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TrxLoE npanonirotlrCALCULUS €i 1980byMichaelSpivak,SecordEdition o 1991tlo qv €Lhvn{i-yk6ooui(IIANEIIDTHMIAKE: EK OXED KPII|ID, AnP .nq1991 naoot dz6oorl: 1991 Aeircat;?i6oot;NodFPQnE Taiq dx6oot:Iclol|dQros1993 Titooq dxbool::€nxd$ploE1995 IIdF]nId,{boon: Ioi\'ros1999 ISBN960 ?309-13-8 MsidqQaortftr6 ft aw),r,{d:Ar6orcbq ftow6nouloE,Moftp0ltx6s Add,{Ioa rov Mo0lporui6v Erlontpovr|'|i€nrpdl,€ld:Alpirpl]S KoaayLawdl{rtq, Mq{d}.IEAdprpou,EnixollQo(KaonnxrigMootFffnid! IlovertonlFiou
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flPOAOTOEETHN 2q EKAOIH qopfq6rr o rir),oqroD9rF),iou Moueinovnol,?,6q du'ro6eo6'rper€vo rirovKdfl oav((l\{io ErsoyoyTi orrvAvdl,uon,,oqo$rprloLtionoEirqr ouvnerDq ons rqpaS6oeq6rouor orouSosriq6rouvri6nldosrrnv ptlavrKi '!I€up,irouAoyropot rtrolopooirora eivorxoorEpofavqni,aovoqv EupdJnrl Kolouqei(ovrorTr,€uraio6]'0xaLm6 nolforiq HnA. M€rd ar6 13 1p6v1a,IoD eoivsror.rol.iopd vo qlldEo rov rirlo, otr)d tp€loq6rovd),l€q a?,ldy;qr#po ard Tlq6rop0droelc rnEnh€6poe r0rv,runolpa$rK6vKol sn, 6ao).[drov. Ylldprouvrri)paoro pr9),ioq€loprordnopspflisorayla ro},)d 06!dra,roDornv rpdirTl 6{600rl€itqv unorwl00i:no}.r(Cq ouv'r€roypdv€q, opor6poperl ouvdl€u,iropqperprl(onoirlorlro!rD).d'v,oopoiopolaRi€mann, lcolrl lprisil r(Dvolo(lrlpopdrov yrorov uroloTl0!6 plKd)v,6yKovKar6!pq6djv.MEprKd06poro,ririrDq o j(erprop6q rov Suvotoo€rptiv, 8€€rd(owar nrcovd,Dl} d oro nop6vo$l.rpotpoKaLuirdplouv€niortqro),ld v€oripopLiparooe oj(donf€ ourdrd ofuqro,€v6d ,q e6!oro,6rroqn #eoboqNev'ton,o Kov6voq avorrlooovrald60) rov npop].trdrov.Yndploov6uvorou Tpo'I€(iouror roDSimpson, l"rKd160n€piJroD vaonpop).iFara,no).)don6 ro onoia sivdrForlC 6Doro).ioc, rdnou qvdueooo'rd l.ito rpop),rlloropoDrivtrE orrlv opj(ri Kdoe]r€qoloiouKol o eKeivoro 6uoxo?"6repa noueraovi(ovroL opy6r€po. MichaelSpivak
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IIPOAEIOMENA ETH 2lt EAAHNIKH EKAOIH :rnv lrl Ei,).ylvrxri6x6oot dn6 ur€pFofrl(li isoq 6ljoro0ioia ,rpoqro!€ trrprEfirr6rclouq trpo).6'lor'( ro! ou1ypoe6a eixo|.l8oeopioer n€prrr6 vo rpoo0ddoullr l(dr 1n 6LKri]roqaovri. H 0€pl6rorn 6!(0( unoboti tlou 6ru1€6vo dord pdon 6160ffLX6oiwpat|lo 6,roq ro ovd t€ipaq prp],io, [€ (EEavdt{d(€o, ordvroq rot) 6€11r€pou cnr|tclrlr]i orrl ynpnrd Alpl6vd- vd ?tpo),o7iooq 2il el.),rvlKri €K6oot ax6pt Dapnr€poKoeriKov,6rpE'!e vo pd).o o€ rdEt rov (aoroouve€rrK66oipovor rou 65poqeKupioe elq 0{ipo6tou Eup€rnpiot',oD'd eviore xor rorv ougp6),tovotoug ritrolg. oo €nlOrpo oo nol"nvd roIUproe6 6rr ndpadidorrFciov (Aoy!o[6, To! M. Spivak ornl].o,#vo ].d06v q],]d, a€6, Kdvovruqxpriort rov coKiiq€6v rorJ oroDE nporoEreiEeorrrlr66 roD OoorKof Ttl{jroroq rou llavitiou Kprirrlq, r€},€uroio drpo 8lotlioi(roo 6rr t!.2. n tl€v doKroq 4(ii) rou Kc'r. 14 €ivs! qva2.toi(, rl 5a doKnat 2(bxiii) rou K€q. 17 rivar ...o),uorrcr'is {croriv|l)q. \dn rst cni qt ouoias Kdt 6Xt rovo c^i ni. ltopeiE pnopti va trpooq6p€rprd trpoo8ypdq [€r{iepaorl). Bdpata, I svuKd)'uvrl ).d06v oloidc orre oox{6ete roD,rporor$nou €ivar l(ur tsoixov rou 6r6{ioKovroqxor 6rSaoKo[6voD nou, apld ri lpii7opo, Ka].eirorva ;(prioq Ttgs?rl).no€r 61,€€(n).rtvFsprK6v(opdrr0v' ri r.or 6;(r 1600 6r6aKflK0tvoE 1lp6rrl ovd7v0i6n).Ag neprop1016).orr!6vorq 8ril,ooit 61r d,tri(ou!€ roq tropa6i6ows 1()p,p).iodnt).IdIFdvoru'ro7pa$rKd,v ).oet.. o (Ao1ro[6qDro! M. SpivakoneD06v€ru1 KUpioqo€ rpEr6xdrrlyopi€qnou petd;1ouv otqv er:ror6eurrKi6roSrKooiu: flp6rov, ornv ficv€rl('rrlrLllNri (orv6rrlru r6rv O€rrK6vlsyo}lcvov Ernorrl116v(6l)}1mp1],oIFavof xd Krad),ard16,20,27,28,29 \Xapie nKoiq 0a pnopoiodvvq nopo1,€rqeoiv ro ro 9dBardvo or[aivcr 6rr o'ra1op€6€ror6€ E'ttr... n.X.8to].d7ova'yvopiosr ond !6voE rou rlv olopald rtq cn63er[rlq 6n o Woordg cpL0p6Er eivcr dppnr116!). A€ir€pov, orouq 6r6rioroweq Kar Sr6oorop6voo(rov TEI 6t!o! lto trpd,trl dvriyvoon Kqr 6l6ao(o).io fnopodv vo napdl.er90o6vefllnl6ov Kot tc Keqdlolo 8 (Iapdprrtld) Kor l3 (nopdpqtla l). Tpirov, ololq o€pdnovrEs rte M6onq E(,roi&uolq Kor 6oouq€|cr('v rElero qoirov lrtg A' 66optg 6Xouveva (I€pdKlD7rs su),),oyriooxrio€o)v7*pu sr6 !a Iuodpr0, rq 6op€dv6rovef6!€vo... Yndpt€r Kor lrra rtruprrl i(arrlTopio nou iooq ove(rilrloE tov M. Spivak np6rt: €K€ivn nou rldvo Kar n6pa an6 eroTr€).Fat1Ke(eltKdll€e, e€o)p€iava KaLd ttohjrsrrK6 prPIio €]rdv6uonKo! 7w€uFotlxriflp6x],Ior1. E].ni(ouU€,ovsEdprlro oe r!o!d *drrl,opia oviK€r o avoyvd'6tn(, I ,tdpo6oo 2t 6K6ooI va rov 0orl0ioEroloraorrKciororq FopaorL|'oiqroL 016Xouq. l(ol ovoy'dvouFc Fc rsioE ro' 0o rnv cv6Lo$ipov L1166€rEn lrlrt60rr. HpdK),€ro, Oxr6Pproq1991 d. KapalawdKrlq
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o€!E),Ldj8€g 0li.roilpo rou 3e!,0a10oroSeiiolu€e66,pqq)dol 1rorolq ol(€poiol,g, 6n au1]irl mpotoworoirtd! €ivqrUols6Lni,o!'6€v,rdpobtsurri\rl 1I o€rpdrov ropol6vlov. Etor, 1lo mp{i6ft'/fo,o 28 6€! Fnop€i,rordvo lporp8ioov ?rv6F€vo np6rovo avoeon6ro0i onoiouqvo eivoro 3. oirs l'rop€i vo lpoeei I d!a! rp6,ro rou vq n€prdz€L ro 2 r6volro qopd(ed'np€n€ r6pu vq crTrri6.le ro 6rr a€v€nrrpd\yoorov vo I sivoUrpd'toq). Fe (b) Xptorponordivrq! al)r6ro y€7ov6q, ono6€iETe 6n o 1fi eivarrippnroq,exr6qov , = rr,-flo Kdnoro!,q!,o1K6 0pr0f6m. (c) Ano8siqrs nroycvrxritirr o it eivcLripplroq,exr6qcv n = ,z'. (d)
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^3! l'nopoqr€vo K),€iooufE rn oDqirrlorltLo rouqrp6rolq aprofo[qxopiqva outn€prkipoLrle riv 6pop9l an6Setnrou EuK),€i6rl 7ruro 6Tlundpi(olvdntF pornp.i,rorqpr0foi.Ano6liLnorr 6Lvrropoiv |c r fldpto,v jrdvofl.^Lpoof,\'or rov p, A ... p,+ I. ro d.ieoc npd,roroprotoipL.A. A. .-., p, 0sop{i,vroq Anoaeiqre 6rL.ov o r rrovonor€i rrtv -r"+d, id' r+. iao=0' / r o r d r o r o r a( K d p o r odu. ; . . . d 0r o u o r r i v a r d p p r t r o ( . ! ( r o , o r o r L r v o r ordporog {frori o!t6 yovr{€[clroIlp6pingo I?J
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AnoSaila 6n o yD + fi eivarripplroq.Yn66€LEr1i Ssxrvciorsuno],o/i(o!raq rL; rpdirLq6 6rvd,rcLq a' Torror flproFoi. t9. Ano66iqt6 1lv ovro6rnrolou Befnoulli: Av, > - I, T6rs
(l+i)'>l+nr. frari aur6€iv0rlsrpLIdvo 0YIl > 0; 20. H aroioueiotou Fibonacci d,, d:. a,,... opi{Emr('; eiri{: ar=l' dr= l' a,= a" + a" .'I.'fIn>3 ALrrri rl o{oioloio,nor)qpXi(er an6.ovFibonacci t€ rolrel, 1,2,l,5,8,...,ovoKo}.uaortKr (1175 1250), O Fibonacci undosle 6rLdvu !s apoplridvqrp6p],rpoor€rrK6rc Kouvdho.
28 Ercayayi (EDriplKoDd,r6vro lrivo, Kor6rr opxr{6qsuldplKoDv0l, tJvt8vvouac;vflKdrvo[p7ro 6uoLfl.To n),i0oqd, rov lsrd an6 8DorrivEqKdo€Karvoipyroq€utdprourn8pL9€p6rav (su?upLdjv priva€ivaLa, r + (l" r, yroli 6vu(sL,ydpL noD1€wroiwoL ro 11-0016 KouvEIL6v '/€wrfrqryLaKde€q€u7dpr no'JT€wielKerov rpoitlorip€!,o Urivo,rqr oK6fqrdoe(€uldpr rorTew{fuxe rpLvErlo prlvsqsivn rd,poEori o' dvo{i}.}.o(surdpr.ro n}.'i0orrot' sv6roqEpdvrovororsl.'otdlov ro'r aaopo'jvoDflivrr)v o'.o).ou6io ci\rolar' o).riocl0oiriolsuro 'Otd.o( un{ipxfloKdroKL;vaq Fibonacci noL,'K6i3&dvodvruTo. rrtv &ryrnoid Frbo-
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rrari 1016€r our66rflv di = . = di Ynoedroup€ 6n 6Ev€ivor6),aro a;ioo, dg no6ll3d' + dr.Av dvnKardorrioo'rtr€ Kd'ro a KqLro drpe(4 + d)/2. Tr yiverorpc rov aprojr'tnKd ... I a,) Ir: Tr yivlrur!c rov "ycotcrpr(6 doo A"- la op,(8r;e at. ... d,; f lari, ov erqvoldpoup€ ouri q 6ro6rxqoio r{60" G"= i/ (No pla rEpinl0)on, 6,10r eivol {alri re).rKd oro8€LKvioupe 6rL G, < A,; eopdq doxnq ro vo 66osr€fLo lrnx'l otr66'l(it ps €ralori, Ko06qKl 6vadrDno €,nx€rprlI0.)
H orflol6nol olrtv rpoitTorjrLsvrt or63el€rouvld€roror€vd!e lrov dlh Eviroedpoloo 0n66c1Et. pE€noloyrioro ro T€./o!,6q 6i, C, < A,yLo, = 2, ono8EiEre, (b) X pnorporor(b\,ftq k.61t G"< A"^t\tr n= 2t. (c)
rro r'r1oior, ;oro 2'' > l?.Eaap!6orEro (b)!€poqorolq 2'opr0!o[q
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ADroi or oprogoiDno6€lKvJouv rll ouvrlorslarq6ro6l(0oionou oroloD€olfs 11ovo prc Eivarro f(r) yrq Kdo€oprep6x olo ft6io opioqupe ouvripqol / taq l.dv€]1016 op ojroi1rt: /. i noporlt:iorc6rLEivoror.prpiqro i6rour ro vo rqEno,v roL6civorro apt€F6a. flro /(b) yLoKdecqprerd6. xln.).Irrv npdql.Kd'rorcq ol"vrFri I (d)?,o KdOc oelc,ivo1ov€rrds.O opro!6q(l) eq propono€ vo Tpoqsiorl6q (r) f(x) = x1, yrorov oprop6(4)n r6vn ssrtv 61suKp1v1or1xrl spdorl"y!oKd06.x"vo€wo€iror.B€poro, 8uvarrioivrrton €ivorI ( ) t ( x )= t ( ' ' t1<x 0 Kalx'< 81. irLKp6rspoq IIqp€pnrr6vroq,our6ro oivol,oeivorroiSLo!0 ro o!vo).o{x | 0 < x< 21.Yrdplfl dllrl al)d nol'l Kol\'ri.To orjvo).o fLqnopa],i'oyi,Lt6rEpo6LoD?iq {1,3,2,4j,?la mpd8srllo, 'i€pr6xsr oKprp6q rouqr6oo€pLq oprofoiql, 2, 3 Kel4 o'rtpo).i(srrrsniorleils {n I x = I ri
x=3ir=2ir=41
MepLKdnpd,taro yro ro d0po16!a,ro y1r,6}€vo,Kol ro iqli'.o ouvaprlos'D! 0ivorrpoAavsiqouvdr€rq ovrioroLtov r8roflirrDvrou a0poi6taro!, ro! ylvotdvo! Kor ro\i nit).iKo! o p r e u ; v .L a n a p { i b c r y L-r:fvl .a r o } . i c i K o } . ot o 6 c r i . f : d r
\ f + g ) +h = + G + h ) . ro6qto; yero(ti6L1o H on66€Lql€ivo!Iopoxrrlpro xri 7ra 6r,EqoX€66!nq anodeilerg olvoprio€ov npdr0lva 8€iEouuc 611{)16ioolvopflioErqitouv ro i6ronc8ioopropori, KaLrtv i6Lo1turi?roKd00opr0r6oro ns6iooproroi. fla rap{i6sr1!o,ltovo 6siEolrL6 611 (1+ g) +,,r = f+ (s+ r), nopdrrtpoiF6rL,flvEaop!6oou}l'rov opLo}6Kororu8io I .rl,
= (r+sxx)+ ,(x) t(f+s)+ nl(.r) = t(, + s(.{)l+ r(,r)
38
Oep,n tf- k+ l)l(x)= f(r) + ls+ r)(x) = f(x)= ts(x)+ nG)1, {o, n Lo6rrtror(Dv[f(ir) + s(r)] + ,(x) {o1 {r) + ts(x) +,'I(.r)l eivcrAto t6L6rqtorov 8evalqqaportflouadrq ap,ordrv.: oufli\'rt! or66sLqtI,o6rtro rtdv6ions6i{Dvopropoi quiis rt! eiLodocLq ro voTpdqorFe ?rori€ivarrpoqaviq,dr6 rl orLrrli roti 0 opjtiooup€ n06ioopro[oi rtq (f+ s) + ,'rxoe6! Kolrr]sf+ G + /r) sivqrnpoqo!6qro r. op.In n. f + g +,|r yLoflt! (I- g) + lr = I+ (-q+ lr), op.ga n. op.,. fpdQourr.aLl6ro).o1lKd, yLoopLolLoi(. oKpl0dr( 6noqKdlop€l 0ov.r> 0.(Aur6ro xollrin .ivor mqp6, ol).,i ov rpoo6to rpop).iloro g1o1e),€utors rl( "erl,oooqrxiq"nqpoqpio€rqnououv66e'rqv roia E6orredi.ora,0a6;p€r€r' vo rliv€r€.) (d) Ano8'iqre 61' f(4> fbtrsvx>y. (e) Ano8€iqr€ ro yc.rov6q 6 6rr /(x) = xTro,cdo€ x. Yn66€LEI: XptorFororeiore pl16g avripeou oe6io'tt1oior6cprojroigurdpl€L6\,of opr0l6q*18.floLdqofprp6qouvorl Koqrpirn vo rKavotrororjv or /, g, , , ror k ilor 6ore f(x )gLy) = /r(.{)kcv)yLa (d€r x Koryi *19.(a) A,roSEilr€ 6116rvurdptolv ouvopfli6€rE / KargF€pLoqr6rLqeEiqL3r6rlre6: li) f(x)+ s0)= xYl.rrrdOexrcrY (ii) /({) 80) = x+ YTroxdOsrrcLY yro {tv / ri tv I Yn66€rEr:npoonoo€ior€ v(l ndp€r€xdnoEqn),rlpoqopi.e Sivowoqoyrexprpiv€qntfE o1o.r ror /. (b) Bp€ir€ouvsprrlo€r6 f ror g tdtut€q66r€ /(x + .y)= g(xy)?raKdoe.x'rory. .20.(a) BpeilEfro ouviiprtot f, 6'trom66pi,r.ro,o 6 6tE ll!) - f(x)l< ly x . (b) Yno061ou!s x)r7ra xd€€x Kary. (frari qur6ontqiv€r 61r fg)flx) 0. pnopoip€otoq oprerdsu{oio \rojacrproronlc {or rLgdlirq rEprnrdjosrg. O apre!6;c, no! roiv rpr]tbvo\rno! srao!i(o!r{'r olrv qr66srqf,ldy€raL uerpd€rro i61o rov n).Eup6v r;io? rrs sueeids. KoLyraxde€su0Eio ropd],l,rtlrt!' ourriv).tr06rr fxrLrtv i6Lrtr Kiiol c. '" T^ , rXripluuro' XOtor.ronoriouuf ,o ' ,. df\ Fnooo' oir, utot rio, .t oi r. ro ro 00opriooL,ss runLKion65sLqt. npay!oflKd,?rovo6d,ooL,|0lLlo ouorlprlo116dErit eoirqv ys aKoioueioouIr. ovaTKoio vo Kd!ouuE !tl1nopi{poot nou3e!situ6rs{r06)"ouiroLtLor H auoqpri on68€LhonoLao'inarcrpor(l1rt\nououvSrorl00iusrpr{fs{ar a).T€ppLK6c avloLc(ea anqrrouotrp6ru an 6i,olLo npolparrKian66srirt($ o{pr0drq iirarurortLa\.! ' r o 0 i o . . l o ' o i , ' u o t p , ' o! r o , " o L i s , s v r o o r / o , \ ! f o . y , ( " K p r , rrnpoo n o 6 r " ' npflTUuflKo[( opr€!o[q.Er16!on o!16,8oiro! ovu.,,xdio vs !E],€rliooupc rIV E, nESofsrpiq oro i6roPdeoq noL,oKoneioLu'va lsiorriooL,rrs rL_( LA,6ntrE! rov rpsluolrK6! apresvdnrler rtq erLns8o[s'rpiflq civali\,oopol6rorosdro.o).].dos tuir!.BtpoLsI ].erro!€priq Ka[rd'Itpitrr0roldevrpootrorleir(lr pEi6ri] rt rou flnsLpoorrKoi ].oTr6troi. OuiiprloF ?ro poieELo ra70op€rprKd orri[aro oov orr 6Loio0r]oi ror,q oKoroi; lono,oilE r6vo Foi ro ((or yLo ra n€proo6r€pu Eivur {006).o lKuvono,rtrLK6 vq 10 opiooul€ ro loq ro€rtrurr{d) 'trireijooavro oL,vo],o 6i,ovrov qsL,16v npa?paflK6v apr0r6v.Ko1vo .,plooUl€ qsu0si€; ou\r{dtroLqoLl!o).sq€!}6v,no! 'rcprlflppdvot)v, ovdtEoooc rii).o,ro 6[vo].o{(r, cx)ri n o o / u o . { o - o r , 0 u 6 - . f ' o \ o , { o 5 r o o o L , / r r i , - ' l v - 1 v - 7 v 1 - d n c r u o l ,
(al €ivo!TpopFrKri sEKdo€srdorrpo[1/(" + l), l/,] r(o1[ l/n, l/(,+ l)]. (oapr0!6q 0 6evr€p6r(€rororo llEiio oprolorirlq /.) BEPorq, vq ypd\'€!6vovoxplFi frop€i Kav€iq 16'10T la ro /(x), 6rovro x eivororo [/(r + I), l/,] ouni €ivorplq Ka],i doKrtoTl rdvo orrl Ipion rov ypotrllK6v oDvqp1lo€o)v, l(alsm6e on' our6 0a sag nsiosr6Tl plo €rx6vo a4t(6r600 j(i]'Eq ].fq€rq. Eival Eri6I6 6uvor6!,{ropiooule, p' dvcvro?uirro oal6 rp6no, plq ouvdpnton rrouvo rapouodq€l or(p!06qauniv qv !616.1110, 6ll,o6i va ralavrd'v€rar dn6lp€qaopiq Kovrd oro 0, lprlorpnol6vroq rrl ouvrip1rlq sine.trc Keqdlaro 15 0o ll€lefliooup€ su.ri .rrt suvdpqon F€l,snop6p€Eq,Koe6qror Tlt LrglpnonN rad lto rrtv (ipa, €ivorEuKol.6r€po vs plsrpororiooDl6 rI fdlprlotl o€ loip€q 'I!o flq yovi€q.H ypo{prKiirlopdmaonqe suvdp-
4tulat8
56
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txitld 19 qdte sineaoiverdroro trri po l9 (':j(oDpe s].r,iq€'rtv '(Iipo,(ootov opr(6vflod6ovaaror ortpa6orEn lpaarKrirqpdoraq va eivorKa€op6cprl'I FdlpitoI o€ rad,€Kt6(0,16.16 n).soviKrrl[o drr foq ond,ldoocran6 wrKdqloenlorrx€qqq r6r6rt1€q, 6t€rto np6o€Eto rdroL€E o "ay6d. ypoqrKri ACtrtpouFsr6porrouvdprnorl/(.y) = sinl,/.r.B),6nerern nqpdoraq q( / mo lj0lfa 20.(rro voqolpoqioer€ ourrirnT paqLKil napdorqsrleq 0oleiofl vo napaq pilll
f(x) = |
''-
ll
' 9 0 I+ n 0
90 90+ 360
I ^{nx = I , ''' ') no, 210+ 3$ 210+ 120 6rav ro -x€ivorrsyd?'o,on6r6ro l,/x eivorprKp6,ro /(x) EivoL Iapqrlp€iorc ,T0)(, cniolqtrKp66rovrorciva."noliopqflK6'.6r).aSri.orqvTo.rluivorpcldlo apvt1tK6x, nril.lro /(r) sivorKovrdoro 0,qv Kar/(r) < 0. MLo €v8roq€poDso nopallq,i oulils rlq o'rvoprtorl( sivdl I /(x) = xsin th H ypaqr(rimpdorodt ourri(rnqouvriprnone 6rfl oxe8rtrorei np6rErpo oro lxitq 21.Aqo$ ro sin l/x ro),olr6vsrordnnp€qaop;qKovr{ioro 0,!€roEirou I Korrou- I , I ouv,ipqsrt qoptef€ro€i rouxKorlon -.(. H oDl1mprqopd rlq dneLpeq f(x) = xsinl,/.yro),ovr(i'v€rar
tti]ta 20
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tLitld 22
5 8 Ory ra 6c! dX€Lxotrio€).ni6ovo eivornpotloflxd (oKprpriq,.To x().rlr8ponol, unopoir€vq xdvoutrc civorvo SeiloutcdvoKolfdrr llg xqr v oQioou!€x€v610{otpdfl Kovldoro 0 (noucivorro svSLoeipov Ko!!drt. KoLpdLrdrq.€ivoreiKo?,o vo Fpoutrs ro),i nroo,riaq odpiBero'. tut, tropotiouldprio€LqnoL,I ypq{0r(tltot'qncprioraolvopt1o106Ld(EroL $t q ropqordofiE It)v !q 6rqKpivor,F ?poQ(K6q fe ^ I \'. .Y0 nrprdxn, €Eoproto[,d],q roorucio(x,,y)nou rlo,rdorqoi rorca ro (o,b)sivs,iorlFs. O KUx),os unorEi,€iroL rnotivos (:tiro 25)an 610ra oru0io(r, -v)re
rr
r ir
a r _r , y 1 , : ,
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trita 2t
O rixro!u3xirrpo ro (0.0)|(srqKrivaI, o onoiog0ropriroroutvrixdn oqvo €xnpiioc)no; rt! orxotivrro!,).r'lrToL o yot'sdeio; tuKlot. 'Evaq or€v6i or)/tr!ri; rou |ri,r),ou€ivorrl alln'yt. A! i opi(sroroqv 10 oivoio rov orlFeiorv trouro .idporotdrov qnoorios6vroui ondl[o dooFdvo oiltsio sivoLoro0Ep6. ( 'Orovm 6io oltciq rquli(ovroL, noipvoD[s dvavruK),o. ) Av.7roruro],ia,outQoviooD[e To6uoortlsio vo eilor ro (-c,0/ rdr rofc,0), xoLoqvcioporota ro! onooldosovndpoL,|r
4. f patprci6 mpaouioeq 59 ro 2d(oropd]ovrdE2on),oDor€6€1 Kdrroroog 0irol,oyrolofq), y) ppior(€ror 16r€ro (_x, rdvo onlv 6),1€$yTl av Karjr6voav
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'i ri
2a-Jtx ;t1-
x1+ 2cx+ c'1 + )?= 4a1 4al0;3TV
+ x1-2cxl c,Iy,
a?x- all= ts \lG- c),+ f c'1x'- zcxa'1+ al = a1@1 2cx i c1I
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dy=
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:+:= t, 6rioo, = \, a' c' (aeo6np6nerooqdlevo 6lgt66oDp€ a > c, 6mrcr 6rr o2 c,) 0). Bl.in€rerrlv erK6vorloq d).),€,\fiqoro Ijarira 26.H 6ll.s1\ndrvEr rov oplq6vnod€ova 610\,/= 0, dp06rav
trips 26 '(qrrov KaroK6puqo dEovo6rovr = 0, EI)"oEi J_= I
v=tb
H Dfl€p0ohiopi(ETorovdl.oyo,Jr6vonou orrarroijr€rl aragopdrov 8io arooflio€o)vvo €ivsrorao€pri.Ala),4owoSr(orndl.rrd 6tp€ia (- c, 0) Kar(c, 0), Koloov oroo€prior6oroon 1020,noipvoDle, ocv onvoiKqyrov' ovriKsrlo (.{,/) orrlvumppo}.rj, qv
trtu ' ct' Y' - u$=if
rl onoioroipverrnvor1.oi6.€pn jroparl _+_-!=1
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-a<x 0.c 0, oiBqovo fe ror rp6ro ror rroSeLxr[EroL ndworcv6l!o vo pr).d$ ?1oll.rodllEio UEnoll{iq 6lvrcro?trv0e(r, P),,od6!o Korflv oodaslo610,rtL dtL€ o!vr€ro1t;v0q'4006\ro(or oiou. !trdp1ErKdtroro ?ro(!rq noLrKdq Ei!'olaovsp6.on6ro txipa I (Kor10:ti!o 2) 6rL1()ort[siopr noirK6qourEr(7p6\'Eq r r.1 r no' 6ifor-u ,r-d.rg(]7'or ; (r. d| lcr Kuprfo.uv:g o' v-rruTpivr.g i = .cosd.I= rsind. (.r,,v).r6rs 01(onorsodi Awrorp6aoq,ov iva ol[Eio 6)/EL K(prEoLflvd; ou\'rEraT!6ve; norLan6nt no).r\i,o' \ u -otl:\ | , {r d) .\svo'oroir- ; rt rl' rarq = y/x. t\ :t+ l). Tripo,dolo 6rL / Eivarlra ol]vdprrtort. T6rE,#?ovrai io Tpri9qturit6 / 6€ iioirx6q rl oLiilo/ri o].ovrov oluriov P ts troiLKdc orlrErottiv8! (r, ,) ourr€loTl6v€q, evloor5uE no! r{dvoro,o[vIt]v e6iooortr= /(d). MEdiio ).6?ra.ro ?p{iprptlrrtq f o' noirK6q (/(d). d).dE! ol]vftro?rlvcr6i\,orro oulolo 6kDvrov ortrsiorvrLcnol,LK1j( ouvr€ruyuivE; papLirtra (/(u). P)!e !o /(0) npdrorvo 6o0siLSroirspt o1olETov6s6rLe€opoip€(ri]'t npdrro.o€ ovriosortnpoqro (ei'll (r. /(.r))orl o'r!tero!6vl.rpoqrnin(Ipdorootqq /: Eiv(ranldrq06faouppdo€os vs €€opEiror ro . oqvI np6rl noiLK,iolwero/pa!t {oLI , o0\r8€irept. H y p o , p r \rio p i o o a t , 1 , / a " t r o ; , { , ,o ' r , , o i p i L c , . ' L p , / p d q l ^' u' / u o r r ' t l ypflqLKinopdoraorrlq eiioools r= /(9)'. r1o ndpdblr7rlo, oq lroodooLlse 6t | / 6u!dpi! 6! f (0) = a rta oJ.trd A. H rpdqrKinopdoroor]rt r sliooott; eivorlra oraeEpri r= ( sivoron).6sdvoq{iKloe ps Kav.poo 1(d,o,(Tivoa (txilo l). A!16 ro n(1pd6€LlrL( o{Lo}pdQri, ro t8lov6q6Ttor noirK;; oursra?pdvc;pdil,ov tL€KdriDrlo\r3poo$rt1p6no, an).ororoivro npdyjraro os n€prnr6o€Lq no! nepLKl,€ioL,v olpts!pia o( npoqritv opzrtO. H lpflqLKrinapdoroolritq EEioootqr= d iEirlEraLoio trripo 4. H nqler{irpdrL}Lri o v r . o r o L Z Ldi ]o, fq r L q , , , , i , d 0 . r v i 1 6 o ^ ' r o p . L t o f ,.- u,8< 0. Tdioq,osopsiols rrt lpuprKinapdoroort fltqsEiooorq r= cos9. To:xi[q 5(a)6€i7v€r ro ripoq no! ovrlorolji'i oro 0 < d < 90,[p€d os !oip€q].To txipa 5(b)SsixvEL ro !6poq nouo!'rrorolrci61090< d < 180(€8dr r < 0).Mnopsirsvo El.a?i€Is 6fl DE!npoori0'\'rrlr tradoio' v6oortleiaov d > 180i d < 0. EivorciKoio va nEpL?pd\ioup€ rrtv i6raTpaqLKri
68 @epiAn
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r= cosd, I = rcosg,
or Kopr€orovaqouwrro-rudv€q ro'r rKavonorcrlv rnv €Eiooorl
x:+i=x rloroiq nrprypde€r;vov riirLo (np6pl,rltd3-16).(Avtrorp6grog, eivcrler<doapo 6rr,erivot Kop1EoLav6q qv.{r + f = x,16t€qur6ppioK€su}1erqlpdvoe tv6golleiou rKovororo6v ror ndvosro ypdqllo 1!( €Eiooslgr= cosd).
NPOBAHMATA l. Av 6[0 orl]Eio6louvnohKdgoDtrelo?liv€E(i , 4 ) Kor(4 , 4 ), 6EiEi€ 6iLI o,6ora6rl d leroqi rouq6iv€1or orrdq otdort - q\. d1= \1 + 41 - 2rt4cos(ot Ilord I l8ol8rprKrirng olpooioi 2. Il€prtpriryr€ 10 l6vrKiilqpoxnlprorrxdqq tpoqrrllc nopdoroolqqq / o0 rohxdg qv ouwErqylav€q, (i) rl / €ivardpflo. (ii) rl / .ivor reprrlri. (iii) /(d) = /(d + 180)(or uoip€c). 3. tt€6rdo1etreTpoglrdempasrdsaqrov mo Kdro€Erodroeov: (i) r= asin9. (ii) r= asec9.(Yn66€rqrlr Eivorro),rian).i., (iii) r= cos2r,Kd,rl rirrl (iY) r= cos3r.
1v; r= lcos2dl. (vi) r= lcos30l. 1. Bprrie clroioere yro fl6 Koprcsrovdg ouvrrrolF;vecrov onfEi(,)vt0rv lpo0rK6v (i),(ii) xcr (iii) rou l]poplriloros 3. ncpcordoeo:v
4. fpuEx$ rapaodoe4 69 5. (a) (b) (c)
xls6dor€ rq 1paelrll nopdo.acrl roDrdpdlott,oit r: | - sine. AEiEre6n dDni atrot€l,Eietriq€ Kal qv lpoqrrcrirapdotool TrlE r= I - sind. Ae(te dn pnopaiva neptlpcAeiat6 qv eliooorl
t+f:\FT7-v xcr or:pnepdvecrlcrpopsi rnicng lo 1t€prlpogsiror ffI6 nlv €Elooorl
("+f+Y\t=i'+f 6. tXe8dorErq tpaelKrapa6rdo€6 !ov etopivov4rodroEov: (i) r=l-0,5sin0. (n) r=l-2sind. (iii) r= 2 + cosd. 7. (a) trE6rdo€ rn lpoetni ropda.a('l lou l4l'{oro, l:2atcas20. O) (c) (d)
Bpeir€pia €Eioo6! tro fl! Kopr€onvt{ tou aDw€Io$rsv€g. A€ifts ifl €tlur I oDll'oti 6lov lov onF€iov P roD llllpatog 6 noD.rctrvoltolodvq ol6orl 44 = d':. qpoano0eiqrs.vapowC\rErs10 olila rov KapirDl6v.ov oroiov ra 6nP€idP txor.ororofv qv effoool d 4= b, 61lrlvb ) a' rct 6rw b 1 u'z.
KEOAAAIO
OPIA
H 6v!,oroxouopio'Jei!,qroroodinolErl mo oltovrLKri,KoriooqI ]I1o8DoKoho ()16{}.rtporov AneLpoon{6 Ao-rrof6.Xr61oqouroi ro! KsQo?,aiou Eivoro oprot6qrov opiov, 0a qcxr|rioo'rtL€ 6prDS opLop6oDr6'iou ea opit1o lLu ax6soqopiir' 6vov60r(1100rr{6 ooule,8cvcivorrt ).dqrt(6plo' o).).drt iworo lLosouvdprtotqnoL,rsiv€Lo€ KdnoLo 6pro. H suv{ip'rror/ reivsLoro 6proI Kovrdoro a,ovtnopoip€vflaapoD}'0 ro /(-{) 600Kovrd 8dl,oupe oto I orcurlwcg ro ,rvo EivoL opKerdxovrdoro, olld 6XLioo pEro, d. An6 IlC;81oDvapr{o€rq no! olrSldoaFsoro lrrifo l, !6vo oLnp61'Erp€rEl5ivouvoro I xovri 6r() d. apaqp€iors611,av Korlog(d) 6cvopi(srdr,,.orro,l(a) Eilsropropdvo (p€l.d0oqrp6roD,ral6€rxfll rd),r 61l rt g {o1n I r€ivolv oro / xo\rli oro o. Aur6 lLori, pTlrdrrlv ovd?Krt orov oprop6paq,onoKl€ioop€ vo sq6fioo'r!' rrtv rlri rnSo',vdpntorlc o1o a qpKEiro /(.r) va sivurKovrdoro I rLo x xovrd oro d, al,ld 6tt ioo le n a. yLaqv ruri /(a), oLheKoveEroursro spdrrrtpa Arldrs, lsv s!6rdesp6JroorE ov ro /(a) opiqsror.
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'^o vo on',{ovioouf€ rov ,oxuproF66r, rt rEiv€Loro / Kovrdoro |]ol.L{6trp6,ror / a, nopdt€.oLan6 lLo !d0o6o olcSiaartqol'vopflio€ov nou 6evovaqdp0rtK'oro Krad],aLo nol] Ka0errdnoprordvorro R. }t0 4. :rjtaovo r' auri rr t;0o3o, (rDlpaaiqou!06rjosuoei€q, on6;vo orl!0io.x rrq oro rrts dl).rts. Iro tlito 2 6ivo!!LErro rdrolo P6].r fLos, f(x) ErK6va-t103io SlaaopEflKdC 6uvuprrtosE. Aq eeopioouFgrdJpo|ro oL,uiprlort / nol, rt lpoa,Ki rnq napdoraortlord(er t' ourriv oro :xril'o 3. Aq uno0;oourr€orL (rrdF ro /(x) vo opioKr.ol xovrd oro l, aCron$ rdoa oro ovorlr6 6L{ior ro, nou dtoure ox€8rdo€r610:ripo 3. Aur6 s_Eaoaol,iqEror flv
5. Opu 7!
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L
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oeopioou!€[6vo rouqoproloJqr oro6rdolruqA rou t:tiparoq 3.(l'qur610 6rdTpstlo 6Ld),iEote ro le1q),fr€po6rd6tltio nou5o!),€i0r'0o InopoiootEawi yL' qur6vo 5loi,6Etdorntanounspdi(srro a.) Av 6ro),6toujrc dvalLKp6ftpo EoL't€onoro8ri,!or€ fLKp6t0po vo 8rqld(oru€6vapr(p6r€poA', q d 600 8rdolrtpqr'(tlrigo 4) 0o,rpamL,oD\algoe, p((p6 l(r av ndpolpem ovor:f6 6rdo{rlfqr, !no1l0€1ar 6n rdvro 0o L'ndpX€L xdroro ovol1166LdmlpoA noueo6orrlEi€r. Mro mp6fora onrlx{ epttr,€iosivdr8uvtrniIe pdol q lpaerxri nopdolool qq /, d,id o' ouriv rlv napriorol to 6rdoqFo, ,rpir€l va qe6rc6r€i dlo\, (oroK6puqo dlova,roLro oivoio A orovopr(6vnoriiovo.To 7qov6q6rLro /(,t) ovircLoro r6ruv 10I €ivoLoro A , orlpcivar6ir to p6poq 1rtqTpa9rxiqfiapdoroonq rou ppior 0, Yravq 9sFarooofp€
x ] s i n - l< e , ap(€ivq ororrioouto 6rr lxl I (t {il?sppaoaivsroror ooq}. q duvdprtorl/(.x) = sin l/.x (trifo 9).fr' oufl1q ouvdpn]ort Ti),oq,oq eq€rdooulE 6evrolrirr 6r n f reiv.l oro 0 Kovrdoto 0. ADr6rso6uvqreipero vo ro6pe6116evro16er
5. Opn 73
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ro /(,r) - 0 | < €,61oL6.rowoq Io rdeeqpr0!6€ > 0,.6rrfroporip€vo Kdvou!€ ,{ opK€rd l.or + 0. fra vfl ro a'ro66iEour6 our6,8sv;tow€ nspdvq ppo6r' avo€ > 0 }lo.o u,{pd onoion ouloriq l/{.r) 0 | < s vo prlv4aoaa?,i(sro1, 600pLKp6 xr ov onoll4ooLue vo I pipurot ro \ Korro r uucx{ivcr6cv/ropoiucvu cruaon orr /rrr "i\fl, I 600fr{p6 Krov qtrriooulcvo 0ir,o1 ro Ix | 7rqri,av A 0ivoronoro6inors61do1rt!o noD rEpLdX€r ro 0, L,ndpl€rK{inoroqopr0u6!x= li(90 + 360r) ,rouDpiox6rola' u!r6 ro 6LdorntaKor?roro onoio6,routs/(x) = L To idrosrrr8iprro jrnop€ivq rploLponorreei{ttiso l0) tLs va SEiqour€ 611n / 6€v r€iveroE,rdvlvdlapre!6rovfli oro 0 fro va 36i6our's o116trprnsrvq ppo{p0,?roo,!o1o6i rolerdrpu Soopdvo I, K{inolov opL0r6 € > 0 r6rorov 6arsn l/(x) 1l<evdt4v1o1nel, | rlooprpo ,,1or oncLrrloo, F:vo, ,o .r H .n;\oli 6o,-ifl rrqonorovSirorc ti to opl8sdl 6rl),obri, 600prKp6Krav (rtrioou!6 vo eivcLro lx,6evrropoipevcelcoeai.ioouFE 61l l/(.{) I 0. 1616rl / reivfl oro I Kov(i oro a: 'rpaytaflKd,yro vo PsBoro0o'i!€ 6r' lt{.{)- ll< €, oi}oupo€ivqropKer6va orarrrioourelx- ol< a, oeoricr.rr6olpaivcr 6Tl
i
o0 l/(x)-/ ro! prop€ivo loq 5o0Ei: H ouvdprrtorl oro 6prol xovrdorod, l1\r]1ordosopre!6€ > 0 pnopoDfs / ou1l(],iv6r va Kdvo!!€to "fG) tl 0. Eoro r avoqquorK6;opr0f6(r6oof€7dLoq 60r'1ir<s nopalrporipE 6rLorf6! orqpLopoi x yluroli onoiousIt /(x) 0l 0 l]rdptsl Kdroro6> 0 rdrorori'6r€,tlo Kde€-x, ov0< lx d ODttdplouvtr , 4 > 0, r6ton 6oiE, llo
rdosx,
dv0< lx- ol < rr,t6telt(x)- /l 0 tdrcta 6orE,lta rdee r,
t eA r -l < + , o v o (l r - a l ( d r , t d V r c r c v O ( l , r - c l ( 6 , ,t o t el g ( . x ) - a l < ; t=min (rr, t1),Av 0< lr- dl < 6, t6terl 0< lx- al(dr rat I T6pcoeroqre rat ot&ro,dpaot 0 < lr - r | < t1 toxfoDv ll(,1- | dD. (ro 6prs0'!6 rriroD(I1i9o 17)opi(owar6pou: lim /(r) = 1(ri = 1)onlloiv€r li:n/(.r) 6rL1lo K{io€e > 0 unrip;(€r dvot> 0 rfrolo 60r€,-rrorde€x, qv0 N,T6rE f(x)-
tl<e.
H avol.oliaF6rov oproF6rov ouvleroFivovopiov'rpdravo€ivorqov€pi:ev6rl ouvoiKrl (0 < lx d < dDsKqpdqflroT€1ov6(6Tlro x eivorKowdoro a, r ouveliKt(,x> ND €Kapd(aro ?Eyov6E 6rr ro x €ivo!tr€yd?'o. yrori rl Aal€pdloal€),iyop6vo1p6voorq 6proqr6 rdvo Kor0?16 Kdr{o, Korsro drr8rpo, yevrxi arloooeiardso qn6rouqopFto6Eyiv€roreqvEpri ov xoraldp€r€rov opro!6rov ouvl0ropdvovopiov (ro! €ivoLooiyKprlorno oilfovrrKd).noDlfe aor4o€lEfidvo 6' ouro{rqrouqopropo$e 6ivovrororo llpopliporo, rol) n0p161ouv Kor6liQopouedD,ouE r$noDq opiovnoull€prKdq aop;q€ivorlpiorpor.
T]POBHMATA 1. Bp€ir€ro 6pu ('Ol,o au10rd 6pro6rlovroLlsrd a,r6Kdrrorouq otry6pprKoiq l€raol!pepom€eirs dn6 ro tu! O€opisaroq 2 hdeopa dn Edpetsno i Fanopoiq, uiprl o€ KdeE rspirroorl, fI lrr€ir€6IoEorov K6rova ro avoa6psre.) xploqro,mlo6vror
(i) lima-r+ , . . . i,l m. - x ' - 8
(ll )
!'-R
(iii lim ---+ ( r v , r r m,-!
lv) (vi) \ ,
x
y
lima---l-
!''
x
Y
'rC+h-'ta !iq) ;-ji.
h
2. Bp€i€ ra eli( 6pr0l (l)
llm-
(ii)
t - ,fi=j lim:-------:-.:--:
.... ,. l- v4 rrrrr uT----l-.
l
3. t€ Kd0elldflr6 rrqer6levs€neplnr6osq,Ppeir0dvad drorc 6010l/(.r) - /l < E,yLo l(d€ex rou rKavonorei'rTlv 0 < lr dl < d. = (i) x", 1= a'. f(x) I (ii) /t.tt= a=l.l=t l
( i i i ) | ( x ) = x a+ - '
a = 1 , 1 =2 .
(iv)
/(r) = ---1-
(v)
/(x) = \-hl, d=0,1=0.
, d = 0, / = 0.
(\it f@= \G,a=t,1=t.
5.
'Opu
85
oro I]p6phpo 4-17,€l'rdar6'/tanoroiqopr0rofqd 4. flo Koo€prd otr6flEouvoplioeLq L,rdpr€L ro oprolim /(r). oro np6prlpc 4-19. nq oL,vqprriaole 5, (a) KdwEro i6Lo7la 61,8q \unqionourorollidmLpaE€Ko8LKd (b) To i6ronp6ol.nla,av lplorponolriooDue qlloioo en69. oe louv os tLroohoiSo on60 o.!ti 11 alrrdqrou xoruliloLrv lrs 3 > 0 '(ar KdoE 61ror oDvapriosq/ rar g ;xouvltv €EriqL8ldrtlra:YLU 6. Ynoearouts ?roKdo€x, or0
lr
2 l r ! n { : I | " . T d T cI t ^ t \9 /
2 / t:.
c v 0 < x - 2 1 ( e : , r 6 r el e ( x ) + l < e . xd0e,{, fro xd00€ > 0 ppsircdva,> 0 rdtotod,or'.11o (i) uv0< Jr 2 I < d.r6rcI / (x)+ s(x) 6 r; Yn66e(n: Tvo € Ko).o6pro€ivoLro lim l/rl=0 Kdvt€ trin dtrl€ppo d,sr€ qur{ vq sivqr rl p6v!,r},rlpoQopiororl Xp€r0(o6ao1E. 33. Bp€ir€ru €{ir 6pL0l ... {r,
.. .r f sinrx [mt-*
)x1O
,... ._ , stnr
lln
Im-'
'-*x +)
(iii) timr4r+ .r- x. -. . .. xil + tin:.x)
tivl
llm-
r* (x.r srn.rr-
34. Arcd€iEr€6tlliq! /(l/x) = Irg /(rI 35. Bp€ir€ra €EiC6pro,oovqpfliosrIou opreloJa = lim(siLl)/-x(U
tm---'
(ii)
limlsin -i-
( '/1^,^ ao
36.opiore ro (lim /(.{) = 1D. (a) Bp€irs ro lim |ry" + ...+ or)l@nx'+ ...+ 4). (b) Aro63ills611lin (r)=lnn "f "f(-x). lc) AnoSeitrE oTLLm f tl/xt=lirj, ftxt. = € Opi(o'rpE ro lim /(i) vo otroivEr611,i,ldKde€Nurdpxsr6vo, > 0 161010 60€, (Zo.rpo{pisr€ Evqoj(rifa 'rouvo 7ra Kdo€x, ov 0 < lx dl < ,, 16r€ /(x) > N. rorpL'i(€L!) (a) AeiEtE 6tLiin Ii(x - 3)r = - . (b) Ano6sitre6rL,qv /(x)>E>o ylo Kdosx, xor lim g(x)=0, 16r€lim
38.k)
O) fc)
f (x)ls(x)= * opiorera lim.,fG)= -,
= -, xqrlim/(x) = e. ('H rouldxrorov l,+ "f(x)
nsioer€ro\, €oL,r6ooq 6n €o Fnoporloors vq ]pd\ier€rouqopropo$qav 6sv popr6oooTe. n6oa {fl,lo r;ron onlBo}.qtropsirs vo opissr€, AroSciErs 6rr lim 11-r=-. AnoEciErc 6n lim /i yr @.0v 0 rdroLodrols,?lo Kd0s.x,
( 3 )a vl x - d l < r , r d l o l s ( . {- ) s ( d ) < d ' .
I!v6!d(o!1oq q (2)l 0 urdpxsr 6> 0.droro6ore,1L0 Kd06 x, 0v lr
al 0. MLo 6!ora on66€Lqtpnop€i vq 6oeei .rro rrv repinroort /(a) (lKqrn(.y)ov.r< a. Ano6eill€6nI / sivor suvErriqoro a. (b) Yro0drou!661l ! g sivorouveTrie oro [d, b] KorI I ouv€tiq oro [6, c] l(or g(b)=/r(b).Oalour€ b]Kqr,,r(x). ororo[b,c]. f(x)vqEivdrs(.y)tlo.yoro[d, Aeilrc 61r I / €ivqrolv€j(iq oro ta, cl. ('Eror, Eio ouv€t€i(oL,vqpris€Lq troporlvva (suyKol.l.t0oivD.) (a) Aro8€iEr€trlv oK6),o!01napq),trafirol) Oeopiporoq3 yraItv (ouvriXsro and rq 6€q!i,:Yno06roq€61,lim-t(r) = /(d), xa, /(a) > 0. Tdr€'r'rdplfl 6vae opr0!6q 6>0 droroq (G1e /(x)>0 ]1o xd6€ x 'lou Lrovonoreirrlv 0 < x - a ^-r) 1l0 xdoexoro[0. l] npdTfqrr.np6ruqn 6)d oiyotlpo6rv oivqro),ri8Eru6Il f(l) < ^x)tLq nd00r oro [0, I ], on6r€6€v]rropoitL€ va 6rc];{orge.v= I . oii€ lno polrr€va d,aidiouue 0 < _v< l yrari /(_v)< /(x)av r sivoLonoroodinors opr0!6q!e y < x< L ADr6ro nop{i601?[o 60i1v€r6rr ro Oe6plpa.] 6ivor ort[alrrKd nro ro1Lrp6 flr6 ro Oe6plto 2.:uXvd ropuQpdqoufs ro O€(iplpo3 ).dyowqq d1r,pro ouvrX{qoL,vdprno! opLop6{o 6vs{lsrord Erd6q!o (troipv€L q !;-rrorrlrLtil ingr o' a!r610 irddrtpo. :ov ofo,Pri7,orilv olrortp6rnrarov tl,ro06o€{Dv rov tpr6veroprudrovfoe. orFnrpdofqrq eivolr€icioi Erdqoprrrrigqi,o€oqan' our{irov 'rpotloip€vovo€oplldrov.neprTpriqou!ril orun€preopd lroi o!!(iptrlorlq-6Zr !6!0 xovrdo' 6vaonp€io.ai.Ldo' iva oi6'.).Ipo dL{ioryuodror€6(rooo},rrdr'r6r6'!rlr€q lroq ouvdpltoilg€ivqrndwor€noii rro diotoi,o \" ono8€rj(roiron6 IK GonrK6E, r6r6rnreq. Kordouvfn€ro€iy(lrnol,i nro ro/upiq.rro va oxrolpoqrioourc 1!;(prsrl6rnrq rov Osoprfdrov 1.2Ksr3.es €idyours oirropo l(dnora.onouSoiuoL,lr€p(iopsru.ol)d eq 0ol01os( ra oloqipot]p€nprira rdnor€qonLdqysvrKsrlorL! ourdvro! 0€oplfdra oF.opHNtA{ Av ! / sivoLolvc7rlqoro [a. b] KoL/(a) < c < /(b)16I€ undpxerKdnorox oro [a b] droro d,ort /(r) = .. rnorEEu Eoros= t - c. T6ren srivor 6uw7iq.xorS(a) c > /(b). 16rrumipZ xdnorc.r oro [a. 6] r€roro6ore /(x) = .. Ano,\Er:HH ouvdpllol - / sivorolv€rrieoro [d, b] KoL- /(a) < - c < /(r). A'16ro -/(i) = c, roLrar oivfl Oc6plfo 4;rcrqr 6fl undptsrrdroro x oro [lr, b] rdrorodro10 6rL /(r) = c. I
98
Oep
"ru To osopriFolo4 xar 5 tq(i 6€ixvouv 6rr n / naipvflkdorrrlri qvdf€oqoro /(d) KqLro /(b). Mropoils vo n€liloule KdnoK6!aKd,6repoan' qor6:ov c r(qrd aviKouvoro [a, bl, r6ts r / noipvs'KrieE rLfri ovdl1€oo oro /(c)Kor ro f(d). H on66€4n€ivoronhi av, ro Oeopriraro4 Kqr5 oro Suio1ruo[c,d]. c< d, 16r€srt6qEaapr6(oDte lta,Topd6c,yra, AvaKEao),or6vovra(, av lla ouvdp tort ouv€:(iqo' ;vd dLdortBoroipver560rql.q, nqipverKorKdesrLtri dvdFod rouq ourriil pLKpri 1 oulvd onoKot€viKsuo!ioL,Ooopliporog ).6iraL 0€6prtpoEv6rdteonq TLpiq. oEapHMA6 Av ! / eivorouv€j(iqoro [a, b], 16r€I / eivalKdlo QpoTlfvrloro [d, b], qpre!6qM rfroroq60r€ /(r) > M' q rdo€:roro [o, b]. 6t).o5i, unripj(sr KdnoLoE AnorEEnH ouvdprrorl- / eivaro'rv€tiqoro ta, 61,dpaon6ro OE6prpa2 undp){€r 6vqq apr0F6qM'rdroroE 6olc - flx)< M''ts KAgE x oro [d. b]. Au166poEsrlFqiv€r6rr M'ltsKs9t x aro [a, b], on6r€rnopoip€vo odooupe M = - M'.1 flx)> To O€{rpripqra 2 Kor6 !q(1 beirvol,v6n uro ouvdprlo! / olvcXricoro [a. b] sivar ropoypr:vq oro [o.b].61).cbri. M rtroroqd,orrl/rr) a M ttKdOt t'ndp{d;vocop,0,r6q .{ 610[4 b]. Ilpd,porL,oaoiro Osripllo2 0€0u6v€L rrlviraph] Ev6q opLoloiMrrdronD 6ors/(x)sMrroKdo€xo1o[a,b],ro60o€6plpo6p€poLdvsLrrlv,ropqrlsv6Eopl vq rdpoq\. M = nax( Mtl, M: #roLou6ors t(.{) > M: Io {{iosx oro [a, b], pnoporipo
M.D.
€ropsir^7 Av I / 6ivorolvezi( 610[a, 6], r6te ur(ip7€rrd,roro/ oro [d, b] rdroro66rs oro [d, b]. /Lv) - /(.x) yLoKd€Er sro [d, b], rou otfoivEr 6rL /1y) < /(x) Io Kdee.roro [o. b].1 Trbporou d;(ort€orrt hdo€oritaq rrq r€rprftdvssol]vir€r€qr(lrvO€opnl{irovl, 2 xol l, pnopoiuevo apTiooul€v qnoS€rxviou!€ npdlpola. l€prKd€vbraeapovro qv a > 0, orl)pHrA s Kde€e€rrK6E opl0F6(dIEruro rErpq-fovrfiipi(o. M dD.o )-671a, = nir€ unrip/€rxdnoLoq riroro6 tnote r' op(0p6q .r o anoiEEH O€opoip€rlv ouvdprlct /(x) = xr, rou sitqr pdporoduvrzrie.IaportpoifE yropri vc rxrppcorsi 6ft ro (troit8vo ro! o€oprlgotoE a 6I€rpro Fdootlg /r (o opr0u6q pi(o' olpo(veLdtl ! t noipv n)v Iru! a. H ord6erlrloutoi rou ye'1ov6rog rcrpolcrvLrri dvoqopre!6q tlo rlv / EivoLFLoaft).riouvdr0loror' oEopriforoq4. YrdpXfl ,rpoQovd( poivcror b>0rdro(oe60rr/(b) > a (6,13)( olo:Iita 7) ,rpdlpoIr, o! d > l lrnoporip€ vfl n{ipou[rb = d, 3\16ov o < t [nopoJ[6r,oflipouusb = L Agoi /(0) < e < fib), ov (oro [0, b]), dxours s9ap!6ooL,lero Os{ipruo4olo[0,b] oDpn€poivolitr 6fl yLsKd'roro.x = = a. 6rr].., r: 0.1 /(x)
To idro qKprp6(tfiltipitgo grop€i va Zplorponorlfti ./rq vo onoE.iioupe6rr dvqq orTtx(ig opr0!696Z€rprot-oori pi(o,iro rd08quor(6opr0p6r. Av orl outpoivErva €ivqr nrprrr6q,Unoporip€ vo ,r€litorrlsr.(in ro).irepo:rddr qpr0[6edlerprq ,-oofli piqo.frq qpr0g6to t1€rlrlr-oofli pi(q.r, vq ro ano8€iEoutE, nqpoli poipeanl{bq6rr,ov o 0s1rl(6q 6I)...ovx'=a,16r€(-x)'=-d(crer6ior€ivqrftsprtt66),dpoo-oal€rrIr-oorr pi(o x. o roTuprou6q. 6rLyro n€prrr6, {{i0' apr0p6fa 6Xe1 [1o n-oorri p(o, €i\'ur
7. Tpla6[ord"aqmpi1paru99 1006v0foqrs rrv np6rool:1c[ioo:o1 x,
d=0
ir€r pra pi(o av, 'ivo r6prrr6q.Av 8roru'ioe€ir' qur6vlov rp6no ro otrordloofu om8i1€rar f €ydl.ny€viKeDorl. oEopHlr^,Av o 11sivorri€prrr6E, 16r€Kdo€€Eiooorl . { "+ a , j . x 'r + . . + o o = 0 drsl [to pi(a. vo pr).cF'loo' anod||:r npopsvijrq edloL,,rc l-c In o' v{ipr'tor +ao' f(x)=x'+a,,r''+ KorjroplKde oo od),ojr0 vo arobsiEoup€ 6rrrl / eivorpsprK6q aop6E€8tltri aopiqopvnnKi. H SnLoorrLdri r6do€ivqr6rr ]ro u8yd),oI.xL I ouvdprloitIo iqelno}.'jt'€ rrlv s(x) = i" xaL,oaoi or1 eivoLrep1lr6q,o!fli rl orvdprlort 'ivor eErrKriyLo!Eld),c o€rrKdx {ot yro pEyd].o opvrlnid x. To t6vo rou tpsla(6paore]Lq vo e6oour€owriv lrtv opvrl'rLKri Eivorl.{7rtd}.yEppa. 6raLoerllKi660 o€ eqoppoyri, H rcrril].1Iq ovri)'o1 rrlqo!,vdprlot( / Ei\flrvd yp{iuorpc
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0. 'Apa /(d) : 0 Kor lncivolro orov rcrpo' r(apiq :apo!!L€ 6!oq.6rL KdrLnpdra vs nitTf1ivflolpuPd, (aoi 6!r ZptorponoLrioclpe !o lJpoift ro ouaLoDlKrlL!oipTraL6l6rlrrr rouR, KqrdE! zpErdleltlL|1€TdioqilsTZoq TLcr rriortro olFeio. Eivor qov€p6 61r FnopoitLsvu AruidqoL'lctvov apr0!6 a ro! vo sivoL r'toin.rpoq or' 6io ro olor/€iorou A (1ta napd8ELT!c. unopo,rtr vl1drai6;ou[r o = b). aild 8sv si\,orKoL1600qavEp66!r fnopoLifr v(16Lol.i\ot4e *w d.ri(roto. uparyaq u, uno06ool]p€ 6rL ro A onol8],EiraLon6 6iouq iouq apL0foui r > 0 7LaroL,!otroiolq .r' < 2. A v o c p r 1 ] r 6 \q4 S e v ! i p x s , 6 E !0 u l n i p l s 6 v a q€ l d l r o I o q o p r e p 6 fce l a i L l r s p o co r ' o i d ro oioL.lsia rou A lLo Kd0e-v> f noLr0c;roipvupE,es lnoporioofe ndlrors !o Bpoifs Evoar(ota prKporEpo. TdJponou ;xoutE uvoKaiiiusrrtv nidv],l paq,eivaLorc66! aovep6noLdrp6o0!rl rd,6r n r o r ' a T p a T L c{ru v u p L l ] l ; , 7 p - a ; o " a o r - .T o l o v o n o r i i o p r v u r u r o , , c . i r o r r o ' q . ELor!n6oo!lL€Kal](1pd Kol vo q I lprtolponoLrioo!pe. M flrrd aoloiouf(iorr oro Etr6r3vo
NPOBAHMATA l. f1o Kol]€lrd on6 rrq ollopflio€rc nou oKo).ol,eoiv,€).':yEr€noLiq sivo' dvo i Kdio) r p p o l ! , v r : o r o [ . ; of . o r o o r i , . 0 6 r , \ f i o , t , . r o r a o , d , t r o ' p r o ' r r r p i i o . f l r ' ] v sidzLor! rLprirouq.(IIa pol\pEiarE 6rt ptr f 0t1lnoporo€ va d/€Lau!6i rL; Ldr6rtl8c flK6la nr l1\,3E! sivar ol,v'7iq. KrlruK6!u KL,1vro drdorlla 6 EivflLKisLor6.) (i)
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fG) = xr oro R. /(it) = x'oro t0, -). tt . 0 ru Kdroro.r> a xor t(x) < 0 y'o K'inoro r< a. Tr pJlopoitevo noip€yu rrlv /(,r) ylg 6),qro -x* a. *(c) Elffdoc. dfol(l)s,io ,rp6qlro 1ouxr + .i-v + x/ + 6rovro x *or ro 6€v€ivor _vr / xqr ro 6io 0. qro [a. 6] roL 61r /(a) < g{d}, aiJd 10. Yno0iroup€61rit / Korrl g eivorouvet(eiE = g(b).Ano6€i6i€ g(.r) > drL {dnoro -xoro [d. b]. (Av rt q7r66e(iooq f(b) 7lg /(x) d€v€ivarnoii oivtoprl,t6r€ 6€veivaroud nou0d np€n€.) 9. {a)
ll. Ynoedrowr6rL/ .ivqr lro otjvriprnonouvsXieoro [0. I ] ror drLro /(x) ovi rc€r610 6fl /(.r) = .r tro xdnorcvapL0r'6 .r. [0, 1] 7Lsl(doEr (kdvrcavooxrits). AnodeiEr€ 12. (a) To np6p).!!a Il 8€iXvEr 6rLI / ritrwl rrt8roldrvroro! rEipol6vouotolXrijrtr l4 (o7,6xhpt Tpatpri).Asiqr€61r! /,rptn€r vo rdrv€rKorrrlvd),1! (rr 6roK€roUpdvrt) 6L016vro. (b) A,ro6silr€ro sliq tevrr6lspooro16l.colr0: Av I g €ivaroL,v€rleoro [0, l] (or = = = = g(0) g(l) I ri I, 16T3 0,9(l) 0, x. s(0) /(x) = g(x)1roKdroro 13.(a)'Eoro /(x) = sinl/x yro.x+ 0 Korf(0) = 0. Eivo'I / ouv€-tiq o1o[- I, l]; .dEi€r€d n f rKovonorEi ro ou!,rdpdolorou OsopriforoqEvdrdleofiTrUi( oro [- l, l] tc di),q l4ra, ov I / naipva6noTltderdnouoto [- l. l], naipvsl r(gr|(d0€flIri avdlroii roug. *(b) YnooarouF6fl Ev6rdpEolq I / rcovoror€ito ouurdposporoDOeopripolog Trgriqxar 6rr I / raipv€rKdoetlftl ldyotia qoAi. Aro6€i810 6r rt / civor *(c) fEvrx€iolc srtlv ftcpinroot nou I / ,roipv€rxri0r rtlrli neffpoogiveqt6vo qopds. 14. Av / Eivurfro ouldpqol suverriqoro [0, l], edrou$ | / ll q p6rro1l llpri ltE l / I oro [0. 1]. (a) Atro6rie1€ drL1lo (d0€apr€p6ir iroup€ 1".r | = l. | . | ,r | . *(b) Ano60iEr€ drL l/+s /(r) yLqxdoEx. *18. (a) Y,ro0dtoDUedrL rl / eivorouveXieoro [4 t], Kordorox oroLoodlnolsopr0!6q. Aro6€i6rs6tLulldp1sL dvodnleio orrt TpaerKri nopdotoolr1g / noueivoLto nl.tordor€poolo (x.0I !3dLl,o ),6]ro,urdptsrKrinoro/ o1o[a,6]ldroro d)or€ ro (J,,/0/)) vs €ivor< a,r6qv o,r6o10oit rou (.r.0) n o,r6orootto! (x,0) 0116 g'n6rc (2, JIz))rta KiQE z dlo [a 6]. (A€ir€ro txifo 15.)
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l
AEiqrE 61r,o iSroqal,r6qLorupro!6q dsv'iva, ovoTKooru.d ooor6quv ro [a, b] pEro (a, b). ovflKoruorflo€i ^€iLr€6rr I np6loor lozr€r ov ro [d, b] avrlKoroorooEi tl€ro R. :r,q tr€pur6o€q(a)xqr{c),odroutes(.{)rrv t'ixr6rrt on66ro6lrou (.{,0)on6 oljrtio rrls /poqrnistrapdoroorq qq /. Ano6si6r'6fl gLv)< g(x) + lx - y g Koroljrnepdv0r6 d I sivorsuv€trig. 6It undptol]vop,0jroi,r!Kor.rroro [.l, b]r6roroLdrorert on6aro6rtro! G) AnoSEiEr€ (-r0,0) an6ro (xj, /(r )) vo sivo,< an6rrv an6oloor 1ou(rb',0) on6ro (xr', x,'orot0. bl. /(r,)) yroKd0o.!', **19.(a) Yno0iroup€ 61rr / €ivaroL'vsriqoro [0, l] Ko, /(0) = /(l) Eo1r,rr] ruraioq qpLep6g. Anodeiqrs6fl Dn{ipjt0lrdroloq opr€p6qx riroLoq60r€ A!orl(69 /{\j / l x ] . I . d n u co a . v,,o o . o : t i u o l b / , o n 4 Y r o d f rtl: o . u p , , or3rn o"\r'iprr6rl s(x) = f(al(x + Ii,) 1re0 ioxusov s(.{) + 0 ?r0Kd03 (b)
Yroe61oL,!E 6n 0 < a < I oiid 6rLo d 6sv'ivaLiooi le 1i, ]Lqrqvdvq\,$uorK6 oprep6r. BpEirepro ouvdprTloTl / nou vo sivaralv€irlq oio [0. l] l(ar vo rKnvonoLsl rrv /(0) = /(1). a].id vo t'rlvlxovonolrirqv /(.r) = /(.r + a) iLo
*20.G) Ano60i(r'6fl 6€vudptsL ouverrirouvdpqol / oprotdvroro R nouvo noipvELKdo€ rlprioKprod,c 5u6aopdq. Yn66€Leit: Av f(d)= f(b)rwd 0 y,a rd€E;ros "r(x,) ,f(r,) /(}1). 6Lloon6rorpioaLoorrirara(r,,x,),(r,,x,),(x,,,;!).i/(.r) y
H riiL6rlioI13iooqmpaieverleL ou| exr6qrligaroq, oii"d oDTli,ivaL ortv npotpolrK6_ rlfir lra ono rrg opEliqqq fLfl !o o).oK1,rtp6ooL,!E rov Kord).oy6 luq pErtq 0oqLK6q r6r6rtrEqrov npoTpqrLK(ijv opr0!drv.36!, anqrrotr€KdnorcaiorperrrtiSuov6lq np6mo11,
108 Otpi)n nopdldvo lro r8rdqro 1600onl.i nou0o drpefte!'q oLs0ov0oipr aqc).riqnourrtv €i7qps mpq0l6!,er.B;paro,n 1616T110 rou €),olioroud\,oqpdlporo!6€vEilqror'oLio€ra 1600 oo(i)u600rrlv neprTpri{poupc oro xdro-Kriro,a, loXi€r'/rorou! pilloiq qpr0to6(Q. fra nflpd6€r'/tq,qv A €ivqrroouvo).o6iovlov plr6vopr0FdJvx ro! rxuvoloLoivr1v xr ( 2. y nor vc eivartivr,:9pdIIfl roD,4 Kor[rKp6rspoq r6rs 60vundptsrp4r.j' qpr0p6q ]i iooE an6ororov6inored).),op4rd qpL€p6no!€ivqr dvoQpdyparoD,4. Mdvoora6roxd0o }ivfl xdoqpdTon6ooonou6aiasivorn Il3, or).d'i6n pprox6paor€ o€ 0;orlvo onoSciEours rn divofi rlr. oulrn).np{i}\'ovroe flS onoSsi€€rq no! sitof€ nopai€iveroToKsAd),oro 7. etoptrAT.r Av rt t ciwr ouvr7iqoro [a, 6] xar /{d) < 0 < /(b), 16r€I'ndpt€rr.dnoro( opr0F6g x olo [d, b] droroq6orE /(.{) = 0. Ano,\o:rrH trr6dsti lqq e(volan).69tLqouollpri aK8oon roureprypdptraroq ro! flvanriopr0f6x oro [d, b] p€ Lalt oro ri7,oqiou KE$qioiou7 0a €r,rorioolg€rov prKp6r€po Opi(olt8 lo orilolo A, troueoivErororo tttilo l. o( €Erie: ,4 = l]c a < .x< b, xorr / €ivaropvrl '('i oro 6rdortla [o. x]1.
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rIpopqv66.A+ 0, oQol]ro a ovrlKor 610A olrlv 'rpqllorLr6rtrq. llrdpX€rKdnoroii> 0 rdrorodor0 ro A va rdprdTeL 6),0ro ot}trio x nouLKovororonv llt, d S x < a + ii' 0!16 ,rpoKinTer 0,16ro np6ptn|'o 6-15,daod r / €ivo' o'rvrxriqoro [a, b] {dr /(d) 0rdrolo6ore (trrito 01'o a d-< x< a + d 2). Td}pq. urdpt€r Kdnoroq opr0r6er{0oroA noo /(.{)< r(ovo'Ior€ilrt! d - ,< xo< o {7rarid.i,r6Eto d 6Ev0qriruvro r,ltizro.odvoqpriTlarou ,4). A!16 oruoiv€r6fl n t €ivarapvnTlK'lo'o]d'(),nporo 8rdornrq[o, x0].A].id, ov xr sivoravoqopr0!6qovdleoqoro d l(or oro d + d, r6re I / €ivorenionqdp\rtflKrioro 6rdo1!f0[x0,-rr].Enotdvoert / EivorrpvlnKri oto 6rdorlpa[a, rj], or6rs ro -rLqvriK€L oro A. A].).dour6avriKolrqloro-fe7ov6!o1rro a sivqrdvodvo Qpri'/Uq rou A rl aptr(ri pcgur6€eoqrltr /{o r < 0 rpdrrcrvo rivor }.o!4qotdvl. A6 unoodooulre, on6 qv dl.h Fp,d, drr /(o) >). T6rs undpXor dvo{qpr€r6et> 0 rdroroqdrorrt(-r) > oyroo t < x < o + t (t,(iFo l). f rorro oKdpoqopdidpollre6rr urr(ipXsr dva;i0sroAnour{ovonor€inlvo- 6 0 rdroros160r€ n/ vo eivdrdvo qpoypdvlolo 6dsrruo (a - d. a + l') (9l,6nE ro txipo 4). Ano,1r.r:rAaoi lim/(x) = f(d), un{iprEr, 60rr, '/rr1ri{io€x ?LoKdorr > 0, dvo6 > 0 rdToro
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c v l x - a l < , r . t 6 r el / ( r )
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ov lx a < 6. r6rs /(x) - /(a) | < L 'Er6ro1, c16,{6.epa, dr, ov lx a 0 16rorc{iot8 n / vo Eivor qpolp6vloro{a - ,, d + {t).Aaoi a sivorro e}.dlL6ro dvoI pdlla ro! A lrdptfl Kdnoro 6n n / €itorQpaludvrl oro [d, xooro A aourxovoroieiqvd d< x0< o. Al)16ortpoi!,Er ,\01. apr0p6(Fea < rr < d + t, t6I€ it / eivqrapa,p6\al Ar.ld, sv rr €ivoronoroodrirore ror dlo [x0,xj]. T6r€I / Eivorqpa-r!6vldlo [a, xr], dpoo xr avixEroto A, rpliyla noD 6rr ovflrio€raroro y€-i,ov6q 6rLro d €ivqLdlo qpd,Uolou A. ADli I avriqaol SEiXv€r pLql€notdp€ro:I q,t66t1irito( olont}'d rro06ler6n a = b. o{i'np€m v' ovo9dpoup€ 6rdorlfo (d - 4 d + ,)I rl,n0qv6rnro a< a(6o1en / va pnopsivooprot€ioe Kdttoro qv lprtorponol{ooltLc rnv unopErt ev6q,> 0 rdroLou a = o p,ropsivo sroKlslor€i6poLa, 60rr I f va eilor apalsivn oro {r: a5 r( a * tl}. H o,r66s(rl6sveivorrt eior n),riprq- lipoulrrdvo 6 n / sivorgpqtEvrl oro [4 r] 6tL1 / rivcr 9po'/tdutoro [o, b]. nopo].courd,]t6vo tla Kdocx < l, 6Ir qva?Kaonxri dvafrl(p6€nrt€ipruat(p€rd(€ror !,4npoor00ei. Yndpr€rd> 0 *roro 6o1EI f !o €ivd ppoTfdvnoro {.x:b - d < x < 61.Yndpl€r6vo x0olo A iEroro601r b - d'< xr ( b.'Eror n / Eivorqpaypdv!oro [4, ri] KraK6!doro la,. bl. dpon / siva,ppaldvn oro [d, b]. I xatoq6iyoDte o ivo ldlvqopo. fro v' atrodoiloltr€ ro rpiro onou6cio0e6p1gc, oEapllAr,] A! rl / EivorouvETri.oro [d. b]. 16rr lrdplet ivqe optoIdc-v oro [4, 6] lfroLoq60r /(.f) > /(r) 1io xd0rx oro [d. 6]. 1no$r=H :dpo![€ i6n 6n r / .ivor epoyldq oto [4, b], nouoruoiv€r6fl ro o[vo].o [/(x): :r oto [a b]] rivorrppo7dvo. Aur6ro sivoio eildr€p0ov6!6X19,dpodx€r;voEldlroTodvoqp{illaa. Aaoi a > /(r)./ro.x dto [o. b] op'rsivo d€iqouls61ra = /Lv)./ra xdnoroy oro [a 6]. Aq uno0;6orFc ovrioero6rLa * /LJ,)tro xdos_voro [a, ,]. T61er ouvriprilolg no'r opi(rr(lron6rrtv .. s(.r)=
I
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ro! EElro!dl.ouq6ev€ivor nor60. A,r'qv rivorolvrtrii oro [d. b], qqoi o nopovoFoottie dr,rln),cupd,o eivarro €idzroro,ivoirpdlro rou {/(.r):-r oro [a, b]1 0U16orfoiv€r 611 -/roxde€E> 0ljllqpl€Lx olo €. [d. b] lr3d lr)> Aur6.p€rt oapd rouorllqi\,€rdrr 11o{d0sc > 0 wdpj(sr.r sro [a. b] !E s(x) > I ie. olo oro [4, b], ,ro! ovrrqdoK€( Aild ro r€t€r,roiooruqi!'er6n n I 6€vrivaLapoyudvrt npolToir€voosripnro.I ro oDvo).o lov qusu(6vopreg6v^- 6ov rapri:rnv opri otllo rot' K0qqioio!.6dloou€ 6!,!6).ou. Ilpdrelror rdlpo vo a,rod€i 0 0ivorl,r> € yLoKd€6,oro N. ' E r o r r < l l E y r s K d o € n o r o N . A l i d7io our6ortroilsl6rlroliesivoL{tvoqpd.rfuyror nouavlado{', oro ou6prud2. I M1uofvropr]rofld oro K0qdidro6 0d Ssiir 6 ro qrorElsolo roDOsopriluroq3 BdBoLo, ro OE6prlFo 3 6EviravSLoedoLpo rptorironori0nKsno}.Idqqopfqolrt ouq'irrtort. . rci|I rt o- t,rri.ol) d -0 .opo6-rlflruiruv rooo6 tfovIrd iorc rpohcuivo,va ra Ed'oouts, KdnoLs ondrrlErrrpEn6rav. tcv psplKi8lrorotoid Tr' uurrivrrlv o pio pnoporloiL 0!16 ro onorileopo6€vXprlorlorouiorlKs nordortv on66€LEl tLcvo roluprororitrc Kanato\, eropiuaxos,a)").6ov I nioq onoroDSinorE ;781Kiovrorsi.Froovo6poprio' 6).eq trg cro0ai[r:Lq rou 61oL,v Eurut6q,rdro,oondrl 6€v0o EovdliveLoq rdrpo,EnLPd],i'rcL. 6l8q rq r6L6lrtr0q rov ,rpo?palrrdJv opretdrvroDproIptLoorei. EXou!€nLobrarL,ndrosL pLifo /pL,ooroiF,. Ar6.ddr\or ripu.d/r d).).u vdtaro.
I{POBAHMATA L Bp€i.€ro e).dlroro{ivoApd'ttroxor ro pi]lolo KdrrD ApdIIfl (ovDndplolr,)rov oK6 iolorDvoL,v6iov.EE€rdor'oK6Faroui oLivoroizouv !€?rorori €i.rixloroorolXsio (ijtti.. E(srdor€rlv ro sid oro dvrDppd?lori ro p6]LoroKdroqpd?to6L,rr0uiv€r va av'tKEL61() o'ivo),o).
o {+,.."NJ
712 Oryi)w {r I t-:"oloZxqrn#0}' (iii) {.x:x= 0i x_=llr 1lor.dnorc,610Nl. (iv) {.x:0< xS V2 Kqro.r €ivolprr6e}. (ij)
(v) Ir: f,'+x+l>01. (vi) [x:rr+x-l rE > : 7. Ano8ciirc 6n, ov rt I €iva' ol'vsxiq Kor /(.r + r) = /(.{) + /(_y) l(I Nd(]s.r ro' _v, 1610l]rdpX6rf\ru( opr0p6qc r6.oroq 60r€ ,f(.r)= c.r ilu Kri0€x. (H an66s'i't €ivoL d[s6l ov duvdldoer€ro onocidopcra 6io rpollouf6v(Dv npoFiftdrov). MLa n),1pa,sapit DndpXorvt|i-oUyryr|( olvapflioslq / nou 'i:ovono'oiv lrv l(.\ + ' )= f(r)+ f(f)?'a '('ie€:{ Kar-v,alld 3€vrnopoulr!oro ono6EiiouFe Idjp(l npdt!(Ifl. 0!16 ro anl6 €p6rrtlo onaLr€idvvoLEq nou o!vrieoq Ssy oLr!flv1rr;rvr(r rorf o 6w - p o ' n X r o h o , r d e f L o . H B lo ; / p s { i d " o L , , 0 o. , 1 n . . . | , | . . , / . . r ' d r ' . - . d t u r ' { , , : *8. YnoedroLuE6lr Ei\aLprflowdprlol riloLa 6010 / /(a) < /(6)ijnon:a < b (:x'ir'o 6). (a) Ano6ciile 6r, rc lim/(r) KL rc rtr{ipror! {o, ru lui. Yn63rLi1: /G) 1'+ ppioKsraL fr(ri i]Droro np6F),r[a o 0r]1610rEad].qro: ( b ) A n o 6 ' i l r € 6 f l r / 3 € v ; l s r n o l f D n 6u i p s o r v o o u f l r r a ( l o p o ) . o t i on p o r p z € r a r 0n6 ro [po0tl]ro 6-16). (c) Anodsiir' 6n, (lv '1 / rKolonor6ila oL,InspdoturaIoL,OEopriF(roqEliiLd|E6rtq Turtq, r6rE civoLorvsrriq. *e. Av = sup{l/(r) :xmo / ' i \ r o l r r o o u v d p l r o ! e p a ] I 6 \ ' r l o r o[ 0 . l ] . € d r o L , f sl / l [0. l]]. AnoS€iEIeLdr6rlleqr1q | | ovd],o7ecr' {'rri! ro! IlpoP).ir(ro! 7-14. r0. Yroedroup€6fl o > 0. AnodeiErs61r Kde€ap'ep6sr rn{iaurdr Kord FovulLxdrp6rio oflt ropqi r = fta + .r'. 6nor k eivoroKiporo;. xoi 0 < {'< o. ll.(a) (b)
(c)
Y r o # r o u [ s 6 f l . r . o : . d , , . . . E i l o r f L { r q { o } . o I , € i a o r r L k { l v { r p L u r L d r | 0 L'ndpltsrKdno,on f' o, < r. Yno06roup€6fl P eiwr dva x(1vovLx6ro).i,,(D!o tT,,rypolut\o riotl o iva KiKlo. Av P'sivdr ro 'ryclpu{trivo Ko!ov1r6 noiilovo }1. Elnidolo ofLep6 rlc!pdrv.60iir€ 6rLr drapopdflvdFeo(Ioro efpad6vro! KiNioL,KdLro 'r!olo! roLr P'Eivar L761rpt on6 ro Fro6 rlg iiLdqopdgoldirtoi:' oro tr0(ij6r roL, KiK).ouKqLro elLpoij6v ro,, P (rp'toLFonoirior' ro txttfo 7). 6rr dvo xo!o!r{6 noiitovo P sTl'}pa!!d!o o dvav n iilo Anod'iqls L,ndpxcL pr r:ppoiirivooo r 0 qp/ior! 6rqla}o \4oqx oro,,1{or ,yoro B }rcsupA )0 ljndp'lerxdroro, > 0, ilor 60r€,ylo 6),0ro.r, / oro ,4,
qv lx - yl < d,16.s| ,/(r) - ,fcr)| < ..
pnop€ivq EivurouvsXri6 Ei6o|'1€ 6fl tio ouvdprrtort oe 6lit titv Eloeio,ri o€ Cvadvor1.r6 j(opis An6 qv d),i,n, I ouvriprtol vq €ivoL olor6topes suwtrig €r€i. 6r{io1ruo, = .a npotflrvr oror6lop$a ouv€tiE oE xd6€ xL€ro1,5 6rdolr q. Aud 6ev0o ,npeft /(.{) vo foq€xn).ioosrno]. - €ivorro i5rcnpd7tqrou €pQovi(Erqr 6ravpordF qv Fraoovriprlsrl eivorqpoludvl ,rdvoos avo6rdonlpo- KoL0o drprm vo odlTrl6o6F€ ornv Dno\yia or,\cXric ofl KoeroLVLIiqo, vdprtot oc rkroro 6rdortFoEivatcriol6 KoroFor6Fopoo our6,npdrsrnp6rova ovr4rroriooujrstvo ).€m6oI p€io. o' a!r6. fro vd ro ono6siEou|]€ 'Eor0r r0r(a,tou[e6lio6Laoflituratd, b] Kort6,'clp€xoLv6ro dKpo6, korrio ouvdprrlsn mu eivdr oov€jariE oro [d, c]. Toro e > 0 xor oquro€dooupe 6rr oLe(fq 6tiorpordoers / (i) (ii)
ov x Ko,y 0ivq'oro[o, bl xq, li /l 0 rdroLo6orE, uv x Koly 610[a. c] Kar lx
_vl< ,, rrjre "f(x)
"f(.y)I < s.
Ano$r"H En'l8i r / 0iva1oDv0liqoro b, Dndprsr;vo4 > 0, r;roro ri,or€ llv .Y b 0{o1 ., /r=
{trrro l Tror< 0. Aur6ano6€LK-vu€,
. I(h)'t(0)" ;:i 0 I!rtv rpoyparLKornro,
rtor ' '
liml9) f(h\
l'$
r,
t ( 0 1_ . t .
qopiE I 6r€rd ropriYoToqKot rl ap,otepi rapdToToq, (ADrd ra 6io 6pro ),Ilov1o1l'0pLKaq
tXitta 11
avriororxs, rnq / oro0.) Av d + 0,r6reI /'(d)undplsr.KaLtdlLoro, : ovx> 0 f'(x) = I, 0vx < 0. f't.tr'|= t, qveup6oaoT3 lrlv ropdoaiv€roL o€soEGivol€rjKo).! H an63e1Er ouro$lou npdypolor ./o?opraq mpoordoeq rov / {or /'aaivovrqroro otl!'dpqols).Orypoqrxdq lpoppLKriq rrrips I l. fLs rt ouvdprnorl | , t ' . o v . t 3'
:ripa n
To t'tripo 20 8Eitverq oovdFlol /, pa(i ll€ fl( 6rdQop€q mpa-r6yorrE ntq. 'Evo mo 6loeo1lolll(,j ropd6El & 6iv€1oL ofilj ritv 0l(610o0rl oDvriprqot, qq Fd),lov onoioft ypoarnlnqpdolaotpoiv8rolmo t1iFa 2l(a): ^ | J r x )= i
.rr.
x20
.r.
xso.
Eivor€[Ko],ovo 6erKov€ig 6rr f'(a) = 2a' av a>0 f'(a) = - 2c' N a LA!r / l{ovono1oi nrv /(x) < lx | ",oros'iils6rLI f nopol0r?i(rro, rro 0. 1?.'Eor{D 0 < B -ylr {o1 /(0) = 0, 1or€! / 6EvnapoTo?i('raL oro 0. *18. Ofrou[s /(:\) = 0 a dppr]ro.y,Ku I q Ifl r: p/q 6noL,10 p./q Ei\rurovd?0J?o. AnodeiEr€ 611l / 6€vnopaT(D7iq€laL oro a yLqKqvdvo o. Yn66€Lh:IIpoqov6qflpKsi ur ro otrodsilou|L.-rLo dpptro a. frarii A! o = , . ardra,... ro lEKoijL{6 ov{ttrru7EivoL + prrr6, ro'r a. eo(,ipsioro ro h) h Ko06q xar !o [/la flx)]ih lft ]ld n= 0 . 0 0 . .d . 0" . ,d , - , . . . re. (a) Ynoe6rotue 6rt /14) = s(lr) = l(a).611/(x) < s(.r)= r(;t)7LaKdo€x,Ka'6rL t (o\ hto). A-odrii'' ofl I s ropu?uliiroL f la) = s1d)= h1d). (b) Asiir€ drL 10 ou[rapooua ic! ,oj(ncLov ]iopaicill/olrLsrl! !tr6€'or f l a ) = E @ ) =h ( s ) . 20. Eoro / FrdroiuovuprK,ioL,vdprrtor0a dolp€oro €n6lcvoric9di(1ro 611I f i|aL nopo'/otiorut.H sednrorLalloI't! / oro (a. /(o)) €ivalrt lpo$rKrtnapdoraortrrti c(r) : "f'(a) (x d) + /(d). EroLrt /(r) - -s(r)eivsLnotlov,rprni ouvdprro,l d{x)= f(x) riiir dEr6rr nv _/(.r)= r'. r0T' f'(a) (x o)- /(a). E7o!}L€ dlr) = G a):,xorov /(r) = i, 16r€d(r) = (x - d)1.r+ 2d). (a) Bp€i13 rlv d(r) 6ro! f(x) = r'xor 6€ilre6n irqrpsiroL p€(x d)'. (b) Ar' 6,r, aoiv'roL',r{ipro!v opK€riq€!6Eilsrq6rr r d(j!) SLarp€ill1l ndlrorr rrs (r a)'. To lrirLo 22lLoq6iv8rlro 6raroo!rLKian6SELqr:olviooq, or arosic( or nopriT,h).tq tQonr6lsvl iitv eo r6Ivolv r] nflpdoraort os 6Llo ft TpoQrKri otlria rt 0Qorrolivrt#uvELrl ]poqrKlindpdorqotf6vo lia qopdKovr{iolo otp€io, dpa auri r rouri 0d npcnrvd 'ivol (Ertr).iroilliD.rra vq 66o€rsurfl ouorrlprion65srtt.or'trrsr6ol8 npdjroor1 d(x)
J lr)
f(a)_
f,Gt
T6po drovrriorD01( oqriqep{Dflio8lq. rrari ro /(x) }'€ /(o) drarp€iroL (:r a); rLqri rripT pro rol.',rDv'r!L'i'iouvdpfltot , rdrora dror€ /](ir)= d(.y)r(x a) ro x+ drrLori 'ivor limr(.t)= 0; rrari sivsrn(al= 0r froria!t6 ).ivrLto trp60illa: 2r. (a)^3iqr36rL f'ld)= Iintflx) - /(a)li (x - d) (A€vL'ndpx€L rirorl1ro 0o8'lt66)
9. Ilapriyayor 137 (b)
ov /(x) = s(x) 7la Kde€.{ o' Arod8iqr€6rr q nqpd?(Dloq €ivor(Ionl{i 1a,6rnmD: 6voovorxr6ddorrta rou mprdl€Lro a, 16r' /'(d) = g'(d) (ADr6onloivsr6rr yLo rov !110).01,016 rrtq /'(d), fnopeircvo oyvoris€Icro /(x) ll0 x'iocotTxeKprFvo va fl.l,vorio€rE ro /(,r) yLo614ourdro x ro'rr6lpovol) ,(+ o. ololK{i 6€vJrnopeirs
+22.(a)
oro x. Ano6EiEr€ 6rr YnoeftorFs61,rl / nopa?o1i(cror ",. . _ ,, fl,'
ht
fl\
hl
r6rvaoto 6voqopL0!6q 6€vlsrq Yn65€LEl: Oullegift dvoroL16o).1€Fp1K6 ov rou trpoo€doouus Kor rou aqolpdooDpE rrv i6Lfl noo6rrtro. P "rsmr *r(b) Aroisi€ro. n16yEvLKd,6 fG+ h) flx k) 'f"' G t = t;i .n: i . : n h+k *23. Atro6ciqr0 = 6rr ov rt / sivaLdprro,16rs /,(,r) - /,(-r) (rLo vo ftpropi60r€rn n sivdrrs.)KdvrE oi?ruort,060r€ s(.x)= /(-,r)' pp€ircq!, s(r) ro1/E?t6lFrt6€irc dvaotrtU0l 124.Ano8eiere r6rs f,(x) = /,( x). Kor nd?,I, xdvr€6vqorlpo. 611ovn/ sivolnspLrrri, 25. Ta npoPiriporo23xdr 24livE 6rLrl /''ivordprloav! f €ivolnEprrfii,KornEprrrri ov I / sivordplro.Tr tropoi|rc iorndvva noils yLflrTlv/1'); 26. Bp€iftirv /"(x) av li) /(x) = x' (ii) /(r): x' (iii) /'(x) =.rr (iv) /(r + 3) = rr. 2?. Av s,(n)= x', Kor0 < ft < ,. ono8EiilE 6 s'L\)-
\n
K)l
- r:l;J" *28. (a)
BpsirDrnv /'(.r) av /(-x) = l.r i. Bpeker1v /"(.r). Y:rdpr€rr't/"(.{) lLq rdes
(b) Avq).io1c6[()10rrv / av /(x) = x" lLo i> 0 (or /(x) = - x" ],o x< 0 *29,Odrouf E,f (x) = x' rn.r > 0 xcr,f (x) = 0'7lox < 0. Ano8€iqr€ 61lI /u "'rn'ip1sr (Kor'lpeileEtdvdiro ?r' olri!), olld 6rr r / 'r(0)devundpr6r. olr'0ol,rotoi rouL€ibnizKo06vo €ivoLtrd ovo6Lq30. EpflvEiorero ropoKdro6ci?iLolo [p6Frr]]to. rnr0i6rtl(drorouorolxsioDnoDstaoviorrtK€ oenpollori)!€vo ..
{ll
d^" -=
dz (.... I) -= ol\'f . ,
tr
-
I
dLI(xJ+cl
av z=dlttl
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738 Ilapdyayotrar d,od,\p(ipard
(v)
dz
d"g
*
z=r+ c.
(vi) (vii)
(viii)
df(x+ a) | df(x) | t , = , a r l " =' ' , d, df\cx) ,." ddft.cx) - ' d^
G)
^ df\) a' ,t| (J) ,ly
s+=rfi)""'
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s d-h KF-OAAAIO
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To 6€6r€po o€d'ptpq€ivolxrdur6pla E rKi rspin (,6I ev6quro)'oylotori0'16ro 'rpon-
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,t . t,-r "\
t.^I(a+ h) ;.rij h .. a+h
I(a)
a
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\f + d' la)= f'(a) + s'(a).
140 naptiyuyotKato)'od'4pritpau
{ lnd,l:H
trt I I L gt tat = tin ' ,
",\a t h) (lts)(a) --i_ "tta - h)- shJ hl-Ifta)t
gta)l
h
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ftal
.. atd+ hl- a(d) i.i h
= f,ld)+ s,(d).a O dnoq ]ro qv naprit0rlo]rvodvou 6rv €ivar1600un),696000a€ul6ravmy€iE,eivqr 6toq €Eqrp€rud oujrp6rprx66, Kolil on66€rlrlonarrEi[6vo avo0r),6o),lrpprKdrdXvosps, noupaEilerxarcr ti),),1n€pifiroorlQovEilprioLpo 6vaEapr€p6( 6€vU€l(pdllsrorov il i6ratuo6qrd npool€0eiKqra9oLpoo€i trr'o!r6v. €ropHM^rAv or /l(ol g€ivoLnopoloyiorl0qoro d, rdt8 ! f
gelvol €,riolg,rapo,oTi-
(f .et,@1= f , ( a ) s. ( d + ) J @ ) .s , ( a ) . : 1 7. , y 1 o=1 1 i ^f( e ) @ + t t ) - l f e ) @ ) - i l,,' d f- (s+ h)staI ht ftutqh) h _ 16l I td+ntEto;:dt h
nt- Etdtl+IJtar nt- Jtatlgtall t h
= l i m / t a + n ) l i q ' "ro tht n . t i mp { a )
etat + t;^!i!i_!)iJ9 ,.0
h
= f(4J c'(a)+ f'(d) c@J. (nqpqqp€isr€dlr lpnotporotioopEro OE(iprUr9-l lrd va outrEpdlouts6rr + 1')= .f(a).)I 1,4 .f(a :€ pu sdrxlir0pinrool,ro oedpltq4 qn).oflor€irm dilgovrrl((i: oroplfv^! Avg(r)=c./(x)KoLIf€ivornapo?o1iorFrlorod,r61EIgoitalmpoloti8'@)= c' f'(al = = AflodEEHAv r(x) c, on6rEI r. t, 16r€on6ro O€6plta4, E@)= (h f l(a) = h(a) f'(a)+ h'(a) l(a) = c' f'(a) + 0' f(a)= c' f'(a). 4 noporrlp€ior6, sr8rxdr€pa, 6fl | f)'la)=- f'(at, sroptvoE(/-g)'(a)= ( f + l- 8l)'@) = f'(d) - s'ld). (l)6rdrpa,80u,ro),oiooup€lv fLavo8eiqowe nl XprlolF6fillo rov doovdlouFsKoroqdp8l qx6to el6LK6v napdyoyoJrsprKdrv ol'vopoiosov. ororHMr 6 Av /(x) = x'yLq (d,roLo!q!mx6 apr€F6r, rdrs I f'(a) = na' t:rlKagEd ano'lfEH H on66€rqrl0q Ivsr Ie €nolo-rri rrdvo oro ,. flo rI = I naipvoulreon}.{ircro OE6prpa2. A( uroodooule r6pq 6n ro O€6pnrorox'j€rTrar,6rlq6'1 6r', ov /(x) = x",
10.napqyl)foi 14! ' t'(a) = ra' lto Kd6€a 'Eororg1x)= = = ;"'. 4u 71r) x, rl ro6rtra -{'* x' . r, lnopeiva ]paeei s(x) =/(r) /(x) yLakdo€.Y' dpag =/
L An6ro O€6prluo 4 dn€rar6rr
E@)= (f . r)'la) = f'\a) r(a)+ fla) . r\al =rdrr.a+a!.1
= (r + l)o",
yr0xdo€a.
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I)',o,= ela) eI lE\a)l' AnoatEBIlprv r(ovypd\yoqrE ro
*,r- (j)r"r
c )r'
h ttp6tt€lvq orloupsuro[lo6fl auq n roprioraorl6XEL v6i]pa EivoronapqirTlro vo €];yBo'rtl8 6rr I (l/g) (a+ r) opi(€roLylo apKerdIrKp6 r. flo rov oxo,r6aD16opKeiva rapornpriooull8 6uoldvo npd,Isrq. AqoJI geivaL,ord rlv ur60eot,nopolo.fiorplsro d, olturEpaivoDls on6ro O€6ptga9-l dfl rt gsivorolv€ iq oro a. A{poig(a) * 0, 0116 ro O€drpruq 6-36n€rar&r urdpl€r6> 0 r6rorodrole8(a + h) +0rft l/' |< t. EnoFvoe.ro (l/8) (a + It) il.r dworo]lo oprerdFrxp66, KorF ropoilr va ]pdvoutE
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(noporrtp'ioft orr xprtorlonorioo!€ dih llfl aopdrn ouvdrnornqs oro a.) I MropoiFs16poenKo).o va PporjrLr rov'ycvrK6 rlno ?rofltv mpdlolo nrt).iKou. Av {or 6'v sivo,ld,oiIspoe).KUoflx6q, civaLortpavnKdq, Koreonpdner vo rov arorvrrovciooute. oEopHr',{B Av oL/ {o, gEivo,nator,'1iorroqoro d Kols(d) * 0, r6rEn /ismpololi(€t ! \ ..
s t d \ .f t a t . s t u l
\s/.
AnorEr:H g= f _Aqoi f
tc(a)1,
(l iE) t:lorttE
tt\' / l-l(a)=lf \' \8/
-
l\' i(d) C/
I , d t . t' e- t t o t 7 , " ' . 1 ] - | , " '
_ f'la) cta)
fld)(-s'la)) lE@)l
lta\c\dl _ fia)gtat I k(d)l' oK6foouvapnto€L€fro nopdSsryfo, EitoorD16paos 06ot vo ropsy0)lioouf€ FspLKdg \: c v / r f r. = xl
" 0v /(;)=
|{r' - l)(2r) (,r : o r r J .r r r = (r l) (.\ - l) r(2r,
(.r rJ:
.\: | ll
,rd
/'(rl=
.=(-
ll2r) = 4 "
(r' + l)'
l
(r
t
rl'
l)\'.
n o p q ,q p f r o l o f i r o . L ) . i r o i o n o p d br,/ p a p r o p c i v c / e r . < u . ' t r ic: r
/G)=x
qDo1K6 op10u6 ', " = +,1,o Kdltolov
dporo Osdprtfu6 roxisrrtnKd Korlla osrLKoiq Kor'/leopvrlaKoiqoKipalo!|C. Av 0ptrtvsioourErrrv /G) = xo oov /(-{) = t, Ka, rrlv f ' (x) = 0. x oov f'(') = 0, {6re ro OsdJpnro sivorovo'j/xoio 6 roXisl KaL?lo , = 0. (H tdEnrcptrlv0iouFcD TLoribsvsivqr ooa;qron6C0d'np6n6vooproreiro0'rcr, oereirriavdl.uort.ro0. 0 L6evdr€r;vvo,o.) rrova npotopiooutsoK6panro ndpoorrlvnopaT6yLorl, npanel \rolvopiEoulcrl( nopoy6r,o!g(rirorov sr6rrdrv 6uvqpliosov nou0o lel€rriooup€op76r€pa. Mro or' oDldqEivoL
10.IIapayiryo4 743 rl ouvripqot tUirovo (sine).fro rtv d'pqed anoKoli\youte, xor 0q 1plorgonoulooDpE, ra eliq: r0piq ar66€1!rl, .rlaKde€d, sin'(o)= cosd = coj(d) sina -fu xdo€a. nol,I€q(i),l€gouvdprtio6rc. flo Auri I rlrlpoaopio tdE errrpin€rva napala)liooup€ mpd6rrtlro, ov f(r) = xsin" r6r€ = + sin't' "f'(r) = .rcos'r -'tsin'r* cosr* cos-r "f"(.r) = - x s i n x + 2 c o s x c(x) = sinrx= sinx siDr, s'(x) = sinxcosx+ cosx sinx = 2sinxcosi, g"(.{) = 2[Gin.{)(- sin-r)+ cosr cosr] - sinrx] 2[cos],r '(x) = cosrr= cosr cosr' r1r) = (cosr) ( sin.r)+ (-!in.{)cosx = - 2sinxcosr, '"(t)=-2[cos'x-s !' Iloporrlpeidre 6tl g'lx) + h'(xl = 0' plo KorG+r) (.r)= sinlx{ cosrr= L 'Ono40a Kdfl nou €)dxro'iqfa( exr).r1ooer, mpi|l€vexqv€iq,roj(6er loli g"(.t)+ I"(r) = 0 tranapd68iyporo rou€i6oFrllrondvo,EilallevaKdvoDpE 6iosuvaprrl!6vo |]31lv6lrsvo o€ov.M€ Fdonro o€dplgo 4 Jrropoi$ vo owrp€rrorisouts xqr oDvoptioeqroDr6pldIouv rplnld yrv6!€vo'ofilv rpdln ourdFopsi vo Tiv€rg€6[0 rp6,louq.ouFlo€i€ dfl ro / . g i €iv01Fn oovtltorl rou 'Is'h). ll'8)'h \ rcr f Av brddqoulE,-rumpd8erllo.qv rprirl an6TlE6io popedg, dloutr€
( f c h l ' ( x= ) |f.El(r) h(x)+(f s)(x)h'tx) = lf'(t)s(x) + f(x)E'@lh(x) + f(x).s(x).h'lx) = f'(x)B(x')h(x) + f (x)s'@h(x) + f(x)Elxlh'lxl. Av6roL4o|lsqv t .G. l,) 0axaraLitaf€, eoorK'i,oro i6roonordltsru r€ 6raeop€nK6 ro ev6ip€oo prila. To r6hr(6 outndpsopa€ivqr on6luft sugltrprx6 xql €tKolo vo rd€e H (t . g. ,)'cival10 doporopq iov 1pr6v6povron noipvoqeov ropclrrryioorrpe p€rrEdu,€qEio. fia on6flC /.8 ror1l l(orllv ,rol.).onl.oordooujrr fra ropd8sryto,av 'f(x) = .r'sin'cosx rdro xrcosxcosx+xr(sinxX-sinx). -f'(x) = 3x'?sinxcosx* frv6psvqftprsoor6povoadtper6onvcprrioerE cvrLll€I(j)(ovrol !o rov (610tp6no.fLu napd8erTlro, lsv od'rper€vo ouvo!{ioel€6!oro),beIo vo Esiqetrrcv luno
744 llapdyr'tyot rat dnd.4 priryaru \f
c h.k)'(x)=f'{r) s(x),(x)k(x)+ f(x) s'(x)h(x)k(x) + /(r)s(x) r1x)t{r)+ f(x)s(x)h(x)k'(x).
(!s €no10r7ri) Mnopeir€fl{6!o, vo npoorooiioc.E vo ano8ciEeT8 rov ?'vl{d rinol g
(I.
I t(\t
LI":
I tn IInl. 'xt ... lttt.
rra vo €iraorco' 0i6rt va napo?oyi(ou!€ nr rpoltLonxiiEv8rdq,pouoeq olvoprrioErq, npin€LonooirirorEvo 610'110 6vov ino 1lontv(/ . s)'(r) 6uvopnioclr'D! /' ,.oLs'. r,o vo pElluml]oill8 6n r / " I 'ivo, nopo'/oliolln 610a. adiv'rol oprrKd).orL{6vo L,roea ooril]8rrtv rapo1o1161!6rrtro rrt( I oro a. Aaoi I olln€preopdn]e / o s Kowdoro d sEoprdrolon6rrt oupr€prqopd nte f Kovldoro s(a) (6rL{ovld oro a), qoiv€rsLeniorq loTLK6 votlno0aooute 611it/ nopo?(,Jtfqsrol oro s(d).npo?taflKd,0aoro8triqoufc 611,av oro a l(oLr / nopa.roli(€ro r oro s(a), 16r€n / ' I nopoyoyiEoroL oro I s nopoTroyiq€rar (f as ) ' @ ) = f ' ( s ( . s ) ) ' E l d ) Aur6i o tEdrpErrxd ortrovrix6qdnog. literdL Kavdva;t4EAlroi'ae, nr0a\,6\^trsri l.to oiv0€61oL,loprliocov 0u frToponos vo Ld"/sr0r {or (q},uoi6orsuvoprrioe(ov.oportp€ioio 6n ! (/ o s)' €ivorornv trpder10yLv6pEvo rov /' KoLs', al)d 6rLoKprpdjq our6:Tl / npdndrva uoio.rrorei oro s(() Kq,r s'o1o a. IIprv €rrrerpioor,fcvo anoSEiEoup€ our6ro 0cd'pr1fo. 0o doXolrtooljr€ p€frprxdq€qoppo-r69. Aguno06ooutE 6fl = sl"x /(') As )tptorponoLriooL,pc. rpoooprvd.ro oippolo STrarlv ouvdpqonS(.x)= :rr.T6rs, ,/=sin"S. Etrordvog, 610up€
s'(x) /'(.r) = sin'(s(.{))
= cos/ . 2x. qv To qnord)"€opo Eivarr€1"€ioq Srsqop€rLK6 :
0rri\ q! nrpinrdot
= "f'(-l) = s(sin:{) sin'(r) 2sinxcosr. nupalrtpciors6fl arr6 Bpi6x€1or os ollLQoviq(6noqnpdn€,fs ro onor6),€opa nounoipvo' l-r ov iprivo'r. / - , n . ..n ror Xp1o;-o'o.toor,.rrro\ rirToTorTrvot-iro,. Av '(u1c]I1vorl6ol'0 6vos1l,K6oiFpol,o,ro S, rro vo ovotdooups fltv Ksrpo./ovi(ouoo, oDv{iprton.6evxpsLd(srnr rar pe7di1elrioxlol 7Lov(1prop€iKov€iq vo ),{vErnpop},ripora oav Kr oD16Topiqva ?pdq€L €r61Kd ou[0o].arru olvoprio€Lq,n ar6pa,Xropiq vo oKprpi oivo€or nou div€r rrlv oivrorro oUvloi(el{0veiq qv vo ovolLi€r Tpdqerritv / / nopaTotiosrq Sivovraroav doqorl ?ro Froriroro 6Lo, Bdoooro !!dL6 xou Or €'r6ll€veq vortrrKiEqdaKIort- ovosoprio8r€ aldyKoiova 6ol,ii\ysrE],i?oKoln{ivooro7opri,rpoq OsouKdvr6ro, olld npocno€riore vo ovonr[!.Ere rr']! EDIdp€Lr roLrvr1?pdqsrs fltv f' a!6ooq!6),Lqooq6o0€io oprotdqrI q /. To 'rpopiirora ot]roiroDsi6ouq €i\,flr16(]0or d nou,ov fld riiq0urdorsrov Kov6lurrtqA).roiloq,ijs\'rperd(sroL va oKEarEirr Koe6lou. ' T6rc/'(r) = cosxr 3xr = 3sin';rcosx "f'(r)
10.Ilapqf(i)'t@\ 145
f(t)
_l
/(x) = sinGin,{) , f ( x ) = ! i n ( . r ' +3 x : ) /(r)=(rr+lx:)rr
/ (r)=ms-(-r1 = cos(sin.{) cos,t "F'(-r) = co(xr+ 3x') (3x'+ 6x) "f'(r)= -f'(x) 53(x'+3x)" (lx'+ 6x).
Mro ouvdprlon6noqrt /(r)=sin,r,=[sinxrl: ro! eivqrI 6rjv0sotrpriv olvopnio€ov /=SosinoS, pftop€{sniorlqvo nqpol0)llo0ol ro! Kov6r'qqq A}.uoi8aq. ApK€ivo €lplesi !€ sqqppoyti (ovei(6tlrrsrpuU oiv€eol/ o so, otlloi!€r(/ og)o, i / o (,eo ,). Etor.q!
/r {) - (in:.r: proporll€vs Ypdvoufs / = (so sin)o s. ,f = S" (sin" S). H rqpdlotoE rle xooEtrdqond nq 6io EKqpdoflqppiox8rorf€ 6rn].i €papFolrtrou Kavdvort( Al.uoi6oqro l6vo o!(rtfliorJroolt€io €ivarov or 6io eKqpdoerq 0a o6tyrioouvo€€EioouonronEu'rolo}lopoiq.B6Dqro,6noq sivor.rv{ror6 os xde6slorKErotdvo !€ napololio€rq,€ivsrnpo1rp610po vo ;(plorpnouioot'F tlv deir€pl:
/=s"GiooS) Mnopo6rrr6po vo Fpoul18 rrtv /'(x). fLs vq E€Krvriooul'€, onB€rdjvouFe 6rr rl np6rrl ouvdprlol noDnp6n€L vo ropa,otiooupe€ilar I 5, dpoo fllnoq]rq rlv /'(x) dpli(er !e
/-c)=,( ).l Mdooofitv ropivoeot trpin€rva 66oou!ero sion', ttv fl[i rrlEE€irrptgouvdprloit!, sino S, oro.r. Tror op;ai(oqrtpdqovr(q
trd=,s#:l (n rapdvocot6€vrpsro(6Tov rE),rK{i). npdnerr6puvq nol}.on?,aordoourE a!r6 ro Korldrr r'tEandwtonsre rrv rop{iyoyortSsin. S oro.{ aurdro pilro €ivorajxoto - dzoutern krirr noolmpo $ vq ro ovflFrroxiooute.tav t€hr6 onor;oivoeot 660ouvopnio€ov, ).€oto, i,(ou|re 'f'(x) = 2sinx: cosx: 2j' ho Et6psvorapdE€1114, 6oD),06o!p€ uErov iErc1p6,ro.Yro€&olpt
X0)piq{qv vo tpcrooleivo tptivolF rry / 6qv oivo€ongoI' o t rpl(i)vouvqprisE0)v, 6flI oplmrp6tepn0orivort sin,dpot nopdolootnou0dPpoipe?rq rt! t'(r) 9l.6trorrF opliq !o (.r)= co(
).
746 llapdyayorrard.odtrlpriparu pdloutr€fltv ruri roul, k(x) olni 6rv€ivordll,t on6ro Mioo 6rlvnqp6v0€olnpdnErvo 'Aps sinxr(alJI6nounaipvorJp€ on6rnv sinGinxr) av anol.ri\youu! ro np6rosin). I napdoroor)fitq /'(r).rivEroL
Mnoporirr€ rdpo !o l€XdoouF€ ro rp6ro sin ornv sin(sirl{':|0o npdflsr!d ro).t"qn),qord, oo,/'auro no, clo' F, l]p,| (r: -r;rpo!c r"l\ rop{i}oyornc our{iprt6t( noun Lprt Ttq oro x Eivflrsinxl q),)"dour6Eivorrdh;vo rp6plnfo no! 116lt6poutEva ),rlvourr€: cos-rr' 2-r' f'(x) = cos(sin.{'1) Ti],oq,vo or ropdlolor jrrprrdrvrfl").rov ouvcprrioeov rou €ivaLorvoiost(rov sin ror S, {o0drqxor xdrorovfl,iovrpln}.6v 6uv0a60ov. nreov6vvu(P}.6,rer6D odooE 6fl or onovri o€rq€ivororDord_cov 6Ir, npooro€riorevq Tpdvei€ lv / oovoivosorl: ov /(x) = sin(Gin.{)r) r6Is /'(r) = co((sinr)) . 2sinx. cos.r = .co(sinx) cosr /(x) tsinGinr)l' _f(x)= 2sinGin.{) = sin(sin(sinr) = ,cosr /'(.{) co(sinGinx)).cos(sin.r) "f(r) = . = /(.{) sin'(.rsin.r) .t'(x) 2sin(xsinx)co(x sin.r) [sin.r+ xcosx] /(.r) = sin(sin(xtinr)) ,r'(x)= co(sin(rlsin.{))co(xtinx) [zysinr+ r']cosr]. O xav6vuq1lorrc ol)vo;o€rE looodpov(i l(srrsproo6rspov) ouvopnio€ov €ivors6l(ol.oq - 'rdvro(voEpd) pd).r8nop6v06oElq qpri(owqqon' ro 6sEi, foGo(hok\, Koropliors vo ovd,610rov utro].oyLof6 ornv nspoytir.yrol prcqoir0eorlgprrp6rupou cpr-
f'G(r(t(x))). fro rapd8€LJ,tro.0v
f(.{) = sinrGinr(.r))
Lf
=s"Gin.(s'sin))l
16ro
/'(rr
,i )..nr.cocr. 2 r i n t , i \n) , o . { s . n
/(x) = sin(Ginr:))
"f'{r)
[/=sinososinos = sino (s o Gino s))l
= cos(Ginrr)r)hinx' cosx' 2x.
/(i) = sin'GinGinx))
ro eo€iq,av rp€rd(€ro4 [ourn],rpdrors
i6rs /'(.r) = 2sinGinGin4)co(sinGinxD co(sinx) cos{ M o'rr{iro nopo8€iytqra oqvtln6po€po, rp€ld(€atcdvof6vo np{iylo'/ravoTivsr€dvqq (ropa^/0J?rorrlqD rd).sroq sqrioKron.Mropeireoiyoupova rslsronolt0eirefs rLqa6Ki' oeLgoto rd),oqrou K€qoluiol,,r6pa 6poqTipeen lsydtl onlfri vc cro8eilouletov Ko!6v0rtg All]oi&rq.
10.Ilapaydtyrorl147 Or €tt6lr€voL oulloyLdpoi, uv Kor 6€varotei-oiv ard6€r6n.umd€rrliolv t€prK{ion61a rqavdotqto nou 0o Fopoio. vo j(pnouonoula€t Kqv€iq,xae{irqKorleprrdq un6 rL; 6usxo, ),irq nolj 0' qwrurt6ll(e. Apli(ou|'l€,PdBolo,tr€rov opLof6
= tirU:!!3!-!LJl:l!'4 17 . g' 1o1 I(ck + hr- l\EtaD h Kdro! 66{i0o 06},a$vo eteovror€iI r0prtoToo! ?ro (tv g'(d).Mrqdnovrlcivatvu ro
+ h ) ) - f E l a ) J = t i n f E ( a +h ) ) - f @ ( . a ).)s l a +h ) - s ( a ) tr-r inf@@ h
to
h E(t1+h) s(cr) Aur6ro r.).ruraio6€lolri(erdoj(nfo,ral$oiv€turqK6!sKdi r€poov lpd\toUBE ,._(I o E)la+ h) | | o e)\a)
t1l'-----
r,
f g t u )+ l s ( d +h ) - e l a )-l f E @ ) t , . _ e t uh+) - s ( d ) g l d+ h ) - g ( d ) i'ti,' , To 6€lr€po6prorivol o nqpd?ovTaq g'(a) noL,€dLorus. g(d + t,) g(d) = I Av 06oou|1s (7u vo €iloors o|(plFsiq 0d'np€n€ vo ?p{ivouFk(I))).161€ro np6lo 6pro€ivor ,_[IB(a I kJ\- Iletatl ta
h
To 6proaur60d np€nsioo( vd €ivorI /'G(d)), lLori ar6 q ouvdlercr1q g oro a, ro .t nlloivor oro 0 6ra\rqyoiver 10r. npoyJrdnKd, gJrop€i Kovsiq, Kor0o ro Kdvout€KrElsiq 03 ),i10,vo xdvfl ro oD).},o./ropd au16o{prpi. YrdpX€rri5rl,6!rDq,;vo rpdphts, noD0o dXclel(r €oEiqroportprioelqv €iorero €i5oqiou ov0p6noD nou8Evdlorpeioro rue).d. AK6fo xqr lLs t + 0 0o rnopor,oE vo eivors(o + /,) - s(a) = 0, nFriyjro'rouxdvsr I 6roip€olror rov tro1.).o,r).oorqof6 lllg(d + l) - g(a)vo jrnvter dworo.Eivord,io0rd 61rev6rqq€p6por€ pdvo1lo Fr(p61, ul,).dro gld + t) g(d) 0olnopoio€vq Eivor0 lla oootillot€ pLKpd,r. ((iTldroro,€ivorvo tl(oufErrlvg O EDKo),&epog rp6nogyrcvu orLFp€i oroo€pio!vdpr!o!, g(.{)= c.'16\€E@+ h) - g@) = 0 TLql(do€}. !' owri! rlv n6p! nroorl,l / o s EivaL eniorqoro0€priouvdprlorl,(/ o s)(x) = /(c), dpao Kov{tvoq rits A),uoi8orroi(isldr' olio€ra: t f o e l ( a \ = 0 = f ' t 9 l a ) ) .c , k \ . YJtdp;(ouv 6ll(r(xorIrlo(0epdq ouvoprio€rcg lro lrq onoi€q ro€(d + ,) - g(a) = 0,1lo = ooo6inor0frrpd,E.fld nopriaerTpo, cv a 0, 1 or.,vdprrlort g 0ojrropotoEvo €ivoln
x':sin1'x+o
!' qurrivqv ,r8pintool,g10)= 0 6no( onodriloteoro KeQdLoro 9. Av o Kov6vuqqe A).Irdi6o6 rivol o0)or6c,0o rpdftr vd ;(ou!€(/ o g)'(0)= 0 7roKd0E napaTo/ortrt/' Kar cutd 6eveivcrxar16oonpoQ0vaq. Mrqon65sr6t10! Kov6valrtqA),uoi6o6 lnopeivo liv€L a! po),eriooDFr ourdq1l( rSr6rpo,rEq ouvopflio€Lq topLord,o).).doivor€u(o].6rspovo eyxdro).si\rouF€ on).68orri!' lrlv npoad?rat,Karvq lplolloroltoolts ivo rdlvoolo. 6ropH|l^t(or{^\ori^:rEA\r:D^!} Avrlg0ivar ropoToTiorlno1od, xorI / €ivornopo?oroto o. Kqr tio'ln oro g(d). r6rc rl / o 8 ,rqpqyoTi;rror
148 llapdyayorrar d,o .4priryaw (f . E)'b)= f'G\a)) E@1. AnolEEfl Opi(owe jro o'rv'ip1r)ortd orqe(iq.
d(i)=
t I@ls hl I(g(a\\ . E\a+ h) 8\a)
0v p{d r fir
p{a)7 u
ovs(a+ l?) s(o)= 0
f'Gtd)),
Aloro€rlnrd,npi,relvo €ivorKq0ap6 6Ttit d sivqrouvEl]ieoro 0. Orqvro ,! €i\,arprKp6, prl(p6, Tog(d+,1) - g(a) €iva1 snioitE dpqovrog(a+ r) - g(a)6€v€ivoL II56v,16I€ro ppio(€ror (ovrd (L oro av €ivar!!66v, r6rs ro d(r)rooirorte /'G(a)), d(r)€o /'(s(a)l Kdfl ql(6pql(ol.irspo.Mrq KorI qlvdlsrq filg d sivor! xop6rd61,19qq or66€r(lq,0a 36ooulrs ro1lrpr6pori. [n rpooeKnxrljrerdepo6tolroi ro! 6td160ilttxorj EapoDrs 6rr rt / 6ivo1nopa?oo,rrrtoro g(d).Aur6 rtloiv€L 611 ! L\I ( " 1 ' \ \- i \ c t d t i . LI(gb)
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'E1or, av € > 0 urdptcr Kdroroeoprgp6!J'> 0 raroLoE 6sr€,yro Kde€k, llr
a!0< ltll'l lI"rat'e"''tat. F|o\K/'
.19.Am8eiqre6rrav rl /r'i '(orI g'")(a)uriprouv xcr or360,16r€u'rdpterrl(t o s)r')(a). Av n€rpaforrorrir€Liro 0o n€roI€ir€6rr d€v€ivqrqp6vrp vo qvoqlrioout€ 6vsv oiro.rro rnv (/ os)'"r(a).rro va 6eiEer6 €'rorfv{,q6n r (/ os)i'{o) on'ipI€r,0o npinfl vo oKeer€ir€ xdnorovl'o x6 roxoprop6 ra rnv ( / o s)'{'r(a ) no',vo tnop€iva ono8€rxr€ire €rayoyri.Aonlrior€ Kri oov: (t o s)"(a) u'dptsr xor eivor 6vo doporoto6pov,o xa0dvqg on6rouqonoiouE €ivoryrv6lEvo6povntc popei(...'. | g ralon d,or€ 20. (^) Av f(x)= a;a'+ a,,x" |... * 'o, 0p€irelllo ouvdprtroTr s'= /. Bpsir€n d).),rlpio. (b) Av
rat=\+\+ +4' Pp€irE lra ouvdprlo!s uEs'= /. (c) Yrdpx€r ouvdplqsTl
/ ( r ) = o , x ' + .+. d o+ rdrorodor€ /'(.x) = l/.ri
q+. ++
154 napliyayotKatd,od.np(ipard r?r6rorc6ot€ A€iEr€ 6Il u,(ip1 rol.lowFrxriouv'ipqorl/ 9q0ro6 (a) /'(x) = 0 yLooKprp6( ' I aprepoie x.
(bJ /'(yJ - 0 yLf,Ko\dva r. ov orl Livorreprrr6q. (cJ / (yJ- 0 yLuoKprpdJq ivu ,r.sv o Ir dvoLdplo,. (d) /'(r) = 0 yraqKprpdiq * spr0poie,ov o t - * eivarn€prrr6q. (a) Adu' 6rr o opr0rdEa €ivor6u!li pi(o lrtq nol.uo\.rprriqouvdpltotc f ov 6Tlo a /(r) = (r a):g(r) ylq Kdroronoluov0lrxriolvdpqol g. Ano6€iqr€ €ivar6rnli pi(q rnq / ov {or f6vo av €ivorlolt6lpova piqarylqt karrlq /'. (b) [6r€r f(xr= dtt'1+bx+c(d+ 0)dt€t6rnli pi(o;T1).i€r o!fli rlouveriKTr y€otr€rprKri; Av rt / ropo,o?i(€roL oro a, dorod(x)= JG)- f'(a)(x- a)- /(d). Bpsir€rrlv :€ ouv6uoord d(a). rr ro np6pinto 22,our66iv€Lsra dD.r ).u6t lto 10 np6pLtpa 9-20. .24. Arrr6ro np6plruo 'ricr loqi Ie 10 I]p60l'rl|rq3-6. Tg1o)d,, ... a, (or 6r,..., 4 loo!dvoropdroi. (a) Avor-!,..., x, €ivor8rol(€rpuivoropropoi,orooeiqr€6rrD'Idp]tEr 'IoluowfrKri j+i,rsr oDvdpqol / Pooloi2r l, rdron 60r€ /(x, = f'(x)='lrs = = (x) a, Kst bt. Yn65e(rlr Ouprt€sir€ ro 11p6p).lpo 22. f f'(x) (b) A€i6r€6rr urdpxsrno).uovDprKri ouvriplior] / pq0!o',2, I $ /(4) = a, Kor = b, Kdo€ t. /'(r,) }lo 6rr o Korb Eivar6606ra6ol*q pi(.q frsErol-u{0vlFrxri( ouvdpltnte /, Ynooaroulr€ qiid 6rr o a xar o b d€v eivor 6rnld6 pi(€g, otr61[ frmpofllE vo lpd\youF f(x)= (x al.\x- b)c(r) 6nous(d)* 0 Kors(b) + 0. (a) Ano6siqft6rrros(o) Kolros(b) fuouvroi6ronp6oruo.(Ou!!e€ir€6rr o a xor pi(€q-) o b eivor6rd6o;(rxdq (b) Ano8€iBrs 6rL'rndpxer Kdronsopr0!6qr p€d /(/) Io rdoe/ oro A. 0 apr0p6q /(x)I6?erar I FiTrclr iui rrlq / sro,4 (Idp€qK6ra6fl I / (d10rq p6]roiri lrtElrurl oro A. olo or'uEioxD). nopoqp€i6rs6rr rltdtrorn flIi rnq / oro A 0a[noporio€vo€ivdrto /(r) lLa rolld 6roaop0rlxd -x(ttrito II g' d).LqL6-.vro, uroouvdptrtsrl/ rnopeivo iX€r,ro}.)d|loaoperrli rte €ivarro noti !io. Ev8rue€p6poor€ rLxdotp€iq p€1ioIouolo A. qv xorn jrd-rrorrl olvri0oEyra qv nrpinrool nol,ro A eivqrivo k).€ror66rdoqpo [4. b] av I / €ivor or\ s1iq,r6rs ro O€dlprta7-3€Tndrqr6n I / dj(srrpqTpqTlxd slo pdlrolt flFri oro [d,
bl.
eivoro o oprop6qrou e).o1iorou r!q / oro A oQivelqrosoaq.('Evoqnr0ov6q oproFde ak6).oD0oq: I / dtcl cldtrolo oro A 610x, ov I / d10rN1loro oro ,a oro x.) Eipoor€r(ipa irorporTro dvo0€6pl/o 'roD6evpooilsrolrov orlv irqpEl o),otiorrdv 'Eo1(0 pro oEnpHn^ r o!,vdpqs! opLogdlloro (4 b). Av x trivoroueio peyiorou(i / €laliorou) TLarrlv / o1o(o, 6), ror I / eivaLrapotoyiol[r] oro.x,16r€/'(.r) = 0. (hp€r6voups6n 6evlroIio0rarI rapaloyror[drnra,06r€l(avn ouvdj(ero rr]q f 06d],iq or'lg€i0,) Ano,rrr=HEq€rd(oup€ lilv nepi:rrorolorlv oroio 1 / 6IErUi}loro olo x. (To t;(iltr 2 nou ndpouord(Er rlv orL4 r6donou Kpfo€ro riooi orr' rtv 6].n o,r66s(n- or xop8dE ,r€ploivqn6orlFio opror€pdrou (x, /(x));Iouv (lio€E > 0, {or ortopodqnoumpvoiv our6i 6;(Er 016oIuEiod€irorou (x, /(,r)) 61ouvx).ioer6< 0.) Aloiutlrd. o oul.).otroF6t oq cqriq: qpr€p6g A! , €ivoronoroo6{ro1€ droroq66t€ 10x + , vo ppioxrroloro (a, b). 16rg
triw 2
flx)> f(x + h)
IL H oqpao[ar4Empayrityou157 dqoi I / ;Ztl fdloro oro (a, b) oro x. A']t6 ol]'oiv€l orL
/(x+ r) - /(x)< 0.
'EroL, ov, > 0;t(ouf'
flx+ h)
f(x)
n
h
I ,J - r/,1 .. -,-.!,:::-a l Lr /{.\ ;rt h An6 r+, dI).I nl,supd, av, < 0,6ro!p€
f(x + h) dp,r ''
/(\ ' i)
f t x t> r / \ l' >
[.
A116rrtv 1,n60'6r1, I / eivaLnqporoyiorlrt oro x. dpo oL,rdra ariloopLqnpinrL vo 0iva' i o o K a .o a a u ) . diqo o ! r r q r / t . r t . A L , . u o q p o i v . 6 t .
f'(") < 0 Ko' /'(.{) > 0, , 0n 6noL,6nera, 6rL/'(r) = 0. x oaivsroroc oq (66o1ss,o t!r6ij€'6rl H nepirroorlorr'tvotroioil / d)tcr')dxtoro 611) pro o€ ?poltLri).1 :14ta J
ro (d, b) !s ro [4, b] [opor]p€ior' (:Zripa3) 6fl 5€vlropoiu' vd ovrrKoroofliootisE (ellT6q uv tpoo0dooul86rLquno0dosrq rov 6po61110 .t olrtv crpdvrtol rou OEopiuqio,c (d, b)). Ppio'(€r0loro En0r6in /'(:{) €EoprdraL }6!0 otr6rrqruiq rrtq / Kovrdoro.r, EivsLole56vnpoqovd! pro ro7lp6cprl a{6oortrou Osopriporoq l. Apzi(oul0u ro rTdqfnopoitE vo n{ipoup€ 6vovop10!6noL,€hyEirororo tlrifu 4.
'Eoro lrtq opr0!01v no! nepfl€rqroro ne6iooprotroD / plaouvapqort,{or A ivo ouvo).o 'Evq (.iqrio!ou) nt! oro A or x oro A ldl€ro, drtsio lo'flxoi , uv / lsyiolou 7,o /. u.io rnv urdpXsrKdnoro6> 0 rdroro6orc ro.r vo EivuLolpeio petiorou(3).oXioroL,) 71o / oroAn(d-r,J+6). oEopl| ,{? Av I / o pi(€r(r oro ((l. b ) Kar6I€L10rl{6 [;?1o1o(i E].dILoro)oro .{, xa' I / 6i!ol nopa?0J1i6rtnoro,r,16r€ /'(r) = 0.
trird I
Axo,\Er:H Od'npsnE !o siooorc ot €don ve 6€irs ?tari oulri €ivoL lro ortri 0Aqp!o/ri IoL,oeoprif0roq t. I To olridrpoao ro! OEopii[aros 2 oaa6q devroTusr- fnop8i i] /'(x)vo€ivor 0, aK6!a {r av ro.r 8EVsivflr ol[eio ronrKoi rsTiorot' ri E].oltioro! yro lrt!' /. To anio'lo!€po n o p r i o , r / !d o r \ " r u .o n o - f o ' v o p T t o| . / r . \ r r . l o L J { 0 r- 0 . u } . ) . d 1 / 5 L \ : X f r pf1loro ronrK6 ii eidrroro noL,65vd. To irsproo6rspoiooq 6Lq6€6o!6vfl oqdlporo 6rov An0rpoofl{6 Ao?rof6,;Xotiv 076ot proq olvdprtolq / Kovrd o' ivq x oro oroio ,/'(x) = 0. To ot6rro I' .rt oufrT6p,qopd no! K{ivoI€orrtv npoTlTouIevtnqpdlpqpo E€lvldrolr{too ypriyopoor' uLrroiqroli 06},o!v alriorpoao rou OE{Dpri rof K6o[o unioior€poun' 6,rr EivqL,trou0o ro snovu?dBol]tL€:ro
158 Ilagiyoyor rar dnd"4 priryan rurro(2 Jrr Lortjcr- I ouv0iKr /'(i) = 0 try opKsilLo va civaLro x or]pEioronrrorl p€'/ioro'ri ol,oliorourrtg /. fr' our6voKpLF6q ro ).6yo,uroosrolfsKdnoro€nu),6ov opololio tLovo rcpL?pd\rour' louqopretois.rnourxavonoro[v rr] o'rvoiq /'(r) = 0. OPI!MO!
Xpiorpoorlp€iolroq ouvdpllolq / ftIs ;vav opr0!6 x rilolov 6o1e
f'(x) = 0 O flp10p6r /(x)iil'roL 16rrxpiorprtlprirrq /. oL {piou8q ruis ltq /. Koe(t Kurli1ol oK6paopL0foi,Ei\'oL r€irxd oDroinoDrpirlsL vf1l8icrioougs lLovo ppoipEropaTr6roKaLsidlLoro6oof6vrlgouvlipfltolq/. fLoroLre fltirtrol]q,n €[p€61Ir]ejr07iot1grorTlg e],c1ioqqfltie rLdqouvriprrlqqeivor6voon6 Is nronqpd€svo 0dloroolov Ar6lpoonK6l\olLsp6Kqr6€vundpxa{arrd ovlippitot 6rLro quroi rou €i6ouq npoo)"riporo a7oriv1oloro (Ialpr vd iioer Kovsiqro np6rosrcor6 rou,ri Aq E6sTdoou!0 rp6rq ro np6p?,ItarIq eipsotqro! lsliorouri rouElo1ioroDrrlq f a' d\rdK],€ro16 duiolr]rro[l1,6]. (T6re.av I / €ildr ouvr iq, lnopoirl' rouidlldrov vo rir'o6tEpaporor6rr undprsr!ra tallorrt KaLrld E].dtrorl rrti.) rLo vo 0vrori6orrL€ ro rpio ei6norp€irDv: dyloto {or ro €ldxrororrtq /, rpdnErvu s€0rdoous! (l) To xpiorfq olp€ia r]q / oro [d, b]. (2)T0 {iKp0u ral b. (3)Ta or[sio x oro [d, b], 6rou ! / 6€vnopoT(0yi(Eror. Av.r €ivar6rp6io!61ioro! ri oilp€io€),a,{iorou ntq / oro [a, b], ro x np6nErvovriK€Los pLoon6flq rp6rqKldo€rqnoup6irq svoqfpopE: 7rari,ov ro x 6s\rovrir€ror! 6€i.r€ptri oIlv rpir! o!{i6a,r6rs ro r lpiorercr oro (a,61 Korrt f 0ivo1nopolo.riorrlltoro x enop':vr'rq, /'(r) = 0 o'16ro ordrplpo I, rr our6 orfoivsr 6rr ro .r o\rrix€rorrv np{ifll ofd60. Av urTdpXouv rol.],{iolpeio o aL,rdq rlq rpslc{dT]]lopi6g,ro vq vdE€L Kuv€iq 10ldllolo Kflrro €idj{rorofltq f i6oq EivoLlidlaLogK6noqav 6!oq urdptoljv}.i-roKpiorfoonp€iu, l(dr ).i?osnfsia oro onoiart / 60vEilor rapo?o?iou!, rt 31d61Kooio ?ivsrorrsl"€ioq dreol: ppioxerxsveiqro /(x) lla {dosx trolj,KovotroLEi rrtv /'(i) = 0, xol ro /(x).rrq {d8sn oro onoioI / 6evsivurnapoyo}161rr1, KolidloqIq /(a) KoL/(r). O !€./oiir€poe or' auro[Erolg opr0lo'jgeo eivorr] pdyLornfltri ntq /, KoLo prKp6rEpoq 0o 6ivarrl tdXroflt. Nu d!'don).6nopd6€r.rro: ppoiFsrrtpdTrorrt Aquroo;ool]ps 6t106r,orp€va xurfitv€).dllort rLpi rrlqouv{ipfltanq f (x) = x3 x oro bLoolrFa rrol -- l. 21.| ,o \0 optroo'r' Kdrtoq. f'(x) = 3t' t' dpo/'(r) = 0 610!3x1- I = 0,6rlio8ri 6ruv
, = vrl,,J ,i
.'/1.,3.
oLcpr0poi/1,'3rar r,['3Fpio'tovrsrraror;iooro[ 1.2].dpurl rpi6q oudsorov 1,,oVq,ri(,!/ro r t\ eiot ro, !']ioTo, \ar TouclsXioro'-itor
(l) VfJ,
\-l,a
H SErhspIoId6onspl6lsrro d(p(lrou SLqorrllqroq
(2) - t,2.
ll. H oqpao[at46rapayrityouL59 H rpiryl otdno eivar{cv!i,./Lori ! / nups}ofisror nowoi. To 1s).rr6gtilo eivor!o
,/(\fi.!r) = (r4r3),- \,4/3= + !4i3-'n ts= - +,fr1, /{ \r rr -{ v4]' r rr:r --{vi r : F = 0, "41=+"[]. ,f( l)
=6 fl:2) ' trvoLipovrpoorr rt €/.oxLortrrFrtervqr ; v r/ .1.oro olgcio 14., xqr I riyroTrlnti Auroi rou €i6oll!rl 6Lo6rl(ooio, dmv rivor€errni, ewoni(€rt(ivror€rrl [6110rrlror rylv eid1rort rrpi Fro(duveloigouvdprrlolco' 6vcx).eror66rdoruq. Av 6[oe I ouvripqorl rcu Eterd(oull[6rv eivorouv[l'le, 'i av (nlits ro ldroro ri ro rldt(roroo dvoavorrr6 16r€6rvrifsol€ 9€0oror on6nprv,orir€rq ro ov ur(ipjt€ril irdoqso i o 6).rlqv eu0€io, rou naipvouf€f' oD iv rnv !€yrorl KarI €ld;(rorl rrpi. on6r€6).Eqor n).Ipoqopi€E 6ro6Lr(ooia fnop€ivo pl oil!oi!or,! rinolo.Kor r .r 6po!. pto olettl(]i Euorpoqio Unopsi oulvd v ororo).[vsl rlt qiot rov rprrudrov. :ro Krad].oroT lioap€ oxprF6q6vo ,c(o,onpopLIFn. druvano6siEoFc dr..ov o r .:vor{iorroq. r6TE I o, rdprnorl
'+.. +oo , f ( x )= x "+ d ", x '
:zita 5
axfl sldxront Trrrio o).6K),rlpI rrlv€l]0€is. ADr6orlpoivsL 6rr I tldrrolrl pri noipverqr Trtv ]ld dvavopr0f6.xnour(qvo,rorci 0 = f'(x) = nxt | + (' l)a, ,x'-r + ...+ a, . Av Koroa;pouf€ vo ).rloourro',r'iv rrlv €qio{rorl,Koror,yKpivoD$ rq !6q rou /(.x) yr' poteiser qv trtE ould rq .r, 9pioKoure6v1o( €ldrrsq rrpi Evosx6uoropdt. Ioog pporiF€ Ag 10 6eL7po. Uro0ioor,po 6' od).oup€ vo Fdllo1o{ql ro sldlloro, ov ufliplouv, /,*' -
I
,
", 'Eloup€ ro! sivqroprofivrl 6to qvoL1l66ldorlpo (- 1, l). ". (t
2^ Il'
dpo f'(r) = 0ldvo yrax = 0. Bld:rouleopior4 6 ,Io.x xovldoro l, ri oro - | or rLp6( yl / ooedrq6€vtI€L pilrolo. M outrivrrlv rorrf(.r) tivovror oooorinor€ $y(i).se,011616 noporipnortsivarni'(olo vo 6eiqoup€ 6fl I / ;ler s)dXrorooro 0. noporlpoille trn),6e uproroid xor b, !s Gtriru 5) 6110aundpxouv -l 0lLo rdo€.ro dvoSrdo1rtua, 161r! f Eivora Eoloaoro 6rriolrl, Iro ov /'(x) < 0Io 6Loro x o dvq6rdoq!o. r6reit / €ivorqoivouoooro brdorrlpo. Ano.4rEHOcopo$!€rrlvnepbnoroq /(.r) > 0. t6rodn d Korb€iwr 6rnarlg€iotouduonilratoq pea< b- T6reurdpX€r(dnoro-r oro (o, b) lrE "... - I\bt
I@)
00
Aild /'(x) > 0 rra xdocr oro (a, b), dpq
rJ!:J!\" b-
d
Aao[ b - d > 0, dn€ror 6rt f\b)> f(a). H or66e(t rou nopiogalog,6rav r, aeriveralos oa(. I /'(.{) < 0 y1orcd€e Iloporrlp€ior€ 6rr,ev6to qvriorpoqor{Dvnoprop(irovI ror 2 roXiouv(xotsivqrnpoeavj), ro awiorporporor Ilopioporog3 5.v roj(6€rAv I t €ivaroiEouoo,eivar€iKo).ova
t1. H ortpaoiaxqqnapaytityou163 SoiFe6n/'(x) > 0ll0l(lio€r, ol]d pnop€ivorsrriEr{or ro6rrtro}1oKd,mlax (osopsiors yLo!'a ndpoutspro Ka),i L66olLo rrt rT,Ipopopi€q To l6proFo 3 fdc ,ropal opKer6q va o1s6id6out6 ro).)d ortp€io.Aq ndpoupE, ropdoroon ouvdprrlsrlg rpaarxi lloq lopiq - x. 'Exouls = x' oK6ufl rn ouvdprtot /la lr1o aopd, /(;{) /'(4 = 3*'- I 'Eroup€ - Jf,l, rar tropoiur vu i6I olrL€r66er6n /'(x) = O rra x= r/t,': l(q1x= I> 0 rpoo6ropioouFe ro np6otto ro'r /'(.{) yLo6Ia rs dl).o x. Ilopon]psiot8dr1lx' 0l(prB6q 6r0v lr'> l, ,r]
r> \,4i3,ir< - /tl {ipdSrr- I < 0 a{prprijq 6rqv
'Apo,! elvoln[iouoa x ( ra /
-vlr'3 0. T6rcro f'(o + lr)i lr npan€L ILKp6r. Etro|;voq: f(a + l) npan'rvo rivol6'r1{ri 7raup{0d [r{p6 } > 0. '(o1tt / (a + l) rpincl !'o eivrl qpvlnxi ?roqpx€rdfL{p6 ]lr< 0. Aur6ottaifer(n6pLota3)6rrI/EiroaiqouoflosKdnoLo8Ldornra6'q1droudKor!/ EivuL aeivouooos Kdroro6Ldor!!aaprorspdrolr d. Enopdvoq, n / 6xsrroftK6 tdxroro H 0r6aeri! ]rd rrlv nspirnDot/"(d) < 0 eivqL6!010.f To os(ijpnra5 epap!6EsraL ort ouvdprrt6r/(.{) = r' 'Elot]i1€ '/'(x) = 3r' f" lx) = 6x
.{. no'rdroutsidrt lsisrrloer.
i
:la KpioLpa ortr€io v4, t *o' .,/-tr:,,;r^r,.
/"( \,1/3)= 6Vl''3< o. t"(\-|'9:6Vl-'3>0. Erop6voq,ro - Jl/3 eivoLonr€io romxo[ reliorou xcLro r,{ J eivarolpeio rornxoL] Av KoLro Or6prllo 5 anoasrl(viEror nol,i tplior[o yLdnolL,ovL,lLKEq ouvqpnig8q,r1 r,o 6'irrprt nopdt0r'r,oe ro ,6v olvaprioEov€ivor1600no),,jn)"oKr rou EivorEuko],6r'po rEsrd(oLrlr ro np6onlortq np6rlq nflpoyrijyoD. Ax6!o. ov d sivordva(piouo orllEioritg 5 6Elqq /. ftropsivo ol,rLo'ivo lori,Et J'@) = 0 t ouflivrtv n€pinrool,ro O€drpqpa 6ivsLxotld n),rtpoaopio: Eilor 6DVar6 ro ll vo €ivarollEio lonLKoip€7iorou,oll€io (trripa 2l) 0'16flq ouvapflic€Lq ron,KorsiorioroDli rinoroun' !o 6io, 6noqqoivsroL = =.{', -{', /(.{) /(-{) /(.r) = r'. : 6)..sns rEprtrrdo8Lq, f'(0) : /'(0) : 0, oD.'i ro 0 EivarorpEioronrfoi lsriorou yLo qv np6rl lJIl8ioronrKo[slflIiorou?ra rI6e{rcpt,Korlinorasn rabio]roItvrpirl. Arir6ro ort|13io 0(1EeKaeopiosr opy6repo. oro 40 Mfpoq. oDroFdroE EivorEvSrap6pov va otlt€Lr;jooutE 61rro OE6pIpo5 qrodEuvLler 6vopsp1K6 rlvflorpoaorou EouroD Iou. oEoFHMA 6 Yno06rour!6rLI /'(d) undprer.Av ! f dt€r ronx6 sidltoro 610 d, 1613 > 0' av rt / dx01rolT1K6 litloro oro d, r6re /"(d) < 0. /"(d)
tl. H o4puo[a146mpayityou 769 AfloiErlHYro06loute6rr I / 6ler ronrr6 e).dlrorooro d. Av /"(d) < 0, r6te I / eo eirEKarro'rL{6fd/roro oro d. otr6ro o'6prtuo 5 T6rcrt f 0o rirovoraecprioErdnoro 'Apo 8Ldorwodpo ur6 ro a, dpu0osilu|Lsf "(a) = 0. droro. rpilln vo€iraL/'(a) > 0. fro rrlv nrpitrr0r6rt rorlxo! tL6?iorou eplaE6toor0 fc ro! i61orp6no.I (Au6 ro lLspir6ov.i6rpoaorouoeopriloroq5 'ivoL6.11Ko].'jl8po r,tropo'jFvoci (ou Ir:ronp6otfo > Ko1< 6rvlropofr voovnKoraorooofr, trr > Kol 0, I olldprlol / 0o ci!o, olverriqolo [4, ir + .l] (sr nqpo./oyiourt oro (a, a + n) (o u\,riorortoq 1atup,6!oq ol,rt0en'ryLoop{o.d rrKp6 , < 0). An6 ro oc6plfo MdortqTr!!i; L,ndpx'r6wq aprolo; (Yjoro (ll. d + ,|r)liloroq riiors
f\a+ h) h
f ( d )=
f'(d)'
T6pdro dr lEivE,oro d {o0d).ro,'r rsiyEL o1o0,trori 10o, avri{croro (a, o + 1]} oqo'iro lim /'(x) ',ndpln, dn8ror61r + hJ- Ih\ r,t,t ti^ t(o ,im/(o r Lm/{.rr. ,o n (Eiv(lro{6nl!o l11360€18 rra ouorrtpriE-d-an66nin lL a'r.6ro ro},rrroiooriuo,noDro nopoL'oLdoopE 'rdrrDq drDn11.) | AK6fo Kr ov rt f ivar navroi ropaloliollrt auvdpfltol.cir,or3L,vor6v I /' vo rivqL (IoUvs.lrq. A!16 ou}tpuiv€I. dv 7ranapd6€r./ra,
i G)= {*'+' :rlpqovo6foq s€ ro oE6pllo 7, oq .rpoqLKri rapdoroortrt( /' E6vftropeirord va €tqovrol€iqouverslooovKLo!.ri nougoivErororo rtirs 22.To [lp60i!fo 55onqTpq, ,r],Ipoeopisq acirr't\ron65€rqlev6qd],l.ouKorvo60€opri[aroqnou6iv€rr€pro06repe6 7Lu .rru€vioruortrouOsrDpri rt 6!vdpq or / ' . KqLro I] p6P],rpo 56ro rprtorFotrorEi lioro( 7. ptts, proy€viK€uort To cr6frvo€€dr rolrOropipororM6oIqTrjriq,rapouord(€r 0v6Lqq6povrupiog).67oTov erpopgoltirv rou.
:/ita22
oEcpHr\rA r (Toouapr{r\lA \rE!l{rri\rtr!ro} c^u.Hy) Avorl Katg EivolouvEleiq oro [a, b] Kor napoTo]iorf8q olo (d, b), 1616urdpl€r{vuqopr0!6qx oro (o, b) rrrolog6o1e
tf(b) I@)leG)=ts@) c@)lf'(x)
170 llapdyayorraroior),4p6paru (Avs{b ) + s (d), Kdrg'(x) + 0,ol,fliit ro6rtrq}priecrorKor f(b) J@) f'(x) s(b) e(a) e'6) naporrtp8iors 6rL,av s(x)= r ?roKdee x, r6res'(.r)= l, KaLnaipvoup€ ro OE(iprpa M6orls Trliq. An6rnv dL],rln),€upd, av Eaflpjr6ooupE ro O€rirprpo MdonqTLpriq orLq/ Korsroprord,PpioKoup€ 6rrtrndpr{ouv.x Karr oro(a,6) p€ f(b) f\t) _ f'(x) . 4b) s(t) s@' p our6vrovrp6roeoEivorioo. al)d rinor€6€vroqEfiudror6rrlox {aLynouFpioxot]rr€ Alrtq ornopqrrprloErq TrrriEIou iooqSrprolptorivqvevriroor6rrroOe6prlpa M6orlE ori]vnpolfanx6rlro6fogopx€i6vonoli Cauchy €ivorfdlt"ov6[oxo)"o v oroberi{r€i.
h(t()= f(r)tc@)- s(d)l s(x)U(r) /(o)1. orvrrrico.o[o,b].napoyrD, iorpr't oro(4,b),xoL T61E I ,-€ivaL h(a)= f(a) s(bl s(a)/(r) = l(r). An6 ro Os6prpo iou Roll€;r€ra, 6rr ,,r'(x)= 01l0 Kdnoro,{oro (a, b), ro otroio oIFaiv€r 6rL
0= f'(x)k(b) s(a)l E@tf@)- f(a)l.l yLo rpyol€ionouitp€LoE6pqors To O€6pllo MaortqTqriq rou Cauchyeivorro Poo1x6 popaiq opiov rr]q vd orodsiqouf'dvo066p!!a noD8ruKotriv€r rov l,no).o11a[6 ' /(-t) hm:, , o gijrl
Im "fG) = 0 K0'lrmcG)= 0' I ourrivqv n€pinrooq,ro Oe6plfq 5-3lqq 6ivard1prtcto.KdosropdloyoqEivor€vo 6p,oourrirrnq ropqiq {ar o unolo}Lof6qrov napat6lov oDlvdororreinolli 6ou)'€rd. rolld 6prac'rflis rrtq lopaiq proporiv16povo Av 6ros qdpout€Kdnoreq rupo'flir1ouq. Dnol,o?roroiv siKoIo. 6fl olruHMA e(l({\oN^!Tot LHopnL) Yno0droutt€ lim/(x)= 0Korlims(r)=0' 161€ro lim/G)/s(x) ',n'ip1n,Ko, enior! uroedrolp€61,ro llT/'(r)/c'(.r) ulr(iprer.
(x)
, It4 lrmI ;j; c(x) ;ri; c(.{) (Ilqpol!p€ior€6n ro os{i,prl|10 7 0ivarplo sdLKi nspinrood. Aflo,\rEHH Dn60€ort 6fl ro
ri,rdprer, repLfr€lorotrrlpd6to uroedo€,E: fim/'(x)is'(x) (l) undpn€L 6vqErdorrtfo(d A d + d)r6rororirors01 /'(.r) Karslr) vo D,rdplouv -rroKdesx oro (a - ,, a + 6) sxrdg,iooe,ylo x = d. (2) o qL,16 to Srtortfo s'(x) + 0I€ rlt' moavri€EoipsollKern{i}.r,ror.{= n. oro d. Av opioouts Al!6 1ltvdlh n],€upd, or / (or g 6€v'Jrori00vrolKavoproldv€q
11.H o4paoiaqq rapayrityouI77 r,( rporl?orilEvsq r4ts rov /(a) Kots(d)), f(d) = s(o) = 0 (ond(ovroq,o\r1peraor€i, oto a. Av a < x < a + t, t6rEro O€6plpoMdorlETrprlq r6rs or / norg sivoloDv€xeiq Korro Oe6plfo Mdolq Trpriqrou Cauchy€9opt6qowflr Yloflq / l(ots oro btdoqlo [d, ilp6rq d t< x < d). EQopp6qovroq €ivorooor6q-rra .xl(Kr6vog ovriororloqro1upLop6q 611s(.{)I 0,lLoriov rirovg(r) = 0 eourriple ro OE6prlFa MdorlsTqliq orlvs, Pr6trouls 10 {dno,o xr oro (d, .{) !€ s'(x,) = 0, amladoKovroq !e qv (2). rdJpo,Eaopl6(ovrqq d, oe6pn!'oMfortqTuris rou Cauch)}la rls / {qrs, p}frouFs6ft undpl€r;vaqopreF6e oro (4 x), raroloq6orc 0l/'({!,) 0l sld,) = k(r) i/(ir) r1 f(xJ f'\a,) 3(*J C1"t ro r rsi!€, 610d, ?tari ro a, aviK€roro (4. il oao[ ro Lim Ai)d ro a, rsiveroro d Koedrs /'(r)/s'(/) urdpt€r,fruroL6rt . '/ ( . {I /'(o.r L a. ' m8 ( ' \ J i . 'm qto\)
,'rr) lim'. " " 8\J)
lrq lcntopdpEEq oulourou (fro trloa{6uoqopd,Kot'orlp0 rov o\royvdrorl va oDpnllpdroet KorLfoTto{ rrl( or65E(lq.)I
IIPOBAHMATA rLqouvaprioeLq nouoroloueoi!, Pptir€rttd?rorrtxaL tv€rdxrort l. rra Ka€sIrLdon6 ro oTlp€io rol, SrootriJroroc nfri oro 81oo.irorono! oqqDroi€rwiolFc,JlpioKovraq q r$;q o' aurdro olf€io f3 rLC ora oroio n ropdyoyogEivcL0, ror ouyrpivovrnq .4ra( or0 dKp0.
= (i) "fG) x' r' 8ir+ I (ii) f(r)=i'+i+l (iii) /(x) = 3rr 8xr+ 6fr 1 l i v ) " fG )= . ' ' .
orc [ oro I
2. 2]. l, L].
*"t+,+l
*"[+.,] *"1-r.+]
("i) fG)=-1.
oro [0, 5].
ro! npopfi 2. T6po or€iniore rn ypoqrxri nopdoloort ro0€ltldco116rlq oDvoprTio€tq taros I, Ka! Ppsir€6),oIa olp€iq romroi Fg1iorouKor 6).otiarou. J. LXc5rdoT.fl, /poqrKic ,iopa6tdalrc ldv cqic 6r,vootfoLr0v:
ri)
I
r
/rx)
(ii) f(x)=x+: (iii) /(x)=:, (iv)
/(i{)
= -,
I
1
172 llapdyayorrat dorT4pripau
4.rsi
A v d ,< . . . < d . .p p L i rrcr v c l d x r o r rnL l i r r q 7 ' , r =
1,,
- ar,.
'(b) I pa Ppcircrtv ,ldxrorrl flFi rnc /trr I lr q l. nr,ra €ivqrdvorp6p).!!o oro onoioo AnorpootLrdq nolop6E li=Jec porterio€r kd06).oui oru 610oriuaro ovdp€oqoro d, rt ouv{iprlol / rivar tpo$rKi, dpo ro €}dzroro npoqov6q su!0oivero€Krinoro 0116 10a,,xorsur{irivoLorpLp6g Ia onF€ia6100 'Oporq, €ivareiKo),ovo pp€irErrlv andwlorl ov I / 6€v€ivormpotoliorll. Eq8rdo8r€ ndjql13rapri],1.€ror ro /(x) (a06qlsraFoivoL,lrE ar6 dvordroLo6rdorrtuqo€dLtro. .(c) 'Emo d > 0. Aei(t. 6fl n Udlloritlllri qq
/" ', '_' l-l i
l*l-r,
il-",
eivor(2 + a)i (l + a). (Mtropeir€vd pp€ic rrlv rqpdyoyoo€ ro0dvd016 rq 8looiripora( 6,0), (0, d) Kqr(a, -) roprord.) fio Koo€pldon6 rq ouvapflicsl!,rouono),olooiv,ppsirs6l.a ro oltrsio rorTlrcot !€-rig1oui E).otiorou. x. x+3,5,7,9
(i)
/l{) = {
(iD -/(x)= (iii) t(:r) =
0, rdpptro( I lq, r = p,/q 6^oDp/q avd.tap x, x prlr6q 0. x dpplroq. I, x = li n TLaKdnoror oro N 0, ql.h6s.
(iv) /(x) = 'Eoro 6. (a) orl'€io (.{0, /0) ro ommdou Kor , rl lpae'rJl nopdoroonqs = mx + b. 8p€ir€ro t iroL dJore /(r) I an6mqsrlp€roEi(d,I0) Karfx, /(I) q!fli rlv a,r6oftol0i!ar vo €ivqr€)dloTrl. (tn$rdror€6rr €l,oi{roronordrvro! ro i6roge ro vc c),clmoroLriooup€ ro rgrpd-r0)v6 rtc. Aur6 Kdn{r6a{iotrorei roDqDno),oyrolorlq.) (b) Bprir€Eniorlg ro x-l.qtpdvovloglJn 6vrv 6rr rl sLro€io nouidpl€rsr a,r6ro (.x{, y0)Karro fa /(x)) sivorK{ioern ornv a. (c) Bp€ir€flty an6orqorlrou (x{, yJ s'I{t rrlv I. (Ourno}.o1loloi0o Tivou! qv urooforr€Jrp6rq6Tt6 = 0 lrrd rQaplrdor€ suro).6r€por ro qnor .6oIdouc - b).) on] ypoqLKindpdoroort rlq /(.r) rnr Karro orltsio (.xo, r lu?xpivorsIr ro tlp6plrlp04-22. (d) O€{opcior€ rrv eu€Eiq p€€eiooonAx + 8/ + C= 0 (np60l,rllo4-7).A€i€r€6 sivarA a+ Byo+ Cit/A4 E. t or6orqor rou(xo,/0) 016qv evi6y0)euo€itr 7. To rpoTl?olovorp6F),Ifourorvioo€ror rnv oqrig0p6rnot:riordEivort ojtdqrlperoqi rov rprru(6voru€ic)vqe / rdr rnq fr; E. Odpvoulellq rDosioqnd10ol!.io (0.d) 6rovopr(6v o dEovo,Korl€rd r(so oro (t, ,), 6rrDqoro :Xrito 23.AroSeiErs6fl ro orvol,rK6lTixogliv€roL€),d:(roro 6rov or rpdir&vo e6p€r€prq olvdprrlol oro orrlvLx6r lovieq o rqr B sivorio€6.(OuoLKd €tqpd6r€ro lriKoq duvoptrio€r1or,.r, 6rou (x, 0) EivoLro otl€io orov op(6v1rc deova.H 6ux€Ko}l!6vl ypaHri dro Ij(itq 23 ,rpo€iver Fu rvql").oK.rrrilEofElpr'oi
ll. H oqpaoiat4qmpayrltyou173 on66€lEqKoronq 6io rcprnrd)oeq, ro np6p),iUaLvsrorInpiq va tperoor€ivo pprir€ ro onp€io(x, 0).) pe 6oopdvnrepiusrpo,ro rolpd,olo iIeL ro 9. Arod€iqrs6t1,at! d),c'tc opoo-r(bvrc €s0o56v. !cyol,6r€po 10. M€IoEnrov op06vKul.iv6pov!€ oroeEpd6?Ko% pp€ir€our6v p€ ro pLrp6r€po (oD!,r€p slpdvowaqrqr rq rlpo8d rov 6 o odo€ov,{inoEmo rlpo66v smgoveioc
txiro 2a).
ll, OpeoT6vro rpiTdvoFr !,rorLivouoo lriKo!,qa flEprorpdpcr0r 7ip6 qr6 tiq o,r{iflq {dg61sqrleupiq rou ror ropd-f€ldvovop86KUK),IK6 K[tvo.Ilor6q €ivaro liTrorog 6o\rdr6q 6f(o( ooloriloD K6vou; 12,AJo6 i6polorn)droufo Kor, ovrioror;(ord|volIar Kde€ro(tlifa 25).noL6sivolro ,rou}r,ropei, p€ro9€p6Fvtop(6wu, va orpit€yol{,r€po5uvor6priKoqlloc oKd}.o6
txira 24
I3.'EvdEKtinognp6x6Lrar va oXe6rqo0€i oro oriFo €vrig(]Jxlffon1op6o(tj(d6ro26),R pdorKri(yovioq orcivsgxor O. O Kinoqnpgn€r vo aX[lopro[dvoeppo66vA.fronodq .rNe rov R Kol O (oeoKlivrq)ro priKoqqq r€piqpoqli 0a€ivar,.dXro1o; 14. A€ilr€ 6.r ro d€poropo€v6q o€fl{oJ opropoi rcr rou avrLorp6eouroD sivor ro'r1.'iirorov 2. t5. Bp€ir€to rpo,rd(rotr€ro p€yol"i1Epo €lDa6dvrou fropei vo eylpoosios llllxiK).m or.rivoqa, !r rlv pio friat fliv{o oqv 6rdp0rpo.
tLittd 25
16. Mro opoi lovia xrveirL rmd pixor'!itE 6uldrpou €v6gKUK),ou oKrivoqd,6roq qaiverqroro tTrilo 27. nor6 €ivol ro t€ld.i16po auvor6vFrixoErou A + ,8 nou oltok6,lr€r on6rrlv tovio o K6Kl.oE; 17.O oLKo),6yoq Ed rpdr€rva 8(oolioery1oLipvrtKUK).rKof ol{polo( aKriloe I km. Mnopei vo Korrrll.qoioerg67prrilv andvavrr6ttfu !€ ro1ilrtro 2 km^ i vo n€pnorioEryipn-}ljpo !s 4 km^ i vo (onit),qrioer l(qd ivq ldpoq Kor vo n€pnqrrio€r rnv ur6),orq 6n6pori Pxrlro 28). old 6ro6pofliftpdftor va €lll16q€L ztirtd 26
{i) (ii)
No 6' r oooto 6r'volovrlproo6rrpo016ro rorio: No rcpdocro,rd!ow,ooo ro 6r,vor6v ,flo?priTopo:
yrDvioFra66s).i6oe It. H Kdro6€E1d 61n).{iv€1al dloL60r€ p6}.qaxoulrd rrtv apror€pri ,r},tUpd rrtq oeliSog,6]I(,(Aoivsrororo l1ito 29.Av ro nl.droq1outopflof Eivqrd .rnE) Kor! oE).i6o €ivor]10).6 snqriKrl€,6€i6r06ir ro sldtroro Fri{oE1rt6(reprpdrpori qvo8inl.oorqeivar3V3 di4. 19. To !(iuo l0 dsiTverlrl tpqqrxi nopd61oort lnc,napaf(ittourrtc,/. Bpcir€6Ia ro oill€io rorr{oi U0fiolou l(qr €lottisrou qq /.
tripn 28
txita 29
l'l4
lTctpiyayortcoro)od1pJ4tnru '20. Yno6iroDfr6fl l €ir,fl,l'ru roirrovritLKri ouvdpritol./(r) = r' + d, x' +...+s,). (piorfo KpiorfEq1lAiqrrr 6, l, 2, 4. l. 6rtf0i( ro l. l, 2, 3, 4. Kal avriororneq Fs I { " ' r a o r rcI 7 o u0 r K ni u p d o r o o r lnc / . E f x u p i : o \ r u g r c ' r o r F - u a r q n d o - o q r o r n
*21.{a) Ynooirou[e 61rr] noLuovu[rKri ouvdprr]ot ,t (r ) = .r' + a, ,.x" +...+c4dt8r Kpioua orrpEiora 1. l, 2, 3, xnr /"(- l) = 0, /"(l) > 0, /'(2) < 0, l "(3) = 0. :xe6Ldors, !E dortaKpiFEL{ fnopeirr,r] ?poqLKinspdorootrr'tq/ uL,r6q rLq nhpo,popi€;. !E 0dor (b) Yndp/€r noilovuprKi ouvdpntol p€ rrq idl€qL5r6rtrEq, o! 10 3 5€v sivflr KproLtoortfno; q(or noil t€rrx;; r,po!!6q, 22, II r pLtp{tll,€TlttpdQr{rtmpdoroortllac prtrrtqotjvdplrtort ro,.,voi.r/iovapoqrrt\r-prlpnQnnJ.7rv-oroK.i.-\o7rar'typorp.{rlnup no).rrDf uf L{6! 6u!oprios(dv). 23. (a)
Ano6€iqr0 orL3!o nohovu[,{6q ouvaprriosqpe Poep6m Kurr, avliororxo, litvovrol ro noir,nrar(,J-rlaopdq. (b) fro Kde€r, Kor, 660r€8L,onoitro\urrKdq olvoprrioEL,c lioeud,!rr IroL, no,J r6!vowa'mar(t,,,)eopdq. *24. (r) Ynoedro!|o6ft I no).ur0vr41rKi oL,vdprnot/(x) = r'+ d, .{" + ... + d0 dxsrt1rpr06q t KpioLp( orp€iuKaL/'(.r) l0 ]ro 6iq Iq Kpiouqot!€io x. A€iErE 6rr o, t sivoLnEpLrr6q. (b) fLq Kdos,, 6€iqre6rr udpr€L uLono)"u,rurFrxri ol]vdprrtolf pooto'jr1!r r rp ' r,rJofl' io.o\ . n I .:vo r.p rrd:. (c) Ynoot.o']rE61rrl noiu(DvrirrKri oL'vdprrort/(.r) = .r" + a, ,.r" + ... + di 6xEL k, ort[cill ronrKoulE?iorou{or k olpsiu rorlKoi eluliorou. ADiile orL tr = tj + I flv o11€ivaLdprroq,xar & = ftLav o, €ilaLn€pr1r6q. qx6poLoL p€ft: = k, + I avo, sivoLdpfloq.xqr (d) Eolo r, * , k, lp€Lq & = t,avo no)"ur0vtuL{r1 ourdp , €ivorn€prrr6q. xar t, + & < ,. Aeilre 6n undpr€L lrs ntot / 0oefoL],,fe kr ollreio ronKorlfsrioro! Kd r. olleio rcntorj erqXi oro'. \ro5..-t: nuor- d o (.{ d,) (l + :{r)'7raKflrdil,r).o apr0!61.
;orrur;or /r^r-
ll
2s.(a) Anodei€r'6rL. ov /'(;r)>M r,o Kdo' r oro [o. b], r6rs /(6)> f(a) + Mlb- a). (b) Anobeii:e6rL.or, /'G)<m y1or.d€'x oro ta, bl, r6rs /(b) M> 0, ?Lo610ra x oro [0, 1].AEilft 6rLL,ndprEL 6voaL{iorruo prixouq li4. oro oxoioLorf(rF\ f >Mia. 27. (a)
Ynoo;rolr's6r' /'(.{) > s'(.r) 7ra Kdosx, KaL6r' /(dl=s(d). dsiirE 6rL I.o,6n /(a) = s(o).^€iqrE 6n /(r) > g(jr)rLn.r>a /(x) > s(r)7,(1Kd0e.r, x o L/ ( r ) < s ( I ) r L o x < i r . (b) A0i(rt ft ,r,u nopdSsrtuo 6n {r!rd ro oriFntpdoporo SE!rqtLlow X0rpigrlv un60rol /(d) = s(0). 2 8 .B p c, ' o ]' , , , - . ' L o t . r t o,c/ j , u , , , o n . L . , (a) /'(;r) = sinr (b) /"(r) = x' (c) "f-G)=x+r'. 29. EydJLorier6rL,uv daiooup€6vo0dpocvonio€r€v6rirqvaKivtro,r6rEeo6tsrndosr
'. H o4paotarrlgmpa'VitTou775
:/ittd 3t
or,rdro nsLpofanx6 &6oudvod€vnsprTpds(l ) = l6 rj n6drall€rdar6 r 6€ur€p6irntd. 9€rrq oufftprqopdpap6vnoupi;(vowdrnpofta rdvoi npoqrd Kd.o.A'16rrtvdr,n iielpd, o v6Fofs"(r) = 32 roIU0,nd!,rorsKorsivornoli aaoaiqyLovfl €qtyio€rrrl o!snrpLpopd rv6q|]{ipouq ro', aaiv€raLan6oro'oSrinor€ ri,\yoq, aprF !e onoro6rinore l(i ruXtirnro.flq vo on7,ouor€iooDle ro npd-/pqTo, agoripq(0vriooupc vq !€rp{ip€rq ri'yrlrpo_c ro n{iv0rqn6riv cnredv€Ls rou E6d$ouq o' suliv lv rEpi,ltorort or laj(irrlrlq !ivqr ocflKii tro stifotq roL'qw0sivo,v riorqpvt!r\is 7rood'uotq,iourderou!. {oL6tq ro o6tmo 'r6etouvoripqovoFcto v6lo s"(, = - 12. (a) AciErE 611! .' sivoLr!( Fopqriq r(r) = - 16rr+ dr + B. (b) odrowoqr=0 orov rino rqq s, xor perd orov rino rrq s',8eilre 6rr r(r) lbr -r.r ! \. dro, ,0 LivorToivoq otr6ro onoio09riv€r0r to o;Fo rrt /povrrrto-r7t-i0 |\orq civo"t oXilIr0!ettvonoioo,0ivetor. (c) EvapdpoiF{i}.lrro(rpoqro ndvo)[€ roXl)lnro., ,rd6roro 6arr€pd],e,rro, 016rtv etrradvsro rot €6dao!8.Il{too \rt)d 0 ovd0eq {To (11600 vn}d' olloivcr lino6 eivor.o rd oro i'yoqD).nord Eivorrl lqrrhrlrd rou In rpovrxrioflllri norl p(iv€r oro pdlloroivog: noL{i€ivqrI ctlrtdluvorirou €reivrlrrl olrtui; n6rE 1160110a sivorn lozirilrd{ou rirov0olovor6oerom 0q#osr lovd oro d8qqoEi d6oqog; l) Koru'r6yoviod (tlirLo 3l), o'r6re 30.'E!oBl,'i$apdi).sra'onoro 63oaoq rs raTri)rnra dTeLprc xomr6prql ouvLorioaraxirlloq t) sina Korpla op(6v1q ouvrordjoou = - I6t: + (usind)1, cosd.Hon6oroorlrour(1)an6roa5oeoqum(oi.ro1ov6!os(t) c!d,rl opr[6vflorol'itIr0 'iqpqtdla orogtpdiot !3 r,co\o. (a) Anod€iirc'tn n rpo'[dlou F),iforoerivoLrro,Topo0o],ri (Bprir€ryloiorlro'ros ppioKowor K'itutpovr'fll orrlFri r. llar E€iEre 6rr o'r(i ro oru€io ndvoo€ lro mpaFoiri). (b) Bp€irs1rl ?ovio d nou p€?roronorsi rrlv opL(6vo on6sraol nou6rav[eLro olrifo laxpr v(l r6sfl oro 630$oq.
*31,(a) A6ot3 mpd88r?taouvdprrtonq/ lLo rrlv oaoiavo urTdpyer ro lim /(.r),6jrL lb)
6!oCm lin /'(x). Ano6€ii1€6rr, ov ro li! f(t
Korro lim /'(x) ',,(ipjlouvKo' ra 660.rdrr lin
f'\x) = o (c)
Ano8siqre6fl, o\,ro lim /(x) Dn{ipxE'xo' ro liTt"(x) llndpr.r, r6re lim ,f"(.{) = 0. (t.r.E: sniorqLiT ,t'(r) = 0.)
(AsiIerrioit( ro np6plrlFo I9-15.) rrlv J2. Yro0droutc61r / (oLI ei\'qrdio nopoTotioUs(ouvsprrlocr,rou r(cvoirororiv (6qpiq€gntg 0. Aro6ei(critr,cvarcr bcivcrdLc6o /g'-,f?= /,'(0Ltog(a)Kor g(b)d€veivorKorro 6rio0. 16r€g(.x)= 0 7rqrdnoror qvdp€oo ortr d ror b. (OuoLrd qv €voi),{i(oupt 10 i6rooupn;poopo 1016€r IrE / roLg 6rlla5i, or piq€!rorv/ KoLg oL urv Irq 6€.)Ytr66€(I: o6ityrt00ir€os avrirpoorlL,no0dlovrq! 6n 6rqX0rpi(ouv g(x) + 0 yLo6l,0ra x qvdpeoaotc a xcLoro D:ov dvoeoproldq6€v€ivoL0, lrdpx€r Kdr ro).i oL'o.o).olrKd rou unopoi,r'vu rdvo't-sF a' rov. 61r lllx)- fly)ll.AnoiriEr'6flr / €ivororo0€pi eisfliqovrqrrv /'. Iu$piv€re r€ ro np60),rrol-20. rd6€oqo dIo x qv utr(ip7er orooepdC dror6orE 34. M(q ol)vdprrlon/ sivorLipschitz
( * )l / ( . r ) / ( _ y ) l < cr - I
"
e ?ro610ro,r 06dvo6rdortFa po an6ro x. H olvdprnortf EivarLipschilzr{iEne ss 6Ldorntaov I (r) roru$ yLo610ra x Kory oro 6Ldornpa.
176 llapdyayorraroloKlnp(in$ (a) (b) (c) (d) G)
rdEEoq a > 0.orox,6eiEre61rdvor otv€zriqoro.x Av rl / civqrLipschitz a > 0 o€ 6rdoqFa,r6rs n f €ivorolor6lopqo Av rl / 6ivorLipschitzrdqsorq ouv€triqolo atdorlpoou16(p).dreKsqdloro8, [opdpqpo). ldEoogI olo -x lo:(ri8L olo x, 16r€! / sivorLipschirz Av rt / 0tvor 6ro9opiorprl ro qvliorpoeo; rd6€o(| oto [4, 6]; Avt / s(vor6roqopiourloro[4, b]. €ivdtLipschitz oro Av I eivorLipschirzrri(rq a > I oto [d, b] 6€iqre61reivaro1oe€pd "/
lo,bl.
6tr,0v 35. Aro8€iEr€ 12
...++.=0,
oa+ atx+ ...+ o,x"= 0 7ra{dro,o.r oro [0, l]. rtrr r notuovlFroi ouvdprlo! f.lx) = x' lx + n 6€vil€r nor€6io 16.Anoa€ierE pi(Eqoro [0. l], oveidprtro 016 rnv Trti Iou |n. (Aur6 npoKiflt€rajKo)'oon610 proovdlrrKi ard6t(n, vtrolE6Ld' pc Tor.rRolle.Eivurlprlorpo,o9or,6rioere Oedrpl rl ?poQLXri ,rapdoto6!rnE f", oe crs rnv /0 Kat1!v /r, Kdrvq 6€i1€noi 0pioKoror ot6q !' q!r!g.) Yroftroul€ &r I / €ivorouvElrieror rspq,oTioun oro [0, l], 61rro f(x) o\'liK€r oro t0, rl ?ro l('io€.x,xdt irl f'\x)+ | 7u xatu x olo [0, l] A€i€r€61rDnriplEt €l(€tii6l x olo [0, I] riroro66ore/(x) = .x.(Toprodnp6p)'npq oKprF6e ivoqoprelr66 yivsr,oro np6pLtpo7-l l.) = 0 llo Eio aKplF6q rrtv 38, (a) Anoorilrr 6r rt -f(r) = .r' cosxLxovonol0i "f(x) rrjraqrou x. rpoxordpr(nKoi (b) Aro60iETs ro i6rotto rtv /(x) = 2r: - xsinx cosrx(MeprKoi unoloTrofoi0o eilar lpliorlor droT.vo ftpropioouvrrq ftoavignrproldqrov p(d,v rnq /.) *19. Atud€i(r€ 6fl. qv n f €ivor 6io eop6qnapqyoyiourlouv'iprnorlpc /(0) = 0' r oto [0, l] rro vo /(l)= I KoL/10)=.f'(l)=0, rrtte | /"(x) | > a 7torcd'toro yivoupenro mpooloru(oiiivaxrvtlldnoorqEdtiu lovo6loiqan6otoorlo€lovo6Lsio -co,roti(rr rolitllo 0 ar {dnotdXoovLKri orr?!i, atrr /c xporo.Ksr0pti(Erxar >a yLu oto Kdro,ox Yro8nEr Atro&:qrr.6fl1 / r.{)\a Lnrrdx,vol [0. +1. rl x olo [i, U al),r6q /"(:t) < 4 yro,.ri,Tolo '(x) = li x1loxdoer>0'(or drL/ €ivorf l( ouv,ipqorl|l€ .f /(l) = 0. 40.Yno06roup€ = 1]lv y> 0. Ynd6crtrl:Bpoi1t s'(r), Anod€iir€61r flx!) flxr+ /0)7Lq Kdo€x, 6no',s(x) = /(x/). '41. Yro0erouf€61ril , (Kovoror€i rnv
f'lx)+ f'lt) s(x)- fG)=0
6n, ov I / eitdt 0 o36io olf€ia, 1613il / €ivol0 7roxdtroloolvdpTrlsIg. Ano8€iErc touE.Yn66erEn: XplouororEiorEro O66pllro6 oro 6uioqpc ovtipeori rl / civarr-aopdenoputo.riowit{416n /(x) = 07u n + l6naopr{2. Yro0drou}'c61r nxd r. Anoirii$ 6tl /'"'(r) = 0ltc xdnotor. {J. toTu o,. ....d"- tlloio orlNiooro ld. bl Kardoro
err)=II(x.r). Yno€droulr6rLI f civcra + t gop( naic'/olioriritxor 61rro P eivarnoluovurLrri
q; raPaYriYoa177 1t H o4Paoia < n, droLdjoreP(x,)= /(r,)Io i= l....,l] + I (B16r' p60)rouvdprrorlFa€poi rn 3-6).Aciqrs6rLI,o Kdo€x oro [4, b] urdpia0lc oro (o, b) 610rd,ore '.-' (r + l)l /t,rt- rf,rr q,rrr'1-+ rnv ouvdprlo't YI65EIEI:O€op€ior8 P(x)]. Q(x)t"r(r) P(t)l Q(r)[/(r) o0 r?+ 2 SlaqopcrrKd otfsio rol] [4 6] KarlpnorlonorEi Asiq.s6fl q F FrSevi(€ra, ole 10np6plrud 42. F0):
44. Ariodeiqre 61, l.',t'n I
l" < i
'"'
8
(rt:piqvc lroloyicere rn \,4; !;xpL 2 6E{o61Kd vnaial). ysviKeuol oslrjprllo.oq Maolq T'prl;: Av I / €ivo1 rlv sqiq rou 4s. Anob€ilr€ lt{pri IoIr I ndpXLL oL,vt;(tt, {o. rooa?dYioruq oTors.br {o' ta I m/t, r..imfr , rr ado/ol'v. r.drorox oro (a, b) 161010 60r! - liq,fLy) lim f(v) ' '
f ' t' t t '
"'
b'0
H on65sr6rl oaqe{i npensv qpri(sL{dnoqdror:dL,ri eivarpLdrsrp,![dq ouvEnero .oDO€opriroroq M;orq T,Fiq lra.i...D. 46. Ano6eilr€6rL'rootFnapaoloroL,osopri[aroqMaoIq Tuiq rou Cauchyprop€ivq -rpa9er orl popert f(.b) c(b)
f(a) = f'G), c(0) cG)
rrq np6o0er€q Dnoedo€Lq 6r s(b) + s(d) Kor6rLrt /'(x) KorI s'(x) 6cv ov xdvorJy€ €iv0rnotdrrur6rpovo0 oro (a, 6). *47. AroSeiqre oro (d, b)' oro [4, b] {qLnapolo?ioLpeq 6rr, ov n / KoLI s 'ivar oL,v€,(Eiq (d, x oro b) (4, Kdnoio oro b),16rs !,rdp1El Kors'(x)+ 0'/to Kdeex IE
f'(x) c'6)
f(n f@). 8(b) c(x)
litt a!r6 Ynd8srqt:noltor).oordorenpd,rd,l1ova 6sirEn nf.oluflrLKd 48. TL6€vrdfl Ka)d}1€qv ropdKdrolpriorl lou Kov6vft1o!,I'H6piial;
. x- r + x - z = . . 3 n ' + l . . 6 . r ^ un-2 -J=l'llt=, rrm '') x' 3x+2 ,' (To 6proorrtvripdygonK6rrlra eivqr- 4 ) 49. Bp€ir€r0 6plfl: , _ ot a n {
(ll)
Lm-
s0. Bpeift rrlv /'(0), ov
= s(0)= 0,s"(0) = l7 KoLs(0)
!7 8
llapdyayot rot d,otc),qpritpata 51. Ano8€iqrrr( o{6lou0sepopq6g rou Kov6\,orou L'H6lital (Kourd60vororrri {droLo oL'orfl ofl KdKolvonp?Lo €trrXEipltq) = I, 1611lim/G)i s(x) = I G) Av liq "f G) = lin cG) = 0, xoLLirq/'G)/sl, (ofoioq.a0oprqdn rs qprorspd). (b) Av lim /(x) = lln sG) = 0, Ko,lim /'G)i slt = -, r61slll .fG)icG)=-. -&,iovro.{ - oovrlKoroalooDitlEx - a'ix - n); (ru, orLoioq ]ro Av lim /(.r) = lim s(x) = 0 xo, lim /'G)i €1, = I 16rslim /(x)ls(r) = I (oFoior;lLq - -). Yn6SsrEr: o€op€i6rrrolinl /(li;r)id(l/.y) = = (d) Av 1im/(-{) llTcG) 0 {o, lim f'G)i c(x) = e, r6i€ lim /(.y)i sG) = -. 52. YndpIEr!ra dliit !op9ri rou Ko\r6voro! I'H6!italrou Zp€rdEsraL {d rnpondvoqr6 = I, oirepprKou; lEraotrlnrroloir: A! linr/(x) = lims(x) : -, Korlin /'(r)is(t r6a lim /(:r)rs(i) = L Ano8Eihsro oq eEiql (c)
la)
| o Koo"L
0 L^op/, ilo, op o.rdq o riroLoq u6r-
[ o. !9 c'(x) :Erpcpp6orc ro oerlplpa MdoitqTr[iq rou Cauchy iio fi]v / roLntvs oro t.1,.r1 11.,vu 6ciE6rc 6rL |.'..'.t1..':.' ' 8\^) 8\o) ' (b)
fYnr'd
(rlori utropoDts va Kdvoupe r1v un68eo1g(x) g(a) l0J T6p07p{i,yx€
i6t
rlt
c(.{)
c(x) c(d) f(r)
ftat.
ft*t
s(i
fla)
(?rari lnopo[rro !o rtrooioolr€ 6 tro pEldlfl x, /(x) pov€18o I rrrl l'
c\rl
ll.
c@) c(r) /(d) + 0;)KqLoL,unE-
) t , o 0 n K L , !0, y d l u . \ .
53. fra vo aLrfrLnpoeei rt r)"it06parov nopo],ia76vrou Kor,6\ro ro! I'H6pilai,XpTtorponolEior€ro np60;.!r'o52yLuvu aroSeilere peprrdq cK6rorcprrr6o€1q rtq ox6),ou0rtq yo ErL),6EsrE yE''rK'tq nporoorti(un'ipXolvr6o€snoi),6q5l,var6fltT8e ro! 0o np6nEr p€prxEq p6vo,ov undpXouv, ror oaqn6LaQ6pouv: Av limr/G) = liTrc{.{)= { lKo,linr/'(.r)is'G)= ( ),r6rEli1n /(x)isG)=( I E i j 6 r o [ ] l n o p s i v u s i v o , a r i a ' ' i d - r i € r i - ! , K o l r o { I f n o p €vi os i v a0Li - i -,r(oLro( ) lnop€iva sivorI i - i -€ *s,1.(a) Ynoo;ro'rrs6r, oro [o, b]. Ano6€ilrc6rr, ov ro el,drrI / 'ivq, ropq7oTioL[r] oio ritq / 610lo, bl si\roloro d, 16r€ /'(d) > 0. roL av €ivororo b, r6rE rrl rLroian6 irv on66€Lqrt roL,osriptiforoq1.) /'(b) < 0. (ou Enova).'iDsrs (b) Yroe6roL'ls 6n /'(d) < 0 Ko1/'(b) > 0. d€iErs6rr f'(r) = 0 r1o,.drorox oro (4. b). Yn6ssrqn:osopEior€ro s).drroroqq I arc td. bl.rrori npinErvq llpioxtlsr Kdnouoro (a, b)i AnossiirE o ov f'(x) < c< f'(b),16.6 /'(x) = . rro rdioro .r oro (o, b). k) (A!16 ro anordi€ofu€ivol113)016 oav ro Oe6prt!o.o! Darboux.)Yn66€LEr: Mols,pavr€proouvriprrtorortv onoiuvo Epopp6Esrol ro fdpoq(b).
IL H otlpooiarrlqnopaytityou179 55. Ynoedor€ 6fl n / eivcL8rcaopiorploe xd,roroI rdorruorou ,Tsp,dl*a , d\d. 1 f' srvqLoouv€jarlq oro 4. (a) To rl,EDprKd 6prdlirrt'(.x) Korlim t'G) &v Frop€ivo nflip1oDv{or Is 660. (Aur6€ivoLod,6qrlo d.oqpdnapoD,ollioro O€d'pitFo 7.) (b) Aud ru llov6nl,€L'po 6pro,6€vInopo6vvo !,r(ipXo!vxor ro 6rjo,dx6u0(qr !€ rlv sworo1ou1- , Yn66srlrl:Xprtorpo,lor€iorE ro Oe6pnBo rou Darboux(np6Binta 5a). .55, Eivors[Kolovo Bpo6]'E lLs ou\rdp.rtort / rdrorc6oreq l/ lvo eivcrncpalo:yiors1, cv6 I / 6jtr. fra ropd6er,fa,ptropoiFsvo ,rdpolp€/(-r) = I yrd x prlr6, KeL I ?la r dppllo. X' curriro rcpd6nlgo1 / 6ev€ivoLxovouv€trif,xr our6 /(x) = 6€v€ivolarr).io l,lroorl: Ano6gihs611,ov I l/ eivarnapqlotiorln oro a. Korq / Eivaroirvexiqoro d, r6re rar I f eivo rapo?101iowrt oro s. Yn66E(n:Apx€i va Ei€rdoou!€ f6vo ra d lls rq onoiof(ai = 0. rrori l' ol)I'ivr.lv trEpinoot,Il npinsr vo €ivar10 / l'(d); 'Eoro y+ 57. (a) 0 xor, dpfloe.Ano6ei6rs 6rr x'+ J."= (x + y)" p6vo6rcv x = 0. Yn6Serqt:Av r0'+ y'= (x! + y)', eaopu6orcro oe6prro rov Rolleontv f()tJ= x" + v'- (x+ v)'rdvo oro [0,&] (b) d€i6Te6rr, ov y+ 0 Korr nrp1rr6q, rdr€x'+ /'= (r + y)'p6!ro dro! x = 0 i y. x= +5t. Xplorsororeiorerq u6€o6o rou npopl.iroroE57tLav' anodeiEer€ 6rr, ov o r €ivqr dpnoqKfl f(r) = x',16re |(d0€€qqrroufvl rtq / r6pv€rrrv / p6volro sopd. *'59. Am6tiereox6!o ,I|oysvLrd61l,qv rl f' civorq06ouoa, 16r€Kde€Eqanropaqrdfvfl qv / [6vo }l'o aopd. *60, Yflooarouir0 6ft /(0)=0 xqr I f' €ivar oilouoc. Ano6ei(Te61l rt ouvdprrorl g(x) = /(x)i x sivatoigoDoaoro (0,6 ). Yr6Serh:n poqqv6qeonp6rEL va KorrdEcrr qvg(r). AnoDsiliedrLsivor0€rrri €Qopp6(ovftq ro OE6pttoM6onqTuiq tro rtv / oro Kordl,l.nlo8rdollla(0a 0otoiffL vo 0utooaors6rLI un6€eon/(0) = 0 eir'or = I + x'). ouoroorrKri, 6r{rqeoiv€ror0,16rn orvdprnot "f(x) '51. Xprlouonoreiore nopoy6rouq 6t|, ov , > I, r6rc 1la v' a,!o6€iEr€
( l + . { ) ' >I + r x r o
ttitld 32
I <x I Eivororapoirrtrova 6{erd6ouj10 o! ro 2x sin 1/x cosl/io tiv€rqr< - 1.rrqopretoiqx Kovrdoro 0. Eival Kdrcrqpo],r{6repovo rr€krrioou!€ritduydprror sCy)= 2Giny)iy cos/yloy+0 0o€dloI /ro fEyd)"o rs vo q;poorrrovsLy)< J' Aur6ro rp69hro €ivaLro],[ l,6nr6. ro nrd onlovrrKdKojrpdrtroDg(r) Eilolro cos/, nouprdv€rrqv rrtri i, o).Idorir6outFoiv€rp6vo 6ravsin/= 0, Kdr8ev€ivar{o€6}.ouoopdq10ovrtg rt i61olrop€i vo dI€,r,r,iE< - L H npoqoviqnpoodTrroro' oDr6ro rp6p).rtpa sivolva Fpoil0 r( romKds].dXLsr€r rLp6glr'tgg. AuoruX0lq, eivdra6ivorovo lLioouF.rtv oEiooons'Ly)= 0, oJrorftiroL sroffvoq mproo6reprl erorpogic. (a) d€ih€ 6rr, nv s'Cv)= 0, 16r€
cor - hiny,{- l 2r' Koroupr€p0vor6olr
, u , - t " n, ,\ 2, 1) 2 . '|I (b)
T(ipoSEiqrE 6rL,av s'Ly)= 0, rOrs 'r:' sinY=
41v'
K0ro!tn€p0!dr3 orr
= 2r v' lcLv)l r/4+t (c)
XprorporoL{hvroq ro leyov6q6111z+ y'12la t' > l, 6siEr'6rL,a! a = I, 16rEr1/ 6EveivqraiEoDon oe xovdvo8ldorrtrudp0r un6lo 0.
ll. H oqlooia qi Eapafd)lot' l8l (d) xpnoruoror6vrqq ro rsyov6q 6rLtirn(2+l)/.J4+l= I. deilrr:dt. nv o" *inoro6ui61rtfq o > 1,161e r1/ eivcro,:iouoo 1rlpoqrd 100. **65.MrooDvdprrlol a cvudpl€rivsqopLep6( d > 0,rdroroq6ort o0louoaoto / ]d?eror flx)> f\d) tL\ I 5< x< a. f(x)< flc) qut6 nopartp€ior€noq 6€vonFsiv€r6n rl / sivt( oiqouoooro Srdorruo(d a-. a + tf tra nap'i6nypq,rt olvdprtol nouP),amlsoroIzilo 33€ilor oliouoooro 0. ai.;.dEeveivoroiqouooo€ l(qv;vqavorzr66rdorrtF ro! vo ffpri7sr ro 0. (a) Yno0iroutls 6rr n / €ivorouverisoro[0,l] xar6n I / sivqroiio!6ooro a 71u xdo€d oro [0. l]. Anob€i€r€ 6n r / EivoranEoudo oto[0. l]. (n€i6r€rp{i}mrov va eour6 osq 61r undp't€rKdrr rou lp€rdqsrola,Id60'h.) Yn6lsrirl: r1o 0< 6< I,ano60itls6lrlosLdrrororlq f orolb. llflpdrervoel'eoviqeraroro (b)
6rr I / eilar suvozriq, €Eerd;ovrqg Aro6siqrEro uapoe(a)xopiqva urooaoErs ./rq = rd0r l] ro o'ivo),o 56 > xde€ 6 oro [0, /1o l.r:/(y) /(b) .v olo [r, ;r]]. (Aurd ro g,poqroL,npopliloroq devlpsui(€I0r7rc rc uadioraa.)Yn66erir1: Ano6.iE{drrs! = lx:b < x < Iirdipvovraq ro sups,. (c) Av I / €ivoroiqoloo o1oo xor I / €ivornopoyrDyiorurl oro a. anodeiqt€ 6rr (0lJ16 > 0 €avore$rolo). /'(a) (d) Av /'(d)> 0, omd€iqr€6nit / rivara'ilouoqoro a (snrorpdvrs orovopLor6 ro'r /'(d)). (e) Xprorponor€isrrro (e) '(s' (d) ?lg va deiietr,Zopi( Zpioil rcu O€opipro; M6onqTLpriE, ofl, qv I / EivorouvE/if oro [0, l] '(o' /'(a) > 0]o '(iioro oro g1ot0, ll. l], 16r€ rl [0, / eivarqnEouso (0 Y,loedroup€ co [0, I] xo,6rr /'(o) = 0?'a K'i0€d o.o(0, 6iLrt / eivorouv€;(,iE l). Eqqp!6sr€10lrdpoq(e)oq ouvitprlot s(x) = /(x) + E{ ?rava 6cii'r€ 6r' - €. Opoio:q,6eilte {tr1 /(l) /(l) /(0)> /(0) < e cErfli(ovracrrv /r(x) = cx - /(x). tw,r€p'ivcr06rL /(0) = /(l). H oljlKrxpLdvnolrri onddn_Lrl rou 6fl, tra ouvriprtorl!e ndpdlotoprt6ivnp6ner !s eivororoo€pi,dIErno),ldKoLvd olfEiq !etro otr66t1qlro! H.A. Sch\rar,tro! tivor iooq I np6rrl ololnpri ar66€rqlnoua60rlxs.Tolidzroro!, o 6r hup-/6! rrlq iror AeirEro rv0ouoL66€g v6pr't8. 7p,it+qro! oritv 'rdponouri[al] rnq BL0i.roTpoaiaq. *.66. (a) Av / Eivorp1aolqosprisuvripqot. 16r€{d0e ollleio €ivorottsio romKoi p€lisrourrlE/. Eivqr,tol,n'Il0sv6vq ouf0oiv6rl(dIl ldroro.0|.6!0xr ov n / 6€veivororqo€priolvdpltot:7ro nopdSelruo. ov /(x)=0 ?ro { 0. A7ro6€it1e, 6Foq, /(-r) 7ra rprlorfonor6wo(ro flp6pitro 84 6Ttov €ivqr ouv€triq oro b] Kqr Kde€ srlp€iorou [d, b] oivoroiurEiorotrrxoi I / [a. 16T€n eivsr oruo€pi ouvdprlol. Opoio,(1roIiln€pinrooitro'lrioi lstioroD, / el,oliorou. !6pa6fl Kdo€qpeio €ivor€ir€ro7ltl(6t)dtrolori iorrK6ldtL6rotro O) Ynoedor€ rivol lorrKa€trdxrors.idriiLiu r1v / (ilopiqvc o:ror),eioutE 6rr p€pr{dorltL€io romKd !d1Lo1o).Ano6siq$ 61l rl / eivaL oro0epd oq €81!: Eoro f(oi< fl6). Mropo$!€ vq utrooaooule6ft ,/(C,) fl\) ttrl l"t'tu- ^, t(t) f(a) f(b) h-a
f(d) -2 f(x) - f(a) ' x-d
'Elor4re ),orrdvdvcvroo6ivolooprorrdrlg rupr6rtr0;.
Mrqouvdprqon/ i;7srqr xupri o dva6rdoqlo. qv 1lo d,r{,ro( b o' qur6ro 6rdornpa }l€o<x f(b)
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fl/ /(r) = x', Kolo o EivarsKEparcq ri o awiorpogos€v6qADo.r(oiop10Fo6, r6re '. EivorsiKol.ordrpova €t6f6oDl€6rr aur6qo niroq roi(6€.,ov o d Eivol = ax" f'(n rutoioq plr6q opr0F6q:Eora)a= n/n,6.n01)n €ivol 6vagaxdpalog,Kqrr quorK6q cpr0p6qav f(xt = xd" = Qtt\',
r6re, ofli-iov KovdvoqC AX.Dsi6aq, ' r ' 1l f'(x) = -1"rr';' = L.,
r "1ri"1 t-a - t r'ttrtrtr't- rt
=m ' ,t-o t. Av Ksrr6po ftouls €vovrfro 11orlv /'(x) 6rov /(x) = x'Kor d pnde,I IEMq rrlq - !€Xpr o.n$16 6svq6poq[ ofr€ Kov ouvdpllslq t(x) : ./ Ita dppl@ a avoF6tr1,€rar .tvdwou €v6qsup06lou oov ro xf1 Or ovliorpoqeEnd!'roeouvoprrto€r60oavapflIeofv dvrovoorov op16p6roDx'ylg dppnrod. npoytoflKd,ora op€ooEen6|revo rcEQdl,o1o, ro),MExar ortfdvrlr(Ce ouvoprrio€rE 0o oprsroiv [€ rn 9orie€lgrov ovrrorp6aovrouE
IIPOBAHMATA rlv f-'Trc rri0eFiom6 ng froD oKoXoDeo0v 1. Bp€ir€ (i) /(x) = xr + I (ii) t(x) = (x- l)r (in) _f{x) =l x' x plros t-x, ropprlroq. ,x>o -. ^.. l-t'. trvJ J{x, =lt _ x!. x( 0 t + a,,....a, lx. (v) /(r) x = a . i 1 ,. , n - | - a , + , . x= c\ tdt' (vi) /(x) =.{ + txl. = 0 . o,aror... (Xplopnorofpe6€{o6r{inapdoroot.) (vii) /(0 . ara,or...) ( v i n/)G ) = - 1 ,
-l(x(1.
12.AvrioqorptE ouvaprfiotry207 2. noprlpdvrsflt ?poaLKiropdoroonrrq/ rdrov (i) r/ EivoraiEouooKarrdvra 0errK'i. (iD I / Livo'aiEo,os \oUrdvroopvrr,\i. (iii) r/ eilor qeivouoo{qr rdvro eerlKri. | l \ ) q / ' , ! o ,q 8 , v oo,o r o , ^ o v , oo f v f , , r , t 3. AnodeiEr€ 6rr,ov I / sivoroilouoa.r6rEro i6Loouppuiv€r xoLp€rnv/ ou!dpiioeq. e0ivouoEq
otoioqyrq
4 . A v I / K d r s E i v oari l o u o e eq i,y cxLa rq f + c t H f ' c . H f " c ; 5. (a) AnoSsiirE 6rL,av n / KaLrs sivoLl-t,r6rs Kt1r rt /.s€ivor l-1.Bpsirsrrlv (/ds) orvoprrioarr,rvfrrur H ordlrrtot 6rr Eivo, s. Yn66€rErt: (b)
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Bpeirsrrvg 'oLrvcprrlonr1q /
6.^s:t,'d, q I(n
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, ov s(r) = I + f(r).
c i ! s ,l . ' o r , o , / d v o u \u d
b c r 0 . \ o . 0 0 - . r .r r \ /
7. re nordSraoliFoio[a b] €ivaL1-l oLouvqpfio€Lq nouoxoioD(]oivr - 3.rr. (i) /(.r) =.!' (ii) /(r) =;5 + r (iii) /(.r) = (l +.r') '
1;"1 11,;= l1-! . 8. Yno06or€6rLrt / €ivordra9opiowrt!E nopdl(,ilof'(.y) = (l + rr) Lr. Aeilre 6tr l r s - / , K a v o r o , c i r r v . p i \ ,; - f , r , . 9. Yno0irouf56n I / €ivorfLa I I ouvdpntortKoL611r / 6/€Lnopdy(Dlo noD6€v Ano0tiErr6fl I / €ilqLnopalotiorpl. Yn66srqt:undplsrllfl IlE vi(srornoL,ocvd. nol,l oivrogl 0r66a_Er. 10. H Kard Schwarz nopdtrD?o(. ,/ opio0lx' oro np6p).lpa l0 17. (a) Ano8€ihs6Tl, nv r ?l(x) rndpTn /1o 61,aro x, 1610I 9/i(r) 'riorq !,rdpttnyrq6iq rq x oro r€dioopro[ou1!q / (b) Bpsirodvovri,ro 110q! , f t(x). *ll. (a) AnobsiLr€6rL utrdp-t€r nqpq,oTioLprouvdpfltorl/ rarorod)oret/(x)li+ oavovriorpoqrl /(x) + r = 0yro,.deer. Yr66erqr:^Eiqre6rr | / eKepd(€raL oDvdprrlort. O EUKo).or€po! rp6nogro vq ro x{iveiE,€ivoLvo gp€ir€rtv / Kar o 6ljKol.6repoq Tp6ro!r,rqvc Kd!€rsdurd,civoLva oio€r€x = ,f rcr). (b) Bpsir€trlv /' o'rloprrio€L qe /. rprlo,lonordJvraq r(ardD"r).o esdjpnpo ol,roi rou KseaI0io!. (c) Bpsirorrv f' r' dD.or!6no, otrrdrqropoTrDriqowoq rlv €lio0Jornouopi(€r H ouvdprrortrou npo0l,iloroE ll l.6re6ux\rd 6r, opi(€tq,nenl..tpivoon6rnv €Eiooorl -i,r+J'+x=0.Hnspinroordroeorrricqs€Eio0rotqei!'qrnoii€r6rKi.'Onoq8€irver ro en6pevo npijp),lto, fLo €qiooor auviooq6'v opi(r, mr).syrdvopro ouvlipqor oE ol6x).lpl xrv EL,€siq, €v6 o€ p€prKdq n€proxdq sivol 6Dvor6vvo opi(ovro,nsn).ElFdvo rTspl6o6rDp€g on6Uroot]vopniosrg. 12. (a) nodq sivoro'6io napelo/ioltr'( olvoprrioerq/ rou opi(oworn'nl€?pfvooro ( t,l)fln6ntvsEicoortx:+l= l,6tlolri rour(qvonorciv rtv xr + t/(x)l:= I ( Kdes .r oro l, l); Ilopottlpcior8 6It 8cv urdproL,v ],ioEq opLop6vss dqo 7lo 01610 [- I, t].
202
nqpdf@fotKatoloKlqpd\tora (b) no,6qouvopflio€r(/ r(ovonoroiviitv x: + [/(-x)]r = - l; *(c) Io,de ropololior$€qolvapniosLq/ r(qvo,rororjv qv [/(x)l -3f(x)= x \ ^dEErEl:Oo porl€oiocvo otf6rdocr€rp6ro rr1 Tpcarxrjncpriorool rqq ouvriprnolqg(x) = xr lx. fevrxii,ro vo rrpoo6lopiooL,F€ o€rord 6la6ritoro opiqeror n€rL€fpdvollq,ropsTo-fisrjrt olvdpqorl on6llo outr€xp#tt eEioolol,lnop€i vo sivdrtu).i ).e'Ilrtun60€ot,r(or rnv le),srdlr€rol.ftepqorq rr).oioro rou rpoXoplf6vooA,T€rpoorlroi Aoyrogoi.Av 6fq unoqioobttt6n / eivqrlro riroro nopotoyiortrl).6on,16r€proporigevo cro8ei{oupe dvovriro yra qv /'(,{), oKprp6(dfi,r( oro np6p}.ltq Il(c), napololi(owqqKarra 6io gdII qe €EiooonE,rol, opi(sr rilv f ([l0 6Lo6rKaoio Tvosri oav (mn],6yp6vn Eaoprdor€o!fli q pd0o6o oqv €EiooolI f0)]1 + x1= L Ilapqrrtp€ior€ 6rr ornvandvqoi ooe0s mprdXeror ro ./(,(I6sv fropoioolEvq n€pqrdvoul€ Kdrr {ol6repo, oQo'iundplouvn€proo6rspE( ord Fra ouvopnioer6 noDopi(owor rcr).e7pdva o16r1v eliooor I + .r: = t. (b) E).atqrs6fo6 av I qndvrtoi 6o! 6or,),€isr Kar]la rre560ouvopflioEq/ nou gpriK0r€ oro npdp).lp0 l2(0). (c) Eaop!6or€lrtv i6raolJt]ip60o6o 6Tt! [/(x)]r - 3f(x) = x. 14.(a) :Xptortonor€ior€nsnl,€T fdvl 6rdA6protIo va pprir. r1qf'(x) Kor/"(x) ]ro q o!vqplliosK f nouopi(ovrornsni€Trrivs rioo rte rqio0rolqxr + J,i= 7. (b) Mro 0'16oula(tq / €nql,neai€rlv ozdorl,f(- l) = 2. Bp€irernv /'(- l) Kor lnv /"( l) 7'' ouriv rtv /. 15. H ouD.olti 6).ov ro)votFcilDv(x, y) aror 60r€ 3xr + a.{l - xl + 2J,'= 4 olTllori(€r qC €qalltoFlrte€ue€iog prq opro#q xolnilrl rou s,rr1li6ou. Bpsic 1!v sBisorsrl srqv houd).lorT,i.oro on/.io ( l. l).
ll. (a)
16. O oulFo),rop6q ro! Lcibniz€ivaldloirrpo 06Xprtolos-rra rlv n€rl€?t6q nopqt6yrort.Er€r8riro y j(plorgonor€im1 orr,ci(6coovoiwglsl roD/(r),1€liooort oq rrpoq .xKory nouopi(€rmnt€Ildvo rIv /,duroldroq ovr(Ka0rord rrtv.€iooortro! unori 08rqr6n rrqvonorein /. Me lov ErK6laqoutpol.rotr6n6_c 0o yprieolerov uno).o7rol1( 'roDol.o}.oug€i:
Y'+Y'+rY=l 4' d\ ^' d' dlx. ' di ^v d J+ v + x d ) t .
E= qy'+-, d^
3y'+,
l?. AAoi ipoe oro npooKrivro o or#Bo).roI6E 1ouLeibniz,ed npen€vq lrvaQdpoule Kar qvrrmp6qovouvoprioEov-Av l'l€ lov 0'r!001,10!6 roo Lcibniz-rror( nopoy6Toug d/dx ougFo),i(ou!. lilv rcpriTorlo rlq /,r6rerl nopdyoiocqe / rouppo),i(eror pe d.{,/d/.fpdwe ro O.6ptto 5 I€ our6rov oufpo}.rojr6. H €Eiooortnoutrpoxinr€reo ooSdsiE&dvovoKd9otrdlo-rLarov onoioo oulpo).rol6qrou LeibnizdjtsLl6ooDe ro),Ioiq ord6oig.Oo oElTiq€rar6!o, o€ nor6oIp€in rp6rrervo uro),oti(oulsrlv ( / r)', 6ravXprlorroloLoupe rov or|Bo).ro|6dxldy.IIod €ivdrrl ottaois rounopoKriro urol,oyLofon;
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12.Avtiotpoqeg ouvaprfioery203 lt. Yro€drou'ur ritr / eiwr lLo mpayoryioLpll-l ouvdpritolnouI mpd70)76E It( 6sv
L(x)).A'roIn6€vi(0ro1rou0tvd,'(or dn f = F'. O6roulsG(x) = r/-lx) - F(/ = (Av 6€i6rE6flClx) nopoF).dvoof€ ne ).e'rror6pflee, ould10np60),r!qll'i( /-r(r). LderKdfl ,!oi,[ €varaqdpov: ov i6po'rtelro o0vdprrlolnourl rapdtol6erIE eivdr/, t61si6pou!€xqrliano! I mpdyoy6qqq €ivar/ r.Al.).dn6q0oFnopo6o€ Koveigva pavtEVELrt ouvdplitortC;n€prypdqo!p€ 6io Sroqop€lrxoiq rpdrouqoro IIpoplrilura 14-18 xqr l8 l5). 19.Ynoedroure 6n /l eivarlra ol,vdprrlor!s ,"'(r) = sitr'Gin(x+ I )) ror t(0) = ]. Bpeir€
(i) (r 1'(3). ','(3),
(ii) (B 6no'rP(x)= ,(x + l). 20.Bp€irs rivovrino ?rorlv (/ r)"(i{). i(x)) u,rdpX€r, Ano60ihs6fl ov ! tqr 6Ev'ivor !rt6dv,16r€undpr€rrl "flrt"f ( -')'(.x).
"f 22. (a)
Amdei€r€6rr uo on6o'rooKorlro goivouoaoovripr4oltipvov,arro no),6o' 6voolt€io. (b) BpEire8fo ouvrr€iqu'jqouo€q ouvqprissrE/ Kors, rdror€q60rr f(x) = g(.{) gKprFdrq 6r0vo r sivoroK;pflroq. (c) Bp€ir€proouv€XiaiqouoooDvdprltol/ xar praorvel! poivoLroo olvdpllor] g, oprotiv€qoro R.nou6svrdlvovroL,rouorvd. *23. (a) Av r / €ivarouv€xiqouvdprnoqoro R, xo, f = / i, 6€ie1€6rL u'(ipl€r rould orov avo .r rdrcrc dlor€ t(x) = x. (Ilord rl 7€ouetpr$iolfooio qq otvsirle/= / ) (b) Adrors SdaoporqpodsitForo 6!v€7oig/dro,6aft f = ftKstf(x\=xt:3 ;vo aKprF6q flpoorqo€ior€vo Krivs1trtv / qoivouod{or ol)l1lx. Yrd60rErt: eEirer! ?€rotrsrprkri €ptlv€id rou Fdpo',((a).M1o,Il0qvdrwocivorro loXi€r
far= x.
A€iqc'jrrdv rl / sivaLoilouoq ouvdprroyldr6t dt6rE| = f I, 161't(x) = x yra6Ia to.r. Yr66eril: Av '.orrl7€oletprKri€ptrlveiqsivor€ue6oS n€rorrKti, rt on).oior€plord6€ret(nepinou 660tpoFp;6)eilar va ororloio€r€fl( trEpurr6sErq/(i) > .r l(a' /(x) < r. .24, no16qouvaprrioerE dlouvrnv r6r6rrtrorl ypoQrKirouqrdpdorqol vo sivorKorrdl,r nopdoloon olvdprtorlq 6rov qvok).dor€i oq npo-crl'l ypqQrKirapdorool lpoqfii qq -/(rlv rcvr1610y6vro')r (c)
25. Mroouvriprrldrl/ ).iY8larr+e0ivoD6aov /(.x) < /Ly) 6nor€,r< /. (rto vtreilloor€ mo qKpr9Eiq ed Jtp.,ro vo oDBqovrisoupEvo eivorirdsrruo ro n€6iooproloi116 /). Mrq pl-o66ouao suvdptlorlopiqerar!€ 6loro rp6ro.npoooXirMsplKoiou,lpoq€iq ro (oilouoo' qvri ?roro (Fl'tqoivouooD, Korro q,vnoiorq o EouooD t prtortronororiv 1l0 106tK6luq (aiqouoqD. (a) AnooEiErE 6t(, av I / eivqrrn-q€ivoloq,oL)d 6rr oJiouoa.rdr€ n / €ivqr otao€pios Kdrotodrdoqro (oulolteil€ qr6 rc ].oyorroiTvro: ro (6 qriqouoqD 6sv€ivolro i6ro!e ro qrl-afEouoq'). (b) AnoS€iir€6rr, ov n f civdrnopoyoyioLF txdr lit{p6ivouoo,r61E/'(-x) > o-ru xdo€.x. (c) AnoSEiqrs 6fl, dv /'(.r) > 0 yraxdo€.r,r6rs I / sivorprt-Q0tvoloa. '26. (a) Yroe;roDls6n Ano6€i(1€6tl t(x) > 0lla K{io€x. Kar6rLI / €ivorQ8ivouoo. r^6pa rE ouve'/.46qgirouoa crrvdprqoq g, tirorc drolc0 < s(.x)< /(r) ?ro l(doeir. (b) AriErc6n lropoitl va ro KoropdpolFr iLor iorr I g vo rl(ovororrixqr rnv
Iims(.r)r t(x) = 0
204
llapdyayotrar d"ot),rlpripau IIAPAFIHMA. NAPAMETPIKHNAPA'TA'H TON KAMIIYAON
,zita I
trito 2
To nspretdlrrooL,roitot, xegq),oiou dlersov or6j(ova d6oErdpeqono€ kdfl ttouonp€F penoti opdid€lqdvrol d€v€ivo (or' ovritKi! il lporpuoi 6ooIE0116 nprv:lro r(opn6),rt rsp(ioruon({inoroeolr,riprilol( (Elitd l). M€ dr}-o).67u,pnopEivo litv livorqr vo ri}, rcprlpdvout€oo! llJ 6ivo),or(dvditprio! (r /(x)). OuoL'.dUrop€ivq ?iv€lsrvo t! nsprtprivoup€ oq\'10 oilolo rov olltoi0)! (f(x), x). fro rup{i8€L7ps, I Ko[ni],I ro! tl{poroe I sivql 10 oivo),oti),0' u €ivorciloloo, ilor Kd0rx 0iva1u(t)^trt ivt ttova6tK6l, 6nl.o6ri.o n ouvripqot 'Eror t = u '(x). ro Fova6LK6 olFeio roDr.DK),oe$o6e ro x eivor uerpdJq oow€rorudvrl ro oilpEio (r, u(u L(x))). M€ dl,)'o1,6yra, ro ruxloer66c €ivorn lpaerKi rqpdoraot rlq / = u o ui. A€vDrdpt€r rp6roc vq 6o0€itu olu)'ururi cKqpqorl pEoo€v6Eldvo rinou, rlq / (66ir6ro Ilp6pl,rllra 6 lta lrlv 'tro l(au ttpoo6ytroylIoD o4ralod, al,ld n rapaF€tprKli tropdoirol Fopei vo1plorFonornoci onoroD8riroie ttprifFqloE, nooo 7rdrovono),o1tot6 rinoq rne / 0o;6rve.flo rrqpd8nlfa,ro np6rorrp60l,nlaroDoro),ouo€i n6Sva 6€i1v€r rq saa'rt6treveE o' dvaKurl.o€r6iq. Ppo6u€
IIPOBAHMATA 1. Eoroix = !(t), /= u(t) Frqnopalsrp1{rlrloprioroon Kapri},Tlq KoLuno0Crou|'ls 6n n u sivor;vonpoe6voo' dvq6Ldorntq,6oft p€poE rrlg KotnrlltEva DlvortoT pdqlld rrq,f=uou-r. (aJ Arillr 6fl oro otFEiox = u(t irouusdn
f.@=!!J) u'(4 Arol.ou06woqq lpqqri roDLeibniz rpdgoule €v8€rKrrxri dy E= dt. dx dx dl (b) 'EXouUe rniolq 6fl u'\t)u"\!) - u'(t) u'(t) (,'(t)'
2. oc0)p€i6rstLa suvdprlol / ,roD opi(erar rErl.€rudvq!6so qe EElooolq r" .i" = L Ynotrolior€ro t'(rj $ bio rponouc: (i) Mi6o ,retr).E, pdqg ncpoyoyi6€0)C. (i,) orupdwoqqv nspoF€rprrirop{ioroorlr = cos'r,y= sin'r. peu xoLu oraqopiorllgE, 3, Eorc)x = u(l),y = u(r)I rapo$rprKrinapdoroolKolnu),ng
206
napdf@fotKatd,or),qpbpara Kordor(0P = (xi. /i) orloio ror 0r1na6ou. Aeilre6fl ovro onr€ioQ = (u(r),',,(t))srrt on6ro P KorQ Kauni),r€ivorro slrirqro oro (x0,y!), 16rr rl suo€iarou 8rdp1Eror KoIfl0),Iqoro Q P1ilo6). To {3rodnordl€opo sivorKdo€t trpo{irtv eeo,rro[dvrln]q oltsio an6ro (x!,y!). roxierov ro Q rivoLToridov poKprv6 Lro nopdpqta rou Krqolsior,4o!(ltiootL 7ra rt /poarKtnaprioloot!ioc ,ropdd€rtlro 'ropol€ouvripqotq / or no),rxdq ouwEra1ldveg. Autdeeivor6vd€r5rK6 rprxriqrcpriorooqqr 0 ro x{ieex oro [a b]. EivarFo],rx6vo ourFo].i(our€ os aurof rou fle €medvsl€q nEpri,ap0dvowar rq R(/, d, b). noporylporlpe 6r, ei6ouq ts l(orlq rpiyo)vo, {o06EKu no d ,}.soirouiaialeo!€rpr(d o1rilqlo. opeoy6vrc o opr6l6q nolj 0o 6oooqr rel,rl(doav €Ipaidv rou R( ,f, a, 6) 0o ovojroor€iro olox,l ripapaqE f a|o la, bl.Lrlv npq-rfatrK6rtro, ro ololcl.ipouaeo oprorri oK6toKorTlq iI ouvoprrloerE nou Eev Lxovororoiv o0v0{(I /(-{) > 0 yra K'ie€x oro [o, b]. Av / / rrl €ivor! ouvdpltot lIoD61ou!eo't[6rdo€roro IIiIo 2, ro oi,or).ipotm0o nqprordvEr qq sn pdvElaq Korro €ppa36v 6uQop(iovdp€oaoro €t0tr66vqE sl.oap6qoKloopdvnE sr0o36vrrou R(/, q b)). dmovoor.roodqq €'rq{ivnoq(ro (o}.y€0prK6 H r6dorou BpioK€rar rio{o qn6 rov !€}.),ovnK6oprstri, gaiv€Ioroto l1rilla 3. To 6uiorqfo[d, 6] 4(€rSrarp€oEi sEr6og3paunoElosrripota
t4,r,1 F,,il [r:,r,1 [r,,r.] tritld 2
gdoorr,rvupr€;r.rr4. lr. rr. rr. n lro 1016o,roiolqroXier a=4 u ( f, P ) > . . . >u ( f , n = u U , Q ) . 1 To oEdipllanooedla$ vo om6€iEo'rle €ivara,r).iouvineloouroi rou hiHroroe. 'Eoro oropflM^ 6fl Pr (clPr €ivar6rd|16pioerq r rou[q 6], {qr / praouvdprlonqpoypdln sro [a, 6]. 16r€
L ( f .P \ t su ( f , 4 \ . AfloaEEIrYndpl€rlro 6rqrliplo! P nounrpr6Xer rdoo rnv P1600ror qv Pr (n.jt.I P vq nepriX€r ro onp€idrrlq P, *or rrs P, dxpr86q). tijr$ovu pr ro A{pso,
LU, Pt)sL(f, P)= u(f, P)< u(f, P).a A,1610O€riptlr I inetqr6rrl({ie€dvodoporsloU( /, P') €ivorivo dvorAp{ilUa./ro ro o6voi.od),ovrov KriroaopoLotrdrorv l(/, P).Iuv€r6S,Kd0E dvod0porolloU(/, P) €ivor peroLir€poli i6o an6ro ,tdlroro dvo Qpriypo dl.ovrov (dro ooporotrirov: supll, ( /, P): P oropdplot1oula, bll < U(f , n, 1lol(d0€P'. ADr6,Ferl oetpdlou, srpoivEr6rrro sup{r(/, P)j€ivor6vaxdro)apdrpa7ra ro oivo),odl.ovro)vdvoa0porolrirovfitq/.I!vsr6q,
suptl(/,P)l< inr{u(/,P)j. Eivor ,rpoeqvdc 6rr outoi or 6io apr0foi ppioKowardvdu€oooro \dTo ror ro dvo 'ioporotoqq f.rlu dtr< flE 6raFpioe(:
L(f , n <suplL(f , Pll< u(f , n, L(f , P)< if\\u\f , P)j= u(f . n.
2t2
llapdyayorrar oToK^tlptiltotd
7ra{d0' Erd#prorlP. Mrop€iKd).),rola vq ollr0ci
supil(/,P)l= inl{u(/,P)l c ourjv qv repir 0. Av P= [6, ...,t,] Eiwl Fa iroFdptorrou [0, b], r6tE(xrriFo9) ni= ti_t K$ Mi= t rcotsropdvoq L (' 'f , 4 =
), r,_.(, r_r)=lD(4 6)+rr(,-tr)+... lr" r(r,- r. r).
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l" r
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rc utf, P);
(b) Yno06roDtrc 6fl t,-t, j=(' yu rd€s t. Anob€ilt8611Ulf, P)-L(f, P)= dtf(b) - f(s)1. (c) A'ioseiEre 6n I / €ivoLo),od,rpdrorpr. (d) A60r€';vonopdit€r, ouv,ipqonsoro[0, ]l,nouvo eivorqoDv€ to prt qeivouooq j(rig€ dnErpo ottL€io.
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Ay ot tDtdio alitd AacE,Eouapi{ttut atu ia opef: ypawd( Aa. A€, rdr rt|rdrtr,l, acE, cflpdtuorytoratarditoa dptqt1 tapdt^tlioTptit+avAb, Bc. Cd Ki.n..rar utuA;rcDq6n t/ourioEalJdoda^B-BC,CD,Ktrt.,KdL6ttot,tuptaBb,Cc,Ddxi"t.tiratruptjil\i.qorn aKbl.blcm. cl\,Id!, tna ,aptj Aa tut oritt(Toq.ru oD! ,\pttooDttcld rdpd,t,1?,ldlpdlr/rd ,n., vh€, ar 8aqioaap) 6u ro iidtu; ani)v tuy tupd^^ti,olptjt+(Dv ddtt(t!&aL rat tu ,iqa,ra\au:dvet tn dr.tpor,tottpitottit,jtLott,$oi).67a1tutAaixotetot/rtlpatrttvo ozita AKbLcMdD,to xEpt"/typa\ftw o/ itd AalbncndaE,KdLtu Katrtu).dypdt4oaxiua Aab.dE. tu ad ttt ro tt^Lo,Adtitor t,ti"/otto6t\tu.. { I o . / , o r : I i j o q o o o ' , ) E t y c l p s , r L r i v oorr,d r o r r p . y " ; p c L r . i v oo 1 1 ; o . r v o rr o d€poloparov nopuihior,pdftov Kl, Lm, Mn, Do, 611,.(on6 ritv ro6fltra 6lov rrDv prios(,rv), ro opeo?6vronou opiirroL on6 rro on6Irq pdosrq!o!q Kb Karro dopoLofo 6Lo! rov L,v6vAa, Srlaii ro opeoldv,o ABla. Al].d our6 ro opeoTrijlLo, €rEr5i 10 n t . d J - J A B , r o r i e f l u r d r r . ; u n o v - r s r . n d n l r p o v .7 r v r o , f . { p o o p o o , o o l o r o 6Itort 6oopdvo€!Pq66!. KaL Enol;\,oq (on6 ro AiIIo l), ro rEpryrTpolfilo xal ro c77e7p{ttpfvooXripotivovror rsl,LKdro iva ilro trsro dLio KoLax6!q nrpLoo6r€po, Toe\&d!€oo Kotnul,6lpatlo o1rilo ofl civorrcilKd ioo !s {qofva qn6 ra 6io. O.E.A. *22. Yno0arowc 6T1 r / 'i!aL o'iqouoo.To :xrirLo l6 !,roEfi{\r!0, 6rL f"
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AnodeiEr'6n, o! r1/ sivoLo}.o{Lrtpd,o,r!orc [d. b] x{tLD < f(.xJ<M,tnt 6ic ra x oro [a. b]. r6rs
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to aroT6].eoya ovoltti(€torOs6ptgaqe Maonq 11oxd,roroi oro [4, b]. AtLrr5 Tuiq 7lo ro O).od.rtp6uots. (qr llrl 0€tlx4oro[d. (d) tufmpdv€rsdrr 10;{6€r to i6rcov n g Eivorol,ox).npri}oLFl bl. (c) Aei_Ere 6n I lio a't6tq 6$0ouraquno06o€rE 7rdtrtvg oivorolotaorrr.ti. 25. :€ oDr6ro np6p).ruoo€opoiFsntv tpaarKi napdoroorlpio( olvdprrlolqos no].rK6q (Ke0d).aro 4. tlapdprrtlu).To tlifd l8 6oiXlstilo K!r)rx6 rolda p€ oDrr€ra7t6\,€q 'otuv rouloffo d. d PaoLKi7(Dvio n $rpcirar oeorrivro (Kcad],aroI 5),r1sm$dverq A 16portl! rEproxriA routXiForoq l9.6nouI Kofntr:1ereppa66v I Oeorpsiors T rtopdorool or'vdplnolc oL\Tcro?.ri!fq tlq otvExoiq ).I r r!or n ypo,pLxri oL ^o).rKdc /. ^eiqrc'trL: A= ep.pa66\)
t[' tJa.f t0)'d0.
*26.'Eoro / fio orv€Xri€ olv,iprnonoro [a, b]. Av P= {ro,.., r,}sivqrlro Srqp;prorlro! bl. opiEou!'€ rov opL0!6 ld,
t vt -l-t
,xittd 20
+ t/(J _/r-il.
orrl Ko!,riUq c/16'/po$t6!'IE o opr0!6q1(/, P) rap(old10priKoS [ioe no].uy(0vLKriq qq (plin€ rdvo on6 ro Iliro 20). Opi(our810 nopdoroort rlq !1ixog / TpuaLKri / (ur6 rLq 6layepio€tr P rrlv rpoijn60'oil oav ro supremum rov 1(/, P)1lo 6leq 6l la, 6rLro aivo),o6),ovour6vrov opr0y6vsivaldvo apuyfdvo). (a) Av n / sitol tpopttnoioro [d. 6], Seiirs6fl ro sriKoirlt / Eivorrl ondorool rcmqrjro'l (d, /(d)) Ksrro', (b, /(6)). 6fl 'rrdpt€r6nl6prol P = {4, r, 61 10l] (b) Av n / 6Ev€ivortpotsLxi, qno5€iq{s ^d)) [a bldror60r€ro/(/,P)vo€ivdru8lo].r€poqn6rlvqn6oros!rou(4, ro llp60).ruo4-9). 0'16ro (6, ^b)). (OotperqorEir€ (c) turrEpdvor€6tl leloEi 6lov rov oDvopflio€ov / oro [q b] le /(a) = c '(ar 'tpopfll.ie oDvripqorlisivor10 lt|ip6lepo.( H, 16 flb) = d, n fri|(o( qe outpoflxri o).]d anc),ftrorr(d ln€pdcldrnopo).otio,61ri(i €uo€io€ivarrl nto oivroprton6oraong€1ol$Srjodttsiov'.) (d) Yno06or€ 6rr I /'oivqr gpolFdvrloro[a, b]. Avrt P.i!arllXqio drapiplorlroD 61 3€iq1o 61r ld,
L & E + ( ' , p ) < r ( f, p ) < u ( \ - l+ o " L P ) .
yno,\tEH: XpIorlonor0iot€ro O€6plpaM;onqTLprif. 1e) r,oti ro76ersup{r(,,/l + 1/ fP)l < supu(f. P)};(E!xo).o.) i,o, o,l (l) Aritrcr6pa&, suplttI . Pt' ia\Ln,J| + | l- . Pil o,Iod€rKrio\roc l"
ro !,ixoqrnq / rdvos'roro l a. b j ci!a,J, r/ t l
r li*
n J t t r l' r,-_i*'
o).oKi,rlp6oLtn oro [o, b]. r61e Yn66srql:Ap{€i vq o€iEoure6nov Pf(r P"civar6iotl,;(qi€i6Lsf€pionc.
230 flopdyayorrat oiod 1pipow Loxftr6n l(/. P') < u(\-l + Ul?1. Av rt P mptrj(n ra oit[cia Ko1.l( P {o, rt( P", no trloxdort frru\i t( f. P') KtJtr t(f, P); (s) 'Eoro 31.!) ro rrl{oE rrtc?paa,{iq rap{ioroorte rrts / n'ivrDon6ro [d, x] Kor (a, d(.{)ro r',iKoq.rts euosioq ro! l(o, ro! (x, /(x)). ^eiqrs6n fsraql ld))
IT# =' Yr6SErEit: Oo Por0locir€av K{ivsi€6Ln?,{rprior'trou OEopri[oro,c M;orq TLpiis. 27.\.4oo,vdp,,tot(opo,rivto,otd.bl).i&,o,ri,tporo{orvriprlol.orrnriplu 8,orip,olP={4,...,r,lrou[a.b],rirora60r€livaEivororoo'prioe{d0e(t, rLlfe qe s 6ro l, urop€iva rivoronot3dbrirols). (a) Ano8siqr'6fl. ov rt / silol o?,o{Lrtp6oLtrt oro [o, b]. 1616yro rdos c>0 ' ! d r r , . ,L , o , r , u s , , u -or , v o p , r . q\ , / t * l '
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Ynooirours6rr yla {d0€E> 0 undpxouv Kluu{orre olvopnioErqr, < / {dr fb rb . >/r:ror.q;orsi ' L . A 1 0 6i 4 r " 6 n I / L i v ao. ; ! o { ) . r l p ; o . L n . l.
(c)
Bpsilsfro ouv,iprlo! / noDvq ltv eiloLrhloxr,rri, u).),dvo rxovoroLei rt! J, f
= L( f, P) lto Kenoto8roudprolP rol, [4, b].
+28.Ano8€iir€6rLo! rl / EivoL orox),rlpii'olutoro [o, b],.6rs tLoKd06€ > 0 undpl€rfro |, l, ouv€rrlcouvriprrto!s< ,f p4,, s. Yn6Ssrqr: ndprE'rp6roproxiuqrorni "f J, c< ro ouvriprtonr' auriv rtv Ldr6rrtrq, Korlsrd lro oulsrri. EvooXri!o0o oort0oioE I671010. 29. (a)
(b)
A€iEr€61l, qv sj KoLq €ivor x).rfdKoreqo',vaprio€Lq610 [d, b]. r6re ro i6Lo LoIUn {dt lLd rlv sj r s,.
Arooeilrs, topiq vo xprtorronoriosr€ro o€rhpruo t, u.' / t",*ol= ts rtsr.
(c)
Xplorronoisiorsro l'ipoq(b) (daLro llp6phra 27)7rova Sdro€rE frq 6vq].loK.,Krior63c1qrro'rooopriroroq5.
olo la bl. AroS8iqrs611unriplerivaq 10. YnoodroL,fs 6rr r / €ivoro].ox).qpdrorsl t' I opr0p6q x oro [4 b] rdroros6meJ, f = J, /. Asiqrc!' 6vanapd6eLTpq 61lr€v €ivoL nd\rud'varov0610r,Eo, Lr ro \ o'o ld. bl. *lr. O 6Kor6qaL,roilou 61() rpoPiitqroq €ivqLvq 6Eils,6rr.uvn / 'iva, oloK).rtp6o1prt or rcl).d ort!€ioro', [d, b]. vq €ivorouv€r.ris la bl, 16r€r / np€nsr (a)'Eor{, P= {t0,...,t,j lrq 6rq}6p,or rou[ir,b] rE u(f . P) L( f , P) < b d. AroS€i€i€6iL /ro Kdroloi oivaL,\4- m,< L (b) ArobEilre6tt undprorvoploloi dr Kur b, pEd< d,< 6, < b KaLsupt/(x): a <x 0 ov / /(r ) > 0 ra rdo€x olo [q b]. (a)
A6orsivo nopd60Llpo 6nou/ (i) > 0 7rqxde€x, KaL/ (r) > 0}lo Kriirolox oro t4 bl, Kq,J"/ = 0.
G)
Ynoefroup€ 6 f(x)>01loK{ioe r o1o[o,b]roL6nt / eivar6u!€ iq oro .xo,oro [d, D], rlq ro o'mio f (&)>0. eroaeila orr,[" / >0. Yn66eGq: Apx8iva pp€ift6voKiro d0po'otqI (/. P) no'rvo €ivol0errr6.
(c)
Ynootroursdrr rl / sivaroLox).lp6orplo1o[a, ,] xfl /(x) > 0 rrs Kde€.x
*33. (a)
cro ta,6l. Aro6€i€rrd J / > 0. Y,r66eLEt: Oqj(p€rodxcils 10n p6pl.rtfq31. " Korl{i}.roro,ar|16q ivqg}.67097ro EivaL rov oroio outn€pr?dpole ro np60tlla 3l o' aur6qrq aoxio€K. fr Ynoo€roull[6rr I / €ivalouvrtiq oto [o. b] xa,.J,/s = 0 r,q 6].sene ouvE-
t"
r€iqouvap io€rsI nouopiqowoloro [o, b]. Am6dEre6ft / = 0. (Aur6eival EiKo),ourdpl€rtrd rpoqovriqg rou InopEirsvo 61d;qEr€.) Ytro0drouf s 6r I / otvorouvsr{qoro [d. b I xo, or,/'7 g = o 1o er"iv"q ",q ouvsl€ieouvaplio€rqg oro [a b] no! rxqvororoivrrtv €mnfiov o'rv0'iKn s(d) = s(b) = 0. A'rod€iqr86n/ = 0. (Aur6ro soriosritv ri\urt1e1ov6!qror€i€i 6voorou6aioLrillo orov ].orlold |r€ropo].6vp].ineoq Brp].ro1po9ia.rtd mpdtroFrtc.)Yr666rlrl:Ksrq),rl€r€o€ ovri$ooqq€Krv6waE p€ rrv unito€orl rou 0q 6ro,,6q€18, €loprdrsrqn6nt s'rrn€preopd /(&) > 0 ri /(,\) 01lo Kde€ 6up0o]'iqou€ lo r(ordlM€Mi'Kqr n,' oro [a, D]- EorcrP |rlc6rap6prsrllo![o, b]. l.Il.o rup xarinf.ru np / r(or6!()10opi(outEro 14- xaLm," ro qvg, r(orro rr4 l(qrl', ylo rrlv /s. (a) AnoSeiqrE 6rr M, < M"M,' Ku ni> n,'n,". (b) Atiire 6r'
utf s.Pt rtIE.Pt f l x . l l t ' -I . t < u t I , P ) . i7Kd0cdoporopo Z /t.x,)(r, t,'t) KELt.rotliqporcpaRicrnaffrrls / lLq rylvP.To EIiFq I 664v€rrrl y€oFerprKrt€pptveid ev6qd0poiotaro( Riemann.Eivcr ro ouvol,rr6 eppo66vn opoolovituvnoux€iwaLEvFdp€L Kdlo KaLEvFdpeL rdvo cr6 rq lpcarr! roptiorool t1q €ml,eyei ra 6vrl i0)vopooyoviov, ip6ro! p€rov oiroiodXoov l. EqardaErcu aDeaip€rou 6€vlnopo$$vo noi|'l8|'13 orloupd av dvaooyKeKpqrivo d0porotaRiemann eivorFu(p(ir€-
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on ." ""09r, r = rogx y = locy'.
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or].d)erlv r6danou our6vrov riro - eQoppd(ouus Aevundpl€r).6yoqvq oDldrolKavEiq (d0€ vtr ,rEpi,lroo1 rou €peovi(8rol. Boted8l,6}oe,1o €rllK{ 0pioK€rorriooon' our6v,o€ a'. rapdtol"ror orrlv rapdyoyo 0a €ivor o 0ut6ooors6rLo np6roE s(.r) H ouvdpJrDpinlodit oDro$rou riroD, rroud{i&r vo rlv 0Dp6toors. Yrdplsr lLo €L6rl(1i = pqroiq vo opiooDpeKot oplor€il6vo.rrq a. Mropofllo16po rrlor /(.r) x" irsr oqrd'po va pporlfsrtv rapdyolo rrq ouv{iprrlonq/(r) = r" yra xdosop1ef6d ro anordleopo EivoL oxpLpd'C our6noD0oreprpdvoup€: f(x) = r" = e"'v ' t r;,t-!e'
'=!."'-
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r€Ivdolorors rLq€Ko€rrKEq ouvqprrio€rq 0oTivoDv Tu otryeppr{ri I8sir€pri ooqaion pE ',rp€r6 rToD 0d vo rodouv,6wog ),in €!{io{norl va eup6ooore 6r! 6}.0101ro!6v€C !6vo rotioDv.Or pooLKfE r6r6rlteqqq expsivot o,too6'iroreour6qnoD6rarun6et{ov6ro O€opitqru 2 xoL3: exp'(x)= exp(r)' €xP(x+ Y) = €xP(r) exP(Y)' o1€36vtapoKfi]piq€trTl oDvdplton€xp. IlpoyrorlKd,Kdo€pra on oul6exlq r616q1ee BdForflr exp8Ev€ivoLn lovo61rriouv{iprlon f rou tKavoror€irnv f'= f ,Itad N !' on.rivfltv f = ce",T6rEf' G) = ce" = /lx ) 6poetruriq€ivaroLt6vEqouvqprio€LE 616rtra. KoL oropHMA 5 Av il / .ivol nflpaToyiorJrn /,(x) = /(x)
ra Kde€x,
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fle tpoarK6q nopaordosqruv olvaprriosov 4. XIE6Ldor0
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€9otltoFvl. {o}.oitra1urspgo),r|(6 nritovo,on€p0ollr6ouvlFitovo,Kol D,r€ppor'rKi sinch,cosh,Koltanch).Yrdpxouvnolldeovdo\rxidrorlo(al,ld olviooe 61o0d(ovrar Iofise a!d!6oa o' our6en( cuvaprrioeKKoLro ouq0rol6vo rprlovo!€rplKdrouq ro En6l€vo oro xtrifo l0 ovaFdlloL,}l€Tro o\rricrolitfl.Mro ovd.olio nsprypda€mr {'qd},oLorrtv or66€rqlro! 6fl I nsproxriroDl1{poloe lo(b)61errpdYporrsfFoS6v :ro €n6o).oK)'rlpopdrov. L,ro).o.rr6$oi xi 2.Tlod o oDr6€o qvatrdqollp€ t€066ouq ovo}.oYi€q dl.)"{i lrq dl).eq ova)"o,i€q, lpio rpoplilora €lsrd(oljtr€ 1!o Fo€61sp€( tsva 26.Av60;xer€ri6lert)'ion lo np6pLl0onp;nElvan€pu6!o!p€t4p1ro K€q(i).o,o sinh,cosh,Korlanh po 4, ol€SuiorgrrqtpoarKdq rapoo'flio€Le tov ouvopTioeov 7. Aro6oi6re6r! (a) cosh: sinb':= l. (b) tatrlh'+ l/cosh'- L ( c r . i n b r , r + J ) - r . n n . \ c o "-hc Jo r h , ) i n h ) . (d) co"h(rr J | - co\hr.oshy "inh,{rinht. (e) sid'' = cosh. (l) cosh'= sirh (g) tantr'=
292 Eapdyayor ratd.or)"4prigara E. OrouvdpnioErg sinhl(artaohEtvorl-l 01ow{otpogdg tolq sinh-rror tari-r,1ro! oDl0ol,t(ovrtrlpeargsinhKarargtatrl (to (6prolro,,roDo,r0p0ol1ro6 IFn6voDKal EQamo|,l6vtld, €ivoropralavqoro R Karoto (- l, l), avdotorXrt.Av n ;sh rEproprotei-oroI0, -), fuErawid?pogi,cosh_',,rouoDgpol,iqeror peargcosh,rar opigerot oroU,6). Anoi€16t0, fle,ltqpogopie6 tplouronodwoE rounpopl,ripcroq Z,iti 'x) = (a) sitrh(cosb V/ --1. (b) cosh(sinbrx) = ./ -1-r-rr.
( c ) ( s i t rrh) ' G ) = + ' . . vt+r. =+ (dJ (cosh-)'(.r) yrdx> L vr_ I (e) 9. (a) (b)
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Y wa ,b > l i t a , b n. (bl XproLBonordrvioq rrv ropdoroor f'lx)= e'lt( n)ix" , onodeilre 6n ror, f'\x)> e'''ih +l)"' yLu.y>, + l. i.r 6rsr66orsrru d),).!o:r6deL[1
= lg rt'r -.
15.:Z€6rdortrt /poarri nopdor(orlrn; /(x) = c' r'. 16.(a) Bpcirero lim log(l + _y)i_y.(Mnopcift vo XptorponoLio€rs rov Ko!'6vororl I'rl6pual,olld al]!6€q riruva!oloio.) (b) Bpcire ro limxlog(l+ lr;rl. G) Anodriir€6r'e=lim(l + I r)'. (d) Anod'iilc6rre' = lim{l + a x)'. (Mtroprir€ (0)!' va 1(]ndpol€ 016ro p6poc iva !rKp60ite0pr(6 narlviEr.) '- l). *(e) A,rosci:re 611logb= liT*(b =( + 17.:lEiirndrl rrlv (Xprtorfonorriorrio Itp6pi!}ru "f(.y) I lr.{)' 7ro r > 0. 16(c).J 18. Av prur pdn€(o6ivsr d rorqsKurdr6lio rov / p6vo.rorElra opTrxri Kard0solidrt $ I rilEtuL /(l 1 d 100)ror{i !tr{j I Zpovo.A! n rpdn€(oouwnoioTi(crro! r6Ko(61r. xaTd10! uoioTLop6ror,r6xo!,rou !0rpri6rto!€rrioLor6xooovFfpo;ro! l(€eoi,(rio! enofsvo'r1p6vou),i6r0 | ap7Lririxard0eolorEtivcro€ 1(l + di 100),lsrd on6 t '|p6vLo. A! unooaool]p€ i6pa 6rLo rdKo!6ilsioL6udAopdq ro! Xpdvo.To r€hr6 noo6 psldon6, Xp6v,o 6ivor.6r,0lt,-/6;. 6tr (l + ai 100)!,ol).df6vo 1= (t + s,200):,- a! rior o rdro( Kor(rl}(i).lerqr d!6 qopaqoutvdtepo.npdnerln urodrnioorrioouge rov r{tro or rde0r,,ro).oot6. yrori€ivora 2 ovd[r6d Zp6!o.AL,r6Tonoori€ivorueTutr)rrpo ono.l(l + ar 100)".ai).{i6ltrtro),ulsloiir€po. Ar uroefoor,p€ r6pu6fl I tpdn€, (tl orivL,noioTi(er rov !6xo 6uv0tri(,3liu6i I rpdn€louno).oli(cL n6ol 0oTLv6rov il Kord€rotu!' o rdroq 6rv6ruvI'i?opdqro Zp6vo.xor lerd n&ipver!o €ld)trororivo ApnTuo6iov ur,r6y ro\' opr0jr6!.n6ol 0a dlEr ThELpLsoplrrn rordororl pLq; dpalfii tr€rdan66v(1 7p6ror = 1 9 .( a ) O i r o u t E / ( : { ) l o gi 7 r a r + 0 . A r o d c i i r r 6 f l / ' ( r ) = l , . y 7 , a r + 0 . (b) Av /(r) + 0 yLo6),qrc ir, cnodeilte6n (logI f ll = f t f . 20. Yno0dro!'!s 6Tl.etricvo_. druorifaroi. I or,\'dprlorl/. rxovonorci rrlv /' = c/ 7Lo xdnorov0pr0!6c. (a) Yno06rovrqq 6rLn / dsv'ivqt,rord0, rprtoL$onoL€iol5 ro np6pLrt[ul9(b)tra !q 6€ie€rE 6rr -f(x) =le",7ro K{irTorov 0pr0!6l>0. tumpoivolue 6Tl /(x) = ke" tro r{inolor. (b) A€iirE6Tluurdro anoldr.oojrq Lo7iertopii rn! €mtr;.dov Driierorl6Il n / d€! €i!c'rtrol60. Yrd6eri!: A€iEI€dlr rl / as\'lnop'i va livor 0 olu dKpcovo(Ktori .Looritaro( ?rs ro onoio6ev€ivartro!€evd!!6dv. (c) d(lorr fro c)rio[olEpl on6ijsrlr]rol, 6n /(r) = /re."?ra{dnorol.oeopdrvros t i l y s ( x ) =/ ( r ) ' e ' (d) Yroo;ore6r'I"= f E.^.il:t6tt /(.r)=*d!"7Loxdnorol. *21. Mra p(ijrs!'0p76; o!oi( cinrr6vsrur!c pue!6qvdioyonpoqfit onTlLoiaTtt pd(u (a{pon ro droto 610 rt nr0av6rrt!o 6LdonqolqsivorI i6ro.rt olvolLKriar{ionqoil ILU eirar u!rilo|] Bs ror' {rpr0r6rarva!6fov rou onotdiror,!'). At n(r) €ivoril fd(o rrt ZpolLx'iofl7fn r. ur'rdotFai\'€r6fl /{1, = .A(l)7Lu Krinoroc (,routrsprord\rrr!! m0(1!6rfradrdonaon(rv6; dr6Bo!).
294 Ilapdyayor ptitpara t 0. Yno06roulr€ 6rL l"
f ( x )< c + J ,f c ,
a<x €, rl d! x > e. r6te/< e. (Epgrl\,€iorr rou! mpondvorotuproroi! paoolrlg TporprKriE (a)!) ropdoroorliro! Fdpo'rE (d) Anodoiir€6rr.qv x l(ar,y€ivqreuorKoioprofoiKorrr = /", r6rex = y ri x= 2, y=4,it=4,v=2. (e) AEiqr€6 ro o'ivol,o6r0)vrov (euydjv(n, y) te rr= y, onorE).siror 0,16tn ppcirerlv ropriKoLolrSrdor€dvonp6{aFnilrl Kqrlra suesisro'r rdtrvovraL. I&po o{i1oo ++(0rrale}'0xrl:)=s(.y)i.Aro60i_Er0 6rr I g(.r) 0ivq' ,rapoToliorpl.(Eivol {o}.'i r66qva pc).orrioot€ Xoprordrrg ouvaprio€'q, loax ^ lt\l-'-
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r35. Au16ro rp6B).ltq Xplortoror€iro uiLK6loL,Ilopopfliroro!tou K€qaiaiouIl. (a) Ano8cilr€6t|I exprivo|e,pli Ko,I los Koilt. JI {b) Anol"Llrror..a\ ) p = | \sr 6).arop > 0. Tolr
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rar4 er9erwfi owdprqo4 297 17.H ),o'yap6pwfi 36. (a)
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f rrs osrrki ouv,ip'qq oro [o, b], xcr P, I bnf,prort 10! [d, b] o€ a ioc Sraortitnra.Xplsriroror€ior€10 np6F).ttd 2-22r4 va Ssiqtt€6!r: |
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6r! ]ld olv€zri /. dloufs XprlouoroL6vro;ro tlp60),Ifq l]'12, orrpn€pdvare , t i' r I l . lloe/ 0 - ) , . = I rqr g N,prri. I ro g roilt. dXoutclqv qlriorpoql 0vro6rrlr0
0, 161€0qDnipt€dvqqopLep6s a rdroLoq r;rore, 0 < a < & KoL/(a) = 0, €v6 /(x) + 0 yro d < x < x0.f lori; XprouonoEior€ our610rc]ov6gI.oLro Jripog(b)lro vo Kqlol'iEsr€oeii{ono. ' *45. (a) dEiElE6rL,{rv rl / r{qvonol€ir[]J f" f - 0, t6r' /(x) = dc"+ be ?ro Kdnoloa Korb. (BpsircrTp6rar6q 0o Tp{iqov''av ro d Ka1b, oDvoprioerrov ro IIp6Plnpa44). /(0) '(aL/'{0), KorusrdtprtoLpono6iorE (b) Aeitreax6sc6rL/ = osiM + bcos,,r yLoK{inoLo (drla) d KaLb. 16. Bo,i.LJ;.Lqr.r 6,\oorfortc / noL.{avono,orv r.: (a) f d= f'' 'i. (b) /'' = /' '. *47. Aur6ro rp6phta, nouor8riEsroL Fero np6p),rpsl5-30,nsprtpdq8r lLq napouoiaon rrl; €K0€irKriq ol,vripqolq e€Krv6vlqq p' qv un6e€ot6n rl ijrqqopr.i '€iooort Lion. /'= / dr€,fro {rt.tLrt6€v,l(tt (a) Yro8;roulc 6Tl urdpx'r r,o ou\rdpfltorl/+0 r'€ /'= f. Aro8clqr€611 nt ouvdprrlorts(x) = /(-yo+ r) /(a r). f6) + a Ift KdeEr oEopd,vroq 6roL'/(\) 10. (b) dsiql8 6rL DndpirEL rT}' /'= / Kor rrv rLo ouv{iprron / nou rKovonoLEi .f(0) = L rr oL,vripnrorl G) rr'aurrivrTrv/,6Eiqr€6 t(x+,v)= /(r) /cy)Elsrd(ovraq
cG)= f(x + .v)tf(r).
d€iEr€6rr r / sivor1'l Ksr6n (/ L)'(x)= ]ir. a8.'t6ro / Ka.s o' v"x-iqo' roprio..q.ror,;orLlim/(yl - I ms(x)- 4 \ t-fon r l !. t01;t t nfu(pa dt(i tnvs \J / s)tt\ (d)
.
t(x)
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ru 1 er4ffirrt wv,ipqo1 17.H \'oyaptilyrct1
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log I
dl'o .x,r6u f >> C, 1roapro'JwoE F€T f' t' xcr' aldlK{v dfl Av I >> s ror F(') -J. /, AGJ= J, E, ouvdyEtcr F>> G, srivok qDvoptio€ovrard ariEoooa oErpd Aral6qo1€ Ka06watr6ra al(6},ou0d p€FooDE (Idprv €ororlloe,orlpeldivoolr€ trlv KdoeoDvdpqol oir)'69 rotlJrlroq r): FioorrlEnFig qg o€Kdoe (i) I, c', rr + log(/),log4r, (logr)',1, r+ e-r',/ log'. (ii) rlog'rr, .r',log(.x'), e"r', rks,0o9O'. (iO e', r', ,x',r", 2', e'l1,0ogx)!.
{9, Yno06ioqrs6fl 01gr, & , gr, ... €ivcr o0vs1elcduvopaio€tE.AEilit 6rl un{ip16r oovdptnol t, I oroio oDqdv€t tollJ1cpaon6 oaoui{nors &. Frd oDvsXrtE $. Ano6siqc 6'!r o logrl Etvar6pprFoe.
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Kd0euro?,o-rrof6q napqy6'/o!lqq 6ivEr,ori,Iqovop€ro A€ijl€poO€IslL65sq 0€6prlts rou AmrpoorrxoiAoyrstro[,kLdvovrino otroxlipootq. fra nopd6er?Ua, 0! F(.{) = r(losr) - x, rdr€ r(r) = lo$'
).losxdx= F(b)- F(o)= b(toqb)- b la(losil) d],0 < d, b. Trirora!ro6rou €i6ou!onloroLoiworno),uov uLoeElriooup€ rov ouppoi,ro!6 td
F{\)1, F{6)- F(a). M,TopoLiF 161rvo -rpdvoujrr l' .'," J" logl d.r x(logxi r . A' r6sorro),oyLo!6qJ" logxd,r0qoiorr{cdtnv"rpo{rrEia'61' I logtivsr'tropdyoyo( rnEouvdpqoneF(-x)= ,r(logr) x. rev'Kd,fro olvdprnot F Tdrolq60r€F = / ).dy€nt napdpuoa rnE f. Asqaltbc,ttd ouvEtis ouvdprqor f irEr fliwo$ napdloDos,rrt\
F@=I T, a).]do' our6ro K€qdlaro0onpoonoooil€vo!piokouf€napdlouo€( nouvd ypd$ovTdr !s pdol rrqyvoor;qouvop1liocrq sin,logr?,n.Mraouvdprlsl noupropeivo?paqeip'ql)I6v rov rp6,ro,,;-rsrar ororlt |i6nt ouvtiprnor. f ti \' aKp$ol-o'toi)W*, otoqeui,n ouvdprn04 ).itg quriv noDnqipvelar!e 'rp6o0€orlroi.),un),ooroop6, 6uipeortxor of\&ot on6 rrq prlr6qouvopTlio€rg, ire lpryovot€rprxaq ouvopl]ioeqKor1lqavliorpo$fqrcuq,xq0drq xar rq ouvopflioEqlogKarcxp. npdnorvq rovioontoar61!v qpI{ 6rLouvriorDE €ivaro6$vorovo pp8ooivoroltsl{i6€1e ropd1olo8(.fLo ropd6€L?$d,3e\ $dpxEt orot)(t(i)54( auvdpqorlFrdroLo60r€ F(x)=e'
t ' o * u t "'
(6.v Iri.rip€7rdrrc npooo)prvrl dnoto rov lroentonKdrv onriptsrdvq(6toxoLo)o€drpngo no'rrag p€pord)v€' 6 Esvundpxcrdrolo ouvdprrlorl. Kor, Krilrdr6uotap6repo,rinorq 6ev0a oa€porled€r vo itraKpivrrsov irzop€tri 6rr vo ppee€irrq ororrELd)8rq nop{iyousq
I o oprold! rov onoioeo 66oou!€eivft a!-/xcKp'[ivo!, o1.1.{i dl' *rdia oxp'pi!, i routdlloro! 6t' doo dro F idd ! opi!o\r!' Kor{iliiorov rp6nod)or. vq rp iouFdrolv i i rdo6po!ivo;. tulioo i o' oror-/6Ldldhq ' + .di.l.0F(i? o$pflidc,i. dqiodi oKpridni.' trou,xovorcrciv!,q €iidont G{!)' - /" ,(xDG(r)' . + -fo(r)= 0,6nouro t, Eirorpnt'< o\rp.ioa;. Atid trq rcu; d!&t Jrogo(oiori. raroui duvoprion: Fropoiv!o tropol€rp0oiv.
pi9oiord,otcitlpao4q307 18.Xroryeuitdet; (6Evooq !d!c, nop{irt siridu 611Id npoFiiporu o' a!r6 ro {0a{itaro 6sv 6/oul r!no?poqrKd id0r). En€L8i!;p8Uvu tro ororXsL68sLc ndpd/o!o€qsivor1600aFd0a1l,ro vo ppoipe fLd €i\,o1 ou1vd LdlairsparKavorotlt!r{6. Ar ropdrlpriooutE 6fl r oL,v{iprlorl + r') F 1 . t )= ; r a r c t a u - l o g ( l
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F1x) = a'ctanr (oro rd)q0o tnopoi6uf0 !d odrTitooL,pc o' rtl{irdrororopol]iptol. eivordLil 1610pid), ' t" = .{arclan.r los{l+ ir:) arcran i dx J! ", r6rs,pnopoilova oro0av6laorr6n (rpdyf(rl,, unoloyio,rsc ,oJ'" -cta";r ax AL,r610Ksa{iioror€pliler {drLropqnd\oan6}r€e66o!e olorXadi6nq 7Ldvd I]pioKo,rts nopd?ouosi 6o6rri\rD!oror/eLoSdrv (FLo8r{rdrl(qoifl oL,vapriooov nousildl tvooT]i6uv (oioxl.ip/dortt, po(i pe Kdnorouq our|]oiloloU;,ouvrp,io q. xar oufaovi'r o/s8Lqopi vEq?,ov,r6Ls!{o}.ivo,r! rr 6Lq6Lxiroio. Awri ! npo{drdh\i! !s rrqororx'irij5',(olvqpiri osrqInopri !o br{uroiolte€ip€rpia snlXnptluro: (l) H o).oxLripoort €ivoLilo nspaaooLoK6 l]draolo! An'Lpoo0K6A{)./1016, KurKol],\ o , 0 d n p f r - \ u - p f rr d r T o 0 , p o . , r 0 r ou7, r i ! . (2) Kd0016(10lnop€i vfl ofl,(lpsraorri rpo]p(1lrKri vfl unololio€rsi\,o oioKiripofrl, K{i.rDono ouorir(eqno! 6svooqen,rpirTou! vo ouppoulEUrsirs Kdnolo!o16 rouq nivo{€; (tro oLo(ir]pojrdro! nopdder./po, vo rdperc rvo Fdel!( Tvooroiq InopEi (OD61qq) oro onoiovd eiopEirorq!o1{o6rLciorerKqvoivo oloKlrtp(i,vsrt. (3) Or nl6 xprioL!si(raeo60L,, oioKi.ipoortrsivoLoitv npo./fo K6rrtrunoii oIlov , Kdr0€oplir,oro(ror Eqnpp6(ovloro 6ieq rLs owapliosrq, 61r p6vo ons oroLnEr(i)6erq). O!oLr.{i.o rr).et)roioc i,67oqeivm o oroLao,rr6g.AK6!u l(Lar, ;10rsrl! np6e€ortvo (€xdo6rE n6( oio{ilp6voDv (Kal0o e€1do€18 nL0av6rdro Kdnor€q ?,Enro[dpc,sq Fsrdqn6 lov np(iroKaLp6), dsvnpdncivd E€xdoel€ nordn( FooL{d(!c06doL)q. A!16qor paorrdqd€o6o1snoL0soprirqlano'rfu( snLrpanou! vo rKqpdoo!!Erri rop{i{iiir,r! ouvoprrl6ro,,'. fro v (lpliooltrt louocgtrog ou\dpltotq fs 0dorlrL( tropdloUosq iotnov vo o].oK;.lpdjvoLrts,0a'tp0rooro[rE 6vl1rdardiolo Is rLqr0i.,dfo'lafr'K.lltotor ot]voprrto8orv rnopolpsvo onor.ricouucEyov1610roKar{i}.orooni(i,qnopayroriqovru( 6Ldpopc!yvoordqou,apflioslq.O {or{ilotoi nol,6ivoufs nro ddro XptoLtonoLri;ro KooL'popfvo oDrpoionouon(Lrsi{dtroLdf.i,i./rtof To oi!l]oio
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20. O e €ivaturcp\arrK6E 377 ro np6Fil !o 3?ra 6no!rd I 'Fed|iEowororrlv, ! eaor.?roKd0s,. XplorFono,siars (r,o{doc r,6€iEr0 611 o o 6€v€iva1pi(o ILaq vo ono6ri(0rs611oa sivqrt)n€p0o1u6q. siiooolq 000r'o'ir.) dvovf6vo €r6tK6L,ropparrK6 upre!6,sivoLqovEp6 Av Korro flp6p),tpo4 ovoadps, i.ora6i.€l]dost dr!€rpouq di).ouq apropoig o noL,de! uq6rr Inop€i€iKolq Ko\Eigva p/ql c/q" oltolo8irorc c KoL n. Auioi tou eibotrq ot opr0poi voroLoLivrv la > ln (1809-1882). 'rp61q on6.o! Liouvill€ KaL ouo6rlra oro Ilp60hlo 3 r !€IEflihKov opi0loLiq inao6.rtru rou Liouvill€. Kovdvoqqn6 rot)q u'repl]orr{otq ouxvdanoroleirur pov,o).id 6Ldir€poEv6Lo$a rroL, KorooKeud(ovxu1 u' ou16vrov r p6no3svnapouoLd4€L yLonoli Korp6or !rcpporl{oi opr0poirou Liouvjlleirov o' I6voLr"'tdoroi.AL,fi il l(ardoraorlsrop),rielKE roDCario. (1845rrsro).u5pooflK6rp6ro on6rr 6o!?,€Ld va reprlpd\uor o[rE;vav 6roqoperLK6 uneppqrrr6 aprep6,6rL l9l8). noua6erqE, ,((rpie rpop).tlfota tro./,,loi dpr€Foi oivar DnEpForLKoi. To en6p€vo 6io or ,''o fdq u,odTou! O oprno! enlrpanolv vo Koruvoorlp€ rdrorouq rollproron!. s.lg r8isg 0oorK6q IoC 'Evu oivo).oA ldysroropr0p'i op6qrero! onoionpin€Lvu 6ou),6\,oup€ €ivoro eEiqr 6EILo aKoiotoia 6rpoar ro oroLt€iurorJ?ivElor!o 6rarat0oDv nopdAsr'/po To npoQov6q {orlv rpo}!orLK6rIra,),i}o-noi.i.o Il]'oro\rL{6opldruno) €v6qoprtfiorrou ou\r6tousivorro N, ro o,jvo),orov auou(6vopreu6v'ivdL{po!cp6 {ipflovatjorKriiv apre[drv€ivor€triolsaprofiortro: 6n ro ontoio r0J\r 2'4'6'8 'Iooq oogrvrunoolrio€r K{irToq nEproo6r€po ro y€Tov6g6n 10Z, ro ouvoto 6lov riDv ox€poiov(o0TlKoi, apvrrrroi, Kor0) €ivorkL 0!16 opr0liouo, o),i{i Pidrovraq0o 0 , l , - 1 , 2 ,- 2 , 3 , - l , . . . pqoLKdq To sn6lEva8io rpoDlilLaronoD oKLq.rpqeoivlLq rijr6rtrs!rov oprorioLtLtDv tro!ItiIvo!v 6n (1) ou\r6iov,orlv rpotIoflK6rIro €ivaLpLsosrpdqn6nopsSsiyfaro ltr 'rflipxouvno),i nsproo6rcpo spLofriourooivoiu ar' 6oc 0o fnopouosKCVEi; ox€qreiKor(2)6rr ndp' 6).tlol].d,ftplKri oDvoia8EveivaropLepiolpa. +1 (a) Aeiqc 6n. av ro A KC1 ro B €ivorupll]priorfo,r6re€{val(ar io A u B = lr: x ro i6,olduroofu a\rriknoro A ri -xal,rix€L oro 8j. Yn66Erh:Xprsrponoreior0 ro! 6oi1,E\fs ro Z. t1o (ou16€ivor (b) Aeiqr€6rr 1()oilot"o r0rve€r,K6vplr6v op18{6vcivorop10!rlor}'o (ornKriD rrtv 0Eriq on6ds'an: or' oii0Elo opKsrdsvrutrooLox6' rpit6ljrono1€ior€ ,--)
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yG+r+y,r MropoiuE ax6po vo eftrlriooulrs rc 16 + t \G r9apl6(owo( 10 Oe6plto Mdoilg Trlrielto q ouvdpllon /(x) = V.r oro 6rtiorlpc [n,, + l]. loipvoqrE
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= (,,,,, o.o +t) ft,"",*-", -1 2,Jn Onolo8irorEon' olrderLE66oerttploerqprropeivo j(prlorlorroll0€{ yLov' aro8€iEoufs ro napor{iv0) 6pro l },oftrop€pri( on66sL(Iaqiv€ror oe ouq,ocv oal.i o},}d lprlorgl To 6p1o 3 n ' l 7 n 1- l
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.,7,I 3nr+7n1+l 4rr- 8r + 63 , 8 6 3 Xpnortonor6vraq ouliv lrt! nopdorool,PpioroupE €iKo),aro 6pro,roloirspadv j(prlorouoqororX€la: uororioou!€ro aK6tr (or rs dio.rdrr Av ro lima Korlm 6. r ndoXolv lim(4, + 4) = lim a' + lim b . l i n( o " 4 ) = l i n a ' ' [ n 4 Av, €,nn}.dov,lim 4 + 0. 16r€4 * 0 l|a 6),0ru, Ia f€lol"orspa orj xdrroroN, |(ol lin a,i b, = lim d,llim b,. (Av 0o0t ollsvo eiuool€r.L€ioqouorilpoi,rl 6lqrinool rIE rpintgnp6rqolq0d'np€,r.vu €ivqr sx6!d mo no),6n),or1.'onog q 66oap€,pel.€rdprro dpro qE ol(ol.ouoiaq Ic,j= lan/b,),6nor ot.a €poi c, lnop€i vo Utv opi(ovrsrxav lla UsprKd r< N. Aur6 6loe ij6vn€tpd(€ror' ol.rioerc- 0a g:ropoioape vo opisoulsro c" dno( paqapCo€L.fro r;ro1o, - ]rori ro 6protro6oxo),oloioE 6rvo],ld(fl ov ol.).d6oqrs fltv oxo),ou0ia o' Cvav nEropoogdto 0pre!6srusiov). Av Kor1( nopo,rdvo eivorl!0),6lprioua,6€v0o proripeotov x6ro vo to 8rcordrooope oov0E6p!po 6ev0qnpdn€r vo ou!,ovrriosrs r(0lui 8DoKol.ia oro vo 10aro6€iq€r€ f6vor oqq,?lo!i o oprol6EloD lim d, = ./ Eivarre?,€ioq i6roEIe loD! ,rporllo6lrovolE oplouo$q rov opiov,r8loir€pop€lov l!l! /(r) = 1.
21.
'Ampeq ard,otfiieE 37?
H opo16fiFoovdFEoc o1ou6oproloogrov lim a, = l Karlim /(r) = l eivdrsrnvrpdylrarrr6qIq nro sr€vi a,td lra arrl,i ovol,oyio €ivarSuvor6va opiooD!€ro '!p6ro I€ pdoyl 106ei€po. Av / €ivorI oDvdpllol noDI -rpagrrrlrlq naprioraon(Xtrila 4) onot€l€iflr
\i1td 4 o116 rDe ypojluqi[itara roDsv(i)vouv ro sr]peiqorrt ypaqrKri ropdoroonrrlqaKo].ouoioq 6r),o6i ld"j, f(x)= (d,-t- a,Xr - n)+ d", r< x I, r6rE ro a' -rivEruroooEri,Tor€ Kq06{ io r, p€F}.d}vEr.A0r6q o tglupro}dq roipv€r oulvd fll l€yd),o ;.6" = _, a) l. 6rr I {a"} tsiveroro -. fpdqoqreErioilgro6q1€(6uvrrlv i ax6Faldp€p€prlcic Qopdq 6 lim _a" = KorIdU66rr I {-d"}reiv€r oro --. Ilaparnp€ior€, 6pog,6Tlav d < -1, 16r€ro lim a'6ev uflip1€r,oir. KarI' oDrlivrlv €upirspliwora. llapd rrl o v6solnouKdvoFc rr1 FeFu l"yoorrl(wou,€ivcrno olpowrx6vc roprorooge o'r7r].rdrlp€ pdq qv orx6voIrl( oxol.ou€icg ocv orivoLootl€iov |Idvorotrlv €uo€io (tlllto 3). Ydpl€r n d),l.rlpu olCfll ovdir€oo mo dplo oDvcpniosov xcr rd dpu cKo-
378
( 'Arcrptq)aroAou9iqrar (intpq) oerpi; ).or:oLrirv rou 6;1er vc xdvert'aur4l, rrlv e1K6vo. A'rri n 0160rl€ivorh.r6r€ponpoeqvriq, oli.d rofu flo sv6loadpouoo, or' ouriqrou 6jaoDp€ dvaqdp€! ovri vq opioou!€rq 6plo o{o},o!0iv Fe q 0orleelororv opicrvouvaprioeov,lnopoi|]e vo svTrorpfvoup€ rrl 31a61xaoi0. otrorHMrt'Eorofploouvdprloloprop€vto6voqvorlr68rdorlporou n€p#;(er rc c, EKr6q iooqan' ro i6Loro c, f€
= iiT"r(x)1
Yroo,rolpe6n {a"l €ivarg1ooKol,oloiar&old do.€ (l) ltdo€a, vo ppiorelaloro m6io oplopo[rn6 /, \2) Kflqxd"+ c,
c) |r-qa' = c
T6ft yla qv qKoroueio i/(a")] Lori€r
Iryro;= t
yroKd€eoxo],ouoio Avlorp6aoE,ov our6eivflro),ri0€ra flq nopondvo {o,} nourKovororei ouv0riKeq,'r6r€ lim/G) = /. Ano^EErYno€droutE nprlra 6r' lim t(.r) = r T6I€ yrq xdo€e > 0 u,ldpr(Er dva t > 0 , t r o r od o r r ./ r 0r d € Lr . a v 0 < l x - c l < d , 1 6 r 'l / ( x ) - l l < e . Av 1 uxolou0io{a,l fuerlim a" = c, r6rE(Irrifq 3) undfiErquoLK6s N r&oLoq opr0p6q avr)N,r6re o,-cl{d. A'16rov rpdro nol,6rauLotcro d. or16onpaivcrdfl
tf(
r 0r6roLo6orETLa{d0rt>0vaundpter I 0< .r-c c. yroK{i0c, 0o I ripxL fvsqqprer6qr, riroLoqdrorr Er3rK6r"pfl. 0
E.
T6 port oKol.o'r€io oll}K}.iv€roro c o}.}.d,oaoi l/(x,) ll > e1a rri0en, i.{,1npoaav6( oro /. Aur6avlladoK8r r flKo),ouoia {f(x,)} 6 ouTKIivEr !r trlv ur60€ol ripo,!pdm vtr ,oti€1 6T1lim/(x) = 1.I To O€{i,plFoI fos,ropq nolld rapo8eiyttrro 6u}K),1voD6dJv oKo},oD0r6v. rro,ropd6sl?t'o, ol or.oloD6ise [a,i Kol{4] rcu opi(ovtcrcr6 nE
d , = s i n l r 3 +. l
21.
'Atretpe; arolou9itE 379
,).+)i n=.",(.,"('+( npoqav(ijqou?K?,i!ouvoroDqsin(13)Kal co(sin(1)),avriorolxo. Eivor 6poq ort[uvflr6 vo 6roL,fEorrt drd€eoripos ,.aLKdnoraKpLripra,6xL auro[ rou si6oU;,nou va ETlu6wor rr oitKhorl qKo),ori0L6v. YrdprELivq otpavflK6 KpLrripronoL,sivor noii €iKolov uro6€F XrEi,o).).dqnols),€irrl 0dorl ?ro 6io ru {ii}.c onor5).6oparq.At]!6 ro xpLfliproSLorun(ivEroL re p{iol {dnor€! 6wortq nou opioqKov r,rqouvoprrioeLq. €rop6v(Dq Epopu6(ovroLKar ons cxoLor0ies:sLocxololoio [a" ] ].6y€ru1oiqouooov a, - , > dn 1lo Kdes, , rri-q0ivouoo qv di+ > ar TLoKdocr, Kflr riro qpoy!6vl uv undp78rKdnoLoqopr0l6( M rdloloq 6ors d" < M yru Kde€, nop6roror oproloi un{iprouv1la oKol,oLol'saoi!oL,6s(.lri-or,qoL'osq, KULKdro qpa]lav€q. oEopHrr Av I la" I EivollLl ' egivolon {or {ivo qpo}F6vt, 16reI ld, } ouyKlivcr (ro iSro anora).8ofu1o/[rL ov n ld, l 0ivoLtl diio']6o {or {6r(Dqporu6r'l). Ano^FilH To oLlvolo A nou qnor€i€irar or' 6ioDqrouqaprolouqa, Eivar.cEDnooiosoq, dlo qpoytLiyo.{tpo ro A dIfl ivo ei.dlLoro rivo ApdTgca. IoTLrprt6gcore6rL lim a, = a (:jtr'ifu 5). Ilpdtlofl. o! s > 0, undpl€r Kdnolo(a^ ro!'1{olororsi ri]v o - ar < €.7Lorio
4ira 5
a €ivol rd ririlloro rivo Apd?lLorou A. T6rs tlu ir > N dUoLrtE a,2a{.dpcd-c,=d a\<e. A u o o r o d L L r v r h6rr Ll i m o . = a I H w60cor 611l {d,l siva' qpa./rdvr sivel ftFolo an(poirrtrrt oro o€riprtrLo2: a! I {a,l 6c\,€i\'ardvrD$palpdvl. r6re (sirs Eivor! {a"} }r -aeivoL,ooEirs6rr r ia"} ooa6q oro{}.ivn. ME ltv npdjll pond. iooq qavEi 611d8vEivaLxar 16l]0 6ioKol.o vo ncl m!si! ov fro av I {d,l 6!r{iiv€r rl 6r,. rri-qeivoriooaKolouois {d" I €ivaLdvo apo?pdvr,Kor€no!;\,rD_( :ro €16pe!ol(E{pd}.qro eo EpeqvLo roiv liroLeqsxolouei€qI€ no).i AU01K6rp6iro, m16no( 0o boirt, ro spd'rlFo d! olrrLivoD! ri 6Xr 6e! eivflr Ko06),our€rprlt;\ro. I]poq ro nop6v, npoonuoriol', dv06i€r'. vu &irs ovr ol 0u'I(ififl ivaNt6rolo6o1t la" 1l < €i2tLai;> N. T6poovn> Nxolr?> N, rdr6
,ir t=, ADrrirlr€).€urqia ovro6rttc, lc" - a. | ( r, rou 6evr€Ptdl€rro 6pro1, tnopsivo lplornou€ivolnpoqqvdq ponotl0ei'yturq dLodnoonfLaqsuv0iKne(ro xprlriproro! Cauchy) q oiyx),ron orolou8io4. ovqyKqio frcE lto
Cauchy ov tra (d0€s > 0 D'rdptt€r;voq Mroaro),oD0ia euoLK6e {a") I€f141oxo}.oo0ia qpr0l6E N rdroroe rdoem rcr n, {i'or€,-rro ovrr,, > N,r6rc a"- a,l1e TpdQsraL (Auri I ouvoiKrl or! Iopqi.Lim-ld,- a, = 0). oDvi6o6
21.
'ArerpaE ard.ou9leq 381
H oloped rol, Kprrqpiourou Cauchy6yKslrororo 6rr €ivorlKovdvo pepor6arlfll oifll.ior] Itoe aKo],ou€ioq. Merdon6rlv rrpoKoropKlrKri 5outrEr{i rou Kdvo!6,6€v dvoDv ro roi.).dyLova to ono8€iqoulle our6. offlps[r^3 MLoaKo].ouoio av KqL]6vo ov silor qKo].ouoio [4 j oD?Ktrlv€r Cauchy. 'E1oute Ano^Er=H ri6rtorodeiler6111 {a,j eivcLcro}.ou€ioCaucby ov ou$}.ivsr.H an63e(r1roucvrLorp6rpou roluprspoirspLdl&F6vo6vdFrKp6ritvospa:vo8siEoup€ 6 Kdes o{o),oueioCauchy{d"l sivorqpayldq. Av rrdpoDp€ s = I srov oprot6 qg aKo},oD6ia( rvTiroro;ors ppioKor,Il aaLrch). 6r| | ndplcr,(droro la" - a,l 1 t yrLn, n) N. Er6LK6'r€p0, 0urdolrqiv€L611 la" - c"*, ( lllqKd€er?>N. n}.iooDq d}.Io a,, I {d.: r, > Nl €ilor $potdvl. oqoi ufliplouv!6vo n€rcpooldvou 0].6{).IprtTtoKo},oDeio Eivorqpo?dvrt. To ,r6proto.oD Aipforoq pdq €Eooqotiq€L T6r€61, l(dnoraDroKo).ouoiq rrq {d,l
'Apo
M6vod\roorusio onoF6ve1, flouit qn65E(i rou oqrivsralo€ oog:ov irroDJroKo),ouoiq oKol.ouoio( Cauchy oulK?,iver, r6rc rar 1 i6n 1 oxol.ou0ia fraq Caucby ou.rK).ivsl. I
NPOB HMATA l. Eiq),r€e6ore Koodva on6ro 6p1a'!ouqKo).oueoiv:
(i) Iim-+. = I. ,+ 1 = 0. (iD lim---: o-- n,+4
rur lrmv, + | - v, I I - 0. Yr6b-rtl:06 rp[nc rould;(rorov va F,ropEilr vq onodLil.err on limfn + I i n-= 0 tnr
tim!:
O.t,ro6"rir.nt
ntn
t1... k: t s k < r. {o, crdr(drcpa. ytok < n 2.
(v) limit = r,ava>0.
(vD limi,6= L (vi0 iimirt+ , = l. rv ||
m y'o"
,. -, alnl { r x r l r m- -
D " : m a x ( oD , l ytad, 0 > (, , 0 . o ' r o ! o i n ) L i v a rr o r } . i 0 o q r d v ' r p t i ) l u vr o r 6 r d r p o i vT o , r . y n 6 _
6€t(r: To ?syov6g6ir KdeenprirroqeivcL > 2 6iv€rpra no).[ on].i €Krirrort rou roso.ILKp6rpirsr v0 €iv0rro a(r).
Ip 1xl l,ml=
I, pa
l
382 | Anup\t ard,ouqkgKattd\eecE)otete 2. Bp€ir€m ql(6l.ou0o dpra: tU
n ni \ nm -r-" - -
r j r r I ' J rl - v n r a v r f 0 . ti + r- rrr (in) lim " r-
2r! + (_ |)rl
(i") rim!!ll&:!!1] nn-hl
(v) lim-. "_a,+b, (vi) rim,c,.lcl < l. ... .. 2',1
3, (a)
Tr lrlopofls vo tloite yra rrtv uxo),oooia{a,} qv ooTK),iv€r xar av xder d, eivqroKdporog; (b) Bp€ir€6i.eqtl6 o!7Klivo0oeq unorctoloieerI( axo),ouoiog t, - l, l, - l, t, 1, ... (YndpXoDv d'rflp€q,av xar 6 o eivdr ro ,floav{i6pro aurdjvrov uroro),oueL6v). (c) Bp€ir€dr,€g g ouytr).ivolo€q oroxoiouoisc ritEdxoioloiqq1,2, 1,2, 3, l, 2,3,4, l, 2, l, 4, 5, ... (undplouvrimrporo rlieoq 6pLqnouInopoiv vo dXouv aud( or urqxol,ouoirE.) (d) OsopoJteqv oxo)"ouoio t2 Ll2l2I 3 4 t
T f
4. (a) (b) s. (a) (b)
t,?,t,?
t,t,T
t't,-
fro noroiqoprofor\a undpX&qko).oueio nououyxl.iver oro d; AroEstir€6 , qv I1o uroxo],ouoio Cauchydl1d,iver o' 6vq ULa(oKol.ouo(oq {tpLo1,16r€ro i6rooulFoiv€rxdr Isrilvoplrni oro}.ouoiq Cauchy. Aro60i6r86rr Kdo€urqr 0, (e),rl Kar,yroro pdpoE
I + hr > | + nh+ 4!:)
1| >-4!)
E, r,o 1,2 !J.
384
('Arcrpe) aroAou1leE rar (itterpeg)oerpiE (a) AnoSEiEls 6ir lima" = -, ov d > l,otrowoqa = I + r,6nouI > 0. (b) AroStrie€ 6rLlima" = 0,av0 < d < I (c) Aro8€iqr€ 6tL[m fi= I cv a) l, odrovraq i6= I + I Korrcdvowag Iu rlort1,o.Ji "*.it (d) Anolei€te 6rrlimdrt = l, ov0 < a < L (e) Ano8€itrs 6fl lim{,t = L dKotroueiq eivorttdwoQpotudvt. 11.(a) Ano8€iqr€ 6r pLaal1/K).ivoDoa (b) Yno06rc'rI€ 6ir lim a"= 0, Kqr6n rde€d"> 0. A,to6€ilc6t1ro oivo)'orov d" 4er iva Fyroroalol.t€io. opr0p6v 'jfl 12.(a) Ano6€i(r€ tl ,,(log{aLl)-logr 0. Aeiqr€6fl
4 = -xr- c. T6re, " "t
/(xr) .ili7_
trq rdnoroi oro (c,4). A€iqr€6n
J@ f(x) aF'" rIfi-7'&i tqrn€pdvar€6fl
,.-=#R', trh)e,-tn 'flo Kdnorc oro (q xr, ro1lerd 6rr 4r
,.,u,-,=$ff,,,
386
( 'Arcrpt;)aroTotqG; rat (drcLpt;)otLpiE ( d ) ' E o r o m = l n f / o r o [ a , r r ] K a , i o r o M = s u ! l , f " I o r o I c ,r j l . ^ . i t r € 6 r r r fi0oijoqNewton€ivoLsmr\)titts\ ra, c < n/M. G) [016reivflro dnoq]1oro x, L6rflvf(x)=rr-A; A! rdpoLusA : 2 Kolr0= 1,4,noipfollls -\ = 1,4 .{r: 1.4142857 x, = 1,4142136 x, = t,4142136, noi 'ivar 13rtorpLpdq pdrpr!o 706'{00rl(0\ir qioI :t!r,dor€ 6 oapr0r6qro\r ooorriv &xoburdjv\utqi(,rvro!).drLorov6r:r],coro(6rav xdos qopd. AL,r6, oL,olon{d. 8i!dL€EaoefiLo!d!'o on6rlv avro6rtra(*) o\a\ Wn < t. 15. XptoLrLotroLeior€ rl !600;0 Ncvton71avfl unoLo?io.rE npoosT?ron{d rLr piqrqrr,jv nro l(0r(d o!!dpltoeo\,:
= "r(r) tan.r cosx xovld oro 0. /(t = m.Y rr Kovtdoro 0. o r o[ 0 .l ] /(i()=x'+r-I /(jrt : ,. r./ + 1 oro [0. t]. *16. Aroijqiirs 6rLuv lim a, = l. 16r€ (r) (it liii) (iv)
, . ( a + . . . +x " ) YrdSsLtt:Au6 ro np6B).tpa!ordqsrno,,rirLrro np6B).1[s13-41(ouolaorlKdlivoL !'a €r3rKinopinl0ot). 17. Ynoe;rolrLc6n I / sivor ouv€riq xor 6r + l) ,rG) = 0. AnolsiErc6rL liT /G = liq /(x)i.r 0. Yn66ErErt: &ir6 10rpottoLifsyonp6Bl,IBo
*18.YnoodroL,us 61ro,,> 0 ?r{,l(d0€,KaL6n lima" ro,=l.ArodeiEr€6rLlmVt=t. Yno8ain:E6d,xperd(ovrur nflp6porosnrTsrplirro.o oro I]p6, rL'ouldnouSori).Eiou! ptrtlo t6,Ioii !s ro y€rov6q 6n [mfi = 1,t'o o ; 6. 19.(a)
Ynoo;roulc6Tlia"l sivolfLo oL,tKl,ivouoo ororo!8iq otp€i(ovro! [0, l]. Aro beilIedrr ro lim o, BpioKrrorxLnur6oro [0, l].
(b)
Bp0irclLrooL,7,.iivoL,oa aro).orioio Ia, ]ott€ioiv loD(0, l) rfroLuri)orcro lim d" vn lrv arlKn 610(0, l). 20. Ynoo,rorJr'6rLrt / eivflroL,v6riqKol6n ! oro).oL,oio x, f(x), f( f(x)), ("t(/(x))),... "i o![Lirs, oro l. AnoSeiqrc 6fl ro I 'i!ur (oro0sp6orrEio' q! f, 6l].o6ri/(l) = l. Yn66aitr Aio EL6rkiq n€prnrdroerq ;1otlvriSl epqauorsi.
t/ita l0
21.(a) Ynol]drou[ednr] oro[0. l]{dr0< /(r)< l./,ord6sx oro[0. f rivqrouvsziq ll. To Ilp6p1,r!( 7-l I 6€ir!r 6rr ! /;xsr avaoro8ep6ousio (!e ntv opo).oyio ro'r IlpoP).i!flroq20).Av I / €nar drtoDoa,r6Ts]nopoulsva ElorD'rdooupE rLlotroi,l nro Lorupinp6root:rrs xdeE.roro [0, l]. t qKo).oL,0io .r, .f(r), "f(f(.{)), ... aI€L6pro(no! €i\'orava?KaorLKd o1a0Ep6 oitfsio. ord io IIp6p?,Tua 20).Ano, 6riir0 uur6vlov roTDprot6, €qErd(ovroc rrt olrunspLrpopd 11_c oro).ouoicg 7Lc /(x) > r Kor/(r) < x. 1 {oLrd(ovro(ro :xiro 10.'Evo31{i/por}0at]roi ron Ei6orq1plortonorsio Li levoodoro Marhematician's y1ovs xdvEr Miscellany
21.
'Anapeg aro),otA{eg387
Kripu1pourfp rov oXI[drov: (fla rov €noflt fotio I p6vr]on66€rElnouxpEr{i(srorsivo'[0ur6ro :lit'o],,. r(b) Yro06roups611/ Kors €ivar660 oDvErEiq ouvoprlioslq6r() [0. l], I€ 0< rlv /(.r) < I {o, 0 < s(x) < I Io xdo€.{ oro [0, l], ot oroisq'Kdvonorouv aK6!o,6rLr't/ Eivoraqou6o.^eiqr€6n r / KaLrl / o I = sol. Yno06roup€, Dndpxer 6vaqopre!6qI I 6i(o!vivo Korv6oro0ep6orlfsio ! dllo ),6'110, ldroloc 6ors /(1) = I = s(l). Yr66€rhtr:cxlvoiore€nr#yovraq6vuorq0€p6 orlt|€iornqg. +**(c)Io,rri,€r 6T1I / sivtrlo'iloL'oo; ro olfndpootrdrou (b)t;poDqIo piqqv ur60€6rt {qr To 1dXto6!noro np6p?,Ipo20 €ivorr€],rKdnoli rro ZprioLtoan 6,rr SEiitv€I, oroocpo[or]Isiou,FooiqovroL oro vo ndpoupe on6ro nrc orlpovrrxri(0eopriporo uepLKd ftpiar}.d ovrrnpoo(Dnern{ri, o{o}.ou0i€q rrlq ropqiq & /(:r), /(/(r)),... Mlo €r61Kri, aro npdp].rtpo23(7roro oroio loq trposrotltd'rroorl161()1()! 0eopitoloqnopouold(0rol np6phu0). ftr ro e'r6pevo 22. (a)
(b)
6 ov c + l. r6rs Xplorporo,sior0ro [p6p],r1po2-5yLavo SeiqEr€
61 Yfoe€rours611 c < L ArodsiEre
= liT-c'+ + '" o
(c) Yno06roDpe 6 {x,}6ivorlra o(o}.oL,oio f' lx, r,r, Sc',6norc(l.AnoCauchy. 5€ih€6rr I lx"] sivora'(o).ouoio *23. Yno06ToD[e 6n / Eivorpraouvdpqonoro xrdroLfl6ors (-) < c .{ - _v, T1o{,i6ex {or y oro x, 6rou c< 1.(Mror6roro "r(x) ,ftr).r€rdr 1"f o!oro14.) ouvdprro! (a) A'rodEiqrE r1 611 / 0iv0rdlveliq. (b) Anodeiqre 6rLrt / d;(0,ro roLrjdva6rae0p66rtfsio. q\r dKo).o1,0ifl (c) Eq€Id(ovr0e
.r,,r(.r),_i("f(x)I..., '/la flaaio x oro [d, b] snoa€iErs 6rr ! / 6iterrp,iyfarr dvooroe€p6ortf io. (ADr6ro onor;),€opq, o' 6voysvr(61spo ntoialo, 0ivo,1v0Jo.6 oovro dipta rrtqouorol,rgt. Ano8ri_Ere 6rLov r/ EivorsroeopiorrtroL lf'l < l. r6x€r1/ drcLro nol.i dvo 610€sp6 otrsio. (b) Anooeilre6rl ov tr'(.x) < c< I yLo6rq rq x, r6rs l/ l'Z!r6ro0rp6anr€io. (c) AdJole vo ssiEere6Ilrun6e€or/'(x) < I 3s! sivolopK66voropdsEL.i,Io?ra rTlvri,rapqrora0epoiottsio! qq/. Iri Io vq slooqo),iosr rp6ro, ro qvliorpoqoiou trpotToutrvotl. 2s. AL,r6ro np6B),rFasiva1,Kori KdnoLov 'Eoro rq br= 4 bi,r =/(4). Ano8eihc6n, ov ro rotiopiqerar0?16 6,,Flad{o}.ouoio Av b=lim b, uripler, Kor/'EivaLouv€liqoro,, r6x' l/'(b) < L Yn6selqrt: T on6ro b. {or ra 4 l/'(b) > 1,'6Te /'(x) | > I rru 6io ra i o dvo6t('ollfd Llpor dp{srdpstdlo r. EEsTlior€r6po rrt!/ oro [6, 0d ppio{ov.olo aur6ro Drdorrtpa.Tro 24. (a)
b"l. y1flrord a > 0 ro oippo].o 26. AL,r6ro rp6p],IUo 81€p8Uvd
d I € r v 6 l p a . M e d l i o ] . 6 y r o ,o v o p i c o u u s b r = d , , " * L = d ' " , n 6 r c u d p l s t r o
'ArcrpeQ Kqr(tinEryq) oEtpi€ 388 ( aroTouel€q (a)
Aro8€i€rs 6n, av undpt€l ro b, 16r€ d'= ,. (H Kqrdorqol €ivor nop6lol0 ge
€Keivrr rou npopl.iporo65.) X6l9ovof€'ro ttpoq (a),ovDrrdptErlob, to a pnop€ivq ypoqsiorl popAriy,r yLdKdnoLo /. ncprypdvr€rrt lpoarK{ rapdoraol ntq gcv)= /ri Kqroupmpdvcte6n0(cSerr'. (c) Y,ro06or€ 6tl I5d<er''.A€i6r€6Tlr{4} sivoraiqoroaxor61lsniorlq 4< e.Aur6 ono6Er\-v661 6 ro 6 urrdpla(Kaerirq Kor6rl b < e). yro o < I sivaLl!166ioKo}.n. H ov{i}.D6rt (d) Xpitolponord'woq ro np6phpo 25,8EiErE 6fl, av unriplerto b, r6re e I S b S '< e. Agi6T€ dn€rtud.L e d < eir'. Ard r6po l(ororo €qtE0a uno0drougs 6 e-'< o < L (e) Asi6r€6rl rlcuvdpqoil (b)
-
(0
(s) (h)
logx
sivaLq€ivoooooro (0, l). 'Eoro b o !ovol1K66 op1€F6q yrarov onoioa'= 6. AeiEr€6rr a 0 urdplouvdrErpoortlEioaror A,rou l|(ovomroivttv lx al< e. 6n to lin A €ivario pelo),ir€pooprar6orlFio rou A, raLro .IE A Ano6Eiqrs oivarro UrKp6r€po. p{iqlier drr, a! H ouvnoLotdqgoperirou O€opritroroE Bolzatro-Weierstrass A sivor6vodn€rpo o6vo),o spr0!6v,!our€pdl€ror o' dvod,Eror68rdornsu[a, bl, r61Exdnoroor €io rou [a, b] Eivoroprax6sr!€io ro'r A. Aro6€ierrro !e 6tiorp6rouq: Xprtouonor6wae rrl toperi ,roudj(ourrrlEnanobsiqaoro K€ipEvo. Y'66r(l: Apoi ro A €ivardmrpo,u,rdpj(ouv opLogoi ,{,, x:, xr,...oro A . iroKsKprpdvor Xplorporor6vtq(to 0E6pruorov Krp{rnopdvov Aroorrudrov.Yr66e(t; Av
390 ! Aneeq)oKd.otqittK&(dn pq)oaptt ro Ia, bl drotpo0€io€ 8io 6ro6rifata. rou).dxro1ov dla rpCftL!q nEpLd1er dmrpuortFsiorou .4. *31. (a) X ptortonorsioreio Or6pitl10 Bolrano-Weierstrass 6fl, ov I / tLqv qno6ei6€rs €ivq'o'rvEX'iS oro I d, b], 16r€rl f rilar dvo Qpoyp€vrl o1o[d, b]. Yn66€rlrl:Av rl / 6rv sivorrivogpolrdvr, r6rc uflipzouv6tp€io.x,oro [d, b] pe /(r,) > r. (b) E, ot6. XptoLFonoreiorE {o OEd'pIUd Bolzano-Weie6tass trq vq ono8€i4€r€ dfl, ov rl / eivoroov€Xri(oro [d, b]. 161€I / €ivarotor6lop$asuv€Ziqo' our6(p.dnsK€edioro8, nopdpqFq). +r32.(a) 'Eo1(l) 14l I oKoiouoio ll2r21t7 1t141155
t4t2 T -
6
6-
Yroedot€6110 < d< 65 l. Odroup€ N(r, a, 6) 10 n),{0otr(,)voK€ptrio)v j < r, 7rarolqo,ro(ou6 ro a, avi{n orota,bl ('EIoL,N(2,T.+) = 2,xorN(a,
6n +.+) = 3),Ano6€ili€
lim (b)
N(,, a, 6)
D
,.
Mrqo'(o),o'reio xctaveprlp{vq oro [0, {a,luprSr6voro[0.l] ].d7rrqropor6popga 'il ov IimN(r,4b)=D_d 7rc xrior a roL b Ie 0 S a< b< L Ano8€ilr€6rr, ov s €ivotltlo xi.rloxoni ouvdprnotopros6vrloro [0, l], ro, rl {a"l xarqva!€rqrotor6lopqooro [0, l],
Ji
(c)
;::
AnoS€iEr€ 6lr ov rt ld,leivoLotor6popaaKoro\,eftFd\T oro [0, l] {or I / Eilor o).oxirtpdorFloio [0, l], rdlc
't l, 1=m!91J .) !t'" **33. (a) 'Eoro
/ lro olvdprilsl oprop€rno1o[0, l] r;rora 6oTsro lim/(/) vu Dndplsr To rro \d€L o oto [0. ll. fro l(d0cc \ 0 oroijci{rr orr Lfldptouv,rErepoo!6v0 trLiooEF6vooir€ia a olo t0, ll r€ llim f(y) t(d) | > €. Y,r'j6e(l: Aeil1e 6ll ro otvoio ur,rrivrov olpcir,v 6dv'propei\'0 dILr dvooprq{6 oltcio.{. d€ilvowardrr ro lim/(y)6€v0o unrlpxe.
(b)
Anod€iqre 6Il, !E rnv opo).orio rou npop.iUoroq20,5,ro oivo),oruv st!€iov 6iou n / civqrqsuvclriErivuruprfuiorto. ALrdoaovt(i€.r oprorr(doro epdTroorrvi1€(€6. Idr€ I / Forou Upopl,ilqroq6-16.Av I / 6t€r!6vo €nouor66€E civorolwXiq o(16qar6 ivo aprolioruo oJvo).oolleiov, KoLsL6lKdr€po, il / 5svuopri vq ei!,qlftqworjoouvElrig.
KEOAAAIO
ANEIPE:>EIPET
tro npol?oilEvoKrqdtroro floo-rd-/oJr€ flq dn€rp€q aKolouoioq, ll3rtv np60Eolvq lrslErrirorrg ooup€10@opoiogor(i' or+d:+41+o oUr6ro K€9dlarc.Aur66poq6€v€ivorxdll noutiv61orld,€ioc dp€so,7roriro deporva ols qn€ipov10 rlri0o( opr0!6v 6€v6;(€rFlfl r(irpooprol0i.A!16 roD Fnopori,lr€ -!cpr{d q0poiotqro' !o opiooutcrivor s!6 ro dzflpo doporotaeo rp6,rsrioorqvo oprorcipe rrl oori€olaourrirvrov leprKrbv yrova 8ropope6ooule oopoldfrirrDv. fra ro),riFof ruTL o Urtlovlotr6S aur6vrov oprop6 yLorov frd vourrdplouv Kdror€q s),rd8€q dl€rri6rlavqnrL'I€si olo rporlloip€vox0qdl,oro. ..rouorripo! * * a, f ro uno),o7rol6 oopoioparoq a, a, lEpud o€poio[oros, 0d rp€n€ lcol 'mo rpoo€T1io€rq l{ood}g ro vo roprordvouv6L0 KovrLv6q r tiv€roL6}.0xolmo,r€-rdlo. Aurriqo clsuaiog rolrrptop66 roo6uvolcilr' Kdn rropanrivo an6dvovaKordordro oprop6 opiov:ro (irelpo {i0porofo'd, + ar + dr+ - npfn€rvo €ivorro lims".Auri rl npooE.r]l' -rlqd uKolouoio oI oqrlveravurqorrxd q6prororo doporotq, no),),6voxo),ou0rdtv, I €',(oi6ro1ovq prlvdttl 6pLo.fra nqpd6€qla,I o|(o],oueiq [i" ] FnopEi t. ll' a, = ( l)-
l. I.
1,...
6iverrtv Karvoip|ooxo).ou€io s r= a r = 1 , .!=dr + a:=0' 8 = a + a 1 +a t = l , s,=a + ar+a,+ d{=0,
dlorwr€enrKrdonqroo oprolori 7rortv onoioro lin i, 6evu,(ipX€LAv xaLuriiplouv Kdnor€q .o, Oi'oupe ,c npop),iporq9 ror 23-ll), rord(arovo,r6e€urro ro 6 1rrprrtfq "#lielte oxo).ou0i6q dev 0odXouv oopoiopqra. fr' aur6 rov ].610,dva6lrv(rd oro6€xr6Eop1of66ro! o0poioloro6lroq orol.ouoioq vo trsprd-l€(, ouvror6oa,rdrcra opo).o'/io no! va aroKpivirrrq 0d'trp€trs oov ouoLcorrl(ri r0rvonoiov10 oopoisyqropnoporj!vo oproroiv qd fl( irt6rrpo ruxsp;q oro),ou0ieg aro].0!|€irg.
H ol.o}.ou6io oopoiorFtov rl oKo},ol]€io {a,) ).dy€rar 1r,} o! rr}.i!€L6mr
r , = a r+ - . + 4 . t' ourlivqv n€pinroor,ro lim s, o'rupo),iq€1ar ,r€ ) a-ti. ).ryorcootrnrrd. ur d, I o. + ar I KoriiTErorro dopordrorTlqoKoLo',oia( io,l.
)
392
ottpig | AE pe aKolouqiqKat(tiEErpECl H opo),o?io rro! elxoLvr(ioog€ o oDr6vrov op1op6, cvrrKoolo ror olviooq lle Lr,6repodKprp€iq €{Apdo€rqo i6rogo rir}.o6oDro ro! K€qo},oiou €lvotrappdvo(0116 olJlrivTlv tflorlJplvay].Lnooa.'EvodnerpodoooLsluZ d, ouvrio@q ).dyc r drErptoeed.ottoL1 ).6(I (o€rpdD 6iv€rdtQoorto1nolaon rou urrdpt€rUsntv dffirpt orrci.ouoiq Ia"]. A!{i va idlrr drLn la,l eivo, aopoioru i 6X,.lirr 6t vrleoEdll q oapd !
a, o',F) ivcrri 6xr.
Arri rl opo).o7iqcivqrxqrd xdrorovrpdro t6L6ppue!I.1Loti rJ;ttporo
t o, *'r" Koi${8prlnEpinloortouppoli(erivov apr0p6(ripc6€vf,.opEivo (ou,x),iver,r,rv6 6€v oDlpo),i(flqnol.il('edroro qv rt {c,I 6eveivcrr0poio4rl.nop' d},qourii,auniI II Tl)mni y).6oootivar sii(pilotil {or (aoupouavt, r(or eivsr ardesvovo o6rlyioEroE },ollrde M€prK;q ororXed60E aprelrlflxde,rpdE€K [eraqnon€ipovssrp6vsivqrdjr€s.qouv€rclsq rou opLolo6:Eivsr!u o,t?,idokqol va SEiE€rE 6rr. a! or {a,l Kar[6"] sivoloopoiorpeq,
^ +s ^
i ".""=". [poE ro nop6voori( or ro6nlr€66€vri!fi r.dr ro16 ev6ro96poweg, yrori 6Ev6]{0olr 'Iopa&i7uo1oqepoioqr0)vqro).ouord'v(€Kr6q0116ro rerpr]rdvo ,topo60iTpq srq onoio or 6poreivorr€),rxd6l,010). nporoi nopouordooDl€ avqea, llo a0poiotplal(o},ouoia,0o poutl rdrorE:/cvr{t( ouvo{re;o0pororg6rrlroq. Yn{ip1srpla ava?roiaKorLr(ovri ouveiiq a0porou6rlroenouF,ropo|jF€ vo qv ovoq6poujr€o$ooe. H ol(o),ou0io {o,} eivaraopoiorplcv xar grivoov 1 cxo),ouoio {s,l oulx),i vEr,ro onoioougooiveL, orjtqovo p€ ro Osdrplta2l-3, dv Korf6vo av lim_s, - s" = 0. Aurrirt ouv8riKrt e(rlq: fropri vq teft?Lortlsr€i p€ t porioeraqq oplLrlg cro),ouoiaEog ToKprrHpro ror c^ucEy H aKoloueia{d,Jeivaroepoiorglav xcr pr5vo ov lim a,+,+ +d'=0 Av ro1 ro rprtlipro io! Cauchy61€10€opnflKi oqio, l8v €ivaLrol,i tpriouro ylq va qnoeav0o6!€ o(oloDoioc.Mu antrri 6I('( 1lo qvaepororp6qra xdnouqouyK€Kpuivng oDv;neroron xpuilpiou rou Cauchy,Iag nopfl€l tla drrlraio ouv0iKrl qopororl6rqrsq nou€ivol'roi.rjollrowrcri lla va tnv qvae€p0€i. Hlr\oH$rro\ vHlF\tltrol AvIlo,J rivoroepoiortllorc
I* "'=o' AuriIouvoriq rponinl8lo,rd10KprfllprorouCarchyov ndpoDlsm =, + I ano6errrv6c,0r6LuqKa.a,r' cuosioq dq eEiq.A! Iit i. /. rorc L i m o , = l i m ( s , - i , , ) = l i m s , l i ms , I
AuorujadJe. oufli I ou!€iq andt.r nol6 ano ro vd eivor rKorn. flq nqpd8€rlpo,lin l/n=0, n1,1,6 rl oKolouoioll/r] 6€v€ivaraopoiorlrl npdylqfl, rt ofaionoirtortrov
o'tp4 399 22.AneryE€ dproll(ivl/r noDaxol,oDe€i, deixvsr6TlI aKol,oooid {s"}6eveivorgpclprivq:
| + I + I+I-!+...+f
'r2+3-4L * t + 1 +
567E9
>i I2d6!,
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0 d6c,.
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r ....
16
-dva 64utvoK6Ino or66€rqlerou 1pilorlr6nolrioop€ o' 0016ro ropd6Erypo H F60o6oS jrrlv qov€pd)vel rlv avdyKrlllo Kriror€(ttLo va 10 ol(s9rsinordro! ea F,ropoio€Kav€iq Aur6Eor p60oEor ftpoo6ryror1q cu/ovrtr:vnpoFl,qpdrov. 0o dvdturcnorlFivsqFe666ou( rrulooiv sivrotq (ILoori owdq0a lraqd6ocrlrqv dI),1 cr66e(1 rou 6n 1.;=-' | ltn6ev owxlivEr) ol,ld €ivar dnapaiqro va 6dnoupenpt6& F€pl(d ntrpu8€iylcro o€tp6v rou H no olpowrrri a't' ,jleg flg dnflp€qoapdq€i\,grI q€o|r€rpuci sErpd,
) r'
t-r-r'+l+
M6voor ftpulrdrq€q r < I eivorevSlqq6pouogq, lcti cv lrl > I 016portle o€rpdElov oro 0. ADrdgnq o€rpdE vo rrE r€ivoDv I€rproloile, ylali ra lreprxrio0poioFard Froponf€
s , =l + r + "
+/
lropoiv w rdpouv onloiol€pn topqri. Or 6L'oro6q13c
s , =r + ' + l + - + / +r'+/-l
rsz:r+r'+ o6rllodv61ilv J'(l
r)=l-r"'
i -'
l-r"l l-
t
(n 6roip€oyl[s 1-r emlp6tr€&r'rldri6sv Eler{i(oDt€qv n€pitlrc]oqr= I). T6po 'E:reror rin lim r'= 0, aqon lrl ( l. .l r- r-l =r: )- l = l i m = ' r-@ l-I
l-r
lrl N.
.s A).).dr a,rolo'oto 2. ) b orvor,ouor,y,(),rvcr. Tdrc. ro Oldrprfo I SLiXvndTr1 a" Z :, o' }Kliv,. Ko,o,rd 6rvcrdtfrorofyK).rort 6trncrrlc6Lrpdq Z a,.IoroiadXctFo!onr.nr pool'6vo' ieoq €nurp6o0€r{Dv 6p0Jv. To cr riorpo,po nporim'r oliouc.oooi ilor l-crriior, dr. lim4 o, - | c - 0. I Tokprrliproqeoi,KprotEpdqSiverKld,aolpovrlKriKprflipu6noutp!o+ono1o6 o€Lpdq noD;rooF6ri6t f€tsnioer, oqv Karol.ir€q.Av 51o1,66our€ rr.tY€op€rpLKri o€rpri ocrpd.noipvo, LL ro ,r.o6rpovnxo0n o)a ra Z {. \'r Kat Ltotiv oL'y{).,vo,o0 Kprrrtpla(r0poro1p6rlr|rE. o[orHMA ] (roorrHpro ror Aorot) Toro d, > 0 1laKdor, , r(orqqDlloe6oouls 611
IJr, 1Z
o o ' , , i . v , r ' , - L A n d I t v d ; ] . 1r i . op d .u \ r \ l . r o r l o r o p o , d . ; c v
rcrvovL.on doutZ o on"r] ivur.(IspsrqpcioTLCjn rovl6yoLivoro, aroorKd Yr'or,rd vo Ppofs ro lin a,,i d, KorE, ro lima,i a",rl) Ano,\EEH Yro0droufsnpdrro6rLr< I Arq),6youts 6vovopr0p6sp€r< s < L H u,r60rorl 6,n&ll = .1 I 6u I nopI,r \ororoqNr:roroSl;orc FacLtaopati(cr -'r
i,
y r un ' N .
Aur6 lLrop€ivo ypoq€iroLoq d,+ < sa,rft n> N.
396
( 'Atrapeq) aroTou9icE rat (intrytq) otrpiq
ad+ S Jax,
a*,< sa*, 5 slan,
an*,{ sra". Arpori n )
c,s' - a' ) sroulr).ivcr.to xprt{prorqc or,tKprotEEciXvcr 6r n
)".=I*' sulK).ivrr.An6 ouT6d,rcTor I oiTxlrol rqqonpdqI c". H r€pincrot r> I eivaroK6po'Ilo €6Ko),r.Av I < s < r, r6re Dnrip1El ivoq opr0F6cN
o''
>rr'on>N,
nor:olpcivir 6n ax+r> ons'> dn,l = 0, l, ... ADr6 aro6erKv$€r 6'!r or 6por qq {a,l 6€v oD'rd,ivoDv oro 0, dpa I {a"l 6€v 6ivo1 a0poioLpt.I Lovor^ri caoppoyrlrou rcprqpiourou^6you,osElerdooul'Ern oe'pdi l/r:AvOtoourc = a" l. r:. roipvoujrc I
o*, =('*D
'"
-
!
ol
(r+l)!
-
I
,+ I
nl
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h9r!
= o,
' l/r: srryK}.ivcr. Av dvriIr ournv,IlipouFs rl o€rpd
Io oroioono6tlxvr,{r 6n n oflpd )
oroocpdE 0rTrK6q opr0!6c. r6rE Z /", r:. 6nour EivorKdnoro!
-(, liln
+ lI :'=limL=0.
n1. riparaLn ). - suyx).iver.ErrroL6n
lE;=0, xd roDft€rri6noro8€rj(rsioroKsQdl.oro 16(n 0l!65€16l rou silaus660€r€K€i,poo(6tov orq i6l€gr66eE p' qur6qroD1plorlonol{elKov sro Kpl1ripro roD}.6you). T6l.oq,av 0€oplioorJll€ fil sErpd
22. AttttpcE oerpiE 39?
ylori lim (, + l)i n = l. ADr66€ip€r 6rr, ov 0 < r< I, r6rE n
t
€rop6voE lim rr' = 0. (Au16ro anoli),€6porTpoeav6q lolljer l.o16flv - I < r< 0.)Eivall1o (ali dsqq va oro66i60r€ ril'€oooDr6ro 6pro,topie vo :(prlolFonoios€ €v8lopdooq ro xpldpro rou 1,6T ou. Av l(arro KprnipLo roul6tou ircr rcpdorrqoroprnxri ontooio, oav rpayFanx6cpyol€io 6ivoLoulvd otololr8DrrK6. Tva usrov6Krnpq rou (prrrlpiourou 1,6you€ivarro pnop€i plv vo Dnololi(€ralsfxolo, i Kolva llv u?dp1Er. Mlo t€Tov6e611ro lim o"+r/d, n1ooo9opi o6uvopionoll elQovi(Eral16 sEop,rmlri o0tv6filra, Divor6r! ro 6pro0a ouT1t no'reivo! lulopofo€vo €ivorioo Lr€1. H r€pinroorllim d,+,/d"= I €ivoroKprF6q arpoo6l6prorn: ov d, = I / ,), o},}.d n {a.} p'rop€ivo prlveivoro0poiorrt(ylanapd6sryFo, propsrKcrva crvor.lo atcot,:g pa( €u 6ci1cr, c,rotcvo Kprrrlpro n.X.,6r,tr L
tftrt'
/lY
\,+ ti lim_
= l.
rlV
\t
pi0o6o11ovo aroqarv6polr yro Aurd ro rprfliprogii( riapits! tto rel.€ioq8roqop€rrKi rI o$yKl,ronri ori6Kl,lot orreipovoerp{irv 6noEror ro rprrrlplo rou l6you, eivcr prc rit€sTlsDvErEra yrsrn otlKprrou xprrilpiouqqo$yxprolq,ol)d n o€rpdrou 6ro]#youpe ffl €ivol1€l.Eioenpomnrt. of,oPHrfi1(to xPnHPIo ToYolox^HPoM^rot)Yro06rouF€6r! 1 / €ivor o€rlri ror eeivouoa oro 0. -), Kdr6n /{r) - d" yLor.d0€l'. T6rc1 a" orix).lvrr cv roLuovoov !o 6pro |
I'"'=*1,^' urrdplsr. anoalr:H H i,rupqnrou[m | / civaLroo66vottpc q oiyx].rol rn( o€rpde
J,f +Lf +J.f+Tdpa.doo[ I / EivarQ0ivoroaaxouuc(tlrilq 2)
/(,+l) 0, I out(Lrol rrt{ l/a, civsr roo6t'vqFq,alr(im rprdpro rou otoxtlpdrFaroq, a F€ilv 6,rdpqnroD
f-t | - a|t T6pa
t^LrrJ-i-,, #rL,'wp*1
Jt x'
{rge,
a v p= r .
Aur( 6et1vaorrro tim/" llx'dtD:ltrpfirrv p> l, al,ld Er qvp < t. ap'r ! r1r, oqrl,lv.l axprpd4 &avp > L Edtr'tftpa,n l/, ono{ltv€L I Ta xpnip4,rou p€).€nioqFe oq t6po,€qopF6qovtar arolou0t€6, f6vo oElui-opvlt1k6E jrropo{t€ 6!oEvo €EsrdoouF€ Kalrrg Fi4Etrr66aKo}.oD0i% tov {&o axpf6( rp6rc. F€ IlpaY|ralt|cd, aqof
l"=-(:-")'
22. ArerpeEoerPiE399 peroaep€rot oljtotdloc '.ot orrc Fri-otrr6, 61oup.mr ra rreFi-opvlrrr6qaro?.or:oieg (or Kor opvrtrrKosq dpouesivoLfta dl).rl nou ri€ptdloDv o€TrKoiq xfq- or oKotroDoi€q 6pou6,InopoiI. roroErrrotgr\orqpvYllr(oL\ Av Z d" Eivdrlros'(olouoioror, reprdl(cL tnv oKolouoiq o!riyr'or riv vocqctdooLF Z
o,I. nouolot ot 6potutgcivorfi'qpvl-
rtv nreqv6lltovq dlouttsr€1d6or 6)'€qrLq€v6roqdpouoae nKoi.Av qvdjrslqo}'vorioolF€ qv gropoipc rpo r).6xowoqro a1r6yu vo oprlooupe rrl.tpoqopisq apxrri oxo),ouoio, pro xeivr,rv ro:vurolor,€tiv noug cmriv r1v 6rudrruoidustorp6rowar oe ouTktrivouocc
i+
ov I 6ctpdZ H ocrpd) a" Idlcrcronoliroq6ulxtivouoa
a"lrivoro,YK),ivouoq.
(Mr.6t"roqrro rornrd,I ato).ouoia{41t6}ftor ord'6ro)qro0oia{q av I orolou0io { la,l} eivotcOpoiotln. Av xor 6eviloupe ro 6troioFovo r€prlivoDlonolld npdtFordan a!r6v rov oprop6, sno8€rkvioror r6otpEtlKiolpcwrx6g.To er6yevo0e6ptfo lrilv€r 6rr o oprof6qtouldttl'eic4 tilpqorog. 6ev eivm Ilorov, a&lPHM^ Enislq, Ira oerpd r Kdoc o'rol.fioq oryK),ivoooso pd eivarouyK),ivouso. oDF).iv o,roL6ro6, dv korf6vo qv n o€rpd,rou slrlpari(€ruratt6tolq oErrxoie fllq dpo!6 r.o1ilo€rpdrou ojtntoriqerorcndrorq cpvltrro6qtrlg 6po!q,oL'yKtrivow xar ot 6ud.
.
i,
lrroar=r, Av r I
rriruonrito {p'nip,o touCauchy. la" I or,y 0 , { d". -" ( 0, ov4d,>q>.>0, lima,=0 T6ft r osrpd ) (- l)"a-=a
-a,+a.-
d.+d.-...
^no1IrlH To ttrita 3 ,repLypdedIE ol6o€q nou0oarro6Ei(oupe 6fl rot6ouvov'4eoo cd IEprKdoepoioForq: (l) q S si r, > ..., (l) s, < s,ov I dpfioc'(or/ ftp,rt66. # trira 3 frq va aroieiloup€11(lrp6r€(6rjoovto6q1sg,mpqrlpo$$ 6rl (l) sr".?= q, + d,,+r- db+,> r,,, oQoio,,+r> di,,|] Kar (2) A-t= qd - db+1+a+r < q,.r, qQo[ar"+l> ah+r. fu va oro8dqoutsrilv lpilil ovro6rnto,rcpaqpoil€ np{i}ro&r S &FL aeot a,l > 0.
22. 'AretpeE oeryiq 401 'Etor,
elou|l€}la €16tKri r€pirFoot trtq (l), di,l.d av nipow€ un' dvrv |(or q ( t ), (2)rl y€vrrrl Tcptno)on€ivors6xol,tiqv o k 6ivordprroqKoro ./ ,reprtr6q, enddlor:pe I ttroro drore 2n> k @r2n t>t.
T6?s srStuStu-i 0 (l or66sril Tlav o,rl.olorsLlooup€,
I.so\00oLivo.t,,oan^iLpororoilor,.A$oilo".prilloro'r,,Iivcr.,nrio|riv 0pr0!6qN rdroroq{ior€
s " -" Entia?olfEodv Nr ro /r KpdrEpo N f' ourrivrtv 1516rl ro. Aur6 oltroiv€L6rr
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o
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+Pr,-L
4
tro depotolaq rpoo06rcW6oxprF6qr6sougqplnnl.ol)(6pouq60r€ vo ndpollr' ila 6rdl.6}oot€ rov llKpdrepoorarorvoriptrod0poropq4 pr(pdr6po0,16d. M d o ).6^/rtr, 0oroM yrc rov o,roio i'
L=s,+ ) o- a.o.
dt4 d16 a.Dt
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i) .
c-=
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ro rd€r a,b. rric
22. AretpeEoer.ptg 405 Alid our6 6Ev6rvdrol.todq (p)dn€ npdpl,lto 8), Kor 06r€ poq EKnl.riooErnpoypoflKd, oeo[ drv;Xouf€,rfi rinoreyu rlv oDyx€rpu€ta] 6dtqtl rov 6pov.To €fidFvo0e6pqFo 6aiper 6rr to cno*),rogc dDrdrFi rarr ldlrter 6,rov6ev€"y6updp€r I 6dro€rl rov 6pov. oEnpt{v^ e:Av or i *", i 4 ouyKl{youvaro^rrcrq.(ar rl lcil civarororcsirolE "" oxoloueia rou n6pr6t€rrq .tlv6pe\o q6, yta rdo€ (€6]06(r;J), dlE
i*=i"" ir. t'
"1
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.i rr,i o,=1.,",, oo'tKl.lv6r, ogot or {d"} xar {4} eivdt ono}.6roE oDlKlivouose, xar ogo{ to 6prc €v6( sivor to-frv6tr€vo rov opiov. Eroudvog, €ivor YlvoldvoD oKol.oDoio Cauchy,ror) n {prl or[aiv€r 6 'llc kd& r > 0, ov , mr I'cirur op{oivtoq lr€ydl.o,r6rc ,,j
l). tnt.F rb,t-t ra,.f. to,lcl. i=j
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t'l roJ, ts.nrr d\ 6 / 0?laKd€c",,r.o,qd,orero/ /G)dx "o undpX',, o],idro lim /(x) voftv urdptEr.
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tritld 5 .18, X' our6ro np6pl.tFa0o€sll€),6ooujr€ rI (ErDvurrKri o€rpdD
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)2_ t-=4 t5210
4_!=l-1 30 110 !r_!=l! 10460 o-l 60660
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2l
nouqeopoivoepoioforo'roooflirorvnou 0topr6woLondro x. M r .ltr 16./rq.€v6rqe€p6 oLrvapdottE nolj ond clrol;lo.rqltE loppri! opi(ovro. raurLttt
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f(t)=f,(x)+/,(x)+/r(r)+...
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x', 0 l.
Aurdsor ouvoprridrlqsivar6ic( ouvctEiE, o,.l{i I ouvdpqon f(x) = lim /"(x) 6€veivor
0, I,
0<x l.
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/
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4!
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i < l&1.
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M ti1.),1i.6t\s. ln ouyrl[vauod iirJvatoo€Lpti tt, Kitapo ro Aeiwt ttdvroft n 6ErydTaylar oro0 rn; ouviprroq; rnv otu{a opi(Et. lrt lsl6rl rov Suvo}roif at)ni! q Iqpottr€vlv6ro,0d'rov€iKoiovo olok).rtp{irooutlc Mrdnpooexnroi6uoec{riu!6n rtq r(0rdoroottq 0oafloxoosrpdrv Ko.rrr,rvo€rp6vTaylor. Ii\y€LdpKsid.(16Lslr.pivLoro oq 16pq,ortpcio. or osrpiqTaylorrov sin,cos.KoLexpsivor600 lKovonorrlfldq0o 0a).al0:6l]'/xlivoriv Kord6pouqtLo Kriocx. H oerpdTaylorfltq ouvdp'rnorq 1lo {d6E.y.{o1 ropa?oyiEov.aL yLoriouyK),ivsr EUxdprorr, /(:r) = log(l + x) Eivurxdr(Dq],rT6repo i16voyro I < x< Eivorovoyxoioerox6),or:0o qr giorlr rorv6NorooE,pri,v. Av rt l, oird auri I a6uvopio oerpdTaylor tq / 6!}1r,.ivc1?ro r.dnolo.xi,!e lxi > l,16r€ 0oou'/{Iivel610S,dorr a ( lxr , l:r,l), xor o' curdro 8rdorlro, rt ouvdprnornoL,opiqsl0a silrolnapoloyiolprt, Enotivrds cuvszris.Aild oL,r6€i\,o1o6ivoro,yLuri6€vEirorqpalpdv!oro drdorrpa( l. l) 6rouoivo,ian !r Log(l+ x). pro H osrpdTaylorfltq arctanEivaruK6panro 6uov6lrl quivsroL6rL6€vundpirEr quriq qq Tvo osrpriq \oouyKiiver6iov -{ > 1. ox6Fa 7,o-/rKi €Eiltor] 7rorlvdpvrlot no swlrooLd{6 nopd6EL.r,fo ol]Irient( rLUofltprdSorq oDUrT€prAopdc. 6i! rt ou\rdprnorl r ). Lradrfrp"q,pop;q rupflIuliouno, vrprlof. ro, fivarro no ,proro/\yJ - | (l pLo Taylorrrq / Siv8ror0116 loltKo npdTjra un6 no).uovL,!1,fli ouvdpqol. H o€Lpd lsrd rr'tv p1'1= L=l-xr*x.-x,ix" " l+!r
.
Av Ix I > I . I oerpdTaylor8s\rou]{),i\r',.rlori; [016 o6pdro'Fn63,o6N oerlv€L rrl osrpd l a y l o vr ou r c . - o Orc; rp o c a o . o' r o , , o - l : l u \ o { d u K o v c , q r d , o L q ' p o r ' l o r r , c : v o . ndvlo €nrxiv6Dvo,-rrqri o€ pro ovlLtro€llnxiondvrrlsli oufpoivsL fnopei vq Kqro).ri€€L j d . o , . p o i t " i . o r i / o ' v r o a p , i r r o . oI : u ' . r t r . r t v ^ L p : ^ . u o q o t o - . ' r a p,/r.o 6irinot, I onoio6!oq 8svrnop'i vo 6o8siauri lrt orlllii Av kor ro rpdBl,rluooaopd rouEnpotlorLKoiqapro$oJE, Inop€ivu anavq0€icpnvsuopfvo !6vo ov lonooEr!0€io' 6vfl sup,jr€po nioioro. Oa rp€LoorEiioLn6v! oarsp6ooup8 6l,0Ke$diaraoe relsioq xaLvoipTLo ri).rK6, fdxpLva oloxirp6ooupEr! usl,6rl r(Dvo€LpdvTaylororo Ksqd),oro 26. NPOBAHMATA 1. rrs xqo€prd(1n6flq napaK'iroaKoio',oi'e,I /" 1 pprir! ro x(lri orpEio6pro1rr {/" l (o!' ']ndpx€, oio bLdoll!fl noDL,no8aKvioutL€. xol si'r{t6r€ o! I i/"1 o!l{}.ivc, olLor6rLopao o aufli It ouvdprrort.
2j. Opordpopqrl ouyrTo1 rat \wapooerpiq 431 (i)
,(x) = fi, oro[0,t].
(ii) /"(")={l ",'j',*o1o,a],*,-on. 1ii1 1"1";= {, o.o1;,-1. (iv) /,(r) = e "', oro [-1, l]. (v)
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oro R.
2. Aur6ro np6pl.nto,(rl1dro i6rdrpdtparo6r!oq(o110n p6pl.itpqI , o),].dorsuvaprio€lqroo 6€v€ivorKor1600e xoleg.MepLrfquro6eil€r(6ivowororo rdLoc: (i) /,(x) = x'- .x:"oro[0, l].
riit 1,1*;= , *fr]*o 1"1 1"1"1-1f,d
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1e i"1,1=14+j*"* ('1 /,(,)=t+
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Yno6€i€€q: (i) rro xdo€r, FpEirE ro [611010 rou l/ - /,loro t0, U. (ii) rlo l(d06,, ee{rp€ior€ ro l fG) - f .{. (in) Exgpriorero fal.) f,(x\ aav "(x)l 7ro rEl'ito xliidlo ror uno).otior€kord rpoddtyrol ro lfA)fl/)l yro x > d. (iv) A6olE npood Lol ro! I/(x)- /"(x)IlrotrKp6 |x | . (vii)Xpnor[oror€ior€ l1loE€Xoprofli 3. BpcirErI oa pd I aylororo 0 rra (q0elLdon6 ! rqouvqpriorre: (i\
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Anoleiq1e 6Tl(l +.{)/(x) = d/(x) ru .{ 0 - dpo pro iuvqroo€rpd6€vftlop€i vo r€pllpri\v€rrTlvl(ivl6n €v6go6porog rou rirov cxivrlto li1pl q :(povrKlio1l1lri 0, lcorl€rd dplto€ va xLvEitat! (c)
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.: L, utrLpdvarc dn ro Z b"r' or ir). rt y.o l\ | ap,{ctd L'hp6.
-i "Dl I logtt ls. AercrLdrr ro-rpri ) - -, - t'- .oL/{/.,!Lrorloro/opaoorrv a,tnr I tni t I d, o] 7ro0 < a < l, o].]d 6n oro I au]Kliv€rorovlos2(l).
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.; +fq. \ nooiro'./cd1 n :ioo',rL 6!r r ofrpd I(n- Z u i apxL to )a, oL,1 a"r . Ano6.i",tc pnor 6rr flv d- + ... + a, | < s, r6re a.r' + ... + a"x'l < e, o:16ro Abel.(np6p).1F0 1848.)
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436 ('Arctpej aroTou4leg raL(drerptg) oeLp{q *23. Yrooarollp€ 611! / sivoLnapoyrdtioLpl. Ano8Eiqle 61rit ouvdpfltorl/ 0ivurlo {ard oltsio 6prorLloqo{o}.oreisq oL,vs16v ouvoprrio0o!'. (Aqorli6poupE i 6I rapo6'iIudru qouv€X6v ropo76?ov,dlolfs oK6!o;va rdpdAslTpa oro oroio ro KordortlEio 6proouv€t6vouvdpflioo0Jv 6svsivqrouvErtqoDv{ipfitot.) 24. BpEirsrro oxo).oD0io olol().lp6oLpovoDvaprri6ei0v ortv ([r1o i/, ), nououyKl,ivsr Loxlt prijortn)oDvdprrloil/ nousivorI orouqprtro[g r(0r0 srouqdpptrouq.Yr63s1(t: Kd€s/, 0o€ival00'ff6tar6liTo on!€io. 25.(a) A'lobsii.s drL.ovrl /sivarroorol6ropao6proqq{/"}oro[d,r]KoLrd0r/, aivoro).oK),tp6ortnoro [d, b], 16r€ro i6rorslJsr KaLyro rlt, /. (Enopfvoq, rou OcoprilmoqI 6€v1pslaE6rov.) !l0 qt6 rq uno0doErq (b) tro oe6prpq 3 uno06oop€ xard orusio orrv /. $6vo6rLrt {/,,} ou}K),iv€L AeiEre611ol un6lo1n€q L,noedo€rq eqaoqal,iEouv 61rI {t"l npdllo.l 6!7(}"iv€L opor6lopa0ollv /. (c) Yno06rori!€ 6r' oro os6prpa 3 5€v6ex6!dor0noq rt {/,1 oDyx).iv€r oe pla o'rvdplto! /, all.d p6vo61rt /,(&) ouyK),ivsr 1loxrinoro.t,oro[q b]. deilre 611t /" npd?porL oulx?,iva(opor6Bopqo) oE(droto f (ttt f' = s). (d) AnoociirsdTLI oerp{i
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Iolrierro i6Loav rt oi7Kirol ds! sivoLotor6lopQl; Ynoedroup€ 6rrr I/'leivor uuqrolouoiq quv6r6votlvopflio€ovoro [d, b] 'rou rcivELolo 0xardorusio.Yno0;rourr6, 'inrliov,6fl dXo'rFs /,(,{) > /". (r) > 0 7ra 6Ia ro r. KoLyu 6iq ro x oro la 61.A€iErs6 r {f"J rsiv€Loro 0 01()1610pao oro [d, b]. Yr66ni!r Yno06ore roE 6rr, 6rq],6qrE Kord].,,t],1oKoiouoiuotp€i{Dv x, oro [d, b]. KoLcQopt6oT3 ro Oedrprtpa Bolzano-Weierslrass. (b) Arod€iir€ ro Oe6ptparo'r Dini: Av rt {t,i siv,o pro llrl,oiEouosaKo}.ol,eia ouv€t6volvoplrio€ovoro [a, b] rt onoioreiv€rt o.d ortfrio oc ouv€triouvdpflton /,161e n {/,1 r€iv€rorrv / opor6popea oro [a, b]. (Toi6,odnotdl€opo roxi8rav rl i/"] 'ivor Frt-qoivouoq aKo),o!eio.) (c) IoI{0r ro O0d'prtloro! Diniovl / 6€v€ivqroDvExriq;Tr oDtpoiv0,ovlo [4 b] qr610 o\rorft66L{iortpa(d, b); ovrLKoroorqo€i 2t. (a) Ynoedlolpe61r ouvet6!'ouvdpflio€o!qro [a b] nou I I /, ] sivorlio aKo?,ou6io ou./KLiv€r opor6Fopqq srrtv /. Aro8€ilrs 6Tl,av ro x, r€ivELo1o x, r6re I /,(r,)r€iv8r oro /(r). (b) lori€L ro i61oX0ipiqqv un60€orl 6 or /" Ei\olo!vsxsiq; ro avdorpoaorou rdpouq(a):a! I / €ivaLoDvEtiqoro Ia, bl Kcl k) AnodEiqr€ iq r /"(x,) vq'rEiv8rorlv /(x) ri6r€r l /, j EivorrLo oKo),o'roio l13qv L6L6nt r, reivsroro .{, r61EI /, o!rK}.i!crofLol6ropqq orrv / oro [d, 6]. Yn6581qt: Av6/r, undpr€L ivo€ > 0 KaLtlaoKoicll]io r, irord,ore /,(.r") /(r,) | > s. T6po,IpnoLUororeiare ro 0€6pruqBolzano-Weierstrass. 27. (a)
29. t' our6ronpdoitpq n€prypdqours 8LuaoprnKri rpooarTro!oro ol.oKl.ri, lLs r€],EirDq popo sro#!oe, EcveivoroiKorova Tptorsonorioorusn).tpoaopi'qnoL,67ou!€ ri6l pdesryLura oioK),rtpd)paro.
23. Opordpopqq oi'Jr),roqrar \uvapooerpig 437 (a) 'Eoro s pld Kl.rpoxoflialvdpqal o1o[a, b], 6rl],o6ri! r eivororoe€pi mo (1, r, r,) 1lo Kdro( brqufprorl{6, ..., r,l rou [d, l]. Opi(ouperoJ,r oov ro -', dfl ourdgo r) 610',r. civqrr (oroorpri)lLtri rlq s olo {r, . r.).AEiqre Zr {'. op6'r6( 6sv€Eapriirar on6rn 6rag6prsrt 16,...,r,]. Mlq ouvdpqorl / Ldlmar mvovrxororqpivrlorc [d, 6] ov Eitql lo o]rodpp0o 6p$ prqga(otrouoiag r),rjroKordlv onvopflio€(rv {s"} olo [4 b]. dEitlE6n o ouniv qv n€pint0rolundpl€r,Tra xdo€€ > 0, rdroro N r6rc1o6ot€ tro m, ,>Nvaaj(oute lr"(x) - s.(x) < € lro (dos.r sro [d, b].
t.
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Ar' olrriv rrlv ovro6rrllafiqipvoogs
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Or rpdqcrrrlc rp6o0eotqKaLroo nol.LanlqorqofoiFr1oSrx6v qprold,vdloovror or 611o oqrowrrdqleorgerprrdg eplnr€iEg. H rcpit{root itq rp6o0eonE €ivqror},ri(t ilo 2). A'io fr1a6r'(oioprofoiz= (d, b) ror e = (c, d) opi(olv 6vonopol).nl,61po!ro,roDjrsr oqv660qn6q r).[Upaq rouro €lj06tpollorl{palq on6ro (0,0)oro z xorond10(0,0)oro v rl Kopuerion,vav ond to (0,0) €ivorz+ r (! an66rtl qurof rou 7€olp€rprKo6 6€6opdvou o9rivsr0loeoqq). H epplveicIou rol,l.a,r).qorooloi€ivaLrLo no),rjr1,oKl. Av z=0 rl w=0, 16r€ z e = 0 (lnoporige vo66oouFs oeU10tpatti, oD.daK6psKr lra uno]"ollorci an66€lqn - o ropprolde rpoKunr€r, quni 6Evlp€rdE€rar 6'106dt€l iSrl ano8€rtr€i, o,rdfle Il-I9), dpalropoils vd rrepropioouge rqv rpooo { goqoe tn-u8evrxoic pryo6rro6e opdtoJq. Ap1iqolr€edrowoerdo€rrl-ftlevrx6 Frta6rx6oprollljs€ lra s$rnl rop'pi. (I'JFp(vor€ !e ro nopdprqlorou Keqaiaiou4.) qpLoFd fra Kdocuryq6rK6 z+ 0 Fnopoitcvo lpdq,ollc 7 .-z z X cr.rr{vrry ncpriorcol. ro lzl €ivor6voq0€t1K6('rpc1lctrtrli6 cpr0pri6,evtir z-
|' l''1=14 ' l ' l z l=,, &pt o z,/lz ) eivarevaqqa6rr6qoplep6q qprejr6e !€ or6),uq rti L T6po,xdoeIL7o6(K6e d= x+ iy pEl= lal=x:*yllptiqerctoqpopqrj d= (cosd,sitrd ) = cosd+i sind
24.Mrya|wolapr4poi 447 yroKdrrorov opr0U6 d . Apo xdo€prl-!n6ev!K6q apr0l6qz lnop€ivq ypqqei z= . (cos0+t sird ) ./Lqxiinoror> 0 xqr Kdnorov opr0F6L o dpr0f6Er rivor llow6rx6q0006101 t'[ lzl), al.),do d 6Ir qv 4 €ivqrJra tuvor6.rlro,16r€01dl),e( eivaror 4 + 2kn yl k oro Z raoivagon' ouloi( rouEqpreloiql.6ysror6pt6Faiou z . To Elripo 3 6€ijav€r rov z ouvoprriosr rov r xqr A(fiq va pporl!6 iva 6propa, yrolovz=r+ i lnoporirts vo noporrlpiooutr€ 6T1rt ro6rlrd ,{ + i=
lzl (cos9+I sind)
Ztitld 3
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w I ror rdro(ovqproldd. T6r€fvqEfryo6rx6(aplol16( = r(cos0+isino)
rrovonoEiqv z'= w ov l(crp6voqv r"(cosrd+ ^i.rd) = s(coso+ isino), ro oroio oulBoiwrov KaL!6vo av cosr0+ isirnd= cosd+ isind, At6 ntv rp(i)nteqioool dn€rar6rr
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f'(.x)= n'(x)+ i u'(x), 6rou / ouppo],i(flrr !'7o6LKrl nopdToyo, €vdjt'Kor t'ouppoli(oovflE ouvrio€Lq napoldjyoDr oDr6vrr,)vrpoytorrK(ivouvqp iosov oro R. (b) A€iLr€,T€vr(6r€p0,6 /r(r) =,1,(x.)+ irl,(r). (c)
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H ro66qla 1!( aJtirovl6eptaoioqrrousvayxasrrkdrrsprdl.raro' au16ro x€ed}.oro, ovRoraotiEsrar arl6fro npalpoflKdrpd'rrlqrdqtq rdCa.fra va ono6€iLoup€ 6fl u'r{ipt€r 6von),ripsq 6ror€royl6vo o(iFo,eonpdn€r vo 66ooup€Flo 6oei ror ),6n1o!€pri rrprypoqi ( t)-(10)Iou 6laftiotpdloDs6loroq eq €v6qrErorous6poroqn €roli€0'rot 1{ivo'rv0it'(6v €ivarl(orraonKi oD.ddtson. 'O!oq rl rsprypqprirou i6rourou odporoq,rov otort€iov no! n€pdX0l, eivolot' o),i6€ra€lquiq. trl6rd0soi Ua(ppiorc€lor ro oivol,orov prlr6vopr0p6v,xorqn' our6ro oKardpyooro !hK6 rpdr8l vo rqpdlou!0to odtjrsnour€l,rKdeq ovopriooulerpclpoflroig cpt0po$g. fro rov oltil to, au'!6ttpdtt€L vo eqivsrarr€),rioqorrioovo- sv €ivorWooroildvo or pilroi optoUoi. A,T61106 0o ,tpoil.oouv or un(j),ornol; Oqr6po EloulleqnokrliosrapKsrri€lnDrpio 60r€vo roroloFaivou!€61ril l(ordomst unop€ixcr vo prlv€ivar1600an€),monKri 600 qunl I qngpisK€rrrrl unoS€rKv$€r 0€6pn6irle. H o.rpant1rxinou0q uroo.rriooulE orrlv xor0ol(cLri !0q.4cr i6t Iptorlonorl0ri orroTd.rolanxd ]ro vo oprdroivorouvaprrtocrE r.o1orlr?aSrxoioproloi.Awi vo npoorseisoup€ vs rposEropioorup€ rqv d,Iervrlq'jotD durdrvrov swordrv,66oqg€ dvovoprsp6,ro!rL! neprdypaqe apr€ld,601€vo npoo6ropi(Er nLrlpoqr'q foolronKt tolq L6r6rrcq. 'Evo yrorov opLoF6 nap6loLooXdSro 1(l)vrpalpoflxrirvopL0girv ororreigro neprlpcrprl rov npa?po.rKdlv opr€[6v!e odol ror](pttorjeoprepoie. H llopal1iploit6fl xd0Errps]Iqrrx6qopr0l6qed npEn€ vo opi(erarr),1po( an6.o oivolo lov plrdlv aprold'vnou€ivor qr).i Kqrrol.i s),Kuollxi6uvor6.rtra: lrxp6tsporcn' c!16v,ltro6stKvisr ILo €Krr),IK'nKd 6vaqrpo,lonK6q apr0!690o Unoporior vo (Korrr).Lrd6vroq 0o) n6pr,paq€ioov trd ori),lorrion6 prlroGaprofo q. fro !o 1{vsr6tcrqour6ro oXi6roEQopp6ouro, vq npdnEr dvoqrp6no{vq mprypdvouts(ro o vo},orov pnr6vopr0g6vrou eivqltLKp6lspor opeooi 0116 avov'rpo-rfqtu6op1€p6'Xopiqvs svqeEpooJpe orouqrpclpctLxoLlq optogoiq,ro! sn6ntrdqrma tq Jraorltanxriq 11orrlv 6pa 6€veivirrlirom ,ropqndvo tqq eqvrooioe. Av ro ?4'rp6Ks11dr vo roil8r lov p67,o rououv6tr ourov prlr6vapr0p6vnou€ivor!rKp6r€pora,r6rov rlpotfofll(6 apr0g6a, r61Ero A od'rp€n8vo 6t€rqv €Eri(6r6rrlm: Av 10x atnKelolo A Kor/ rivor6voeprlr6qqprell(ig r6rorog6g1€y < .x,16r€ro y ovrixsroro A. Err6q on' o',niv 1Iv r6'6rtro, ro orivol,oA od'np€ft vq ar€rxqr lEprx6r'i],).€e.Aaoi npdrcrvo uJrdplouv xdnororpiltoi opropoix < a, 10661010A 6€vrpdtr€rvo €ivarK€v6pnroiopl€loi x > d,10 oivo),oA 6€v frq rov i6ro16yo,oeoi npdnEr va u,rdplouvr.dnoror rpdftrva sivorol6Klrtporo Q. Ti),o(,qv-x< a, r6rs npdrervcrnriplaivoqri).logp1r6E y U, x < /< a, dpa10A bEvrp6n€L op1€[6e vo neprft€r6vold,roro olorXeio. Av Oooiptioo'r[e rpooo)prvriTvosroiq rouq npayfqrLxoiEaprOgoq, T6rE60v oivar SrloKo).o vo sidTErr€(np6pl,lpa 8-17)6fl 6vc oivo).o A p'dur,q nq Lir6rrtreqeivor rpalLarL{dro orivolorov ptrd,r cp.0p,irrcr evor prrp6nporon6 Kd'rorov,rpoyporLK6qpret6 d. As6o$!ou6n or npoypoll|oioplotoirpo( ro rrop6v€ivolrp!l.aKrotdvoL, q on6SELEri odq,av Kdv€T€ *(ir!ola,npdn0r vo 0€opl00ionl(ireoavivo avoriolto 016),10 rdvo o' q'rl]iv rn ou('ilrlor. Oo lptorp€Li€I,n{ivroe,y1oto aoer€(o€rdlr 6evEErdostE xdnororcpiorplr6r6rrltarou ouv6),ouA. A€vundpterxovdvo().6toqvo 6ro1d(oujre xL d),Io.
494 Ett oToq
I]ponoirK6eopr0l6qcilar dvoorivo),oa, on6 ptroiq opr0poiq,Fe q cii( tiod€pl( 13r6qreE: (l) Av o.x ovircr olo d (or y 0ivor6voeplr6q aprell6c |r3/ < r. 16r€{or o_vornKrr
( 2 )d + s . ( 3 )d r Q . (4) A€v'rrrdplflldyrolo ororleio oro c. Me d).).c16]t0, ov o x ovtl(fl olo a. 16tr undpl€rKd,roro,olo a lley >.x. To oivo),o67,ovrov npq./paflx(iv opr0r6voup9oli(€rarrE R. fLo vo ooqDnsv0utioollcrtt $rloooqia nourpi9€rar oo qn6lov opLop6, aq iiorl}lE dvoor,yKcxpLFivo rapdbflt!0 tp0]F0r(oi opr0poi: a=[roloQ:r x'y'?ro6Io rd osnKdr'Ko1 'Apa ro.ry 6€vovriKEL oro o .p 6tl,a3ria . E + Q. /'f ourrirrv r5r6rr1q. (4) Yno06roup€ 6ir ro ,r ovriK€L oro d . P, KoLry < 0. Yrdprr Kdrorox 610a rr€ x> 0 (or (dnoro/oro B pEy> A.T6rErcz= xy Ei\ororo d .P Karz> w. Ag unogdoouFe 16po6rL l/ > 0. T6re r = xy 7ro Kdnoroe€rLK6 x oro o KoL Kdno$0srrK6I oro p. Eriorlq.ro d 'r8prdj(cr {dnoro x'} x av z= x'y, co:f '(Eroro o ,8. Artrfl8iro a .p 3€v;lfl ra'j/16ro z) xy= y", oo, "o " ",' oroLxcio. I [opaqp€iorE 6r! ro d .B rpoqav6qoviKeroro P ov ro i6rolojri61ylo ro d, E. AD.6 r1v €lclieslot 6)"ovrorvrbrofir{rv roDP. fra va oufn?,rp6oouus otror).1prlver rov rptirot1v lal. oprop6 rou . opi(oupe Oprolrdq. A\ o civaLivo, npaTfafl\d,opre!6q.16r" 0\a-0 't -o' l = l o o ' o ' o ' 0 lOftopo, \r o \ot P tirc apo/foIro' oprelo'.r6'c I^ qvo 018 0 10. o.8- llol .lfl. ova>0.8>0ror 0.0 0. rro qv qn66€rqq orr 7€vrni nspinroorl
50O Erl,oyo; ({oL0u €ivo1,.droqn,o siKolo o! rplorronoLioers'ro npdnsrvo 61o'ipiv€.s mpLnrijocrq €n6pevo 0c(ijp'tpa). I oEnrHM{A\ d r(oL opLopoi, r6rEd ' P= B. a B eivornpdr,lqnxoi rno,\Er:lrAur6ci!'orQav8pd ov o, B > 0, l(oror {il).€qnepLnr6oErq sldyTovror EiKoio.I oflolrcj.. I = l,{ oro Q:x < l} (Ei!'orqovep66rLro I 0ivoLnpqTpanx6q apL0p6q). oEopH[&Av o Eivor€vqgnpoTloirK6q apr0!6q,r6rs a . I = a. 'Eoro AnoaEilH o > 0. EivorEirolo va 6oip€6fl xdeEoroL1Eio ro! a ' I €ivoLxor oroL jtsio ro! a. A116rrt!,d1,l.ln).supd, oEwo€dooup€ 6rLxdrorox ovriKeroro a. Av x < 0. 1618ro x oulotdr(Dq uvi KclKaLoro a , l Av .r > 0,r6!elndpler Kdnoroq prtr6qopr0!6qy oloardroroq60r€{ 0. Av a < 0, r6rs,Eqopf6qovras ro orot6Leopcnoup6hq ono6€iEup€, d.lolp€
a . r = - ( l c l . l l l ) = - d o l ) =a . Tdl,oq,rdOsdprtfosi!'drrTpo,po!'t 61svo = 0. I Oplop6t.Ay a Eiys.t ac.npoytorLK6g opr0!6ql(aLd > 0, r6rs dn = ix oro Qrx< 0, ri x> 0 Korro l/r 6EvovriK€r oToo, oll.d ro li x d€voilo|ro prrp6cpoororleioror Q - a|' cv a < o,rarea-l= - (lal-1). ororHMl Av a oivoravo(npa?lalrx6qapr0lr6g 6loqop€rLx6q an6 ro 0, 16r€o a-'Eivol rpotparLK6q 0prop6q. AfloaEi:Hnpoqov6qopxsivo eisldoout€r6vo a > 0. Ipdm, vo s).6yEoDf€ riooepLq (t) Aq t)ro6dooDps 6ri / < x, xq, 6r, 10 r ovriKeroro a-r. Av y < 0, 1610ro / uvjxeLoro o-r. A\r y > 0, rdre x > 0, dpo ro l/x 6EvovriKEr oro d. Aao lil> li x, an6ra'6 oir€ro lryavrlftloroa.xarro1i/ooq6q3€v€ivol.o ororl€iorou Q a, ripcto y ovrlreioro o-'. lrKp6rEpo (2) npoeov6q d-r+ P. (3) Aqo'ja > 0, undpl€rKdnoroq plr6q opr0t6cx oro o. T61r'ioli,r 6Ev 0srLK6g 0ui(€Loro a-', dpdd-'* Q. (4) Ynooirourrs 6Tl ro x avr1K61 oro o-'. Av n< 0, 'ivdr qov€p66rr undpl€L rdnoLoy oro a-I gey).r yrori ro d-'nsprdtsl Kd'loug ptroiq. Av osrLKoiq x>0, r6rE ro li;r 5€vov'ix€Loro d. Aaoi ro li n 6'v Eivorro rLrp6lepo ororX0iorou Q - d, L,ndpireL 6vuqplr6g opr0l6qy 611oro o, t€ y< li,r. Ara#?o!r€;r,a\rpnr6oprgp"6 y < z< 1ix.T6r€ro li z aviKEloro d, KoL z p.E li z> :r.'Aporo a-' 8svnEprixeL Kdnoro ororxeio. !671010 I npo{rlfdvouva ono8€ilol]F€ 6tr o d-r silqr or' oliecla o avriorpoQoq ro! a oq rpog rov no).?,anl.uo1oor6, Eir,oltpriolfovo !Xo!r!Eo1l6rd0€oipqqdz,lo6voli!!fl, nouri!'o, o no)Lun].uoLoor.xci a\fl).o7o roL,rp;ro' /o; ).ipt-o,o;. 'E6ro rHMxr dpL0f6gp€ d > 0 xqr z dvoqprlr6qop10!6ctLrz> 1. o 6voqnpa?!ot1K6q prtroiopr0soix oto o ror y nou6evavriKEr T6r€L,n{iprouv oro d, ldroroLdjor€ri n = z. ErLn),6ov, oroLleioror Q o lnopoiuevo uroffooulo611ro-',8svrivqrto e)"ti1Loro
28. Karaorewjtav ryayparviv ap0ptitv 5O7 Arodfl=HAq unoetoou!€ np6m 6 ro z ornKrLoro o. Aeoi z
I>0Kdr
z "= ( t + ( z - l ) ) " >I + r ( z - 1 ) , dn€10r 61ror oor0uoi 2 ,t 1 ,i , . 'Apo 6.v F'ropo'iv6)0r vo ppiol(owqroro o. ul(iplEl Kd'Ioro* rdrororiote to v = z' vo = oro o. npopov66,vix= z. EnUTL€ov, ov ro / ovriK€r oro a! ror ro / z'_'vo Filvo\,1Ker €va-{'> x ororx€iorou a, ougpciver riaeivorro e),ri1loloororlEioroDQ - d, rroipvoulr€ Koro\,trKaororoip€ Tox pe@ t' Karaoy le rc !x' j x , Av ro z 6€vovril(ororo d, lnopo'jF vo 6(iroo!p€ av{i}.o7lrr66fl61, pcoropdvloro 6rr KdnoLoq on6 tolq opr0foig l/zt npirer !o aviKsLoro o. I 16}0!6g eEopHxaAv a Eivorivoqrpqllqnx6E oproldelCoL o * 0. I(irr a . o-r = l. 6rr AnoAEEH llpoedv6eapx€ivo rE€rdooulrs !6vo a > 0, on6reror o-'> 0. Yno0irorrf€ prlr6qopr0!6qsro a-'. Trir€ .r 6ivordvoqeEflx6qpr1r6E dpr0!6qoro o, ror / 6voq0€1116g tol//6evovnK€rorod,ripal//>xenopdvocxy 0 xareq€rd(olp61p6K l€purr6o€rq: p, 7 < 0 KolP < 0, 7 > 0, KorB > 0, 7 < 0. H np6t1 drctorap6o t. T' pa rqd = (o, {b"})KaLP = (c, {4 j) opiqo'ip€ opr0s6(l d + p = sup(ai+ Br)" k auorK6s (ro €)dtloro dvo gptilpo lrdp1sr sn6 ro pdpoe(a)).Av opiooupsIe rov i6Lo r6ren €?Id,rigsuol 6lov 1ovt6roflirovroD06!0rp6noro! no),la,rlooLaot6, rl rrottqax6!oqopd dg0)q, roerivorte).eioedloorl, xor6€ou!'lodrurrbloir€po. pripo. rlnopenro ,or),ooraotoiowrorp6qou0d €ivorro 6uoKol.6T€po
KEOAAAIO
MONAAIKOTHTA TON IIPA|MATIKON
APIOMON
Ooenrorpfvoup€ 16paorovoL,v{ort ourpo),Lo!6 upL0Uoii. r(pord,7Lorouqrp,1?rLoflr.oii vroc r0 nat€rdoiFpo).ayLadi),a odjlaro rol, ruX6vElpavroro{v.AK6to.0o o0oporjp€ tou! o{iporougxqrrougptroiqopr(]poiqoavrpoTJrdnKoi_( apr0IoiS0r6lxoidrol],xoL0o noArpiivoltLEro! ri,oirEporp6nope rov onoioopiornKav()1npaylorlKoioploloi. t 0!16 ro KtQdiorocvdla$tp6lLqors yro 6voKoLp6vo sp6rr o: undp7ouv {i],l,ad.ripoE 6Lor6ro1d!oo6irror(s{r6; on6ro R:H or{i!rto! o' 0!16ro sp6rrtpo,yLova xuprolEKrri oou[e,eivorivaLr. fLs nqpd6€rypa, ro od)troF, ro'r nopoL,ordoor€ 610K€qd].oLo 25€ivor dld n].rlpo)s drolorq,!6!0 o6fa, Kflr0i0oLfl6sv€ivarro R. Aur6ro o6!u eilor;vo (ov6lro' rupdSer?po rrsri io (€iloq (d. d) rnopoilL€vo ro 3oir6 ol1!'dvadiio 6vojrorc! npq1p0flxoiopL€loia or npde€rq ( d , a )+ ( b , b ) = ( a + b , a + b ) . l a ,a ) ' ( b 'b ) - ( a ' b ' a b ) . p uurrilrt Ferovotaaia. sivoLoup|]r|]uorfq Aur6 ro siSoqrqpo6ei?poroq 8silvsr6 , tro 'rourpo0liporoq,anolroivr.r Kd,rolq dnorciso!fl'rLKr'i Uro !El,6rI llo€lIo l(d 6p66r07ro r l v o t n p f o t , r o . o v o0 6\ s o - i v ! f r o \ o . a o . o q . Ar l,o'olrtouot-coop' roo,oTroro, R IptoUonoroi[Lrfl orortriardroroL,o;furo( F, ldrc o0 xd00a on' ro R 0d npenEv avrLoroLxEi 6vo(6vo!o' /(d) oro F. o owpol,1ot6q/(o) unoSslKvi',6tr 1 perovo|ooiapnopsivu SLoruro0€i r€ pdon flq ouvaprio€rE. fla vo Kdvoltr€ r.dIrraroro,0olpsrooroip€pra6wora6uvdprlotqnol,i ]I1o y€lrlnian6onorfl8inor''iii! !{iq ala onooro}.rionor r6po ouoroorrKd,0o onoLrioouts rtlv rro TEvLKri f!'vorq(oLrv{iptortqrnouIprtorponot€irol 6ro }roBtIdrll(d.Mro o!vdprt orl, rr' uuT]irt 78vLr(ri d!'voro, eilqr orl,6q6vasKav6vo;nouor€LKoviqsr rdroro npdtrora o€dl,ifl rp,i?paru.rta vo livouts r!n1Koi,6r!,iptrlq eildr proourio?ri0116Srar€roytdvo qein (ovrrKsudvorv KdeE siSouq) nou6€!n0plit€rErjodrqqopslKd(ei7rlp€ro iSLorp6ro ororreio.To ,r€aioopropoipLoqol]vdplrlortq rov o! x€Lpdvov / oivarro oivolo A 6?,0rv o yLoro onoiaro(a, b) qvrixeroqv / 7raKdnorob uur6ro (!ovaar{6)b6ufBo}.i(€1or F0 f(a). Av ro /(a) ovri{€roro oivo).o, r1o Kdosa oro A, 16r€i6F€dn ! / Eivo,rtlo olvdpltot an6ro ,4 oro B. fra nopd6€L7po. . ov f(ir) = sinx71dKdeE r aro R (KorI / opi(srurp6!oyLd.{oro R),r6rsr / sivaL !1o ouvdprrtotan6ro R olo R si\ra,eniortgtro ol,vdpqor]qn6ro R oro I I, l] . a! /(z) = sinzlLo {ri6cz oro C. 16rer / eivarFroouvdpirtorar6 ro C oro C' . tw f(z)=e'r]!d(sgEz oroC.161e!/ silsLpLoouvdprloqan6ro C oro C sn,llL aK6tap10ouvriprrtort onoro C oro lz o1oC: z I 0l . rt ,3i!'o,l,Lo ouvdpntolqr6 ro lzoro C:z+ 0loro {.roroR:0 EI> >E EIIIAE|MENA NPOBAHMATA
l. (i)
l=aro
a r ( o Y ) = { o - r c } x -l . t = . x '
i ; r n " ^ ' - y ' . i 6 t eo - x-' - l - t r - v ) t t ( r+ Y= 0,6T\lo6i, i x= Y'tl r= Y'
r ) ' x a rD ! o rr'i r - v = 0
i
ro / tt€ro -y ornv (N) (vi) AvtucolooreiotE - y= 0. 2.'Eva plipodttotr€i8raiprol d.r = (dcxb.)-'(0,16 rnv(iii))= oclbc ab=ab '= (dc)(b-rc-) 3.it -ldd+ bcllbrd-r){ar6rlv{iii)) tt;) @d- bct/\bd= lad bctlbd) = a b ) + c d t = d / b+ c / d . ' (iii) ob(o-rb-r)=(d.o )(b br)= I, tipoar ' b = (aD)-'. | (a' b-tr(c d')' (.t) ld/b).4c/O = @/b)(c/d) = (a. b t)\ca' d) = ddlbn' c t) = dd(bcr-l = (ad),/(bc).
4. (D r < - L (o x> "4 it^< - '/1. (v) fra 6la ra x,aqo$.xr 2x+ 2 = (x l)1+ 1 -2 -2, qg xr- r- 6 = 0' orpi(eg dgoi3kdr eivar (vii) r> 3i r< ' x > 7 r r i 5 d + c. olo P. E top6voq, I61t, n (l) olvotdT€1or01r qv (ii), c bc(vii) Xprlotrorol6vra(rrlv (iv),o > 0 xar d < l, iror ar < d' (ix) AvflKaraolsiote m c ps 1o d t(or10 d pero b o1rlv (viii)'
5.(0
9. (i)
./1+ '11- ,"8+ J1.
(iiD l a + b l + l c l - l a + b + c l (v) rt+J1+'ls-'fi. 10.(D a a v o > - b r 0 r b > 0 , -aqva5-bxarb-br.orbS0, -o-26ava0. (iii) x - x ' I c v . x 2 0 , -x- I av.ri 0. , (i) x = I l , 5 . (iii) - 6 < x < - 2 , ro - I €ivst , (v) Kov€vq.x (l ott6oraat tou x stri 10 I oov rnv utr6otdoqrou -xotr6 1oul'li1r61ov2) (vii) .{= I, -1.
t 2 .( i ) ( l x l l ) ) - ( . r y ) r = r y = l r l ) l J , l ' = { l x . l r- l t ' . c , r o , i , l x xv c t l r l l v l lxl l.vl' eiwr l(arro 660>0, ro'r ano6errwct6tt lrvl ' = t n v (ii))-lxl-'l (oi't qv t c n a (iii) lxl,zlyl lxl lyl lxl lv-rl
\D= t*/yt. drr lxl=lv t"l iio .'r" (iv) 6r€ruL l r l - l v l< l r - Y l tvii) ix!yi
(v-x)l< lvl + lv-.rl' rtror
z l < l t + 1 1 + l z l < l r l + l v l + l z l A v t o r i e tr o 6 q t o r 6 r c
rpoplrjpara 522 ArawfiottE ot em),eypiva ro z lx+yl = l.rl + ly , rarorx ro1/ 6rouvro i6m 'rpdqro. Errlnl.dov, npdJlgl va 6lerro i6rorp6urilo tlEro x + y, erctx, y, z np6r€rva 6touv61"oro i6mnp6onro(sKr6s ovKd'rorc eivor0).
l. (i)
Aqoi l': = l . (2). (2. 1 + l)/6, o riro6 a),roefsryroll = l. YnooftoDu€ 6Tlo droe o)'rloeh ylo k. T6re
t t-... I t-
r(,(' l)(21' l) . 1t t r.1, 6
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+ r]+ r), _ (r + D(r+ 2x2[i dpd o 10'1060),16€[€1ra k + l.
2 . ( i ) ) . { 2 i l ) = I + I + 5 + . . .+ ( 2 , l )
'-
-1
= I + 2+3 + ...+ 2n- 2(1+ ...+ n) It n\t10 + t\
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2 5. (a)
Aao'i
r+'=-L", o ntros aUedfl tro , - L Yro0drou$6fl l+.+
+."=
1_
rr+l
T6rs
I + r + ...+ r' + r'+r= ': '
+ r,+l
It
I
r+,+ rd(t _ r) lf
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-'"*' = l l-r (b) r '
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S=l+r+...+l rS= r+...+l+/+'. s(l r)=5 /s=l-'"-',
dDo J = -_.-
Aravrfijeryoeen),eypiva ryop,flpatu 523 6. (D
(r+l)'
+6k:+4*+l,t=1,...,,
dtoLrlt
(n+ 1)"_''
,s
-4
(r+l)!
,c
,c
t {6 ) kr*4 ) k{n,
1
6
, ( r + l ) ( 2 r +l )
{,,
.n(n+11 7
424 (iii) A'i6 rrv I L
.- l { r r+l
;
k(r+l)
r(orpd).,oroEivorn€prn6r.Yroeiroupe6rr ro rr Eivori 8. To I Eivo,ri dpno( i mpLtT6q, dpnoei rr p1rr6q. T6rsro , jr,loprivo ypoAcioirsoov2k €i1roov2k + L llr'tv rp6 l nspinroorl,,+ I = 2ft+ I = r6p1rr6e. tq Sajreprt nEpim0Jol, r + I = 2(r + l) = d, ( Ovroe,qlni rl nErpoq. Kor 6nq 6uonEprrr6onq,o, + I sivflrri dpnosi r0p1rr6e. (6norrrtD, an63e(r lodqer ol.].{i€ivorrpriyrLo.r ooori.) qpL0lriv 9.'Eoro B ro olivol.o6lov rov qtioLK(i)v I yro rouqoroiouqo ,o I + I sivor aro A. T6rEro.1eivaloro 3 Korro / + I EivoLoro B qv eivarro l, dparo B irspld1sr d] orq rorg ,p,or\oiqopreuoic. ro oro:oonFaiv-r dll ro A nrpritfro/.o,q ro'. $ror{oiE apL€poig > ]r0. 12.(a) Nor,616rrovoa+b irov pnr6q, r6rEb = (a + b) a eoirilv prl.6q. Av or a KoL6 €ivordpplror, r6re o a+ b 0o lnopoioEvo rirdv prtr6q,oao6o b eo pnopoioevr €ivoLr- a yruxdnoLov pr'tr6opL0p6 r. (b) Av a = 0, 16r€o ab eivorplr6g. Al)d ov a + 0, 16r€o a, 6sv0o lllTopoio€ va civo otroq.ordTr r6i.o b - {obr. o eo tra\ oIroc. Nor.rL0 'ropd6€r7!o, V2. (d) Nsi. ftc ncpdSrLrro, l[ rur \,[. ll. (a)
( 3 n+ l ) 1 = 9 , 2+ 6 , + I = 3 ( 3 " ? +2 r ) + I , ( 3 n + 2 ) 1 = 9 n+ 1 t 2 n+ 4 = 3 ( 3 r r +4 , + l ) + l , 6nsrqr6tl,ov o *'6rorp€irur!e ro 3, r6rsok np6n€r eriortqvoSrnrpeirorJr€ ro 3. Yro0droufcr6pd 611oV3 EivoL0116q,{or doro V3 = p,/q,6noup xur q 3€v d1o!,!'xorv68rolpdrrl.T6i€p' = 3q', dpoo p' SuLpEiror pero 3, {ipoKaLo p. Enofavoq, p=3p'1u rdroov quorr6oprel6p', KordLpa(3p)1= 3q1,i (oLro g €,riolqSrqrp€iroL 3(p')' = g'. EroIdv(Dq arr6ro 3, rou €ivoLdrono. H i6Loar656lqTt roxiErKorl'o ro'rqV5 KoLV6, Ersr8ior sqlo6osq
( 5 ' + t ) : = 2 5 r : +l 0 ' + I = 5 ( 5 r : + 2 '+) l . (5n+ 2)1= 15n1 + 70n+ 4 = 5(5n1 + 4n)+ 4,
ot tn)tTpiva npoP.4para 524 Anavrrlocrg ( 5 r+ 3 ) 2 =2 5 r r + 3 0 r+ 9 = 5 ( 5 1 1 + 6, + 1)+4. (5n+ 4)1= 25n1+ 40n+ 16= 5(5rr+ 8, + 3) + l,
(b)
Karor avriororleqeqrqd'o€r! Iro rolq oplopoiqrrtquopariq6, = r?, Seilvouv 6rr, ov o kl6Lcrpeiror peto 5 rl ro 6, ro i61ooutpflivElKoru6rov,k.H on66€(t tov V4, en!6i o (4r + 2):dlorp€iraL o:roru,xdverTra |]Ero4.{Axprpdjr}l aur6v rov 1670,j or6SnErloD.rl6€vurTopEi va 1plouonorl€ei 1lq vo 6€iE€L 6n Kujrio o 0ivor ripprlroq ov o d 6€v €ivor rflcro r€rpdyovo:6€v d1o0[s 7€v1K{iVd sio6qdl,1orl 6rr o (d, + 'r):6€v propsivo eivorro),).or)dororou a yroKdnoLo q),ld I on6our6qEivoro),Ierv6q, m < a. Xrrtvrpofporlr6fltra, o 1ol(uprair6q 6€rEtqnoLleiqv r).lpopopidrou npop),ildios 17.) Eq' 6oov ( 2 n+ l ) r = 8 r r + t 2 n 1+ 6 n + t = 2 g n t + 6 r : + l r ) + 1 , sivarKoro.i(. Avi,5= ptq txar p xaLq 6cv 6,rgroL 6rr,ovo ftrsivordpnoq,16r€ dxolv{orv661o1p;rrl, r6rspr = 2qr,610ropr 8LolpEiroL Jr€ro 2,ro i6roKolo p. 'Alc, p = qDorx6oprep6p', {ot xoriLavrin n (zp)t = zqt,il 2p't'o "6no,ou d(?')r- q'. I noLdvoq. npoyjro droro. o q crvoopr oqcni6ng.
.H on66e(qnc r1v1h eivoL6fou, ov lpltorponoriooltt tre €€1666€tE (3n+ 1)1= 21nt+ 27n'z + 9n + l = 3(9nr+ 9': + 3n) + l, = ( 3 n+ 2 ) t 2 7 n + t s 4 n+1 3 6 r + 8 = 3 ( 9 r r + l 8 n 1 +1 2 n + 2 ) + 2 . = 1 + ,r. Ytooiloulo 6n (l + l)'= I + r]r. T6rs 19.Av ,= 1,r6rE(l +n)' (l h) ( l h j ( l l h J > ( l + r l t l I r r t . a p o il - r \ 0
= I + ( ' + 1 ) r + , , i : >1+ ( r , +l ) r .
aeoi rlan > 0,rt avro6rrro€ivaL dteonouv6rcloroD 6DovulLu(oi o€opriFo{oq, 61()1 or d ,or 6porroL,etqovi(owororo qvdnr!1forou (l + l?)"sivor0sn'.oi.
(.r+ l)/(r+2). H iKapoor/(/(x)) 6xsrv6rpop6vo610!x+ x+-2. ( i i i ) I l ( 1+ c x ) ( I 0 . { + l / c o v c + 0 ) . (v) G + y+ 2)/(x+ llly + 1)(rts x,y+ - t).
t. (i)
I Kol
6I t - c{.,{4. a, td npirlr vo !!,,) \4ovo. - l. apoi /(,{/ /(,r) o,,vtndicra' 16IDsry1urou?dirrorov 6vqx + 0. 2. (i) r> 0 l(qrpnr6E,i y> L (iii) 0. ( v ) - 1 , 0 ,1 . 3- (i) ix: I |/| + e, KorouyKE(pqravo,.x
€no),r0ei€r rrv llir - /l < e.
r.droro (b) Av€ldprlroondro ri prop€ivo€ivorro 6> 0,DndpI€L -xro! srrd,Ieei' + l l < r , o ) . ] d x o r r l t v l i ( . { l ) > l / l s , x o rs u l l c € K p u d v e L t 1 v 0 0 drcro dlol€,av
0 < f x a f( e , d t e l f ( x r - I l = 6 / 2 < 6 .
(iii) np6Kerror yrorop6Fora rponoroilon: 6ou),i\ne ro (i). !E ro ,/51lo vondpsr€ (iv) Kiour6€ivor p€io (i),oqori€/l0 > 0 Kor€lvor l.deL rpononoitolr ro i6ronpdyuo Fdvol6nap€l r.iroro,s > 0 rrouEv6rog6psr. € >0 undptouv 29.Av lia /(r) = lint(x) = 1, 16r€,yrdrcrioE t,,4>0, drora 6orE, ]lu 6la rq x
o v o < . r < a + r , , l d r €l f ( x ) - . 1 1< s , 0va- 4<x0 r&oro drore . 1 f ( r ) - 1 1 < E l r o 0 < r < d . A v d ( x ( 0 , 1 6 r €0 < - . { < d , d p o lim/( - x) = /. OFoi106, avro lin/(r) urdpt€l, if( .r)- ll < e,SnoF:voE, t6r€undpt€rxqrro _liml(.x) KoL6t€rrtv i6loflFi. (draro0iFr(i,10 r Eivq! -x kovriioro 0 xareErrK6 av xdrl6vo ovio sivorxowdo1o0 ror aprTflk6). (ii) Av /= li4/(x), 16r€tro 6t!d rd e>0 u,rdplErivo t>0 dloro 6g1€ l / ( x ) - 1 l ( e 1 c 0< . r < , . E r o r , o v0 unipt6r 6vo r>0 rdroro d'ore 'flxt rl < E.E,roFivoq. lim/(x') /. H avrio€ltKqrar€uvon trivErqr !c 6polorp6no.(AloLooryrK{i, av 10x €ivorKo\,{{i6100, r6t€ro.x' €ivcLrovrd oro 0 ro! 0€nK6). 34. Av r = lim /(r), 16r€yro xdoer > 0 undpt€rl(ii,rorc( N 16romqdro1€| f (x t - l l < E l(or rivor oqpdg6fl }ropo6}r€vo urooCooupe nr,;qN>0. T6po, av llc,{>N 0<xN.dpq
1 l < c . E n o s C v uh gm. / { l x ) = 1 . H l/(l.rl avrieErtKarEi0uvotliv€rcLgc ncprlrororprSro.
rco^,{lloi l. (i) F(.x)=x+2Trodl,o10.x. (iii) F(.x)=0yto6Lo10x.
l. (i) (iii)
Opuflrdvldvo xoLKdrr,r. El,qroq rrlri 0. Otr Fdlloro. Opclp6rn rriror, olJd 61 rivo. Eldltorl nprj 0.
530 AmvtfioegoeenAeyptva ryop),rjpara (!)
(iror riiorc Opatpdq KcL dvo Kor Kdro. Eivor suv6lro 6n a>-l I -a- llat d-l.KardroL./(r)=a 2 l) Av- l TradLo -4. 6],0 ro oro EnoFdvog, z ro A sivar Kdro Eor0r 7ru epo.tgivo. d = sup(-A). T616ro o sivor6vodv3)epdrua.rrqro -A, dpa,qvrrorpdqowaq rov npotyoipsvooulLoyrot6,ro d 6ivaLfvo Kdroepdlpa?raro A. ErL,Tl.iov, av B €{voLonoroSiror€Kdro ApdtFo10! A, rdrs ro -p eivcr ivc rivor -A, dpo-p > a Kor;ror < - d. Enotivoq,10 -o sivarro 9pd1!a1ro10 B pEtrsroKdroqpdtts rou A. 5. (a)
Av / €fvaro felq).itepoqordpqrcqF€ I < x , r61EI + | > x. A].td / + I < -{ + I < /. EroF6vo6,fnopoiuEvs rdpour€7lq * ro / + l. (A,r66e(nrtE irupqrlq ro', 1: Aeoi ro N 6evrivor dvo ppqTldvo.undp:(Er K{i,roro;rpr..r\,i.
Arawfioeg ot arit'ypiva rpoBTfipata 53\
(b)
G) (d)
nLi0o! , Us-, < -x< r. Ero#voq, uftipler p6vodlq n€nepoop6vo opr0u6E aKspoiov rov ir€yolilEpo.) / gs r < 1< x. T6po6ro?"6qre Aqo'j/-.x > 0,undplErKriroroq apr0t6qr, dtoroE66r€ l/t < y - x. sDorK6q Mlo Korny ,x > I undpter,orri ro (a),6vaEaKdparoE k p€ ,x < * < ty, to onoio 6rtwi\Et 6rt x < Vn < y. ^lol.ihs r+ V2(s- r)/2. croF6voi(unripx€rprtt6( s l1e A?16ro (b), ufltprgr prlr6Er p€ x d, 16r€.r > (k + l)o, Kordror k * I S .r./a,oa cwioeorlp€rlv enJ,ori roDk. E,rofdvoe, 0 < x'< d. 12.(a)
rrtv/ > x }lo 6Loro x oro A, ororoSriror€y Aqoi ro rdo8y oro B rrovo,ToEi g1oB €ivar6vodv{lrqpdylroT 10ro A. Kdrdrory> supA. (b) To (a)6€itva 6rr ro supA sivoldvoKdroapdyporoD-4,KordrorsupA< infB. I 3. Aeoi x < supAt 01y < supB1l0 Kriosx oro A KarKde€/ oro ,, a,leror6rr x + y < sup,4+ supt. Enopdvoq, ro supA+ sup, €ivor6vqdvoqpdyporou A + B, raLdpd sup(/ + A) < sup-A+ supB. Av ro x Kary €lrll,eyoivqr6 ro A Ka!B ovriororta, dror 6ce supA- -v( e/2 Kor supB /< €i2, 1616supA+ sup-B (x+ y) < 6. Ero!6ioq, sup(' + 3) >.r+ /> supA+ sup, - E.
L r a' )
I Itnl
ht
ft't
al na
h
I h
d
h
-ll
t0 a\ar nl (b)
H €qorror6v!6,4rou (4, l/d) eivorI TpoerKi,lopdororqqs t. _. B\xl=-.tx
a1
l al r:'
a
Av /(r) = s(r), r6re I
,. at rat=lgtulp)
x
+2
=rm(d+r)ra:
qo\Xfipara 532 Amwqoeqoeeru),eypCva 2 .. (-zah-h1l '-r ha'(a + h)' H €qo'tloF6lT&o rou (d, 1/ar ) €ivorq ?paetri ltgp'iorool rqE
O)
2. ..I 8(.r)=__l(.x_a)f__ =- --+
-i
Av /(x) = s(r), 1616 |
-2x.l
Tl
2xr
* at = 0, 3ax'z
n
0= (x- a) (2r,- ax- a,\= (x- a(2x+ a)(x- a). :Apor= d r(r= a/ 2.ToaryEi,o( a/ 2,4,/d') ftirar olrlvowiosqn}'€upri oq npoqrov l([roK6pDqod\ova m6 rc (a, Lt a\.
3. ,r,\-t^f(a+
h)- f(a)- tL'[a+ h- '/A ,Hh
h _('/;a+h-'JA\'/,a+h+'fd, _,. F o h ( l a +h + J a ) l(Va+n+Va) I 2\lG 4. ElKao{a:S;(x) = 11 r. At66E(rt:
-.. .
-
"-.,
+ r) - S,(r't .. (x+ h\"- x' .. S.(.x l_0
h
i:i
h
i {'1"-,a,-," ;:;h
=r-l li)*-'r', r
= I'il,' ' = *" r,aQoiliqrb,| = 0 roj > L \l/
5. /(.r) = 0 tra x 6tr oKdporo. KaL/(x) FqoproFvocv x cKdpcroe. 6. (a) r , , , , . E G +h ) - E l x ) , . t / ( x + , ) + c l - l f ( , ) + c l htrth ,.^f(x+ h) f(x) - ,,", ;:T
Aruvrfioegot tmleypivorpofi,fpan 533 I rhr ,,r,r - li- 8(r l,) s(.{)_ | ^c f(r r h) cf(x') ;.:ii tr h + h) .. f(x f(r) = _ c. tin!--j_cr'(^). = 3 . (36)'. 7.\a) l'(e)=3'e., l'(25)=l (25)'; "f'(36) =f,(e) = (b) /,(3,) 3 e1,f.$1\= f,t2s)= 3'(251, /'(6)= /'(36)=3.(16f. (c) /'(o'?)= 3(o)?= 3a', f'(x1)= 3lx'11=3x'. (d) /'(/) = 3r', aud s(r) = /, dpdg'(x)= 6rr. .. , . 8 ( x t h ' ) - p l t )= . . f ( x + h + c ) - I ( x t c ) b. (ar 81ri=rrT-: l$--,-
/(x+cl+b) ,-0 {'-' b)
/(r+c)
h
+ r)
- _ r._g(} :",/"\ ;:ii
i
g(r) _ i^fl :i:f
+ chl
fkx')
h
_ , _6[/(c.r + c]) /(c])l _ ,,_ cl fkx L k) - fkxtl -ir",,-.r-i,ril-------l-
= c. tinflcx+ k\ fcx) =c. f.k4. (t!T Kpivtrrrrou6l€rprolorjfo oDlrilrl o€rpd10)vunol,olrol6vp€eKeivou( lo! npoplisoroq 5 - 14.) (c) Avg(x) = flx+ a),fi18g'(x)= /(.r + a), ard 10(a).ADd g = /,xqrilor f'lx) = g\x) = l'(x + a) 1ft 6lo to .r, ro o'roio orl[aiv€L6fl I f' eivaL mprobrKrl !e r€piodoa. 9. (i) Av s(r) = i', dre s(.y) = 5/ ArId f(x) = s(r + 5), dpdon6ro np60).rua
9la)iaow. f'(xl = s'lx+ 5)= 5(r+ 5)'. (iD f(x) = (x- 3)',dpo/'(x) = 5(x- 3)',6',0,q orodpoq(i). ' (iii)
10.Av flt) ll. (a)
(b) (c)
f (.{)= (x + 2)'i,dpi f (x) = 1lx + 2)6,6aaroro dpoE (i). /(r)=s(r I x), lorE I'\r)= s'\t a x) 0,16ro npdBtnFq8(a).Av = E( + x),n51Ef'lt) = s'(r + x), on6ro npopl,twqal, nps f'(x) = s\2x). Avr(r) = cl,r6rei'(t) = 2cr Kar6rv u'rdplsropr€p6q k dror60r€ s'(r)= kr(0 (6tl,o6i 2d = kcl) yrq6la ro r. (Er' turarpia,oro olFeio ow6, 6€v.rvo pi(o!p€Kotia fn-Irl6svrKiouvriprlol t lra rrlv onoiaI /' €ivarovdtrorlrlq /. M€d ro Kred],oLol7 0a rirov too(€6oo1rK6va ppoill€ n6! 0o 6lola(s o K6oIoEov o fql0,aioe €iI€ 6irlo.) (i) Av s(t) = (a/2)t?,t61Es'(4 = dr, dpas"(0 = d. (ii) [sIt)]r = (a0: = 2a. (a/2)t1= lasvt. O nol0alqro(rdgr€rs(r) = 16l r!661a o0 . 6sutrp6L€rrd, Kol drorndar 400 n66loo€ I = 5 6€r)rsp6l.enro. M€id a,r65 6eursp6).€rro, I roj(iritm 0o sivoL i'(5) = 5a = 5 . 16 = 80n66rqqvd6surep6).eJl1o. H rqliltra riravit lrori ar' ourrtv6rovt=+ 2
21. (a) (b)
Aur6E€ivarivoqtil,Ioqtp6ro6Tpcaiqrou oproroi (Bldft np'jpl,rra 5-9). Aur6 eivololvirorq rou npoF).ilolo( 5-ll 6rav€qapgoor€i ortv n€pinloorl
534
AravrriotrEot tttl€1p4varyoQ,4lan r o v o u v o p n i o e, !o(vh \ = t f ( a + h ) f ( a ) l l h K g lB ( h ) = L s @ +h ) cld)llh 26.(D f" (x)= 6x (ttl) f"(x)= 4x3. 30.(i) tlpoiv€,6rr/'(a)= nd" a\ f (t= x' (iii) trrsiv€!6rLs'(a)= f'(a)a\ s(x)= f (x)+c. (v) hrqiv€Lro i6ro!E ro (iii) ( v i i )t r f a i v €6Lns ' ( 6 )= f ' ( b + a ) w s ( x ) = f Q + a ) . (ix) truaivn6n s'(b)= cf'(cb\'Lvs(x)= f(cx).
l. (i) (l +2x) cos(x+xr). (in) ( sh{) co(cosx). /cosx \
.
.{sinx cosx
(vii) Go(x+ sii-r)) (l + cosr).
2. (i) Gos((r+l)'?(r+2))) t2(-r+l)(r+2)+(-{+l):1. (iii) t2sin((.r + sin.{))co((x+ sin])r)l 2(x+ sin.{)(l+ cos.{). . Ginx+ x cosx)+ (cos(sin (v) (codxsinx)) xrxcosx:)). 2x. (vii) (2sin-rcosxsinx':sin:/) (2xcos.xrsin:xsinrxr) + + (axsinxrcos-r'sin:xsin./) (ix) 6(x+ sins.{)5(l + 5 sitrlxcosr). (xi) co(sinrxr + l)r ' 7(sinr ir + 1)6 (?sin6xr ' cosrr ' 7i6) (xiii)co(.rr+sin(xr+sin/)) t(2r+ cos(rr+sin.{1J.Qx+zxcosl))1. (1 + sinx)(2xcos/
sin?r + sinxr ' 2 sinx cosx)
cosxsin.rr sinr.r
(l + sinx)r
c",t)*,/--L\ |
/ r'rl
\ s r nt _
3x' sin* r,,,,"i * ),.0.{ \ \ mLl.l , r/ \ .- / x'i
""\
xrcosr 1 /
",, Li" ^ l 4. ul
(iii)
{x+I)'] (x+ 2)) 2x'.
5.(D (iii) 17.
6.(D f'(x)= s'(x+ s(a)r.
(11i)f'(x) = s'lx+ E(x)). (l + s(x)). (\) f'(x) = s(a).
7.(a) (b)
A'(t) = 2rr(t)r'(t). AQo[r'(r) = 6]1 a!r6 ro r troro onoior(r) = 4,jn€ror611 A'\t) = 2n. 4. 6 = 48r, 61c,r t(t) = 4. y(t) Av Eivsro 6y{oq q lpovrxr{orrypr{r, 16r€ y(r) = 4'r(r)r/3, Kor;ror V'(t) Arr(t)'1t'\t)= 4n 4'1.6= 384jt6rt'vt(t) = 4.
Attavaioeqoee .e1plvarpofhipatd 535 (c)
np6rrl M60o6oqr A9ot5A'(t) = 2,tr(t)1(t)xatA(t) = 5lld r(t) = 3, Ctletot6fl
- lo'ourto= - '''' -' .rrr= 2t4t\ .A1l), l. 61t ErcFvoq, t/,(tl = 4r4t),,{l\
= qr.e.l = 306tovr(f) : 3. mparrlporiF€ drt fro va eqofldooDFrq 8aitepl p60o5o,
t(r)= A(r)": y'-A(rn
dpa,XplsUono|dyraq ro llpdpl.qpa9-3](arrov Kov6vd.Iq A},Dot6oE, 1 n t t\l a't'\ -"''',"','' 2^17,1}-
l t ^ = --J--,
""
.
=nhi.tAG),A,o =
T6oo.
]
,l 1tI "',11r 11t*re r,lropeivo e(XogrFaw€V€|.
94il-= r''t'Itlv1r1= 4tdr)13trt]'1
= 4A(i),, 3a rl a1-_ v,(r)= .._, . +lA(o A'tt) = -=,
10.(i)
, 'lt1, 4t A,(t\
=2.3.5=30. (/ o ,)'(0)= /'(l(0)) . ,(0) : /'(3) . sinl(sinl)
={,r"}},"r.''rr [n) d(l) = ,,'G) 2? = sinl(sin(ra+ l)) 21. 12.\O Kav6vog rng Al,lot6oqolvsndFtordfl
$,1
tO= tt " at't'l = l'@(x)\'c'(r) | =---:-'g'(x) 3!.xr
n. M(b - a).
538
Arar.rlocrgot tn)rlLttvo npoQtilnn 'f | h-:\
'f t z-\
/ (r) y.0'(dro.or oro(a.b)
epaf(b) - f(a)= nlb (c)
A!
a).
/ ( x ) | < M 7 1 a6 l . 0r a - { o r o [ a , b ] , 1 6 1 6 M = f \ t ) < M , 6 . p d
f(a)
M(b a)< f(b)< f(d)+ M(b a),
rl I f(b)
f(a)l < M(b a).
cos.r+ a yrdl({inoLov opL€p6 a (616rr/(x)= - cosxei\rolliaT;rora /1x)= Kald tia ouvriprrlorKar oroaoSlroa Srloliror€q orvoprioeq 8roaapouv ora0Epd). (b) f'lx)=x!/4+a 7t0 {dnorovopr0l6 a, dpo /(,Y)=rri20+dx+6 Itq rdnorolqaprejronq d {ar b. =.rri2 + xri3 + o 7u rdroLova. dpo f '(x) = x'i6 + x'| 12+ ax + b G) /"(.r) ({inola yla d Korb, {ipo /(x) = x' | 24 + x' | 60 + dxrl2 + b.{+ c yra KdroLouq xcr nro on!i, flx)=x1i24+t5 60+ax'. gproroq d, b Kdrc. IooSivoiro, + bx + c 1lo KriroroDq uprepo$q 4 bl(qr c. 29. (a) AqoLl r"{r) = 32, irouts 6fl s(r) = - 32t + o }Lo Kdnoro d, dpo s(r)= 16r'+ d. + B yrcxdnora a rar B. (b) Elaovdrq, r(0) = 0 + 0 + B KoLs'(0)= 0 + d. Erodvac, a - q, Ka\P = st (c) t' orir{ rrlv n5pirroot, si = 0 Kol q = !. dpas0) = 16r:+ ur. H }6'110lrl 6rov0 = s'(t)= 321+u,Tir=ui32,dponfa-rrsrrl nti llg s elqovi(eror 28. (a)
,fir,)=rofir,)'+,[j) 64
',12
Exeivrtq tpovlKl orrytri,I rol$rrtraeivqrelqov6q0,d,ld rt €flrd rvoit€lvdl KorK'i0s6Tlytrl. To Pdpoq oerp6vo t > 0 6rov 32(6rrD6 rrDndro d6oqoq 0=s(l)=
16l+uf,
ri I = u/16 (Xperdqerdr 16001p6vovo n6oerKdrro600 lp€doqx€ va qrd@r arlv Kopuai).H rox6rtra r6rs Eivor / rr\
s'(r,/16)=32(;./+Lr
(rt illo roXirtro [s rnv oroio opllKdKrvri0rlK€ rpoqra ndvo). Eqopl6orero O€{irptfaM6olqTlp{q ornv f(x) = !t oro [64,66]:
t '60 . ---=-t,{,oro[04.60]. /".. }ra(daoro.r oo "6q 04 2vx Ene6i64 1, x'dx.Otoioq,6€v vatIobNE b"l4> Jr rrdx. riv€roL 2.'Eloupe 6rfr- D'+(tr l)'+(r- Di (r- r)1. -r ., ,. . . "e,r _ .rL 5 2 3 30 I ,- ,. ,J, , -,,, - o ' 1 ' ro' to' ,rL5 2 3
o1tol
rpop)sjyata 542 AnawrjoaEoeem).e'ypiva Epeav(ilqro l,(/, P"\ Ka\ U(f, P") yllopoiv vo npoo€1'[ioouvro brl5 660 {ovld 'ono)C ror dro [p6p]'rtfo I , our6 ouv0nri06loup€,6utr6towaq ivo ]1opK8fliF€7 .o.
l'1
7 .( i ) J n f = 0 . (iii)Jol=3. (v)
/ 6rr o),oK),rp6ortn.
(r'o prqousrnpri on6sEr4r 6rr or ouvopiftrs oro (i),(iii)mr (vii)€ivororo(litp6o1Fq,Pi,6n€ lp6pl.Ipa21.01flpdq1ovo),oK),npojrdr0N noDeivor€pqaproporiv vEiqa,r6qT€nr€rprKiErK6vo, vo €Eoj(0o[v ous lpd,tpnouoror6.vrqErq r66€q rrlqan66eqrlq rou npop),iporoq 21,oe ouv6uqo!6 !E yv0rord oIoKhp6Faro.) 8. (i) l'tt
"1 J,[$+:; -1a,=::
f ,t:
t u r ) . /, i , | l k)
)tt?z
. 6
x't- x'lox=: 2x+q
x \ d x =a
uu^,/(.xr'iv',oToe€pd) I,'ll" *, rtt,o)*-|"'Jrsl" ru,or).,,
t' ... t'^....".t,...
=J eUVy 1" t(x)d'G6,tJ-scv)dJcivqrI ma€cpd). 14.(a) (b)
Epeqv6q,r(/, P) > 0 yla Kd0eslorl4pron P. toop!6orc ro Ldpoq(a)orlv / C raLrptorto,rorciorcro t€tovd(o
t'.^
t'.f'
J , ( J 8 \ _ J ,t 1 " 8 . 24. (a)
Ell(pavri'q, n(b- a\=L(f ,4=U(f ,P)<M(b l" :, vc,r6q. mr6 aJ<J,Jlt)d\< M\h
r)]tt6 4fle 6up0pis€q P rou[4 b]. at.
I f(x)dx EropdvolE, ro I = b--;- trrvonoreir1v m I p { M. 'Eoro (b) rn rar M 1 e),ri;0oqKarFiyrorr, avriororxo,rLurirrtc / oro [a 6]. EnerSti rt / €ivoroDvslliE, noipvslrK rrpdE,? KorM, xor ouvenri,q, rcr q v ntr,i ,i rou fdpoDe(a).
Aravtryoetg oeenlzytlvaryoBltlyuta 543 34. (a) 0.
o); 38. E,rfl6li
-1rl 0 Kor u- cost=cosJ --4cos
\
6
-Jcos
= \412. E r€ror6Tl,oeoi sin7116 iloous cos7116 > 0,
,i",,6=v!-lG/2/ =+. 'in!'1Yl e.{u) r,o(,+ ''v}= cos(x + y) _sinxcosy+cgs.xsiny cosxcosy-slnxsm/
6'
Arawfioeqoeenkypiw npop),fipara545 smtcos y , cosrslo/ cosx cos/ cosr cos/ cosxco!y_smxsm/ cosx cos/ cosx cos/ _ tanr+tany I - tan .xtan Y (b)
An6 ro lEpoe(a)6tout€ tan(arctanx) + tan(arctaD/) I tan(arctanx)taD(arctan/) = r-t Y ' lry
tan(archn, + arctany) =
Aqoi apr€i ro arctanr, arctan y rcr arclan x + artan y*kr+n|2. 1rl2 I, ?r.) 66rs arctao.{+ arctatr/< - ?r/2,dr€ rpdn€rvq oearpaoouN ' eqiooonEdr6 1!v ,!p6rl 01 6r agorpdo€oc ntq 6€i1epns I l. O rploroqdrroEEEdl€rdr onoi€edivowar flo *dto: cos(r, - ,)r = cos(r?r- ,-r) = c-osrtx cos(- n() - sin n-t sin(-4t) = cosmr cosrl + sin ur sin tx, = cos(m+ ,),r cosrrrcos n-r sbmrsinrr. arat"nt t
Or dl),orronor€ld?owortr 6!010rp6no. 12.A,16ro np6ptlta ll dnEtorort, ov D * t, t6!e f' I t' - co(n | , )r]dr l-,sin nr sianr dx= Tl_, lcos(m ,)r
| (tsitr{o- ,),r si!(r, + ,)rd tsin{D ,)r 2ll n-n n+n ) L n-n
A},Ld,av n = r, t6ll l,
I l'
I sinzrrsinnr dx= i I ' I 2t
cos(m+t)xdr
= -{hr - cos(,n + r)7l - t- r - cos(o' + r)rll 2-
Or d},),orTurorono6srrwowalU€610101p6no.
sin(D+ r)rll n+n 1)
546
AtrawflotryoeerAeypivanpop)'fipara 15.(a) 'Eto'rs€ cos2x= cosrx- sin?x = I 2 sin':x = 2 cos':.x L .. = I -cos2,{ , s,n-.x 2 I + cos2x , cos_i= 2 Auroior rinoLdnovroL 0JI6ro ldpoE(a)€n€rbri cos.{i2> 0 Ko!sinx/2 > 0 (oqon 0< x< 1rl2).
(b) (c,
'
19.(a) (b)
I 1 "y. ,n l"l-(os2,t rd,r-i, ax-TtD Jq 2
ar
t f' frl rcoszx d^ J,cox ^dt'J,ttb-
a)+
| -. rsro z D s r oz d) . ?I lsin2b-sin2d). a
arctanI - arctan0= ?ria. lim arctanx- arctan0= ?r/2.
20.1i,n',;oJ.-h
l,;o "=
1
2r.ra) r 0. cosh i) - sinh,= qfx'- t uooisinny > 01La (c)
r (arcsinbl(x)= -nb,( -
))
_l cosh(arg sinh(x))
-
(b). o'rdro Fapoq + vr'rx'
(argtanh)'(,= ',il*#nh.)1
(e)
= cosh'(arg tanh(x)). A',}d, tatrh,+---i.
= I ar6 rc np6p),nro 7 (b)
cosoly
. dpo =1, tanhlargtanh(x) + -----L cosn(arg taDn(.r)) cosh':(arg tanh(x))= 9. (a)
rI,.
Av y = arg sinhx, dr€ x = sinhJ,=
q:gr 2
dpo d-e,=2x, err_2xe!_l=0,
, _" - z x x . , l * \ t 2 ,ipa e = x * r , / t - r ) c , l oe d, ) 0 rt 'Opora,
y = aresinhr = los(x+ 1/-l + l). argcosh x = toglx*.,/r' - l), lt arstanfi.x= log(l+ x) t ibe(t
(b) J- -T:,
"1.
dr = at9sinh b - arssinhd. 0,16 ro np6putq 8(c),
= toe(a + rft + a) - re@+ \/T-+dr. f" I rfu, rt-togta+r,6--r) J_-/ , , dr=loe(b+
Anavtfioetg oe auk'yyivatpop,tipara fr
'Iq
12.(a)
-i I
r
549
r ! ! -a)1. . d x - , U o g (rl b ) - l o 8 ( l b ) l o sI{ a ) l o g|(
x'
lim a' = liq e'a'. AQoflogd < 0,6toog€lim.xloga = - -, 6rfl lim e'r"!"= 0.
.098J)i= 6. -{= e. 1"1 1;. !-.x G)
(d),dpaliq.r' = l. li'! ], = lin e'bs.A),ld, limxlog x = 0 an6to F6po6
16.ra) linlos (l + v)/v = los(l)= l. (b) li!xlos(l + l/x) = lin log(l+y)/y= l. (c)
e=exp(l)=exp(lim.xlo8(l+l/.x)) (*) = liqexp(.{log(1+ l/x) = lim(l + t/x),. q€ exp oro I Ktrr (H ro6rrlrop€ rov oolepioxoE€cpidtoron6 q oDvd1ela vo arflo),ono€i o)g €iie: Iriop€i frd rd0e e>0 undp:(er xdrlorot>0, dtor d)ot€ e-exp/l N.) ra.x>,ry.'Apo le |
(d)
e'= tlitr{l + l/x)'1" = lir(l + lh)= lim(l + lix)', = tin(t + a/y)'.
prdaptlri xard&ol I 18.Merdar6 Evq1p6vo,o opr0!6q rrDv6o).lapiov nouoro6i6sr 60 ,qpiou, eaeiv0r lim(t + d/100x)* = e"r@. 19.(a) qrsov6c f'(x)=tlt tta x>0. Av .xl e . r i n > e J .
\ ( (!) Z r r - : l A o o i ( ' ; i X t 0 ' 1 , c n * I > t 2 . r i , >I l .
2>18.nn>s).
AltdwilEt€ o€ e.lr)"€ylliva rpofi,rjpara 557 8 .( i ) c r = 4 , + 4 . (iii) ci= (;+ l)a,.
s ^=J'noa,,
l. (D I - r l ( n + l ) = l / ( n + l ) < 3 1 r 0 r + I > l / 8 . (iii)
rmi6r+r -i6Tr = tim(6tTT fl1+ timldGfa+f+1=n16=6. (K'i0r 'Lruarr6 al,rd rs 6io 6pra. fropri vo 9p€0€iFc rov i6ro 1p6rc rrou
I'T {vG+l
- vG)= 0 q'rE6€it0n aroPrFIio.)
Ereov6q,lim (loga)/, = 0.'Apo lim{G = [m e@')'' = e0(atr6io Os{i}ptlo
(vn)
(:/;J,< :,/7+ o< iD (fi )',tor i/7 s i/V * = i/ z?,apa
(vi).) lir!(Vr)'= limv2(Vr)'= | (on6ra dpn tv)KoL (ix) Epgovriq, a(n) S logrrxorlim(log? a)/, = 0. 5. (a) A\ o < a < 2,16i€a'1< 2o ii, dpoo I 'yralz >l:
To dprc ,r+r t?tl!+r)!
lim-
3. (i)
l-l
= lin2 zlr"+Dd
eivcr0 pc lzl < 1,qud - yrolzl > l. To 6pr0
W=d ,. u,*^tf civcr< l'yto lzl < \42,dpaI ocrpdsuyxlivcrqnol'r'ro,( ]ra lzl < \,/t.A,.).dj . ocrpd6cv sutxlivn onoLiro4ylo Iz|> V2,dpon arrivao$rxltcrtcEivar/2. (ni) Aqo'i
(v)
I oelpdooFl.iv€r qml,$roE yro 6lo ra z, dpo n oKrlvaglipd,rartg€ivor@. ro 6p1o \mJ2"z't = zlrn ltrt)l €ivoro,r1olzl < I,aX},'i-yw lz > l,dpon orrtvo sql(),rons€ivorl.
NINAKAE EYMBOAON
l al , t o .t,ll na{& y), l5 min(r,y), 15 e, t6 N,18 q,19 19 "!,
txl,6a |xl,64
I r. zsz
6,76
[Trt'l' zs lim /, 78
l+ f('),78 lim /(x), ?8
l'+ /("),78
z,2l Q,2l R,2l
lin f(x), 7e lim f(x), 7e lim /(x),83 lin/(x) = €,83
(o\,o
]g r(x)
).a,,20
fG),34,3e 1,36 A O 8,37
f ' c,31 f lc,31 c s,31 8, ...1,37
= -, a:
supA, 106 lubA, 106 inf A, 106 glb A, 106 tilne, tl+ lin sup,4, I 14 EqA, ll4 f (d),t23 f,123
{a,,...,d,},38
f + s+ h,38
f '8 h,38 /os,38 fosoh,38 xf(x),3e
l l d , a' 2 o,i.a C^,43 A U 8,43 R A,43
dflx)l f, 132 f", t31 f""t33 d' f(x) ,".
t"
J, f(x)dx,2s2 I
t'
I
f.253 f 253
sin",256 sin',256 r,251
A(x),2s8 cos,258,260 sin,258,260 sec,26l tan,26l csc,26l cot,26l ztcsi\,262 atccos,262 arctat262 log,281 exp,282 e,283 e',284 a',284 lo&,285 sitr-6,291 cosh,29l tanl,291 ars sh&, 292 ary tanh,82 ary corh,D2 Nap log, 296 F(i)
.,
300
If 1 , 4 4 r.:,a].(f,s),aa
f t94 R(f" , a, b),208 LU, N,Ng u(f, Pl,20e
If,tor Jf@atnr
n]tr.(f,c),44 i< c' 46
I r.2t3
10 (Etttoc,(a, b), 47 ro avoltr6 Erdsrnpo(4 t),49
P4''340 P',4r,3q R,/,348
I f0)dx.2l7
la bl, 50 (c,e), s0 [c, o), s0 ( ( (
-, d),50 -, d],50 e, e),50
t( f, P),22e 9(x),230 Ll
f 247
ul r. zqt
r(x),330
lol.:oo Ic,j,37a
liT a,'375
566
i'aru; oulLll.av 7, 384 hi J,. .188 lm supr,, 388
l+r",r88
liminfr. 188
I1r; a, b).190
r,418,44:l c.411 t,445 z,445
Rc.45l lm,45l 4.452
tt! tVl, qsz J"(a),459 sin.470 cos,470 erp,470 4, 478 8",418 D,419 Dt.479 eD.479 A. 479
A,,48t)
u,.481 + , 487. 495
.,487,499 0,487,496 1,488,500 - ll,488.496 a-r,488.500 P,489 > , 489,494 < , 489.494 > . 489,495 < .489.495
lol, 4ee
EYPETHPIO
Abel,NiekHenrik. 138 Os6Qnpa Im',435 KAr tprorox,4l2 Ar4rBo{ori,329 AoQoioLU!, 391,463 ono),ift0(, 399.464 ard Cesaro,410 zard!Ahel,435 Tard liQL Tljio; ro! Abel.|ro, 328 ouoid[oQ@, 42i 422 A0QoLotta, A!o,209 ,qJ-r€eo.357.39 an€iAor axolo$(aE, 392 KftrD,209 M€Qrxd,39l MD/q6rxdJ! oAr€!d)v. 443 nEnEQaoFalo, 202,391 Riemann,233 >iypa or'U.Bolqrd( ror, 20 >!'!aQniosor\,,37. 38 AUQoropo. rlirq! Iox.
yr( ro sin,264.265 ]to ro ran.267 ATAaia [rysirai or(lQrr)ol, 474 AxeQoLo-c.21 oopoiorll391 c|".Id,rinu;,399 xdrd Cesaro,4l0 pLq;,391 doQo|oFo olobrtuall rfi, 4tr2 dnEel 374 auEoroo,379 CaNhy,380.503 !L1106rxi,.+77 tibbonacci, 27,2E lit08\1Frni,194 !r4 ar,!omo, 379 Uir{oilolloa,379 lLlu6rxi, 463, oLroLduoQqo zorqv€lld\rl. otrotijlroQqrr421 oarcFtl!,375
Avd6Qoprrdg oaropdg,l9 Avdlml 03 BoQu(d xTdoloro, 3 14 A!€u voriloro; ouuool(o&o; ro! Bemouli,27 rorryeotrsrQrTou-oprolllTrzoi!6oox, 28 rox Gronwall.294 rou Liouville,371 rou Schwarz,15 rou Young,228 rQrYorvfli,63 ro! carchy-schwatz,232 oobrolgrolB€lo orll1o,489 A{dlot6vrcg,203 Avr aldglsot, rlnos rrc, 305 A1rWrao8r0{dEldtog, q!npdooeor;,8 ror rol).on),oorootolr, 8 (FL A6qroro o).o1tliQ(0po). A\'rl"xaQ{11(qos, AwioraoqoE orrydprtor;. 194 AvriorAoqol'fii.lmt oeLQdg,192 onolurol(wis,
qeL0u6g,445 owdQTllort,45I :lJlrusrpio yQaql'lldxov, 165-166 :wopirpl| d0!0r4q rov, 37.38 oxEQsiopryd6lxi, 474 alli0rQoeo( prdg, 194 ap\'4Tfi6 pdQoqpwE,44 dpiro,44 aoul,rl.L€EproE,m, 9l , 95
anEouso, 162 36 Fafu('E Frognol1]o)11rFflriE,
574
Euptttipro yrvdrclotu!,37 .A,ioropqlmo grdE, 182 ./QaFpr"i, 5 I yEiAU0 lLriE,50. 67 du!a[!tvolo.19 op0liorfo.37l x€ao,Z0 oprard-.iurio!1oS.189,46I xlr'Tl f'rtzrl€td;.461 ]lrzt'o. I 12 Ur^.rerolltq, npdnt,50 6€ reQll.5o >1jllrT0tp6les, re7ii r.D,50 KaATrolards,50
oiorruqtur,.50 :ucrro)Lt. Arilxarn(.387
>Zrirootro;/tnq rxd rqe.lordoErdv. 50 ovuqedlta rtimrupl.I 14 .zlEoov 2X0o!rar,dqodyua(hm tf).1t4 6toreto1lrho,489 LoopooqLxd,50S ntiE E6|,0tsr0yNvo,490 rr$fl16C.ioE6),492 rov quatemions. 492 ^ /.dpazrrQlmLxiev6s.49i )cnwrztraQd l5j t(D.tog. lcnwarz. 0lrodrtrorou.I5,28,232 Jcnwarz. drw€Q'l nsEiyololroL,,16t xnvarT. H.A..15.iIt.5t5 rec(rFrO.261 - d\aimtnsrTrEf]. Arcsec). srmp\on, Kqrovosrou,3ls o!"rrorpoqoE rtE (0tr.Arcsinl, naQqyoyog rIE, t43,259 iTo uor,L'!oTaytortLa,340 unoaol,.ro ror!.352 npoog8Tl)to runoEyro.264-265 SlirlinB, rrhoqruu,I82 srurm.OruQtuoori.,{etoll:ror,,272 rupremum.l06 To (€q).26r otrimrDql rI!. (Et Arcran). r drtortLorr!..154, 429 T2ErQ,r TGI Ladqra;,347 ftdQltorou,350
Eapetfipro575 rolxdJ.mtoxor,340 1|lTolo|llorox, 348 Toworrri owdA to!, 36 To!rcrrxoEre7€ofl19,479 Tsr(Urrrq, F6oll'124 o yptaio,124 TelLemiG roQaldryrortg,4?g T4rvowoex0sfa,i22 Tomonl",iQro!S,394 TEmqE ailar, 408 Tror fiE QG0S, llEllroBd\,o,4@ T0orror loyo!,295 TEmrot, ohxl.rQ6tr]l1oE, 397 T8rQotorvqdsro! xwlo!, 176 TerAdQxl.Io 1Q!pnLl.!68 T4LirrtE|oro x. 34 T6xog,rAoiroEuro].otrotrcnrou,293 Topiow6).N,37 TorD{drryioE Ff}rom oqrEio,182 Tontxi t6tortro ror oQior,85 Torrx6 o!!.€ioslaximoxoxrdQqorK,157 )'trion, 182 Tonu{oot|r€io16rioroxouvripntofi,157 )vioro. 182 Torus,B),enE taunQ6)"),o. TQryolni olio6rrFa,63 TQtyoloPascal, 23 TAryovoUsAlnes owaarrloerq,256, 312 o].ox],iQoo! Iov, 312'313 q\.rt('rQoeor rov, 262 TQrtordplor,ldloE i]1E,8 ytaro lmol.ol]Io,350 TlroE Lagrange Tnmqnoa40oliE rou Lasranse, 42
TnnoErolj llnoloi.mx !r rcio! o).ord.lgnUoror. 349 TlroE qg "69Qorotg l{aId p6AI', 328 Yrorc;\o1]0la,3?9 YreQ&rDdEoQdp6E, 36s YmAfuii,59 YmQ&),rxisqanropMl,zgl YnsAFolDtion€iAo.267 YnsaFo),uiouvriprtol,2gl YrisQFo),ud rpirovo.2ql Y:i3Q6oi,!xd 291 o1J\alsirolo, Yndlowo m),uDwlou Taylor.149 (Yno)otlolo, xxe{6, 462 pr1aSL1{6v @olramDd$6QoE dQ!0p{n!. 443 pr1a6$nls @owoolrx6p€aoE or'.vdpqofi,452 oa!.rdoi 6gdEovqE,445 Fibonacci,oxo),o!0ta.27 Fibonacci, O$*o!,28 Fibonacci, TinoErox ytarov px0pdonErcnE ror nhewpori 1tDvnorJlEl.rdlv, 27-28 Founer,>0rQ6g, 268,2?0 Oljouaoioar,opol. 18,29 Vieta(i Viete),./r!duA,oxor .{rs.1o211r,278 Wallis,-yr,lopevo rorl ytaro 12, 331 Weiersrass M-xardAto,a22 Xaaoxrrprorni(odllomc), 491 Youn&ovro6rqroml), 228 ldFoEror N8i(da tn rqv,294