DIepr gésül. CMBUk5. niymn&y ekegay f CaGnuKmn_kMntélIcenøa¼ J. ekyigniyayfa f manfdiepr gésülrtg a J ebi

Átírás

1 CMBUk5 DIepr gésül ekyigerbiiym&yc,asĺas élimit eakñúgkarvipakg}ts&yrkw¼évifikna eakñúgcmbuke¼eyigsikiksßa lkçn DIepr gésül tamiym&yéerievégukm_rtgḿyycmuc eyigbiitüemillkçn égukm_ maeriev ehiybþetacmenatékarknatmélrbehlrbs FhuFa 5- eriev iym&y ekegay CaGuKm_kMtélIceøa¼ J ekyigiyaya mafdiepr gésülrtg a J ebi a ebi madiepr gésülrtg a ekyigkmtǵukm_ a eaeli eay a () a () = () [ (a)( a) + (a) ] ea¼ () = (a)( -a) + a () El 0 eaebl ekexijya gc,as a GuKm_ ma DIepr gésülrtg a lu¼rtaetmagukm_lieneg r g eaeli EleKKNatMélRbEhlé () eay g( -a) + (a) cmeba¼lmegog a () ElepÞógpÞat lkç&nð a () 0 kalna «TahrN_ sikßapabmadiepr gésülrtg = 0 égukm_ eaeli UcageRkam (i) () = cmeba¼ 0, () (0) 0 Uecñ¼ madiepr gésülrtg 0 ehiy (0) = 0 (ii) lim a () (a) ma ebilimite¼martg a ekyigtagvaeay () ehiyeyigehaaca ² eriev ³ é Rtg () = ²si ² ebi 0 = 0 eaebl 0 ² ebi 0 cmeba¼ 0, () = 0 () (0) 0 si si

2 0 kalna 0 Uecñ¼ madiepr gésülrtg 0 ehiy (0) = 0 «TahrN_ 5- sikßapabmadiepr g EsüléGuKm_sIuuseAelI? yk a tag = a + h El h 0 eyigá eká Et si() si(a) si(a h) si a h sih cosh cos a si h h h si cosh h h h si h = 0 h si si a cos a kalna kalna h 0 Uecñ¼ GuKm_sIuuis madiepr gésüleaeli ehiy si() = cos cmeba¼ cmeba¼gukm_ eaeliceøa¼ebik J CYUkaleKcg vacakarsmrmükñ úgkarsikßalkçn limitefqvg iglimitsþamrtg = a égukm_pl EckkMt eay () (a) ebilimitefqvgma ekyigtag (a) lim eaeli J \ a a () (a) ehiyebimalimitsþamma ek eyigtag (a) lim a () (a) ehiyekyigehavaacaerievefqvg ig erievsþamerogkñaé Rtg a CYUkaleAeBlsikßaPaBmaDIepr gésülé GuKm_ Rtg JeKyIgktśMKaléXIja eta¼bica mima DIepr gesülrtg a k¾eay k¾myy TaMgBIr é eriev agefqvg agsþamgacma CakarBitNas agukm_ madiepr gesül a lu¼rtaeterievagefvg igag sþamekit eligrbmkña ehiyesµikña eyigrktmak TMgrvagPaBCab igpabmadiepr gésül RTwsþIbT ebigukm_ eaeliceøa¼ebik J madieprësülrtg a ea¼ CabŔtg a

3 smraybba ak () (a) () (a) ( eyigma lim (a) a Uce¼cMeBa¼, () (a) = ) (a)0 = 0 kalna 0 Uecñ¼ lim () (a) KWwa CabŔtg a a RtTwsþIbTe¼bg;b &ya KµaDIepr gésül ebi micab CaTUeTAlkçN Rcas értwsþibte¼mibitet «TahrN_ cmeba¼gukm_ eaeli kmtéay () = eyigma () (0) (0) lim ig 0 (0) 0 lim 0 () (0) 0 Uecñ¼ KµaDIepr gésülrtg = 0 et Et CabŔtg = 0 lkçn BICKNitéRTwsþIbTerIev eyigcabépþimeaycgðúlbba ak GMBIGuKm_ermaerIevesµI 0 ehiygukm_øüégmaerievesµi : ebi () = ea¼ () = 0 cmeba¼ ebi () = ea¼ () = cmeba¼ smraybba ak + cmeba¼ () = eyigma ( h) () () = lim lim lim 0 0 h h h0 + cmeba¼ () = eyigma h0 h0 ( h) () ( h) h () = lim lim lim lim h h h h0 h0 h0 h0 RTwsþIbT ebi ig g CaGuKm_eAelIceøa¼ebIk J ehiy ig g madiepr gésülrtg aj ea¼ + g, g cmeba¼ ig g madiepr gésülrtg a ehiy ( i ) ( + g )(a) = (a) +g(a) ( ii) ( )(a) = (a) ( iii) ( g )(a) = (a) g(a) + (a)g(a) (a) ebi g(a) 0 ea¼ g madiepr gésülrtg a ehiy ( iv ) g(a) (a)

4 smraybba ak ( g)() ( g)(a) () (a) g() g(a) ( i ) cmeba¼, () (a) g() g(a) Ettamiym&y lim (a) lim g(a a a ( g)() ( g)(a) Uecñ¼ lim (a) g(a) a ma&ya ( +g)(a) = (a) + g(a) ( ii ) cmeba¼, eyigma (a) (a) () (a) ( )() ( )(a) lim lim ig ) () (a) Uecñ¼ (a) a a ma&yaa ()(a) = (a) ( iii ) cmeba¼, eyigma (g)() (g)(a) () (a) g() g(a) g(a) () () (a) g() g(a) lim (a ig lim g(a) a a (g)() (g)(a) lim (a)g(a) (a)g(a) a eay ) ma&ya (g)(a) = (a)g(a) + (a)g(a) (a) (iv) eay g madiepr gésülrtg a ea¼vacabŕtg a eyigá 0 El g() 0 cmeba¼ a -, a + Uce¼ g ma&yeliceøa¼ a -, a + ea¼

5 cmeba¼, Uecñ¼ lim a () (a) g g g() g(a) g(a)g() () (a) g g g(a) g(a) ma&ya (a) g (a) g g (a) kuruëlr ebi ig g madiepr gésülrtg a ehiy g(a) 0 ea¼ g eaysresr g smraybba ak g eyigá (a) (a) (a) (a) (a) (a) g g g g tam (iii) ageli g(a) = (a) (a) g(a) g (a) (a)g(a) g(a) (a) = g (a) tam (iv) ageli smraybba ak «bmaa 0 El () (a) cmeba¼ a -, a + cmeba¼ a -, a + ig 0 eyigá h () h (a) () (a) h () h (a) (a)g(a) g(a) (a) (a) g g (a) () (a) madiepr gésülrtg a Kwa RTwsþIbT ebi CaGuKm_kMtélIceøa¼ ebik J ig h CaGuKm_kMt eaeliceøa¼ebikelma (J) ehiy ma DIepr gésülrtg aj ig h madiepr gésülrtg (a) ea¼gukm_ J ma DIeprËsülRtg a KWa h g (a) (a)h (a) h ho kmtéaeli (a) h((a)) kalna «bmaakµa 0 ea¼ masvúit El a ig 0 El ( ) = (a), cmeba¼ eayma DIepr gésülrtg a ea¼ (a) = 0 cmeba¼svúit e¼eyigma h o (a) = h ((a))

6 h h (a) a Uce¼ 0 EtcMeBa¼ svúit NamYyEl 0 ig ( h ( ) h (a) h ( ) h (a) ( ) a ) (a), ( ) (a) a 0 erba¼ CabŔtg a ig (a)= 0 Uecñ¼ eakñúgkrnie¼ h o k¾madiepr gésülrtg a Er ehiyeay (h o )(a) = 0 ekgacexija a h o epþógpþatŕbmanvifibnþak eyigrktmak TMgrvagPaBmaDIepr gesülégukm_cabŕbka iggukm_rcasrbs va RTwsþIbT ebi CaGuKm_Rbka ig CabḱMtéAelIceøa¼ebIk J ehiy madiepr gésülrtg a J ig (a) 0 ea¼ madiepr gésülrtg (a) KW : (a) tag b = (a) ig y = () cmeba¼, (y) y b smraybba ak b () (a) () (a) eay madiepr g EsülRtg a ea¼vacabŕtg a tamlkçn PaBCab égukm_rcas Epñk 44 eyigá Cab Rtg b KWa (y) (b) kalna y b kalna y b Uecñ¼ eay (a) 0 eyigá kalna y b (y) (b) y b a RtTwsþIbTe¼masar RbeyaC_ya gøamgeakñúgkarknaeriev «TahrN_ sikßapabmadiepr gésül igrkkeßamerievégukm_ eaeli + kmt eay () = cmeba¼ * (a) MeNa¼Rsay ebieyigcat Tuk CaGuKm_Rcasé GuKm_ g kmtéaeli + eay g(y) = y ea¼tamrtwsþibtagelieyig á

7 g(y) cmeba¼ + \ 0, g(y) Kwa () y Et Kµa DIepr gésülrtg = 0 et ebigukm_ kmtéaeliceøa¼ebik J madiepr gésülrtg RKb J ekyigiyaya madiepr gésül eaeli J ehiycat Tuk acagukm_madiepr gésüleaeli J eyigbiitüemilsmumégukm_el madiepr g Esül eaeliceøa¼ebik J CakarsmRsbeKyIgykcitþTuk akćmeba¼ceøa¼] 0,[ehIyeyIgtagsMuMe¼eayD] 0,[ smum D ] 0,[ matmrg BICKNit EleKGacTajáBIlkçN BICKNitéRTwsþIbTerIev cmeba¼ GuKm_ ig g ElmaDIepr gésüleaeliceøa¼ ] 0,[, + g ig, cmeba¼ k¾cagukm_madiepr g Esül eaeli ]0, [ ehiy D]0, [ CalMhlIeNEG Bit tamiym&yéplbukbirgukm_ igplkun égukm_myy eaycmybit UcKñae¼Er g madiepr gésüleaeli ]0, [ ehiy D]0, [ matmrgćabicknitelmalkçn Rtlb tam iym&y plkunébirgukm_ imiµtsbaøa, erievlmab <s imiµtsbaøa ekyigábghajrycmkehiygmbierieveltageay ( Gaa RBIm ) / b ueþeamaimiµtsbaøaetetot EleKiym erbirás ya gøamg CaBiesseAkñ úgepñkvitüasarsþ ig EpñkvisVkmµ imiµitsbaøaelekiym erbirás bmputea¼ KWwimiµtsBaØa rbsélak Heibig eakñúgimµitsbaøa Heibig ektagbghajerievégukm_ y eay sresr y, y t y, z GaRs&ywgaetIGkßr, t z RtÚváeKeRbICaFatuéEkMtŕbs y UcCa ebiekkmt y eay y() = ³ y() ea¼imµitsbaøa Heibig KW CaTUeTAeKlb ( ) ecalehiysresrya ggaya y = ³ ig y imiµtsbaøa, t, k¾rtúváekerbicabubvbt ( Prei) eabimu z keßam ElRtÚvKNaEr «TahrN_ (³ 4) = ² 4, (t² + t + ) =t +, (z 5 ) = 5z 4 t z

8 eakñúgimiµtsbaøa Heibig rubmþerievkw + () g() () g() + () () ()g() () g() g() () + g() g() g() g() () () g() () + g() + g() CajwkjabéKCMYs ig g eay u ig v ehiya¼rubmþagelietaca «TahrN_ rk y u (u v) u ( u) v (uv) u ( ) v v v v u v u v v u u v v y cmeba¼ y 5 MeNa¼Rsay (5 ) ( ) ( ) (5 ) (5 ) (5 )() ( )(5) = (5 ) (5 ) «TahrN_ rk eyigma y cmeba¼ y = (³ + )( 5 + ) MeNa¼Rsay y 5 5 ( ) ( ) ( ) ( ) 4 5 = ( )(5 ) ( )( ) = ( ) + ( )

9 = «TahrN_ rk t t t t t t t = t MeNa¼Rsay (t ) t t t (t ) t(t) t t (t ) (t ) «TahrN_4 rk u cmeba¼ u = ( + )( + ) MeNa¼Rsay eyigma u = ( + )( + ) = [ ( + ) ] ( + ) = [( + )( + )] u [( + )]( + ) = [( + )]() + ( + ) [ ( + )] = ( + ) + ( + ) [ () + ( + )() ] = ( + ) + ( + )( + ) = ( * ) ( * ) u [( + )]( + ) = [ ( +)( + )] + ( + )( + )() = [( + )() + ( + )()] ( + )( + ) = (+) + ( +)( + ) ( * * ) = ( * * ) ( * ) ig ( * * ) eká u 6 u = ( + )( + ) = ( + + ) = + + u 6 «TahrN_5 KNa MeNa¼Rsay y y Rtg = 0 ig = cmeba¼ ( 4)() () 8 ( 4) ( 4) y + ebirtg = 0, 0, y

10 + ebirtg = y 8 9 erievlmab <s ( Derivatives o Higher Orer ) ebigukm_ madiepr g Esül ea¼ekyiggacbeg;itgukm_µi ebi madiepr gésül " ea¼ekyig Gac beg;iterievrbs va ehaa erievti é igtagvaeay ¾rabNaeAmaPaBmaDIepr g Esül ekeyig Gacbþkñúgrebobeaybeg;It ", cmeba¼erievti4 eyigtag vaeay (4) ehiycatuetaeyigtagerievti eay () ebi () = 5 ea¼ (6) () = 5 4, " () = 0, " () = 60, (4) () = 0, (5) () = 0, (6) () = (7) () = = () () = 0 (7) () () () () 0 4 y 5, y" 0, y = 5 4, y " = 0, eksresrerievlmab <sḱñúgimiµtsbaøa Heibig eay y y y y y y,,,, () (), () (),, () (),, «TahrN_TI6 ebi () = ea¼ () = ig " () = ² 6 «TahrN_TI7 ( ) = Uce¼ ig 4 «TahrN_8 cmeba¼ y = kñúgimiµsbaøa Heibig eká y y, CaTUeTA y, y 4 6, - eakñúgimiµtsbaøa ² ³ eyigá 4 y

11 y = -, y " =, y " = 6 4, y (4) = 4 5 ig y () = ( ) erievbnaþak «bmaa y CaGuKm_maDIepr gésüléger u El :y = (u) ig u CaGuKm_maFDIIepr gésüléger El:u = g() ea¼ y CaGuKm_bNþak é : y = (u) = (g()) eti y maerievefobwg et? cmeliykwbiitcama ehiy UcCaTUeTA ebi y = u ig u = ea¼ y = 6 eyigma y 6, y u u, y y u u Uce¼ 6 ebi y = u³ ig u = ² ea¼ y = y 5 6, Uce¼ «TahrN_ rk y u y 4 u u, u y y u u u y u y eayerbirubmþerievbnþak ebi MeNa¼Rsay u u () u (u ) ig Uce¼ y y u = u eyigá ig u = ² u u cmnam eyiggacttulált pl UcKña eaymicamácérbirubmþerievbnþak eteayrkaétcambug eyigrtúv sresr y CaGuKm_é rycknaerievfmµta eay y y u u y () ig u = ² eyigá y Uce¼ 4 4 ebi CaGuKm_maDIepr gésülé u ehiy u CaGuKm_maFDIepr gésülé ea¼ y u u u (u) (u) u

12 rubmþ smraybba ak tag (u) = u eyigá u u u u u u 00 eyigtag u = ² UcCaeIim,IKNa tamrubmþe¼ ekgacbrgikrubmþ y y u u u u u ya ggay UcCa ebi GaRs&y wg s ea¼eká y s y u u s ehiyebi s GaRs&ywg t ea¼ y t y u u s s t «TahrN rk y ebi y = u +, u =, = s s y s y u MeNa¼Rsay y / u / u s Uecñ¼ u s 6 6 s

13 «TahrN_ rk y Rtg t = 9 ebi t MeNa¼Rsay eyigá y u y u u u, 6s 7 s u s 7 / s t / u, Rtg t = 9 eyigá s = ig u u = 4 Uce¼ y u Uecñ¼Rtg t = 9, y t y u s t / s u s s t 6 u / t s t 9 erievégukm_gambøisuit (Implicit Dieretiatio ) «bmaa y CaGuKm_é ElepÞógpÞat smikar ³y 4y + = 0 eim,irk y eyigrtúvea¼rsaysmikare¼ rk y si rycerievefob wg eyigá y (³ 4)y + = 0 (³ 4)y = y = 4 4 () (9 ) ekk¾gacrk y eaymicamácéa¼rsaysmikare¼rk y Er vifie¼ehaa «DIepr gésülgambøisiut» eayerieveligg:tamgbiirésmikarageliefobwg ( eaycama y CaGuKm_é ) eyigá 4y 0 () y y 9 y 4 = 0 rubmþplkun y 4 = 9²y y 9 y () 4 cmeliyagelie¼emiletauccausbicmeliy EDleyIgáeFVIBImu erba¼eble¼ma y eagg:agsþam eim,i

14 y eakñúg () eyigá epþógpþat acmeliytamgbire¼uckña eyigrtúvcmys y 4 4 ElCacMelIy () ageli DIepr gésülgmbøisuitmasar RbeyaC_CaBiess eaeblelekmigacea¼ RsaysmIkar smikarelekegayrk y á «TahrN_ rkemkunráb TiséEßekag ³ y + y³ = RtgćMuc (,-) MeNa¼Rsay eyigá y y y y y y y y 0 y y 6y y y 0 Rtg = ig y = smikare¼etaca Uecñ¼ emkunráb TisesµI y y 0 y 5 9 y 5 5 ekk¾gacrkerievlmab <s tamiepr gésülgambøisiuter y «TahrN_ rk MeNa¼Rsay erievefobwg eyigá ebi y³ ² = 4 erievmþgetot efobwg eyigá y y y 6y y y 0 () 0 ()

15 y y BI () CMYstMéle¼kñúg () eyigá y y 6y y 0 y 6y 8 5 9y cmnam ebieyigefvidiepr gésülgambøisuitetaeli ² + y² = eyigexija y y y 0 y EtlT ple¼kµa&yet erba¼kµagukm_yketmyy y é NamYyElepÞógpÞatśmIkar ² + y² = DIepr gésülgambøisuitgacguvtþegayma&yácmeba¼smikargbaøat ig y EteAeBlElmaGuKm_ y ig ElepÞógpÞatśmIkar 5- lkçn égukm_madiepr gésül RTwsþIbT ebi CaGuKm_kMtélIceøa¼ebIk J ehiy matmélgtibrma Gb,brmaRtg a J igma DIepr gésülrtg a ea¼ (a) = 0 smraybba ak eay madiepr g EsülRtg a ea¼ (a) = (a) (a) «bmaa matmélgtibrmartg a Kwa () (a), J () (a) () (a) () cmeba¼, 0 lim 0 () a) a () (a) () cmeba¼, 0 (a) lim 0 a ()& () : (a) = 0 «bmaa matmélgb,brmartg a Kwa () (a), J () (a) () (a) ( i ) cmeba¼, 0 (a) lim 0 () (a) a () (a) (ii) cmeba¼, 0 (a) lim 0 (i) & (ii) : (a) = 0 a RTwsþIbT Rolle : ebi CaGuKm_CabéAelIceøa¼biTT&l [a,b] igmadiepr gésüleli ]a,b[ ehiy (a) = (b) ea¼ ma c]a,b[ El (c) =

16 smraybba ak y (c) = 0 ([a,b]) 0 a c b ( rub 5 ) tamlkçn kmuá k égukm_cab, ([a,b]) matmélgtibrma ig Gb,brma «bmaa matmélgtibrma Gb,rmaRtg c]a,b[ ea¼tamrtwsþibtagelieyigá (c) = 0 «bmaa matmélgtibrma ig Gb,brmaRtg a ig b ea¼eay (a) = (b), CaGuKm_er (c) = 0 cmeba¼ c]a,b[ RTWsþIbTtMélmFüm ( The Mea Value Theorem ) ebi CaGuKm_CabéAelIceøa¼biTT&l [a,b] igmadiepr gésüleli ]a,b[ ea¼ ma c]a,b[ El smraybba ak y c (b) (a) b a (b) (b) (a) (c) b a (a) 0 a c b ( rub 5 ) (b) (a) b a eyigbiitüemilgukm_ g eaeli [a,b] kmtéay g() () a g CaGuKm_CabéAelI [a,b] igmadiepr g EsülelI ]a,b[ ehiy g(a) = g(b) = (a)

17 Uecñ¼tamRTwsþIbT Role ma c ]a,b[ El g(c) = 0 (c) (b) (a) b a ebi F CaGuKm_maDIepr g EsüleAelIceøa¼ebIk J / GuKm_µIehAa «GuKm_erIev» igtageay KWCaGuKm_kMtéAelI J El () CaerIevé RtgćMucImYy@ J KYrktśMKal a GuKm_erIev GacCaGuKm_ac «TahrN_ rkdieepr gésülégukm_ eaeli kmt eay () = MeNa¼Rsay \ {0}, () = si cos ehiy lim () Kµa 0 Et (0) = 0 eyigexija madiepr gésül eaeli Et mapabacŕbepti Rtg 0 eyigwgbghaja eta¼bigukm_eriev micamác CaGuKm_Cabḱ¾eayk¾ vabitcamalkçn kunic UcKña smraybba ak yk a, bj El a b ehiy«bmaa (a) 0 (b) eyigbghaja ma c]a,b[ El (c) = 0 GuKm_CabéAelI [a,b] Uce¼tamlkçN kmuá k égukm_cab / ([a,b]) matmélgtibrma eay (a) 0 ea¼ma El a b ig () (a) ehiy eay (b) 0 ea¼ma " El a " b ig (") (b) Uecñ¼ matmélgb,brmartgćmuc c]a,b[ (c) = 0 BiitüemIlkrNITUeTAé a,bj El a b ig (a) (b) eyigbghaja cmeba¼rkb k El (a) k (b), ma c]a,b[ El (c) = k BiitüemIlGuKm_ g eaeli J kmtéay g() = () k eyigá g(a) 0 g(b) ehiytamkrniageli eyigexija ma c]a,b[ El g(c) = 0 (c) = k krni a,b J El a b ig (a) (b) smraybba ak UcKña ² 0 ebi = 0 si ebi 0 RTwsþIbT ebi CaGuKm_maDIepr gésüleaeliceøa¼ebik J ea¼ (J) Caceøa¼ 5- tmélrbehlbhufa ( Polyomial Approimatio ) ekegay CaGuKm_maDIepr gésüleaeliceøa¼ebik J ebigukm_eriev eaeli J madiepr g EsülRtg aj ekiyaya madiepr g EsülTI (lmab ) Rtg a ehiyekehaerievé Rtg a aca

18 «erievti» é Rtg a igektagvaeay " (a) tamrebobuckña GuKm_ GacmaIepr g Esül grtg a cmeba¼ * ehiy ektagerievti é Rtg a eay () (a) ebigukm_ eaeliceøa¼ebik J magukm_erievti ( KW () )CabéAelI J ekiyaga mañak C eaeli J Cakarc,as Nas a ebi mañak C eaeli J ea¼ mañak C k eli J cmeba¼rkb k «TahrN_ GuKm_ eaeli kmtéay () = 0, ebi ebi 0 0 cmeba¼ * mañak C b ueþmimañak C + eaeli ebi magukm_erievrkbĺmabéaeli J ea¼ekiyaya mañak C eaeli J Cakarc,as Nas a ebi mañak C eaeli J ea¼rkbǵukm_erievrbs vacabéaeli J kñ úgcmenamgukm_cabéaeli GuKm_BhuFamaTMrg sambaø EleAkñ úgea¼ rubpabécmuce kmtǵacknaáeaykarefviel@rabǵs érbmanvifibireakñúg KWRbmaNviFIbUk ig RbmaNviFIKuN cmeba¼gukm_tueta CajwkjabéyIgCUbRbT¼wgkarlMákCaeRcIkñúgkarrkrUbPaBécMuceAkñ úgekmt eta¼ca ya gnak¾eay ebiekgacrkbhufamyy ElmatMélRbEhlwgGuKm_eACitcMucEleKeGaymYy ea¼ eyigávifibeg;ittaragknatmélrbehlésmmurubpabrtg cmucea¼ cmeliysmaćmeba¼cmenatékar KNatMélRbEhlBhuFa Tak Tgwgñak égukm_madiepr gésül eakñúgrtwsþibtagerkam ElmalkçN TUeTAéRTwsþIbTtMélmFüm RTwsþIbT Taylor : ebi CaGuKm_maDIepr gésül + geaeliceøa¼ebik J ehiy a,b J ea¼ (b) = (a) + (b a) (a) + + cmeba¼ c eaceøa¼ a ig b smraybba ak yk a b ehiykmt M eay (b) = (a) + eyigrtúvbghaja ma c]a,b[ El kmtéay g() = (b) + () + k k k b a (k) Mb a (a) k! (c) M ( ) eyigbiitüemilgukm_ g eaeli J k b (k) Mb k! eyigá g CabéAelI [a,b] igmadiepr g EsüleAelI ]a,b[ ehiy g(a) = g(b) = 0 b ( b a) () a () (a) (c)!!!!

19 Uecñ¼ tamrtwsþibt Rolle ma c]a,b[ El g(c) = 0 g() = () + = Uecñ¼ (+) (c) = M k b k k b k b () k! k! M cmeba¼rkb ]a,b[! ekehabhufa p eaeli El p k Mb () () (a) () (a)!! aca «BhuFa Taylor» WeRkTI cmeba¼ Rtg a cmeba¼bhufa Taylor p eaeli ekexija p (a) = (a) ig k (k) p a (a), k,,, KWa p () ig () ig erievti rbs vartúvkña Rtg a Uecñ¼ matmélrbehlwgbhufa Taylor Rtg p cmuceacit@ a eta¼bica p (b) CatMélRbEhl¾lðé (b) eaebl b eacit a k¾eay k¾vagars&y wgtmhmé smnl R (b) Er El R (b) = (b) p (b) = b a ( ) (c)! R (b) CalMeGogeAkñ úgkarknatmélrbehlé (b) eay p (b) krnieleyigkmbugknatmélrbehlégukm_ eakñúgv&rsiunas 0 KWmaGtSRbeyaC_Nas eakñúgkrnie¼ BhuFa Taylor marag () p () (0) (0) (0)! ehiylmegogeakñúg karknatmélrbehlé (b) eay p (b) eaebl b eacit 0 egayeay rubmþ R (b) b () (c)! El 0 c b «TahrN_ eayerbibitymbugmisuüébhufa Taylor Rtg = 0 cmeba¼gukm_siuuseaeli bghaja si 0,4794 eayykelðógmielisbi 0,0000 MeNa¼Rsay GuKm_sIuusmañak C eaeli eyigma () (a)

20 p si() = cos, si(0) = si"() = si, si"(0) = 0 si () () = cos, si () (0) = si (4) () = si, si (4) (0) = 0 si (5) () = cos, si (5) (0) = si (6) () = si, si (6) (0) = 0 6! 5 5! Uecñ¼ 0, 4794 ehiy R ( si c ),c] 0, [ ElBIe¼ eyigtaj p 7! 7 6 á R 6 7! 7 0,0000 kartaggukm_caesrisv&ykun ( Power Series Represetatios o Fuctios) ebigukm_ mañak C eaeliceøa¼ebik J ea¼ekgackmtés risv&ykuneaeli ()! (a) + ( a ) (a) + + (a) cmeba¼ a J ekehaes rie¼acaes ri Taylor cmeba¼gukm_ Rtg a ebigukm_ mañak C eaeliceøa¼ebik J ea¼cmeba¼ J,! () = (a) + ( a) (a) + + () a El c eaceøa¼ a ig, * ebieakñúgceøa¼ K J, R () a ()! a c 0 () c! ea¼bhufa Taylor p pþlńuvtmélrbehl¾lð cmeba¼gukm_ eaebl eki KWacMeBa¼sVúItp eyigá p () (), K eakñúgkrnie¼ekagukm_ makartagcaesris&vykunrtg a eaeli K ehiyeksresr ()! () = (a) +( a) (a) + + a, K «TahrN_ GuKm_suIusmañak C eaeli tam«tahrn_agelieyigá Rtg = 0 ehiy 5 p ()! 5! R!!, erba¼ si ( ) cmeba¼ * ig

21 Et R () 0, Uecñ¼ GuKm_suIus makartagcaes risv&ykunrtg = 0 eaeli ehiy 5 si! 5!! cmeba¼gukm_ mañak C eaeli ElmakartagCaes risv&ykunrtg a egayeayrubmþ ()! (a) +( a) (a) + + a ekgacsgß&ya tmél EleFVIeGayes risv&ykune¼rym KWCatMélElEleFVIeGay R ()0 eta¼bica lkçn e¼ekitelig Cajwkjabḱ¾eay CaTUeTA vamibitet ekgacmagukm_ eali Elmaes ri Taylor rymrtgćmuc b b ueþmirymetark (b) et «TahrN_ sikßagukm_ eaeli kmtéay () = e 0 ebi ebi 0 0 GuKm_ mañak C eaeli ehiy () (0) = 0, * Uecñ¼ es ri Taylor é Rtg = 0 rymetarkgukm_ 0 eaeli Et () 0, \{0} 5-4 karknalimit RTwsþIbTtMélmFüm Cauchy: ebi ig g CaGuKm_CabéAelIceøa¼biTT&l [a,b] igmadiepr gésülea eli ]a,b[ ea¼ma c]a,b[ El (c) g(b) g(a) g(c) (b) (a) smraybba ak BiitüemIlGuKm_ h eaeli [a,b]kmtéay () g() h() (a) g(a) (b) g(b) h CaGuKm_CabéAelI [a,b] igmadiepr gésüleaeli ] a, b [ ehiy h(a) = h(b) = 0 Uce¼tamRTwsþIbT Rolle ma c]a, b[ El h(c) = 0 KWa (c) (a) (b) g(c) g(a) g(b) 0 0 g(b) g(a) (b) (a) g(c) (c) (c) g(a) g(b) g(c) (a) (b) 0 (c) g(b) g(a) g(c) (b) (a)

22 RTwsþIbTtMélmFüm Cauchy pþlńuvk,ü¾marbeyac_myysmrabḱnalimitégukm_ rybmþ De Lhôpital : ebi ig g CaGuKm_maDIepr gésüleaeli J \{a} El J Caceøa¼ ig aj ehiy () 0, g() 0 kalna a Et g() 0, g() 0,J \ {a} ehiyelisbie¼etot smraybba ak eay ()0, g()0 kalna ea¼eyiggackmtǵukm_brgik, g é ig g erogkña cmeba¼ J El ig g CabéAelI J Uecñ¼ (a) = g (a) = 0 eakñúgkarsikßalkçn limitégukm_rtg a eyigmikitbitmélégukm_rtg a et eyigexijalkçn limité g lim a g Rtg a UcKñawglkçN limité g Rtg a BiitüemIlsVúItm UNUtU El J \ {a} ig a tamrtwsþibttmélmfüm Cauchy, RKbśVúIt ma c eaceøa¼ a ig El (c ) g ( ) g (a) g (c ) ( ) (a) eay (a) = g (a) = 0 ehiy g() 0, g () 0 cmeba¼ J \ {a} eyigá g g c ma ea¼eká lim a g ma ehiy lim lim a g a g ebl a eyigá c a Et lim a g ma g g Uce¼ c lim a g g lim a Uecñ¼ eyigsñiæaa lim a g g g ma ehiy lim lim a a «TahrN_ sikßalkçn limitrtg = 0 égukm_ h eaeli + \ {0} kmtéay MeNa¼Rsay GuKm_ ig g eaeli + kmtéay cos h()

23 () = cos ig g() = CaGuKm_CabéAelI + igmadiepr g EsüleAelI + \{0} ehiy (0) = g(0) = 0 g si EtcMeBa¼ 0, si 0 kalna 0 Uecñ¼ tamrubmþ De lhôpital eyigá lim h() 0 0 bmerbmrylérubmþ De lhôpital : ebi ig g CaGuKm_ madiepr gésüleaeli J\{a} El J Caceøa¼ ig aj ehiy () +, g() + kalna a Et g () 0 cmeba¼ J \ {0} ehiyelisbie¼etot lim a g ma ea¼eká lim a g ma ehiy lim lim a g a g tag t = lim a g smraybba ak eká g 0, 0, () 0, g() 0 ehiy t eaebl J ig 0 BIEpñkmYyé a, «bmaa EpñkagsþaM eyigerciseris ig El a a + tamrtwsþibttmélmfüm Cauchy ma c], [ El () () g() g() g g c g( ) g() ( ) () eká c g( ) g() ( ) () g eay () +, g() + kalna a ig eayyk CacMYereyIgá kalna a

24 Uecñ¼ 0 El eayerbivismpab eyigtajáa lim a g g g g eaebl J ig 0 t t eaeblj 0 mi igma&ya t «TahrN_ sikßalkçn limitrtg = 0 égukm_ h eaeli + \ {0} kmtéay h() = MeNa¼Rsay GuKm_ ig g eaeli + \ {0} kmtéay () = + \ {0} ehiy (), g() kalna 0 g cmeba¼ 0, 0 Uecñ¼ lim h() 0 0 kalna 0,g(), CaGuKm_maDIepr gésüleaeli lmhat 9 sikßapabmadiepr g EsülRtg = 0 égukm_ eaeli UcagErkam ( i ) () = ², 0 () = ( ), (ii) () = / (iii) () = / si (iv) (v) () = ², () = 0 q 0, 0, = 0,, = q p ( p,q bzmrvagkña )

25 9 bghaja ebigukm_ eaeli madiepr gésülrtg a ea¼, ig madiepr gesül Rtg a ebi (a) 0 egay«tahrn_elbghaja CaTUeTAlkçN agelie¼ mibitet ebi (a) = 0 94 rktmél k eim,iegaygukm_ eaeli kmtéay () (0) 0 ( i ) CabŔtg = 0 k si, (ii) madiepr g EsülRtg = 0 (iii) maerievcabŕtg = ( i ) bghaja GuKm_ eaeli kmtéay () =, * CaGuKm_maDIepr g Esül ea ehiybghaja () = -, (ii) bghaja RKb BhuFaeAelI CaGuKm_maDIepr gésüleaeli (iii) eayerbirummþbnþak bghajagukm_ eaeli + kmtéay () = p, p + \{0} CaGuKm_maDIepr gésüleaeli + \{0} ehiy () = p p-, + \ {0} 96 ekegay CaGuKm_Rbka ig CabéAelIceøa¼ebIk J bghaja ebi madiepr gésülrtg aj ig (a) = 0 ea¼ KµaDIepr gésülrtg (a) et ( ENaM «bmaa madiepr gésülrtg (a) ehiyerbirubmþbnþakétaeli o eim,itajrk lkçn pþúy ) 97 ekegay CaGuKm_kMtéAelIceøa¼ebIk J bghaja ebi madiepr gésülrtg aj ea¼ (a) = lim h0 egay«tahrn_pþúyelbghaja a et a h a h h lim h0 a h a h h Gacma Et KµaDIepr gésülrtg 98 cmeba¼gukm_siuuskmtḱñ úgceøa¼ ], [ bghaja GuKm_Rcas si (= Arcsi) madiepr g EsüleAelI ],[ ehiy si, ],[ 99 KNaerIevéGuKm_ImYy@ageRkam eayerbirubmþ () h () lim h h0 () () = 4 () () = c () () = (4) () =

26 (5) () = 5 ² (6) () = ³ + (7) () = 4 (8)() = (9) () = (0) () = ³ 4 () () = () () = h () () lim h 00 cmeba¼gukm_agerkam rk () eayerbi h0 () () = ( 7)² () () = 7 ² () () = (4) () = 5 4 (5) () = (6) () = 6 0 bghajagukm_, () = KµaDIepr gésülrtg = et, 0 rktmél A ig B eim,iegaygukm_, () = madiepr gésülrtg = 0 ekegay () a) rk () cmeba¼ 0,, 0 0 b) bghaja KµaDIepr gésülrtg = 0 et A B, 04 ekegay CaGuKm_maDIepr gésülrtg 0, bghajagukm_ rk g ( 0 ) (), 0 () (0) 0 (0) g madiepr gésülrtg 0, 0 05 cmeba¼smyrimyy@agerkam curegay«tahrn_égukm_elkmtćmeba¼ ehiyepþógpþat lkç&nðelegay () () = 0, () () = 0, 0, (0) Kµa () () ma ; ( ) Kµa (4) () ma ; (-) ig () Kµa (5) () = ig () = 7 (6) () = 5 ig () = (7) KµaDIepr gésülrtgŕkbćmuc

27 (8) () = cmeba¼ 0 ig () = cmeba¼ 0 06 ekegay h(0) = ig h(0) = rk (0) kñ úgkrniimyy@agerkam () () = h() () () = ²h() 5 () () = h() (4) () = h() + h() h() 07 epþógpþat a ebi, g ig h CaGuKm_maDIepr gésülea¼ (gh) () = ()g()h() + ()g()h() + ()g()h() 08 bghaja ebi CaGuKm_maDIepr gésül ea¼ g() = [()] maeriev g() = ()() 09 bghaja ebi CaGuKm_maDIepr gésülea¼ g() = [()] maeriev g() = ()[()] -,* 0 rktmél A ig B eim,iegaygukm_ () = Cab cmeba¼ rktmél A ig B eim,iegaygukm_ () = Cab cmeba¼ rk y cmeba¼krniimyy@agerkam A B A B 5 B, A, B, A 4, () y = 4 ² + () y = ² + 4 () y = (4) y = (7) y = (0) y = () y = a c b c b rkerievucagerkame¼ ½ ( )( 5) (5) y = (8) y = (6) y = ( )( + ) (9) y = () y = ( )( + ) () y = (4) y = () 5 () 5 (5) y = () (4)

28 (5) (7) t t t t t t 4 u (9) u u u u () u u () (5) (6) (8) (0) t t u t t 4 t t u u () u u u (4) (6) u KNatMélé y Rtg = cmeba¼ GuKm_ImYy@UcageRkam () y = ( +)( + )( + ) () y = ( +)(² + )(³ + ) () y = (4) y = 5 rkerievti égukm_agerkam () () = 7³ 6 5 () () = () () = y 6 rk (4) () = (5) () = (² ) ( + ) (6) () = ebi () y = () y = ( + 5) () y = ( 5) (4) y = (5) y = rkerievucagerkam () (6) y = () () (4) a b c e

29 (5) ( ) 5 8 bghaja CaTUeTA (g) " () ()g"() + " ()g() 4 (6) 5 9 epþógpþat a ()g"() " ()g() = ()g() ()g() 0 kmt tmél El (a) "() = 0, (b) " () 0, (c) "() 0 kñ úgkrniimyy@ucagerkam () () = () () = 4 () () = (4) () = bghajtamkmeia y! () ebi y = ea¼ y () ebi y = si(a + b) ea¼ a si a b y a cos a b () ebi y = cos(a + b) ea¼ KNaerIev (a) eaybøatkeßammuwgknaeriev / (b) eayerbirubmþbnþak bþabḿkepþógpþat lt pl () () = () () = () () = ( +) (4) () = (5) () = (6) () = KNaerIevéGuKm_ageRkam () () = ( ) () () = ( + ) 5 (4) () = 5 0 () () = 0 (5) () = 4 (6) () = 5 (7) () = 4 (8) (t) = t (9) (t) = t 00 (0) (t) = t t () (t) = 4 () () = 4 t t () () = 5 4 (4) () =

30 4 (5) () = (6) () = (7) () = (9) () = (8) () = 5 (0) () = KNaerIevéGuKm_ageRkam () y = cos 4sec () y = sec () y = csc (4) y = si (5) y = cos t (6) y = t tgt (7) y = si 4 u (8) y = ucscu (9) y = tg (0) y = cos () y = 5 KNaerIevageRkam () si 4 4 t cot t t (4) 4 4 () cos 7 rk y Rtg = 0 kñ úgkrniimyy@agerkam (5) si 4 ( cot g) () y = t ( sec ) () t t cost t (6) si () () y = () y = y 8 rk : t u u 4u, u = + () y =, 4 u, u = ( +) 4 u u 5 (5) y = u u +, u 9 rk 7u () y =, u = + ², = t 5 u 7 () y = + u², u =, = 5t + () y = u, u = sec, = t (4) y = u, u = cos, = t (5) y = 4 u, u = cos, = t (6) y = u, u = csc, = t y Rtg = : () y = (s + ), s = t, t =

31 () y = s s s = t t, t = 0 ekegay (0) =, (0) =, () = 0, () =, () =, () = g(0) =, g (0) =, g() =, g () = 0, g() =, g() = h(0) =, h(0) =, h() =, h() =, h() = 0, h() = rktmélé ) 5) 9) ( g) (0) ) ( g ) () 6) ( h ) (0) 0) KNaerIevUcageRkam () ( g) () ) ( g ) () 7) ( h g) () ) () () (4) () (7) ( ) (5) (8) ( g) () 4) ( h) (0) 8) ( g h) () ) () () ( g ) (0) ( h ) () ( g h ) () () (6) rktmél eim,iegay (a) () = 0, (b) () 0, (c) () 0 kñ úgkrniimyy@ucagerkam () () = () () = () () = (4) () = bghajaebi ( a) CaktþaéBhuFa p() ea¼ ( a) Caktþaé p() 4 ekegay CaGuKm_maDIepr gésüleaeli bghaja (a) ebi KU ea¼ ess (b) ebi essea¼ KU 5 rk y kñ úgkrniimyy@ucagerkam / () / y () y () y (4) y = (5) (7) 6 KNa y (8) () (4) () 4 / / y (6) y y ( 4 ) 5 (5) () (6) a b c

32 7 rktmélé y Rtg tmél kñ úgkrniimyy@ () y, = () y 4, = () y =, = 64 (4) y, = (5) / / (7) (9) () () y, = 4 (6) y, = (8) 5 6 y, = (0) y y 8, = (), = (4) (5) y tg, = 6 y 9 4, = y, = 5 y, = 4 y, = 5 y 5, = 0 (6) y = si, = 6 (7) y = cos 4, = (8) y = cotg, = 9 (9) y = csc, = 4 (0) y = sec rk y CaGuKm_é ig y eayerbidiepr gésülgambøisuit, = () + y = 4 () + y y = 0 () 4 + 9y = 6 (4) y 4 (5) y + y 4 = (6) y + y + y = (7) ( y ) y = 0 (8) (y + ) 4 = 0 (9) si( + y ) = y (0) tgy = y y 9 rk CaGuKm_ ig y : () y + y = 6 () y + 4y = () y + y = 9 (4) y = 8 40 rk y ig RtgćMucEleGay () 4y = 9, (5, ) () + 4y + y = 5, (, ) () cos( + y) = 0,, (4) = si y,,

33 4 ( i ) ekegay CaGuKm_maDIepr g EsüleAelI El () = 0, eayerbirtwsþibttmélmfüm bghaja CaGuKm_ereAelI M, (ii) GuKm_ madiepr gésül eaeli malkçn M 0, eayerbirtwsþibttmélmfüm bghaja () g() M y, TajbBa ak a CabésµIeAelI eayerbilt ple¼ TajbBa ak a GuKm_suIusCab eaeli 4 eayerbirtwsþibt Rolle bghaja Kµa k ElsmIkar + k = 0 ma sbirepßgkñaeakñúg ceøa¼ [-,] et 4 ( i ) eayerbirtwsþibttmélmfüm bghaja ebi CaGuKm_maDIepr g EsüleAelIceøa¼ebIk J ig () 0, J ea¼ ekiac at (ii) bghaja ebi CaGuKm_ekIac atmadiepr gésüleaeli J ea¼ () 0 J (iii) egay«tahrn_elbghaja ekgacmagukm_ekiac atmadiepr g EsüleAelI J Elma cj (c) = 0 (iv) bghaja ebi CaGuKm_maDIepr gésüleaeliceøa¼ebik J ig CabélI J ehiycmeba¼ cj (c) 0 ea¼ ekiac ateaeli ceøa¼rgebikelma c (v) eaybiitüemilgukm_ eaeli kmtéay () (0) 0 si, 0 eakñúgv&rsiunasé = 0 bghaja cmeba¼gukm_madiepr gésül eaeliceøa¼ebik J ek Gacma (c) 0 cmeba¼ cj Et micamácékieaeliceøa¼rgnamyyelma c et 44 (i) bghaja GuKm_ eaeli CaGuKm_eá g lu¼rtaet cmeba¼, y, z El y z, (y) () (z) () y z (ii) bghaja ebi CaGuKm_maDIepr g EsüleAelI ehiy CaGuKm_ekIeAelI ea¼ CaGuKm_eá g (ii) bghaja GuKm_eá g eaeli maerieveqvg ig erievsþamrtgŕkbćmucé ehiybghaj a GuKm_erIeveqVg igsþam CaGuKm_ekIeAelI 45 ( i ) GuKm_Cab eaeliceøa¼ebik J madiepr gésüleli J \ {a} El aj bghaja ebi lim () ma ea¼ madiepr gésülrtg a ehiy (a) = lim () a a (ii) eayektmruvlt plé ( i ) bghaja cmeba¼gukm_ ElmaDIepr gésüleaeliceøa¼ebik J, GuKm_erIevrbs vamigacmacmucacŕbepti et

34 46 CaGuKm_maDIepr gésüleaeli [ 0,] ehiy (0) = () = 0 bghaja () matmélea ceøa¼ 0 ig () (0) cmeba¼ [ 0, ] 47 bþab BIepÞógpÞat a GuKm_epÞógpÞatĺkç&NÐRTwsþIbTtMélmFümeAelIceøa¼EleGay rktmél c () () =, [, ] () () = 4, [, 4] () () = /, [, ] (4) () =, [, 8] (5) () =, [0, ] (6) () =, [-, ] 48 BiitüemIla etigukm_ () = ebivaepþógpþat kmt tmél c epþógpþatĺkç&nðértwsþibt Rolle eaeli [, ] et 49 ( i ) ekegay () =, a =, b = epþógpþat a KµatMél c NaEleFVIeGay (c) (b) (a) b a et BülḿUlehtuElGuKm_e¼ miepþógpþatŕtwsþibttmélmfüm (ii) smyruckñacmeba¼ () = 50 KUsRkahVtagGuKm_ () = igknaerievrbs va epþógpþat a ( ) = 0 () Et ()miesµi 0 et BüléhtuplElGuKm_e¼ miepþógpþatŕtwsþibt Rolle 5 bghajasmikar = 0 ma s CacMYBitrepßgKñamielIsBI BIreT ( ENaM erbirtwsþibt Rolle ) 5 bghaja smikar = 0 ma scacmybitetmyykt ( ENaM erbirtwsþibt Rolle ig RTwsþIbTtMélmFüm ) 5 bghaja smikar = 0 ma scacmybitetmyykt 54 ekegay CaGuKm_maerIevTI bghaja smikar () = 0 ma scacmybitepßgkña ea¼ smikar () = 0 ya gehacnas ma ( ) scacmybit epßgkña ehiysmikar "() = 0 ya gehacnas ma ( ) scacmubitepßgkña 55 ekegay P CaBhuFamierEl P() = a a a0 bghaja eaceøa¼rvag sbir bþbþabḱñanamyy ésmikar P() = 0 ya gercibmput ma smyycarbsśmikar P() = 0 56 bghaja smikar + a + b = 0 ma scacmybitetmyyktébi a 0 ehiyya gercibmputma sca cmubitmyy eaceøa¼ 57 ebi () a ig a ebi a 0 cmeba¼ bghaja () () cmeba¼rkbćmybit ig

35 58 ekegay CaGuKm_maDIepr g EsülelIceøa¼ebIk I bghaja ebi () = 0, I ea¼ CaGuKm_ereAelI I 59 ekegay CaGuKm_maDIepr gésüleaeli ]a, b[ El (a) = (b) = 0 ehiy (c) = 0 cmeba¼ c]a, b[ bghajeay«tahrn_a micabéaeli [a, b] et 60 bghaja cmeba¼rkbćmybit ig y eká a) b) cos si cos y si y y y 6 eayerbirtwsþibttmélmfüm bghaja ebi CabŔtg ig + h ehiymadiepr gésülea ceøa¼cmuctamgbirea¼ ( +h) () = h ( + h) cmeba¼cmy eaceøa¼rvag 0 ig 6 ekegay h 0 «bmaa CabéAelIceøa¼ [ 0 h, 0 + h] ehiymadiepr gésüleaeli ] 0 h, 0 [ ] 0, 0 + h [ bghaja ebi lim () lim () L ea¼ ma 0 0 DIepr gésülrtg 0 ehiy ( 0 ) = L 6 ( i ) ekegay () = , [, ] tamrtwsþibttmélmfüm ma c], [ El (c) = () () rk (c) (ii) smyruckñacmeba¼ () = ² +, [, 4] 64 rkes ri Taylor Rtg = 0 cmeba¼gukm_kusiuuseaeli ehiybghaja vatagegaygukm_ kusiuuscmeba¼ 65 ( i ) bghaja GuKm_Giucs,Ú NgÉsül ep: kmtéay ep() = e madiepr g EsüleA eli ehiy ep() = e, (ii) rkes ri Taylor Rtg = 0 égukm_guics,ú Ng Esül ehiybghaja vatagegaygukm_ Giucs,Ú Ng Esül cmeba¼ (iii) eayerbi 6 tymbug ées ri Taylor Rtg = 0 cmeba¼gukm_guics,ú NgÉsül bghaja : e,78 eayykrtwmtspak Þg 66 ( i ) eayerbikarknatmélrbehlbhufa Taylor Rtg = 0 cmeba¼gukm_giucs,ú Ng EsüleAelI bghaja GuKm_bþak oep : CaGuKm_øÜÉgeAelI (ii) bghaja GuKm_elakrItFmµCati madiepr gésüleaeli + \{0} ehiy \{0} 67 bghaja cmeba¼ a 0, a a e, rycbghajagukm_ eaeli + kmtéay () = r, r + \ {0} madiepr gésüleaeli + \ {0} ehiy () = r r-, + \{0}

36 68 ( i ) rkes ri Taylor Rtg = 0 cmeba¼gukm_ g eaeli ], +[ kmtéay g() = ( ) rycbghaja es rie¼ tagegay g kñ úgceøa¼, (ii) rkes ri Taylor Rtg = 0 cmeba¼gukm_ g eaeli ], [ kmtéay h() = ( ) rycbghajaes rie¼tagegay h kñ úgceøa¼, sresrcaes ri Taylor cmeba¼gukm_ k ea eli ], [kmtéay k() = ehiy bghajaes rie¼tagegay k kñ úgceøa¼, (iii) eayerbitmélrbehlbhufartg = 0 bghaja 0, eayerbikarknatmélrbehlbhufa Taylor Rtg = 0 cmeba¼gukm_giucs,ú NgÉsü y = e bghaja cmeba¼gukm_ eaeli + \ {0} kmtéay lim () 0 0 bghaja GuKm_ eaeli kmtéay () = () e, * eyigá e, 0 0, 0 mañak C eaeli ehiybghaja () (0) = 0, * 70 bghaja cmeba¼gukm_ eaeli + \{0} kmtéay () = a e -, a eyigá lim () 0 7 sikßalkçn limitrtg = 0 égukm_ eaeli ] 0, [ kñ úgkrniimyy@agerkam ( i ) (ii) (iii) tg () si () () 4 si 7 sikßalkçn limitrtg = égukm_ eaeli ]0,[ \{} kñ úgkrniimyy@agerkam ( i ) ()

37 4si 6 (ii) () (iii) () tg 6 7 KNa ( i ) lim / ( tag = ) (ii) lim y 74 ekegay g CaGuKm_maerIevTI eaeli ehiy g(0) = g(0) = 0, g"(0) = cmeba¼gukm_ eaeli kmtéay () = g(), 0, 0 0 bghaja madiepr gésülrtg = 0 ehiyrk (0) 75 sikßalkçn limitégukm_ eaeli + \{0} UcageRkam ( i ) a) Rtg = 0, () = a b) kalna +, (ii) a) Rtg = 0, () = () b) kalna +, () =, a + \{0} a / 76 ( i ) bghaja ebi ig g CaGuKm_maDIepr g EsüleAelI + ig g() 0 cmeba¼ + g () +, g() + kalna + ehiyelisbie¼etot lim () maea¼ lim () g g g ma ehiy lim () lim () 77 ekegay CaGuKm_maDIepr gésüleaeli + bghaja ( i ) ebi lim () a ig lim () b ea¼ b = 0 ( i ) ebi lim () a 0 () a ea¼ lim (ii) ebi lim () 0 ea¼ lim 0 () ( ENaM BiitüGuKm_ g eaeli kmtéay g() = () )

38

39 MeNa¼Rsay ([a,b]) tamlkçn kmuá k égukm_cab, ([a,b]) matmélgtibrma ig Gb,brma «bmaa matmélgtibrma Gb,rmaRtg c]a,b[ ea¼tamrtwsþibtageli (c) = 0 «bmaa matmélgtibrma ig Gb,brmaRtg a ig b ea¼eay (a) = (b), CaGuKm_erig (c) = 0 cmeba¼ c]a,b[ RtwsþIbTtMélmFüm ebi CaGuKm_CabéAelIceøa¼biTT&l [a,b] igmadiepr gésüleli ]a,b[ ea¼ ma c]a,b[ El 0 a c b c MeNa¼Rsay ( rub 5 ) (b) (a) b a (b) (a) (b) (a) (c) b a eyigbiitüemilgukm_ g eli [a,b] kmtéay 0 a c b ( rub 5 )

40 g() (b) (a) b a () a g CaGuKm_CabélI [a,b] igmadiepr gésüleli ]a,b[ ig A g(a) = g(b) = (a) Uecñ¼tamRTwsþIbT Role, ma c ]a,b[ El g(c) = 0 (c) (b) (a) b a ebi F CaGuKm_maFIepr gésüleliceøa¼ebik J / GuKm_µIehAaGuKm_erIev itageay kmtéaeli J El () CaerIevé RtgćMucImYy@ J «TahrN_ sikßagukm_ eaeli kmtéay MeNa¼Rsay cmeba¼ \ {0}, () = si cos ehiy lim () Kµa eyigexija madiepr gésül eaeli Et mapabacŕbepti Rtg 0 eyigwgbghaja eta¼bigukm_erievmicamácćagukm_cab k¾eayk¾vabitcamalkçn kunicucca yk a,b J El a () = ² si (0) = 0 MeNa¼Rsay 0 RTwsþIbT ebi CaGuKm_ madiepr gésüleli ceøa¼ J ea¼ (J) Caceøa¼ RTwsþIbT Taylor : ebi CaGuKm_maDIepr gésül + geaeliceøa¼ebik J ehiy a,b, J RtwsþIbTtMélmFüm(Cauchy): ebi ig g CaGuKm_CabéAelIceøa¼biT [a,b] igmadiepr gésüleaeli () = e / ( b a) () b a () + (a) g(b) g(a) (c) g(c) (c) g(b) g(a) g(c) (b)!, 0 (a) (b)! (a) (c) (0) = 0 ()= ]a,b[ ea¼ ea¼ma (b) = c]a,b[ (a) + El (b+a) (a) + + eaceøa¼ a ig b,cmeba¼ 0, 0 cmeba¼ c