'' ( ) y ϕ tag z y ϕ( ) am[ z' y' ϕ' ( ) ig z'' y'' ϕ'' ( ) ek)a z '' z masmikarsmkal; r mab s r,r smikarmacmeliy z Ae Be Edl A,B IR eday z y ϕ( ) am[ y ϕ( ) z duceh y Ae Be, A, B IR ( y ) ( ( ) ) lmhat TI5 ek[rbevgerbrbylmyy MN Edl f ( ) GuKm_ f ( ) CacMelIysmIkarDIepr:g;Esül ³ ( E ): f'' ( ) f' ( ) f ( ) k-knarbevg MN ebiekdwgfa ( ) -kmt;rbevggtibrmaé MN MN f ig '( ) f

102 dmena¼rsay k-knarbevg ( E ): f'' ( ) f' ( ) f ( ) masmikarsmkal; r r b' Δ' smikasmkal;mab sdúb r r r a cmeliysmikar ( E ) CaGuKm_ f ( ) ( A B) e eday f ( ) ( A B) e am[ B ehiy f' ( ) (A B)'e (e )'( A B) Ae e ( A B) eday f' ( ) Ae e ( A B) am[ A amegaycmeliyedayelkésmikar ( E ) KWCaGuKm_ ( ) ( ) e f ducehrbevg MN f ( ) ( ) e Edl -kmt;rbevggtibrmaé MN ekma MN f ( ) ( ) e Edl < ek)a f' ( ) ( )'e (e )'( ) ( ) ( ) e e e <

103 ebi f' ( ) ( ) e am[ cmebah am[ f () ( )e e, 788 KNaedrIevTIBIr f''() e ( )e e eday f''() e < am[gukm_magtibrmartg; ducehrbevggtibrmaé MN KW MN ma e, 788 ÉktaRbEvg lmhat TI5 ek[smikardiepr:g;esül ( E ) : y'' y k-edahrsaysmikar ( E ) -kmt;gukm_ f ( ) CacMelIymYyésmIkar ( E ) ebiekdwgfa ( ) f ig '( ) K-cUrsresrGuKm_ ( ) Edl, ω f f Carag f ( ) kcos( ω ϕ) k ig ϕcabicmybit X-KNaGaMgetRkal d I f ( )

104 dmena¼rsay k-edahrsaysmikar ( E ) ( E ): y'' y masmikarsmkal; r am[ r i,r i ektaj)a α, β α cmeliysmikar ( E ) CaGuKm_TMrg; y ( A cosβ B siβ) e duceh y A cos Bsi Edl A,B IR -kmt;gukm_ f ( ) ekma f ( ) Acos Bsi am[ f '( ) Asi Bcos eday f ( ) Acos Bsi am[ A ig f '( ) A si Bcos am[ B duceh f ( ) cos si K-sresrGuKm_ f ( ) Carag f ( ) kcos( ω ϕ) ekma f ( ) cos si eday ta ek)a f ( ) cos ta si - -

105 si cos si cos cos si si cos cos cos duceh f ( ) cos kcos( ω ϕ) X-KNaGaMgetRkal edayekma f ( ) ek)a tag cmebah I cos d I f ( ) cos d cos d u am[ du d b du d, am[ u, du ek)a du I [ tau] ( ) duceh cos u 8 cos d I f u ( )

106 lmhat TI55 ek[gamgetrkal e d, I IN e, I k-kna I I I rytajrk -KNa I I CaGuKm_é dmena¼rsay k-kna I I, I rytajrk I eyigma e I d, I e e eyig)a e e I I d d d d e e e I - - [ l(e ) ] e (e )' e d d l(e ) l l e (e ) e I I am[ I I l( ) e e I I, I l, I l eday duceh -KNa I I CaGuKm_é e eyig)a I I d e e ( ) e d

107 duceh I e e e d e e I e ( ) - - d e e ( e e ) d lmhat TI56 ek[gukm_ f kmt;eli IR { } ehiyepþogpþat;tmak;tmg³ f ( ) f ( ) 5 ( ) curknagamgetrkal³ f ( ) dmena¼rsay KNaGaMgetRkal³ I f ( ) I d d tag t am[ d t dt cmebah [, ] am[ t [, ] ek)a I f()d f(t )t am[ I t f(t )dt () dt

108 müa:getotebiektag t t d dt ( t) am[ cmebah [, ] am[ t [, ] ek)a t dt I f()d f ( ) t ( t) t f( )dt ( t) t ektaj)a I ( ) buktmak;tmg! ek)a ³ I I t f(t ) ( t) tamsmμtikmμekma f ( ) ek)a 5 I 6 f( ( ) t ) dt t ( t) f 5 t ( t ) dt ( t 6 6 duceh I f()d 5 ) 6 ( ) t f( )dt t

109 lmhat TI57 eksμt;fa f CaGuKm_mYykMt;elI IR ehiyepþogpþat;tmak;tmg³ f ( ) f ( ) cos curkna f ( ) dmena¼rsay I d KNa I f ( ) d eyigma I f()d f()d tag am[ t am[ d dt t, f()d, ig cmebah ek)a f ()d f( t)( dt) f( t)dt f( )d - 5 -

110 ektaj I f( )d f()d [ f( ) f() ] d eday f ( ) f ( ) cos si si ek)a I si d si d [ cos] duceh I f ( ) d lmhat TI58 b b curbgðajfa f ()d f(a b a a Guvtþ_ ³ curkna I log ( dmena¼rsay b b bgðajfa f ()d f(a b a a )d ta ) d )d tag a b t am[ d dt cmebah [ a,b] am[ t [ b,a] b a b ek)a f ()d f(a b t)( dt) f(a b t)dt a b b duceh ()d f (a b a b f )d a a

111 Guvtþ_ ³ KNa I log( ta) d eyig)a I I I I I log log log log log ta( ) d ta d ta ta ta d ta d ta d am[ektaj)a I log ( [ log log ( ta) ] ta)d log d I I - 7 -

112 lmhat TI59 ek[ f CaGuKm_Cab;elI [,] curbgðajfa f(si )d Guvtþ_³ curkna dmena¼rsay bgðajfa f(si )d? si d I cos f(si )d f(si )d tag t am[ d dt cmebah [, ] am[ t [, ] ek)a f(si )d ( t)f [ si( t) ]dt f(si )d f(si )d am[ektaj)a Guvtþ_³ KNa ( t)f(sit)dt f(si )d f(sit)dt f(si )d f(si )d f(si )d I si d cos tf(si t)dt - 8 -

113 ekma si d si d I cos si tag z cos am[ dz si d ehiycmebah [, ] eah z [, ] dz ek)a [ arcta z] si d si I z si d duceh I cos lmhat TI6 ek[ f CaGuKm_KUelI [ a,a] a a ()d k> curbgðajfa a > Guvtþ_ ³ KNa I dmena¼rsay a k>bgðajfa a f f()d, q >,q q cos d a f ()d f()d, q >,q q a a f()d f()d f()d ekma () tag q q q a t am[ d dt a igcmebah [ a,] Ma[ t [ a,]

114 a t a f ()d f( t)dt q f( t)dt ek)a a q a q t q eday () f( ) f() a f ()d q f() ektaj)a d ( ) t q f( )d q f CaGuKm_KUeaH, [ a,a] a q etacyskñúg! ek)a ³ a f ()d q f()d f()d (q )f()d q q q q a a f ()d f()d, q >,q q a duceh a a > Guvtþ_ ³ KNa I eday a q cos d cos CaGuKm_KUeaHeK)a ³ I cos d [ si ] duceh I a a f()d - -

115 - - lmhat TI6 k-knagamgetrkalkmt; ( ) IN, d I -Tajbgðajfa C C C C dmena¼rsay k-knagamgetrkalkmt; ( ) IN, d I ( ) -Tajbgðajfa C C C C tamrubmþetvfajútuekma ³ ( ) C C C C efivgamgetrkalkmt;kñúgceøah [ ], ésmpabehek)a ³

116 - - ( ) ( ) C C C C C C C C d C C C C d duceh C C C C lmhat TI6 ekmasivút ) I ( kmt;cmebahrkb; eday d e ) (! I k-curknaty I -curbba ak; I CaGuKm_é I ryctaj[)afa p P! e I K-cUrrklImIt I lim Tajfa 788 e!!!! lim dmena¼rsay k-curknaty I

117 ekma I ( )e d ( tag )e d! u du d am[ dv e d v e ek)a I [( )e ] e ( d) [ e ] e duceh I e -bba ak; I CaGuKm_é I ekma I ( ) e am[ I! ( ( )! u ( ) tag dv e d ) d e du am[ v e d ( )( ) ek)a I [( ) e ] ( I ( )! ( )!! I I duceh ( )! ( )! ) ( ) e d I ( )! e d - -

118 Taj[)afa ekma I e p I I ( )! : I I! : I I! cmebah cmebah >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> cmebah : I I! edayefivplbuktmak;tmgehggá ig GgÁ ek)a ³ I I eday I e e!!! duceh e e P!!! I!!! - -! p p! K-cUrrklImIt lim I cmebah [, ] ekma e e ig ( ) ek)a ( ) e ( ) e( ) am[ ( ) d ( ) e d ( ) eday! ( ) d! ( ) e! d

119 ektaj)a!( ) kalna e I!( )!( ) am[ duceh lim I Tajfa lim e 788!!!! ekma I e am[ e I p p! p p! ek)a lim lim ( e I ) e lim I p p! lim e!!!! erbah duceh

120 lmhatǵuvtþ_ ek[cmykmupøic z iy Edl ig y CacMYBit curkmt témø ig y ebiekdwgfa ½ 5( i) i z z i i i z k-ek[ 7 i i cursresr z ig 9 z CaTRmg BICKNit 9 kmt BIrcMYBit p ig q edim,i[ 9 9 rbsśmikar z pz q z i Ca s ek[cmykmupøic z i cursresr z ig z CaTRmgŔtIekaNmaRtrYcTajrktémø Rákdé cos ig 8 ek[cmykmupøic si 8 z i ig W cursresr z ig W CaTRmgŔtIekaNmaRt z z 9

121 5 eda RsaysmIkar z ( i )z i 6 ektag z ig z Ca srbsśmikar z z 9 9 curkna S z z 7 eda RsaysmIkar z z 9 8i 8 ek[cmykmupøic α i ig β i ektag Z ( α β)( α β )( α β ) curkmtŕkepñkbit ig Epñkimμité Z 9 ek[cmykmupøic z i ( ) Edl IN k kmt BIrcMYBit A ig B ebiekdwgfa ½ Z A B i i S Z Z Z KNaplbUk ( ) Z k k edaysresrltæplcaragbicknit ek[cmykmupøic z i ektag S z z cmeba RKbćMYKtŕWLaTIhV

122 curbghajfa S S S ek[cmykmupøic w 6 i i cursresr w CaTRmg BICKNit ig CaTRmgŔtIekaNmaRt ryctajrktémørákdé ek[cmykmupøic ½ cos ig si z (cos ) i(si cos si Edl < < curknatémøtucbmputém UDúlrbs z ek[cmykmupøic ½ z (a )(a ) i(a )(a ) ) Edl a CacMYBit k currsayfa z 5(a a ) kñúgbøgḱmupøic (o, i, j ) eksμtfa M CarUbPaBé z kmt TItaMg M edim,i[cm ay OM øibmput - 8 -

123 kñúgbøgḱmupøic (o, i, j ) ek[bycmuc A, B, C ig D magahvikerogkña i, i, 5 iig i k curedacmuc A, B, C ig D currsayfactuekan ABCD carikkñúgrgvgḿyy 6 5 ekmasmikar (E) : z z z 9 k bghajfaebi z Ca srbsśmikar ( E) ea z k¾ca srbsśmikar ( E) Edr eda RsaykñúgsMNMucMYkMupøicUvsmIkar ( E) edaydwgfa smyyrbs vamatrmg a ( i) Edl IR 6 ek[cmykmupøic k bghajfa z cos 5 cos cos 5 5 isi 5 cursresr z CaTRmgŔtIekaNmaRt 7 ek[smikar ½ a (E) : z ( i)z k kmtćmybit b edim,i[ ( i)z i z i bca srbs ( E) - 9 -

124 eda RsaykñúgsMNMucMYkMupøicUvsmIkar ( E) 8 kñúgbøgḱmupøic (o, i, j ) ek[cmuc M magahvik z epþógpþat TMak TMg z i z i currk igsgśmnmucmuc M 9 ek[smikardwerktibir ½ (E) : z ( i)z i ( k eda RsaykñúgsMNMucMYkMupøicUvsmIkar E) sresr stamgbirésmikar E) ek[bhufa P(z) z z i ( CaragRtIekaNmaRt 9 EdlzCacMYkMupøic currkgukm_smnl évifieckrvag P (z) wg z z ek[smikar (E) : z ( i)z (7 i) k kmt s z ig z ésmikar ( E) Edl z < z kñúgbøgḱmupøic (o, i, j ) ektag A,B, C CarUbPaBerogKña écmykmupøic i, z, z curedacmuctamge - -

125 K G CaárIsg érbb&æ A;), (B; ) currkgahvikécmuc G ( ig ( C; ) ek[sivútécmybit ( u ) kmtŕkb IN eday ½ u ; u u u u i k ektag z u u RKb IN i currsayfa z z ryckna u CaGuKm_ igtmak TMg é TajrktYTUeTAésIVút u ) ( ek[sivútcmybit ( a ) ig ( b ) kmtéday ½ a a b ; b a a b b, IN a i b currsayfa z ( i) z k tag z cursresr z ig K TajrktY a ig z CaTRmgŔtIekaNmaRt b CaGuKm_é - -

126 ek[sivútécmykmupøic ( z ) kmtéday ½ z ig Edl IN z i z i i k ektag w z currsayfa w w cursresr w ig w CaTRmgŔtIekaNmaRt K cursresr z CaTRmg z r (cosθ isiθ ) 5 ek[smikardwerktibir (E) : az bz c Edl a, a, b,c IRsμtfaΔ b ac < ea smikar ( E) ma sbircacmykmupøicqøasḱñaedltageday z ig z cmeba RKb ZeKyk S z z currsayfa as bs cs Guvtþ_½ edaymiác BøatcUrKNatémø 7 M ( i ) ( N ( i ) 5 ( i i ) )

127 6 ekmacmykmupøic z i ig z i tag α ig β erogkñacagakuym g é z ig curbghajfa α β coskθ z 7 ek[ C ( ) ig S ( sikθ) k curbghajfa Edl currsayfa k k z C is z z cosθ i siθ θ si z θ cos z θ si S K Tajrktémøé C ig 8 ek[ C ig S k k cos cosk i S k bghajfa kmupøicmyy Tajrktémøé C ig S k θ isi k si cosk C CaplbUksIVútFrNImaRtécMY - -

128 9 KNaplbUk ½ p p C ( C cosp) ig S ( C sip) Edl p C p! p!( p)! ek[sivútécmykmupøic z ) z z k ektag Rsayfa, z - - p ( kmtéday ½ i i z z, IN w z z w i w ryctajfa kñúgbøgḱmupøic (o, i, j ) ekeha M CacMucmaGahVikerogKña,z,, z ektag z w,m,, M M M k k k α β S currklimit lims ek[cmykmupøicbir α ig β Edl ig αβ bghajfa α β αβ CacMYBit

129 ek[cmykmupøic z i i k rkcmykt vic ma EdleFIV[ z CacMYBit KNa z cmeba témøtuccageké EdlárkeXIj KNa edim,i[ z CacMYimμitsuTæ ek[ zcacmykmupøicmam UDúlesμI iggakuym g α cursresr Z z z CaragRtIekaNmaRt kñúgbøgḱmupiøic (o, i, j ) cmuc M CarUbPaBécMYkMupøic ½ z cosθ i siθ eday θ ] ; [ k currksmnmuécmuc M cmuc N magahvik z N z rksmnmucmucn kalnaθepþógpþatĺkç&nð& ] ; [ 5 ek[cmykmupøic z iy ig z z θ i i Edl ig y CacMYBit eti ig y RtUvepÞógpÞatĺkç&NÐ&ya gnaedim,i[ Z ½ k etacacmybit etacacmyimμitsutæ Z - 5 -

130 6 ek[cmykmupøic z epþógpþat z 5i 6 ehiy M CarUbPaBé z kñúgbøgḱmupiøic (o, i, j ) k currk ig sgśmnmuécmuc M curkmt TItaMgécMuc M edim,i[cmykmupøic z magakuy m gǵb,brma ryckmtŕkgakuym gǵb,rmaea 7 ek[cmykmupøic z epþógpþat z 8 6i 5 ehiy M CarUbPaBé z kñúgbøgḱmupiøic (o, i, j ) currk ig sgśmnmuécmuc M curkmt TItaMgécMuc M edim,i[cmykmupøic z mam UDúl ½ k- Gb,brma? -Gtibrma? 8 kñúgbøgḱmupøicek[bicmuc A, M ig M ' magahvik erogkña i, z ig i z curkmtśmnmucmuc M edaydwgfa A,M,M' rtŕtgḱña 9 ek[cmykmupøic z i y ig Edl z ;,y IR Z z z - 6 -

131 ekeha M CacMucmaGahVik z ig M ' CacMucmaGahVik Z sisteakñúgbøgḱmupøic (o, i, j ) k kmtśmnmucmuc M kalna M' ERbRbYlelIGkß& ( o) kmtśmnmucmuc M kalna M' ERbRbYlelIGkß& ( oy) t t z i Edl t IR ek[cmykmupøic t t k curbghajfacmykmupøic z mam UDúlefrRKb témø t M CarUbPaBé z kñúgbøgḱmupiøic (o, i, j ) curkmtśmnmucmuc M kalna t ERbRbYltémø ek[gukm_ f() curknalimit ekmagukm_ f() lim () f() limf( f curknalimit limf() ig ) ek[gukm_ 5 f() 5 limf( curkna ) 5

132 ek[gukm_ f() si cos ( curknalimitégukm_ f ) kalna itetacit 5 ek[gukm_ cursikßapabcab égukm_ f () RtgćMuc 6 ek[gukm_ cos cos( ) ebi f() ebi a b f() Edl limf() kmt BIrcMYBit a ig b edim,i[ 5 7 KNalImIt 8 ek[gukm_ lim m m m (m ) f() m ig IN * Edl IN * curknalimit limf() f () 9 ek[gukm_ k KNatémø f ()

133 KNalImIt 5 KNalImIt 5 ekmagukm_ ( ) f() lim lim ta cos cos(si ) f() Edl cos cos( ) f(), currklimitégukm_ f kalna 5 ekmagukm_ 5 currklimitégukm_ f kalna 5 KNalImItageRkam ½ k l( ) lim ta e cos lim si lim l(ta ) K - 9 -

134 55 KNalImIt ½ k lim( ) lim( 6 ) 56 curknalimit ½ k lim lim( a)( a )( a )( a ), < a < 57 KNalImIt ½ k ( ) lim lim( ) 58 ek[gukm_ ½ f () ma skaer curkna limf () 59 curknalimitagerkam ½ ( ra DIkal ) lim - -

135 6 curkna rycknalimit S 5 ( lims 6 ek[ f CaGuKm_kMt eday ½ f() ektag k S k f( ) lims curknalimit 6 ek[ curkna P lim P 6 ek[sivútécmybit ( u ) kmtéday ½ u u ig u u ) cmeba RKb IN * curkna lim( u ) 6 ek[sivútcmybit ( a ) kmtéday ½ a a a, IN - -

136 k ektag v u Rsayfa ( v ) CasIVútFrNImaRtrYcKNa v CaGuKm_ KNalImIt limu ig lims Edl S (v ) k k a b ebi < 65 ek[gukm_ f() ebi curkmtćmybit a ig b edim,i[ f maedrieveli IR 66 f CaGuKm_kMtélIceøa ] ; [ Edl f() l a b ebi - - ; a,b IR ebi < < curkmtćmybit a ig b edim,i[ f maedrievrtg 67 curknaedrievégukm_ f() l(e e ) f () e ( )e 68 ek[gukm_ currsaytmak TMg f ( ) () f'() f() ek[gukm_ f() si cos si cos k bghajfagukm_ f migars&ywg rkltæple eligvijedaycat Tukfa f CaGuKm_Gefr

137 7 ek[gukm_ y l l k currkedkmt égukm_e cmeba RKb kñúgedkmtćurrsayfa y' ( y) 7 ek[gukm_ f () cos k curknaedriev f '() ; f''() ig f ( ) () edayefivvicartamkmeicurrsayfaedrievti égukm_e f() e si ) kmtéday f () cos( ) ( 7 ek[gukm_ k KNa f '() rycbghajfa f'() e si( ) edayefivvicartamkmeicurrsayfa edrievti égukm_e ) kmtéday f () ( ) e si( ) ( f () ( )e 7 ek[gukm_ k curknaedriev f '() ; f''() ig f ( ) () edayefivvicartamkmeicurrsayfaedrievti égukm_e - -

138 kmtéday f () l ; > ( ) f () ( )e 7 ek[gukm_ edayefivvicartamkmeicurrsayfaedrievti égukm_e kmtéday f! () ( ) f () (cos si )e ( ) 75 ek[gukm_ k curknaedriev f '() ; f''() ig f ( ) () edayefivvicartamkmeicurrsayfaedrievti égukm_ f ( ) marag f ( ) (a cos b si )e Edl ( a ) ig ( b ) CasIVútcMYBitkMtélI IN * eday a a b ig b b a K ektag z a i b bghajfa z ( i) z ryckna z CaGuKm_é X Tajrk a ig b CaGuKm_é ( ) g TajrkGuKm_ f () - -

139 76 ek[gukm_ a b () ϕ ] ; Edl f (taϕ) f '(taϕ) si ϕ acos ϕ f sμtfamacmybit [ currsaybba ak fa 77 curkmtśmikarbþat b wgeßekag RtgćMucEdlmaGabśIusesμI 78 ekmagukm_ f() (c) : y 9 edaymierbiedrievcurrktémøbrmaégukm_e 79 ek[gukm_ f() e a b f ''( f() e Edl a ig b CacMYBit k KNa f '() ig ) kmt BIrcMYBit a ig b edim,i[gukm_f magb,brma esμi e Rtg 8 ek[gukm_ f() KNaedrIev f '() l Edl > bghajfaf matémøgtibrimamyyrtuvkmt - 5 -

140 8 ek[gukm_ g () a bl markab H) bþat ( D) masmikar y kmtćmybit a ig b edim,i[bþat ( D) b wgrkab ( H) RtgćMuc A ( ; ) 8 ek[gukm_ f() l( ) k KNa f '() ( cmeba RKb [ ; ] bghajfa 6 f' () 8 K edayerbivismpabkmeimakmtćmeba RKb [ ; ] currsaybba ak fa ½ 5 9 l l( ) l 5 5 [, 8 ekmagukm_ g : kmtéli [ edayerbivismpabkmeimakmt bghajfacmeba RKb [ ;] eká 5-6 -

141 8 f CaGuKm_kMtélI [ ; [ eday ½ e f() f() f() k KNa lim ebi ebi etiekgacfayägnacmeba GuKm_ f? cmeba Rkabtag f? cmeba RKb ], [ bghajfa f'() K KUstaragGefrPaBéGuKm_ f 85 cmeba RKbćMYBit < a b currsaybba ak fa ½ b a b a lb la b a a b < b a b a tab taa cos a cos b [ ; 86 cmeba RKbćMYBit > e < currsaybba ak fa ½ 87 cmeba RKb ] currsaybba ak fa ½

142 88 kñúgtrmuygrturmäl (o, i, j ) ek[bþatḿasmikar ½ ( d) : y M CacMucsSiteAelIbÞat ( d) magabśius r Edl r < Pig Q CacMeNalEkgé M erogkñaeligkß& ( o) < ig ( oy) kmt témørbs r edim,i[ctuekan OPMQ maépþrklagtibrma 89 kñúgtrmuygrturmäl (o, i, j ) ek[eßekagmasmikar 8 ( c) : y M CacMucsSiteAelI c) r Edl r > Pig Q CacMeNalEkgé M erogkñaeligkß& ( o) ig ( oy) ( magabśius kmt témørbs r edim,i[ctuekan OPMQmaépÞRkLa Gb,brma 9 plbukbircmyvic maesμi kmt BIrcMYe edaydwgfaplkunrvagcmytimyywgkaeré cmytibirmatémøgtibrma - 8 -

143 9 ek[á ra bul ( P) : y ehiy A, B, C CabIcMuc sisteaeli ( P) ekdwgfa A ig B CaBIrcMucma GabśIuserogKña ig ehiyc CacMucmaGabśIus r Edl < r < kmt r edim,i[rtiekan ABC maépþrklagtibrma 6 9 RtIekaNEkgmYymabrimaRt m kmtŕcugrbsŕtiekane edim,i[épþrklavagtibrma? A 6 o 9 RtIekaN ABC myymabrimart 6 m igmmu kmtŕcugrbsŕtiekane edim,i[épþrklavagtibrma? 9 ek[á ra bul (P) : y M CacMucsiSteAelI P) edim,i[ AM matémøgb,brma? 95 ek[ igcmuc ( 6; ) A ( magabśius r kmt témø r ( P) : y ig ( d) : y M CacMucsiSteAelI P) kmt témø r edim,i[ d (M;(d)) matémøgb,brma? ( magabśius r - 9 -

144 96 ek[rtiekan ABC myyedlmargvasŕcúg ½ AB cm ; BC 8cm ; CA 6cm [ Edl CM cm T 5MA MB M CacMucmYyé BC] ektag curkmt edim,i[ T matémøtucbmputrycrktémøtucbmputé T? 97 ekabrivtþ_myycar ikerkaesv kam R 8cm tag r ig h erogkñacakamfasátigkmbsŕbsékane kmt r ig h edim,i[ekae mamadgb,brma? 98 curkmtḱmbsŕbsśiulamgrtgḿyyedlmamadgtibrmaehiy GaccarikeRkAEsV myyedlmakam cm R 99 kmtćm aygb,brmabicmuc M(; ) etaá ra bul 8 y 5 ctuekanekgmyycarikkñúgeglib y ehiyrcugrbsćtuekane RsbCamYywgGkß&rbséGlIb k kmt vimartrbsćtuekane edim,i[vamaépþrklagtibrma? kmt vimartrbsćtuekane edim,i[vamabrimartgtibrma? - -

145 ek[gukm_ f() l k curknaedriev '() KNalImIt limf() PaBéGuKm_ f K curtajfarkb eká - - ( ) kmtćmeba > f igbba akśbaøaé '() ig limf() ryckustaraggefr l f ( ) ek[gukm_ f() e kmtŕkb IR k KNaedrIev '() KNalImIt limf() PaBéGuKm_ f K cmeba RKb f rycsikßasbaøarbs va ig limf() ryckustaraggefr IR curbghajfa e f() cos ) f'() ; f''(), f ( ekmagukm_ k curknaedriev ( ) ig f ) ( ) cmeba RKb curbba akśbaøa f ( ) () ryctajrk ( ) sbaøarbs f (),f''() ig f '() (

146 K TajfaRKb eká cos f() e 6 f'() ; f''(), f ( ) ( ig f ) ( ) ek[gukm_ k curknaedriev ) cursikßasbaøa f ( ) () TajrksBaØarbs f ( ) (),f''() ig '() K TajfaRKb IReKá ½ e 6 f ( 5 k-cmeba RKb currsaybba ak vismpabagerkam ½ l( ) -ekbiitüsivút ( )( )( ) l P P edayerbivismpabageli currktémøgmé K-TajrklImIté P kalna - -

147 6 ekbiitüsivút ½ U si( ) si( ) si( ) si( ) kmtćmeba RKbćMYKt FmμCati 6 U k cmeba RKb currsayfa si edayerbivismpabagelie cursresrkeßamgmé K TajrklImIt lim U 7 ekmasivút ½ S ( k ) k IN k cmeba RKb * p p currsaybba ak fa ½ p l p p cursresrkeßamgmrbs S ryctajrklimit 8 ek[sivútcmybitkmtŕkb IN * eday ½ S ( 9 ) lims - -

148 k cmeba RKbćMYBit a ig b Edl < a b a b a b a lims currsayfa b a KNalImIt 9 curknalimit ek[ f CaGuKm_kMtéday lim 6 f() ( ) ( c) CaRkabtMagé f eakñúgtmruygrturmäl (o, i, j ) IR k bba ak fa f kmtćmeba RKb KNalImIté f kalna itetacit ryctajfa ( c) magasiumtutmyy K KNa f'() rycsikßasbaøarbs va b, Tajfa f maftibrmamyy ig Gb,brmamYyrYcKNatémø brmatamgea X KUstaragGefrPaBé f g KNakUGredaeécMucRbsBVrvagEßekag c) ( iggkß& - -

149 TaMgBIrétMruyigcMucRbsBVrvagEßekagCamYywgGasIumtUtedk c KNa f () ig f () cursgéßekag ( c) c) lim g(), lim g( ( c ek[ g(), ( CaRkabé g k KNalImIt ) ig lim g( ) ryctajrkgasiumtuté ) KUstaragGefrPaBé g K bghajfa ( c) macmucrbtḿyyedlekwgrkkugredaeva X KNa g( ), g( ), g() ig g () g sgéßekag ( c) eakñúgtmruygrturmäl ek[gukm_ 6 9 f() 6 9 markab c) ( k curkedkmt égukm_ f KNalImIt limf(), limf() ig limf( ) TajbBa akśmikargasiumtutqr ig GasIumtUtedké c) K KNaedrIev f '() rycsikßasbaøaé f '() X KUstaragGefrPaBé f ryckusrkab c) ( ( - 5 -