Calculus 2

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Calculus 2.qxd 12/6/07 1:46 PM Page 1
Series continued BarCharts,Inc.® WORLD’S #1 ACADEMIC OUTLINE
Aannd eiqtsu sautimon i ss uSc. hI na sg en/3=ne0aranl =staStemmeeanntss, t/he asenrmieasy c ostnavnedr gfeosr -IaTf hn /eo nbpz-nesredoriivleiesmr gaiten s(do arng aedpo pemirteohtarecirch besnse£ r¥¥aien)s,( ntha‡reenN /o)f toaernn adnui/svbeednr ghfeaossr. TcEoh.gne.v ,i en1rtg+e1eg xraat=t eadn1 -seenrxdiep+so ixhna2ts gw rahidemirueps lt iohefes co1or+ni1gvixenra=gle d1ni-cvee xRrg+,e adxn.2dg m.ay
3 comparisons. Try a “limit” comparison when a series
/an=S. looks like a p-series, but is not directly comparable to it. The initial (geometric) series converges on (–1,1),and the
n=0 integrated series converges on (1,–1).The integration says
•fGoermom /3etarircn s,ewrhieesre. Ari s( nau rmeael rniucmal)b egre aonmd eat„ric0 . sAerkieeys ihdaesn ttihtye •REa.tgi.o, n&/=3 Krsoinot^ 1te/snts2h.Acosnsvuemrgee sa si„nc0e.nl"im3sin1^/1n/2n2h=1. ln]1+xg=n/3=1]-1gn+1xnn for |x|<1; a remainder INTEGRAL & DIFFERENTIAL CALCULUS FOR ADVANCED STUDENTS
is n/N=n=0r0n=1+r+r2+...+rN=1-1-rNr+1]r!1g. It implies If nl"im3aan+n1<1ornl"im3an1/nn<1, then /an converges •aTabragoyuulmot rce noatf n (adsne e Mi nbfaeilcnoLiwtea)luy ir mdinipf fliesereser nieetqisau.balTleith yfeu fnoTcrat iyxol=no r1f .isseries INTEGRATION
rtn/h3==a0±tr n1th=. eT h1se-e1 rciroen^siv fedrrigv<eenr1gcheees aa lnisfd o |pn/r3=o|>s1as1irb. nlTe= hsuea mbs1 eo-r1fi earsn -yd gi1veleoor,mgeesat rniidcf /(gaeboasmnoedlutirvticee rlsygee)r.si .e TsI.fh Tehsnlee"i m t3feosaltlsaon +wanr1ien >dge 1arriovere udnls"i ebm3fyu acl oinnm1 /apnpa>prils1yo,inn gtw htiehtnhe Ik/f3= c0=f]0kk,g! ]citg ]ixs -aclsgok= cafl(lecd)+ af 'M(ca)(cxL–aucr)i+n fsel2lr!]icesg]. xT-hce g2T+ayglo.r •Heuristics. TDhe EdeFfinIitNe iInTtegIraOl cNaptSures the idea of •Area unIdNer Ta cEurvRe.PIf fRis EnonTneAgaTtiveI OandN coSntinuous on eAn'd(xp)o=infts()x.) (valid for one-sided derivatives at the
series at x may converge without converging to f(x). It
series can be determined using the preceding formula. root test: nl"im3np/n=1(any p) and nl"im3]n!g1/n=3.More converges to f(x) if the remainder in Taylor’s formula, adding the values of a function over a continuum. [a,b], then #af xdxgives the area between the x-axis
•Ep-.gse.,r ni/e3=s1. 3F4no=r p4,ca1 r-e1al1 n3u-m1bme=r, 2/3. 1 is called the p-series. precisely, nl"im3n1]n!g1/n=e1. xxRvna]xryg=ing] nw+1it1hg !xfa]nn+d 1ng_)p, i:ap]xp-roaccgh]ne+s1 g0(xxasb netfifiwe¥¥en. Ec.gan.,d t hxe, •RdRdeeiifteeeimmnrmiaatennin nnei n dtss euubgmmyr a .al o Aipfs a sratuht iiettf aiuolbninlmcy,t iiwtwoinhnegi icg fhvh atiedlsude e fa is nuofeimfn di s tuoeoc fn hd v ivas[iaulsum,ieobssn].. oAAisf tahned atrheea g arcacpuhbm. Tuh]laet geadr euap ftuon xc.t iIofn fAis ]nxegg=ati#vaexf, t]hteg dinttgegivreasl f(x) Fundamental Theorem
n=1nP POWER SERIES remainders at x=1 for the MacLaurin polynomials of [a,b] into subintervals, typically expressed by is the negative of the area. A(x) f(x)
The p-series diverges if p£ 1 and converges if p>1 (by ln(1+x) (in Taylor’s formula above) satisfy •Average value.The average value of fover an interval [a,b]
a=x<x···<x=b; and a sampling of points, one point
•zetcbsiAsanetueroenrlr rrmfifimlitmooo(ceCnrsw,tpr: l0 ona yati)nhOsr. ari i dnzeeTtsnl/ee n io3=ehuanNrc nse on1tgrss h e])bh t- wsearVotaiesh sucirsrie1attgnemhi Elngeinydrn n iso. eahet .dRhndasaTIene: bifrc -chc msnGr/ oaet2o=sehoNbsnnela/1eeuN=nsvrsEn1 otiiei1aece]tnlr$r-e u sggNevrtse n mae1/se1m3s=l e+r.gusCv i1arn eeMiann1aeoN sl2naEs ufonden aewri.<rdavne ) hn oeod.aaTo rvf pg Nasetpteehl+rEht,rs ee et1o, et r .afhSnrifocnemairhr tt( Tesst itarnaht gas u gerosllSna tiupmemcls maraeiintrttrteiitta inoieseoattgndesfl, ••TRPpAaItaoanh/n t 3of=ohe p se ptclwepce0o yeoa,r orlw erwnpeaitnnrvehosoceev sau]imiwrrxs en l lsso sterige-eosiefargnrre frl xieri tgssicecee nee sigwosiasss rcnni.len i n ei= ce ertvnx ,Aoishix etue ona h–o irsamrvsegf c0 p oe rrabe+dt(o erf hnoe(egw paierrrncae)lo ese “e osf1.ia(nrc. not– Ali]eueT trsxn¥¥sdn.mm h et-ap uIe,erbn nfom¥¥r/i 3=nee wesc/tiN=)rdbhesn0g; ea te+e0qat ra(er onti ips nrsinicfax n)voe cx” r 2anrw(o exaionlc].eane ,rx –p^sl vrci“ N- osRneaacsw rulbe=o,glcamroeceneig urd0+ve2sb ts s,e+s. Retce erh1ri”q)grsse,f)iu e fe2iacieosssn.te, n r wantwfy cesttriehoerie qthrvimlt.cedeoaihdnesffl ••mcsBnCsf1Reo(oe-aoxer 1cmnlisam)enei]=xnpsc1s2pss un=g/gM =a3u=t=T1irintv0/ina+a3a=ileygny]cxn1nsl Lgo ]]++-tx ]carh1x1 o-u1e-s2nger1 eg+_cfnxi1rTT1f+ng.i+niag.ea1 .c2y^ssyp=.=ie lxeloimrnon/-nnr3=i/+r3t =easc 01yscx1.#:n o<nbxeDeRfnsnf i^+e-1fihfcf,rxo11eii=ueer<"tnneshndn.t1/e03= s tnh:,io0 a]tnatthih+nIenef=rg 1 ftgchx(tonahne)Rn(efc f >)bi^gc/yx0ien eod<!nm.itr1seeT hact.ahrtnrliidyecs RADpfdiumpersrinaepogoi xarcdepirahmxntrm=peecitei0tt g rohm (ieaae cuoiibasnnneiuRnlic–nvditanm1 hni eaporgesc d so)fmeu ipso ;u/ai=m nunaannbcabxntr hi, ni.tiniintn l hsn–Aisot :tx1etetu ewiesii nr/orb=i=uorvne vnnivmfa1urnata afle plth, le+sser ] pasa vuocrliiascveaefmsiDy hrg aln s)] t l ,xccugx:h s sowi.btteui/iuhf=hsi-n hrmbn.Alexo1 eeit :Amnxerfcmoetortp oi e vf-ni[stlas rnaxhue 1avrbilfiemgt[m s]a–t.i 0ci l1tm a-;uo,i,s( o3lxmsnualani] i1xtmn t ]i’gihsn.rcvwm n 3eTiueagt i ulonhlatshuthmu+kheoaet s emer .a s va1mssfce E auDso tlo.tnr3mlhugsaeecieike.gn.si,n, ae ugoeltttselt ihhef ahtif deaessfrt, ••AtmVmaqAvtagtgIihn paeuietmq cvauoelrpttlacyuoseuole enrttu sacgeuo i+tbta pnihim irxgTtentvDtltaeyhihyy ihtuem dlt tee iee ry(el;enr otaaxa ocs vafhagovtgtautriapdeeie veencerml p ddiemrt.de rvt r ira aditenoI c stovtbt( na.hgxehepbMtay e a i tlyIaltoot mia lmuVhh nmt ffcathdeeea Tgiaeacve meit)edgmneseh l i.ipiir f (eanlmsnieoabTcle nnptnaerhgyneihitgl c ttneeariag ee etnogv ntctho vmethiacoirgeafeanrnalc mf eetvg lstl uo sg une parpemmairfigaletonevn r .: lrai as ctetf=sanhyispluEts,tll #g ie oi n fabi.bnto o nogtsbhcie-givnifni r.1tmen t t, i)m ]h iaa ott,atoteh e eliotnatlrgfn hr eh ta ddTivt a #eneitfvetaohatlme inc b=v(lme bcdr .ftoeego veyh)ee rf]s(rrm(=i at Sexany(eticntb)ssneprechghgsD'tm)ncderh(eieaee–tmrt t )xcni s)emMe insit,g.amnsin (eieoet a eoastsitten ninan hhn) ornt..ndgaeaeees)ft •DaTfsaSAdudusnoiix nf]cd ]fudhx ied#tngar gaih==temfxeeaffn2ag ees]#tf#rarn it0uaaeugxtnlftndabs#id]alto.tst i=axniagfnt2mtE dt]hoedtt.tedef g.lgxn o.ddsi,TtnrAu ttaeot.clemt^ hhgdxti,he:ris2 f aeh uf=Sloe“ssari.rseA' eei(F nm’nxcute^th)o:nx i=eamc2i tthsniep2soit,nxeoen ucg=ssxo sira/taen2xxirl xode t. h nfofe^pA ufx tanc ie2hrcnhnhtftv. a uidooionennlfcf v” irtn iitunohelindgees
•nB/3=a1saicn ccoonnsvideregreast,i oannsd. Fcoorn avneyrs eKly, . ifI fn /=3aKan"nco0n,vtehregne s,/ thaenn ctRffhooo>errn c0vpa|e,ox srpie–gnosects n(s|ci<ic)b eaaRlylno.o d tfno Y(etgiho.ie ieutI) h n,p e rcoercaw sawnpse eeirt c ohs(tf iieotvire)nei,ne ely Rs o.,d Crwie sboth eoncirtvcamhehl lr eiegnmndeed an ptychao eeb i niersrt a sa¥¥ad;db ioiusaurosn s(l d uii ooti0ef)f llnn(111+-+xxx)==x2–cx2x2++xx3333+–..x5.=5+n/3=g1]m-=12gn]/n3=+012xgxn2nn+n+]1-1 1^1xx<#1h1g 12 Riemann sum i!=61cos;i-21E+1 21 wgfsxl(,roh xaopop)neshaee t f gogp(efrox tati)sphn ahfets o s iarl(s onx etpt,aiayednc) egfhfrie oeinvxrltda .v tt.Paio Avr letianho c ieuaisn sn s gctlyio a,d pllaeilenenr diedfv i aedastlenoidvg i niemang tot e eetfnha gftcisrsih as fw lpao i ocrft uihuvn rnatsvcr. lietTooiophuoneesf •cMI#noa nebfatt]hinnxe ug govcua]axslsu geoed n xg t=[”hae1,fbo,_ ]rp,et himthe#a ebnafgo v]trexh regaidrgnexet e.isgv raaal ulxxse. inoI ff[ aff,b]ias n sdua ctthga itnaheradet
apsdanaivy retsiinr angcloe restsh.au is(mnEingsqg ua ibissv eoabqulouet,nue entnl.cydge,e. , id fon, /f/=3 thKpeaan1 rnts.iecaAroli nessvsue ermcrigeossen; svoi,e fft rhpgtheoeness .ia tsiTnevfifiqheu iste0e n)ri.mcs T es th hoiiessf c(wn/oo3=hrn1i vc2ehxnrr,g noneow2n,ti)cntl eh"im 3bttyeh2 senst + orl1xval]tiinninmo+g+ 1i ttt1 hegse2t l:,ie ns2seshxnq ounnwa2tlhs/i taynt hx2 tah<ta 1t 1t.h p&eu tEsxr .atg<hd.ei,2 u ,sra ftooiofr Taerhxce=t afon1l+lxowx=+xin–gx2x 3h!23o++ldxx3 5f!35o+-r aggll ==rean/3/=l3 x0]:xn-n!1gn2xn2n++11 ^x#1h •Definite integral.T1he definite in2tegral of ffrom3 ato b •]dSthxiofeglf= uesrltoyeionp0nte+i aftlio #eex lq0xdifnu.]aitttiigaodln t .vya'=lufe( xp) rwobitlhe min.itiTahl ev aslouleu yti(oxn0 )t=oy 0thies somewhere on the MintVerTv aflo: rb I-1ntaeg#raabfls]xgdx=f_pi.
•fpIonoutsneitdgivarteai,ol ann toedfs dta ellc& trhe ea esfisontlgilmo owanit n(egK. ,c¥¥Ari)tse.s rTuiahm efeon r /cf3onifsv enrcgoecnnotcnienv.ueorguess, •fcGuoennocvmteioregnter noiccn e i ptisso wi2ne.tre rvsaelr ioefs .c oAn vpeorgweenrc es:eries determines a cosx=1-x+x2!2+x4!4-gn==0n/3=0]-]21ngngx!2n mThaey lbime idt eissc sraibide dto a esx #iastbf if] xsogdmxe= nuDmlximb"er0 /Si(tfo] cbieg Dcaslil.ed the THEORY
icrun/fi3=ong nd1havnet1n re3desr.sgi dteioenms/1n= ,2 alo1ytvtni 1hne3eirg+fen s tth#i#1nKm3e/=333 asKxftu1i]f3mnxd]gn gx wdgt=#xiht1,ehn. /2=eNcs0ruKor1monf8vr ] .enl.we.r,ggsi+stneh =st th.#KNh eae3rInr] ff o f]rg(xt lhNelgefed+t s x1ss ),e.tsrhitidhaenees ctpxooo "nltyvhneefro g]mfxeusgin=a clftsoni/ o3=r n0a.xn /nN=Ti]0hnxxe -n tsh=ece1r g+ineixn.s+t eOnxr/3=nv2+ea0x l…sna(,y+–s1xi .,Nteh1=.,e) 1 tst-heo1er -i xe1sNs/xe( +qc1uo1–e]nxxnv)!ceerag 1neogd,sf •TBp]s1ihin+nepo x-xmb=g1iipnax=lo -ms1e+rix3ai!e3lp +sx.c+Fox5oe!5rfp- fpi^c„gp2ie-!0=n, t1asnh/n3=xda0 2r]f+-eo] r2g 1|enxgp0n+=|o<x=1n12/3=ng,1!+0,e1ekpp1oox=kp.,ep2o= inssvbainnuunaoottccldrueeuhhm gnge udrar tail peahvseldnap) as d tsdel suet raaotnae hlt n olilSa sbidtnmf,Ree i tdt eihhidde tiseenm# ad topftiebahufffognfe ni]ern nixcarnf tt bgositsfdoel luraeglnxomo.r amwiaonTsst irdihsSn oo aegb#nnai b:d, ab yf f purEt reoa.enlv er Tsccbetsptaihersitelyo ielicote nnhtfneedi tuavse >fnnt egh col0rieeyetsf ai . .l oab o[cTdInalwfaem h, lmebtelaihre] tnue d swlds rw item ao tti hhb rtiihdddeeest •ciIUoiLnnnlv:ostteteeeser]gge egbrdrsrt-aat habiibmnibiaslltie iaeglt,#itry ttevoy soai # faln&o cb ircfsh L iu le]nio£nxncsedtkfgeqede (dgurxx eirsa)na#su£lbttibietlMMmiiegne.ras t:aIet.neflr]A osvbte.rae-g clvsroaaa anollbglutfi.i aln[iatxutiy,oob unoi]nssn. A f[uwa[snas,icbu,tbth]mi] oii,mnn r ogptou hlnfigee inhass •aCiarne phfdtpeuiargsnentrchrgatiaetiebion ounodnt fe w docv riaawt nhre i ivtabchaobeol nuluectati ht nfeacou.nhr ogamIunnesud g e ildnatfeofg.re iAcmvthtan,ae t ki tivhinenee tt ea eagn“gnra rdairann elftad’se”i gsa v rinacasdlol u nseelmtia.im nsoIiufioet ostgrh ubotlisoysf,
!3nI=nKtefg^nrha.l t!esnNt=Kf^nh+#N3+1f^xhdx dgievoemrgeetsri co tsheerriwesi sme.a Ty hbaet iids,e nn/t3=if0ixedn =thr1o-u1gxh ^thxis< b1ahs.icO othneer. ekp^o=2 ph^,pa-nd1 (h“^pp -chk2o!hogse ^kp”-)k+1h •iAwnentletlgi darsea rltiorv eatfhteeirv sre et.osu Atlhntei n fagon rvtmiadlauetreiio vinfa totihvfe e#r aeob fifs foaro nmefu. nfcatniodn[ af,bi]s, aas M Basic Integral Bounds tphoeinnt s w#aibtfh ]gu(gcd)u==aa#cnddf g^g(d]t)g=hgbl.]tgdt, where c,d are
f (N+1) E.g., /3 2:3-nxn=2x/3 x n=2x: 1 , for p function Awhose derivative is f: A'(x)=f(x) for all xin In practice, substituteu=g(t);compute du=g'(t)dt; and
|x/3|<1n. =T0he interval of co3nnv=e0rbg3elnce is3 (–3,13-).x/3 If pis a positive integer, eko=0for k>p. soof mae dfuonmcatiionn ( usouna llayn anin itnetrevravla l)d.i Affenry tbwyo aan tcidoenrsitvaantti ve(as L tfriannds fwohrmat attiios nw h#enb u1=-aua2ndd uu==b#. rE/.2g.,1 u-=ssiinn2ttecffoesctts dthte,
•Calculus of power series.Consider a function given by a consequence of the Mean Value Theorem). E.g., both a b a 0
K N N+1 power series centered at cwith radius of convergence R: All rights reserved.No part of this 1 x-a2 and 1x2-ax are antiderivatives of x–a, which becomes #r/2cos2tdt, since 1-sin2t=cost
••/uEA-cCn/oIhn3=.bofgndaa m1s.vase/,on e rpnlaenru{/agb #s3=clttreineoiimsm1n sn/,3=co ncivab1otn1oet,un3ae n rttt vgv nhwe»neeese.oirsrtntgns tgAh/ a.1e=ea/ 2bAesnbs 1rsscsareaonoseorn1lnul.iru ude3 mct<tsI eeof e+e1 llncyyi3a/tov.}–hn#ne31,,e <rv3abt3rgehn 5nerea>•xsgn11n0.ec 0£3.s/o–bd n4nc.vax(oennnrg‡d=ceioNts1ni,o v.)2n et0horag1arllet8 y sa..i,n. si,a/, f nbaiindnft mSiTdSnf#euuchat]rcecexyxighhfv dgd ar]=aaiattifivl gfv nffdee/veu3=urrta ngne=0tnahcnecit tetnsniiai/rhoo3a=]tein xtn n0eai -ngdnsiis sa +a se cfndeitngn 1lrincdfit]].epfexixessog-r g:ir=henaanctb snt/gw 3=linrea+ha1 bndeo1larine^ue nx so(t ]c-hnox e f–-( cc coR<o–rcin,gRgR nvic-,enh cr.a1+g+l. eRRcno)c)n,,e v aaRennr,ddg beiiudttsst. UCq.uSs.uto$fr4mie.h9ece5urII HkCndSSoAdBBotsNlriNNw.ne t$e--dn7 u#11.sl5 o301d0 .oa::80y fd0 .91As.t2icu75 t3&lt087eho.9o -2s 5rm12 :2aG252te74ra27ldna©NfQptepsw25rotslhuoa orou2esrobtniH2-rciti0mt tgcalsett0oiemkgr:naaac442cnoSeetsaio-r ,nD d ttt2 pnt7uipi ea0uocyacdeden0e, n lpry 5a d 7rm itl®rseon te ,eosBci-r amarawspo Pseassnlriaoet ir9atymdroyh sraCci ikn fe neeDoh. vcgcgmurboaafuh,msr elnreiaoeo d tdn,n rms eseo ti,rya, cn r eInst asbtnbfhptyheloeuyrci, desm iort .na d0n,fip uno1yucrcw0 bmmel8liuditeasdh athioinnoeousrngrt., ddTcckr2eohienhpf]nnaoferreows aiet#rnttc,snliea toeentmednindngxr dtgtgasv 2xaebyte#ao-atry faliinfv.o bv] t1e21Exmla ie ntC.dgoai gidtutvfx,e 2. sx2,e.tag= ewhs(,T rrffe h aohat iithosmrecci xa noohxiitan 2nnl <ytn-sm.dha. t )–aee (a1e1nf Cygtxi,t+y noi,pov hpmiriretCaiet ecn vpfs. boa eusiaelnri tn o eiatxntnren(i g d>goyp erff 1urotraro)enierl.vgn daaor elttaf hri umvvseatnea os lfs,ifuou psaedn aemj, ncuc bitdsiiliysfty o i to netthhhn doifeees)f, •FItI[TIA#haafff#auh e,b]ffbenbofxfii f]rd]ssog,]ex axt=nimahgnmigondesdt nid #erexsax gn np =xAucertf#aaosgar]Aibenaltst dl#tt agiei]i astdvt nxbh nooeutug e n,fseao oab etv]/n[iuhrxdsaatsee i gl,dA mtnubaode a]n o]rx#,ctbi anove.otbgha fanf-i entns]i[ntcvtxtiaA readesg,ugl eodb]corcra ]xufiiat,vs g l lf.a uiasfsnt:os,t ih.nIvtnafieoe nd Otfn ndhe innoarseetfi t cvg hioaanfpenttta iietvvorirnetefvn .uusa :oonl[,fuc a tts,hi tboohe]nnne: •yNndalfsno#tniea1aaer(fdtxr 1lutit dunu1i/ir0nxssair# tdg)ladi o#ulwe1t= lnf.#1oi.l/itg onxhI–Tr gneu 1lhda c/n#tbde rah2t xainuoi.Ftsc. h hb0^aT=Tlmgaeiphhkn p.]tee#e gxhr1 woeAexgofth a iot gns-cohreltahrfem2 it ]1,rgux ubvadoregleanaradtl rl oe ii xnmauee=i.svbxase slep-Seniorodle styfnaeeeft# re e1fo Tfyinnxdreun m t1etcniiprutahdcuriilsnvootl teiaifenpoqdu ds enu nwh r ieictiosnfiisst wter i hoodoslir nn menneoud v.gp fue= xg ttcrht.h1s eh2aei/=destt.,
4 1
Calculus 2.qxd 12/6/07 1:46 PM Page 3
INTEGRATION FORMULAS APPROXIMATIONS
•Banatsidice riinvadteivfien (itaell ionttheegrrsa dlsif.feErainchg bfoyr ma ucolan sgtiavnets f rjoumst tohnaet •OlwRntraxhitt,eitoerx nnrn oalanuls tx ifna,ue xn piscnoitntliyeoxgnn,ro saxm.t icioEoanlsv -xbep,rylyua-p sn adrara t xtsipe oirannoxat.peleg rrf aurnandtcisot inaoranel afmurcansyci tniobxne, ##0213x1xpdln1xxcopndvxergcoesn vfeorrg pes< f1o, rd piv>er1g,edsi voethrgeerws oisteh.erwise. •yVA(r=tc) )l,#eaanb£gtt1h£+. bA,f hglar]asx plheg2n ydg=txhf .(#xA)d bcseu=trwvee# eCbn xpx=alraatman2ed+tr xizy=eldbd hbayts 2(l(edxnt(gt.t)h, TAYLOR’S FORMULA TaTvhnee=r aagepepf ro2]oafx tgih+me alni/et=-ifo1t1 nsf ur]emam- aaniindhs r giv+gahlitfd s2 ]uibfm gfo fhios. rn toThteh p igsoi sviietsin v epa.alsroti titohne.
gin###icexvteoekngxnsdr)dax,x nx=dda= nxinxds=e +kndk+ixess1]f1i kni]nv!nexa!dl0i:dg- o1ng any o###sapx1iexndndx xx=in=dtlxenlr=anvxana-l ]awc!ohse1rxge the elqr(pfidraaurnateocaeigchpodatr enrries aroapeu tnrli oma co ltd.yffmi ouennAanncounao cmmdlmtf ii faeoabupcrlnenato oo ictsnorrtii gi rtmao i (ol nexpelni isl–wnti:sh eec ieIastt)thhr rk acarptnaehi onn acedl loyr ebetcnnghe osore tecmewa foenfdri uitaieoc tlltnofid eeo vo dnnveme bterasn rie sna oh a amaa sptps ouois onarmwuw apmemtoeoarrf arr wo)ontt.fiif hd taA heaasl TENx=h.oget¥¥. e, ,:a#]1pb#3o=xvx01e] 2l gdondixrnx x<tegc1gpo=rdanilvvs-e errag]grneees-s t uoa1st 1 gex]f1alu=nnl¥¥d x i.gn#0p -1cx11o,2mdppx>a1rdi sicvooennrsgv eetsrog.es at •Arerveoalxgav -eo£iannftx eg£air s abas t ,uig hnraragfnasa pd cch aeui rs yreo v=arfee fr v(eC#obxvl)aCovi2slbeu (cid:185)edptt iwafaorb]enaxoema. nguTe thx t1rte=hi+ zseaeu] fdaxrlf gnabad]cyxx ex ig s(2=g,(d exbtnxh(ae]te.br) a,goaItruyefe td(a ttt bhh)i)yees, •nfcT1=aa!ty0f lc,(o niir)st( ’ cspP )oa(nxll(ysx–on)c oc=)manlfliea((pcdlsr) .oa +Tv Mihdfeae' (dcncL t)ht(ahx ude–re icgdn)re epr+eiov T21layat!yinvflo'eo'ms(rc pi)eao(xlxl.iys–ntc)o.) m2Wi+ahl. .eo.+nf •rostMaeufng migdumelepanaoartc in pnhldiat n retiriis ntus ietolteeghnr.me vw Taeailnhtr:hti e s at th Mheroevo fnaus =laguamh anI /tt=p-terhlas01ie pnf tegmhzc egiaod ii+vpdReo i:niewin -mbht.yao21 nstDhenh e m tsmohup.mid pisEoo niatnc hthaes
•F####utcs1ares+dtncchxxxexx2rd=d dinxxxad===recflllitnnnnaincssteseex cccinxxxte++grctaaolnts.xxTh##ec ao1btd-oxxvexd 2cxo==nvalernnctsisoiinnnsx xhold: iMsAxxnAhA-2 aoft+1ht1auch+celb d+t sodB xobreff+ 1ent( wxoxf+cam2am++ri]enxbi flcaxi-Aaat+o+nrk.crc hIg)i]fkkmax.anA(2pdt„+hlklieebe+ b,stq hxBtueh+a kewdrxcreoag rktc1kio.c, u bnludot t b thhe=a esv ufionmlgClm orwaena+inld gsro cDoatsse) eEbI2tso.1 3tgu/a2.nb, ld#1ic#es03hod3 n xcbcv1o3y1e/n +2r1vxgd/ex2exrsg32= /me23n}/2c2o1e3dt/n ox2( o+ [0rc ,2do11in]vave ae2rnrg.gd4ee .ninssc u easm)li wnabcnaeedyr stt ohle egs selie tnt shbtseao gnur na1dn/txsdh.3 a/i2ns. •daM#Ccisc2ope(cid:185)tlliaeoycrndae tmsio=einnn t #aa xb(ot(2)nt(cid:185))=e aPv yl'o](dttHn)igg.m SaYex inllniscSn]tieeog I2nhv+aC. iss y vSSlaenu]lt opagcp2niodttyistde ve. (rtiav) a=txivv'ae(tr )io afab nalde, •wtibRdTnheeanhtertey(e wixrvlrrvo)eeae amerti isln’Rv. sa ceT nicans(hl xfdleaoo)eenn rndr =dms f t o ohfux]rpoen lae ra+(Lnn xx1.ty a h1vi gexnAgar !triMaesnyfrsn ivautng]hacnmgele+L efai1awo ngunr_itrdtefmpih rni v t ohxhap•fa)la o ,.tt( s lfhxy Tc(e–nhxn coire)s)+em = nme1+aiPx1a anpplfisc(rono xeoiodr)snn fs+etts i rifooRni.(nmn unEx (o)e.txfghu o)=.xxesr,, •SiTqfaSnu uih=rtnmaeiecsdgrpbt rvuiisa-osal6ota nlinar ca al’ psstcto a htfr rhattu]ehita[t leiateigoh +n,.in rbtneTe 4t]oereh pffgpe o[ro balaawia,lnt +be2etoysi]sgf.ib iehFnttllhht+otdeoeerdsf s]buSmgimm. S31pimTso1pn+s'osn32 R’Ms u1leon ruthlee
###cxo2ds-dhxxax2=dx=2=1alnslninxhxx++-xaax2+a2=##ssixinndhhxx-=1dx21x+x=lxcnoashx Twhhuesre #Cx,-D aar1ex s-eebn tdo xb=e C]ax=-1-b-a^glD]nx=-x-ab-1gab-.xl-nax-bx-h.b APPLICATIONS to#htf0e taa n]fu uognbddjueac,mtx et]hntrtgoa=wl nxt h]atet0 ogt+rimeme#t 0 ttv0i=]mu0pgdlfirueos.m:E .avg .(h,t )ethige=h hte xivg((h0tt)0 )x= (xt+0) Tlxnh-(e1rx+e 2x+i)s, –a1 <xx1xb<e1tw, .eaeren R0n(axn)d =x ]ns+uch1] g-_t1h1+agtn plnin(+1 1+• xx)n +1=. Sainn2tm er=evhv3ael"sn,f a]n afuog+mrm4bmiue/=-lra0 1 fins]:a=g2+m[2i+of1]h+a2mi/=-01f(aa+ 2+2 bi•h)+f]bgb,
x2+a2 a In gener]al, theg i]ndefingite integral of a proper rational function GEOMETRY with a vertical velocity v(0)=v0undergoes the acceleration 2 3 1+p3 whereh=(b –a)/n.Simpson’s rule is exact on cubics.
_ i
# dx =lnx+ x2a2cosh-1x+lna can be broken down via partial fraction decomposition and –gdue to gravity. Thus v(t)=v(v) + #t -udu= v –gt •Error bounds. As xapproaches c, the remainder generally
x2a2 a linear substitutions (of form u=ax+b) into the integrals •Areas of plane regions.Consider a plane region admitting 0 0 ] g 0 becomes smaller, and a given Taylor polynomial provides a
•C((#####00tToaa1(cid:185)(cid:185)kxmkax//e22nescm2 2 pdio-!snooxsasn2mxa2=iq qt22deidqdvqdn desefx+xi1 i=g=v=n=na1i,lt# #21u210e+e (cid:185)x(cid:185)xi#s0o/n/ 22rrft1 1oeax–-g+rr,22 r2ctxa-accho2ol22orssxo+s!hs:u22-21gdqaaq)22hx2d2doa=lquqnr=t=c)(cid:185)xs4r(cid:185)+(cid:185)i4n2ax#x02(cid:185)s!inax2dx=2 •ssoU#uununbb -[bIssattoM1ii,ttudBuunttuii]Pdoo,fnne#o duRruw -al=i=lnOmlt duaBin2tuPs+>t.]1n)aIE.)f,2, tfhaR1inesgdn, d #euI##af]Ni]3unuf2e2+dT]+x o1g1Endgg -x-[Gandn=,de¥¥dfuRBu]li"am((3nAhhdaa#n aniLdnBdltfleeSed]gdx r wwagdbiittxlhhe [itaDoalbnhearnvyAnete,e ebaa g=rgat]x twoL.hr[ri aaesfS (o t aa,aopi sb cou m)]ogscnD ctforie rh pasvritt ,ppdiaht mhh ir#aeaoniasn eabfg trsdb 6e w s“g ltigeetootiih] c dv oixietsasteni h g ors -ytk”niobD.mn stofpao tupplh]wp nlxeeeaendr grrpr@ e edypftedaou-enx a nnbdvix,scdy iii tcepis icuwAr,ot uholn a=lveia arn i L rdrg t #e(oaetrwgopa dbtiL )ph hot,ghhien^cae(s p hax £ aaxo h)spdxif‡ csb £ipas ffovs .b(uaaaex.tnrnE )Tydpd .t ogheh iigdn.nees, •WcpahWdwIdnnraii(eoss ord=yippasarg )slxm lk haa-fi#(a.sctseclt eu)e biteIFstrcmimhf=i dtD zi]ee oeexFxF nlndnl0git(i=fdt a)fxb+ t,tlxrr) iyi anDn. aA#nx0ggixrsd^,( e vh typathaa0e) hr eDt-ioAe g vbya h(wxa,glpeytrrrpuo im)farirhbosokb,daer xl n uedioiDtndm ohfg W=tfneea ow etntxhr“=e i c0osed D ewlFv +taf heWol (bauvr xr ”c0iDk=a)dt tD n–yiod’Dns ,21xfo i gF wngnaf tl•ent(eeuaf2h idrloiog.ovd (vhrna tyecthalg)-re te,d [ a ayeaw ts,sn iblwmhmsal]eiibiatnetriy’lheesssl. •(bab£ebB(Sasyeex]isso»ctng utam1e x+Mmu+rfifies OMp1x eat0 ag +itp!ouh)pxn ntexhm2rso oo/t-o the2nxaraian,si rc tmn ddiwsrno ade n+itntttqheoihh1.orue ta ian ivftertoT aierrotfrho ioniav]fetnrxh l e ltaa getnobxxht-sr ofotevm iatv en aphtmex lefet,,]u hmi oxsinelrf giee b cmin itotnfsihttiuto htee]nb nnarn+dofo nv euf( a1v xdgl3na3.]t )! dlhbxE=ufeyxig.esdp] g .x33 ( i.rxn,s=go aW -ef)nbto+ai0 oiorr(Pt.O u –h 5|nxxn1 a(|xd=x]s,<tx1 hem013)xdge),.., S•tuSciohnosfeEat unqetnashgu lieewQlsey rtn,i s toecdhufqeUe s saunau. eloa flntl uESelicnyndSeec t q.e0tw NiugAEoioeetn nrrhsasQ Cc e1 laeg q.n s rusUT oeEueathanbatercEsetei Sceroi r ntnNiifhpts ue aa tngeon(&Cecdnf tort )ei sr)oEniy n ena m qrnsSSae ubd fao sweilelErs hrq tleocoiukda sResel enolt eocamded nI we oaEa( mi irttnta heanit Srthiianmeainssrl
0 0 2 4 provided the limit exists. Planar Area Then W=#btA yh ydy. approaches 0 exist.) That is, f(x)–p(x) approaches 0 at expression for its terms, e.g., 1/n (with the domain
To remember which of 1/2 (1– cos 2qq ) equals cos2qq or E.g., #3e-xdx=lim 1-e-B=1 a ^ h ^ h essentially the same rate as xm. E.g., Taylor’s formula understood), in lieu of a fuller notation like:
sin2qq , recall the value at zero. Likewi0se, for appropBr"ia3te] f, #b fgxdxd=eflim #bf xdx. DIFFERENTIAL EQUATIONS implies f(x)=f(0)„ f'(0)x+21f''(0)x2+O(x3) if f has •"E1l/emn,enn3=ta1,royr sneq;"uen1c/ens^. nA=n 1a,ri2t,hfmeht.ic sequencea has a
-3 ] g A"3 A ] g continuous third derivative on an open interval containing n
TECHNIQUES In each case, if the limit exists, the improper integral 0. E.g., sinx=x+O(x3). [Similar relations can be inferred common difference d between successive values:
converges, and otherwise it diverges. For f defined on p •Examples. A differential equation (DE) was solved in the from the identities in the item Basic MacLaurin Series.] an=an–1+d=a0+d·n. It is a sequential version of a linear
(–¥¥ ,¥¥ ) and integrable on every bounded interval, a b item Solution to initial value problem; an example of that •L’Hôpital’s rule. This resolves indeterminate ratios or function, the common difference in the role of slope. A
•Sr(seuevbee srttshiete.u tTFiohonero .raRyn e sfeeincrsttie ogtnor) a,tl h beru etC coohfgatnennigz eeth doe ft foov ramhraiuavlbeal eit sh foeu rsmefdou rmlian #-33f]xgdxd=efAli"m3#Acf]xgdx+Bli"m3#cBf]xgdx (the •VCoolnusmideesr ofa sosloidlisd. fbtyoaprs eitc hi se D idnEe M riivnoavttioiovlnev iiinnngv o ontlhveee d did moenpelenyns dtihoeenn .ti Innvd aethrpioeasnbelde,e tnhites veyax'rp=iarekbslyes.i .o AAn b00or33l.If lxi"maf]xg=0=lxi"mag]xg and if lxi"malf]xg=0 gbeeotwmeeetnr iscu scecqesuseivnec ev,awluieths :t egrnm=sg ng-n1,r h=ags0 ar nc.o Em.gm.,o5n. 0ra,t2io.5 r,
u#a=bFg^(xg)],x dguhg=lg]x'(gxd)xdx(,w ainthd Fmaonddi fgy' cthoen tliinmuiotus so),f yionute cgarnat piount •rScihignohgitc ucelo aonrfv cienrbgteeegisn.rga nadrbsi.t rIafr yf)i,s p droevfiindeedd eoanc h(a i,nbte] gbrualt onno tt haet atphdeamrtp iettnindgicc ruaolnsa sra- xseitsoc tsiuothcnhes agGneednn eiersra alli lnlDyeE a rmwionhr eethr ee d doifenfpliyec nutdhltee n aft rivreas tr–eiaqobruldea etiirso nyd'se+ ripivn(a tt)iwvyeh= iaqcph(pt )e.tahres =ne lcxie"msaslagri]lxyg aat rae) ,d ethfeinne dlxi" maandgf ]gxx(xg=)„ 00=, floxi"rm axgfnllea]xxr gap(rbouvti dneodt •1eCx.o2pn5ov,nee0rn.g6tei2an5lc ,feu0. nA.c3 t1sieo2qn5u,, .et.h.n.e c eIct o {miasn m}a o cnsoe nrqvauteierong tieinsa lti hfv ese orromslieeo non fu ombfa bsaeenr.
aInp perfofepcrita, ttehley :i n#atebgFr^agl i]sx oghvgelr ]ax pgdatxh= on#g t]gha]geb gFu-]auxgids utr.aced out x#a=bfa]axngdd xis d=ienftcel"igmar+ab#cleb fo]nx cgdloxsedp rsouvbidinetde rtvhael sl iomf i(ta e,bxi]s, ttsh. eAn aafucxncisoc tridoivnna gAr y(t op ),iana £knpao£rwebna. Dyien'.'gEd=.es, p –yetk'hn=yda,yte 2xn–at2 ryxev'.' a+Crliixoanmybe'lam+er xoa2nipny pi =net a0har.eps p lidinec apatei no\dnheslt n{atnr eov nsaelricinoaebnaldre}-; o wred.agey.r;, wtgh'he(ae ln)a atttrheeer nllioimmt riitets q eouxfir ifsetdas ,nt odo reg xiaissr te]i.n Tifngoifn irineteist.oe l.T vNhee oa tnreu itlnh]eda etag tlfes'or(m ah)ionalandtdes Laexdim(sctiast,sl l teahdne rNteh eiss u locinmhl yitth )o ants ea|a;t inos–fnieLe s|s< atyeehsef {ofaro nal}llocl wonni‡vnegNr:g . eEIsfv etaor ylLi m,ee ia>tn Ld0
bstya rtth],e tfhuennc titohne gi.n (teIfg rga(lb )is= zger(oa.)) [tEh.eg .p, atuh= r1e+tuxrn2s ytioe lidtss [saim,bil)a. r dEe.fgi.n, itio#n2 hol1ds ifd xthe iisn teglirmand# cis d1efineddx =on Tthhicek vnoelsusm Dep opfe ar pselanbd iocfular to the axis at pis D V=A(p)D p, •Sthoaltu itsi odnifsf.e Are nstoialubtlieo nto othf ea oDrdEe ro onf atnh ei nDteEr vaanld i ss aat isfufinecst itohne rdaiftfioer enacned (¥¥ap–p¥¥ly), trLy’H tôop rietawl’rsi te riutl ea.s aTno inrdeestoelrvmei naatne wcornitveesr gaen, fiit Ld,i voerr gnel"ism.3 Ifa an =seqLu.enIcf e aa sedqivueerngcees idno essu cnho at
SES##0uo.gg1bm=].s,xe t1ig #tng+1uegx+tnlixxoe]x2nxrd 2agmdxdl ax=fxoy==r 21bme21g# u0nu]#lx1sa+1edgsund+1u+a 1fr=1xoe,r:2# 21i2ngglldnxe]uxxdf=ixgnd=it21xel= 21ninl]lnt1ne]+g1gr+ax]lxs2x.gg.2,g. 1lci"m2-a#0rccs4in-1bxc22l=d0xSr2=in.4ag-rucslxian2rc cInm"tegrancd"2- 0 4-x2 •ahlASVenao(nv=dzlgii )dntt=h#hsg0e [ hs ssotqs(2of1u tc–aa1ralrz-e envv /dohohhz lol)uumr]h2mit2zedioeoizga nni=tht.s a tl sCVh c23eoh=rhino,g s.sh#iastdh- bseaAerzs c. ^ tapico Ihtrndssoso ps,l s.ivwd-osE ilet.uhocgm ft.b i, oeora nte tavpoilosym lrua tastmhiriodueindaes •sdesGoBepooaqefrae clmsuountshcahbiteer tcielir ioeabi oannmdselnfotl msye til o(rr uoiIivs tvtnVnattiaii n o-rnlPao t ntpgc)nuhr’ p osaeftd hoalnl ivel rtcyioara nraa rlai ttnundenail eoierlgninl vrnne tag ashDe n s,l tao .odoEha frl rne Iudn gt rt Iie–eeiDnVroin1ais EnDPtel i dsr a.aEvah, el la arliTvgiilsynvfuae h caaleinelun tnsuuei ev.vodr n oeetpAaei lsqqsolv n au ieuaa lnsast leiost .t n . nspilsouioAeotcnmtcil ioau iofenlgtn iy isecpov'tna=aonaitenfilik rnouotaysitnnetl,,. trrwiFonaeohts drgioe eolexrlti;"vteem eraa0lxmsp ip" mpxtir0hnloyxeadx tyuLoeo’cx rHeuitg= x aôgipnnepeotdai0 nta l ar= eneiln nd’ws1t d fi.rreaiiuntlte edl(er 0tlm;hi 0mt,iihs1nx e¥¥aafifi ,tsee0o lxa r1ne p¥¥s/xouxxp0ni)ote=,a nnltbeiatlminkeate xliifi fi naiotld0s.f-e ltt1oeh1/gre/mxa xrriei2nts=haum0tlet, •enrBrtwheq=‡oaaauyu1ltNa nnlitt, shhu dtt aeomhebtn de oba en runevm nnyroesfifid onr,be yneao1d onwMu; t danoor b>ditnithos)e0e .evs b rTe asawa edhsncqimiifisocsue nitiet vo¥s¥rsn en .acasr d geenEfirse vu.i.Ngsee n .Asrd,(g stanciuoefmo s c i|nn,ahrne v a |ncel<tnrtihraedm1gala etiist fftnhia a nrcelcneg>ent>s 1tasMrse, be nstrqofifihtfnsuuaofi.¥etn0r n t;ah coileerfl.
#eg]xggl]xgdx=eg]xg. ] g ditest earxmisin eodf brye vao kluntoiwonn. raTdhieu s afruenac toiofn trh(ez ),c aro£ szs£-sbec, tailoonnagl rewritten ddyt=ky suggests dyy=kdt where y=kt+c. In NUMERICAL INTEGRATION
•Integration by parts.Explicitly, “disk” at z is A(z)=pp r(z)2, and the volume is this way, one finds a solution y=Cekt. On any open SERIES OF REAL NUMBERS
TFTanwd#areiohhiefbetrfeeu h de gi c]r npexetoodronnomgne evtcufeirmleai asnd]telofexi,u a tnd“egrca ;due tsf iot o” xnhiwr rst=e ame e ntugilouundslr ltae ]aea“tdbx gisvi eo grs.i v”inan n li,# ]Va n xdie#btsieuegxue trgwpbahidvdrl-e aiavv tctiht=ein=i#teaodt ebn iuuivngnsavvtr ]tneaxwe-gbadtgghe- rurade#atlthr vnif#e]aoeadx d ncb tvghautsodos.etd. xr h Y aufet.ou ir.pmun r cteodotsdio outnbhnc’eestt aaE[Iinafsp. ta, gfpebar.ri,] osv , sp anulraos#im-na t bd1t1 dee xoe 1tciffw3sl io dnelseieexnefnd=ttd ea sgsautalunr "ciaabdmh b0i f -nliprenti#eo -girtioah1envnt xat-1ns lh3,usc ad,mtl nhopxdbesr+ oe e1ivdlrni bi mlotd"iesfmeigu 0tdp+rsb ao ia#blniol nlt1f et# taxsrh1i bvn3iefnadt lel ixsagmi nsrc oa iditlfnsise f t ef eooixrpntvvi2heeseatnderl. appIpdaVf roiea fe=trrirfhn apaeetmder#“a enizswebanodAtalcalri liecsi]odcz zhuno oelglegfidsoare rrzvsst dho=t, ”bhieln eute #oataam awt xbenh(cid:185)eie dsesert irhnol]g,iez tfkht h gweat2rese od£ at v hzxrrohva.a£ilotsd(u b rliatui),ib o mroon1aofve(,l fe z o t).nh ircaSgeseyn o vcldmoiatr nhornleud2es t(t irszaimi-o)cxos enaeaibssclt,v teiioaocunhss suyiinno'n=ttlieeuqkrrtuvv•yieaaoGyhwi lln'om.,si+e tmoTnhippeslh eoluv h(iyurgtete(i)as”ersao yyln y0)n d =de=,= toswsq1Coufo^(iilyiselttursvh )ekse t.i hittq -noy-(ugonTk(Cta atr hah )td,=rmnei=eeoy waurney lshsI ax)Vothe pilir'sPsuhe+l i t t-anyypihyvo(e(=eeeanta# – )yg) rkhatt=ep thet=no0iakes 0u.t( Dr t(ca–tdfhdEoalo )henus..r /osmhTtl.aau,h=s nCtesitb–oo o etponncrc(n.siita ai vT)udttdiehhseatdeee)rl ••iasiataTGnnhopntrv teefpdaeot nar gwnplhoelvruelaxra=,emzra il oem(sdlbbb i edetay–uofnrt iis a ornnoinf)utnigfei/et gln eneise# n ddb...i tnieraenTedfStrefgh v]ogipx enalrru lgeaaesdltdnlic isnxi oti nisetnad hai tsolerahcg enrtoeo i .tec vn rouTlaian etnnthhemaidnecmvpeoe teiprsfntnslo lytb.gila ceiel rnoad Rytegwnw .ve Tiropaaneeh lrdgguvopri uaaoblomytllulieuae ngedramt hta tshpeevo soaa a uorsdimtntoil,sila ft ynaibttf,oneh oliinyneesrf ••aSCvod afeeo lnrtpunhioaeevetrssee s .trd eoig aArfne/il 3=e nass0ncsa eeuaorn.ttmi hA.eNes:T rsih siens/se rN=te ihaeq0aaensu nsesa/=u3enrmqeac aue0c +o enann.f/l.N=c lc.veo+0eaadn al vunotNhee=b.rse t g aTauteie0pnhs+r eetmi.do f.. +sstahbeaNoyerNfi a e s.nta eshTdd qe dhuii tsiseesn e ncegrvcali afeel ltls uheo.ideesf
one to be differentiated, minus the integral of the product limits on the right were to exist. They don’t, so the integral sections (shells) of the solid parallel to the axis of ] g ; a ] g E graph of a positive function together with the underlying partial sums convergesn,= i0n which case the limit of the
of the two new quantities. The factor to be integrated may diverges. revolution. In this case, the area of the shell at r is If y is a solution to the original DE, then (y/h)'=q/h, interval on the xaxis form a trapezoid whose area is the sequence of partial sums is called the sumof the series.
be 1(giving v=x). •Examples & bounds. A(r)=2pp rh(r), and the volume of the solid is where y=h#q/h. The solution with y(a)=ya is average of the two function values times the length of the If the series converges, the notation for the series itself
E.g., #arctanxdx=xarctanx-#1+xx2dx #13x1pdx converges for p>1, diverges otherwise. V=#abA]rgdr=#ab2(cid:185)rh]rgdr. y(t)=Ya+;-#atq]ugh]ug-1duE. ignivteersv athl.e Atrdapdeinzgo idth reuslee aarpeparso xuipm aotvioenr a regular partition stands also for its sum: n/3=0an=Nli"m3n/N=0an.
2 3
Calculus 2.qxd 12/6/07 1:46 PM Page 3
INTEGRATION FORMULAS APPROXIMATIONS
•Banatsidice riinvadteivfien (itaell ionttheegrrsa dlsif.feErainchg bfoyr ma ucolan sgtiavnets f rjoumst tohnaet •OlwRntraxhitt,eitoerx nnrn oalanuls tx ifna,ue xn piscnoitntliyeoxgnn,ro saxm.t icioEoanlsv -xbep,rylyua-p sn adrara t xtsipe oirannoxat.peleg rrf aurnandtcisot inaoranel afmurcansyci tniobxne, ##0213x1xpdln1xxcopndvxergcoesn vfeorrg pes< f1o, rd piv>er1g,edsi voethrgeerws oisteh.erwise. •yVA(r=tc) )l,#eaanb£gtt1h£+. bA,f hglar]asx plheg2n ydg=txhf .(#xA)d bcseu=trwvee# eCbn xpx=alraatman2ed+tr xizy=eldbd hbayts 2(l(edxnt(gt.t)h, TAYLOR’S FORMULA TaTvhnee=r aagepepf ro2]oafx tgih+me alni/et=-ifo1t1 nsf ur]emam- aaniindhs r giv+gahlitfd s2 ]uibfm gfo fhios. rn toThteh p igsoi sviietsin v epa.alsroti titohne.
gin###icexvteoekngxnsdr)dax,x nx=dda= nxinxds=e +kndk+ixess1]f1i kni]nv!nexa!dl0i:dg- o1ng any o###sapx1iexndndx xx=in=dtlxenlr=anvxana-l ]awc!ohse1rxge the elqr(pfidraaurnateocaeigchpodatr enrries aroapeu tnrli oma co ltd.yffmi ouennAanncounao cmmdlmtf ii faeoabupcrlnenato oo ictsnorrtii gi rtmao i (ol nexpelni isl–wnti:sh eec ieIastt)thhr rk acarptnaehi onn acedl loyr ebetcnnghe osore tecmewa foenfdri uitaieoc tlltnofid eeo vo dnnveme bterasn rie sna oh a amaa sptps ouois onarmwuw apmemtoeoarrf arr wo)ontt.fiif hd taA heaasl TENx=h.oget¥¥. e, ,:a#]1pb#3o=xvx01e] 2l gdondixrnx x<tegc1gpo=rdanilvvs-e errag]grneees-s t uoa1st 1 gex]f1alu=nnl¥¥d x i.gn#0p -1cx11o,2mdppx>a1rdi sicvooennrsgv eetsrog.es at •Arerveoalxgav -eo£iannftx eg£air s abas t ,uig hnraragfnasa pd cch aeui rs yreo v=arfee fr v(eC#obxvl)aCovi2slbeu (cid:185)edptt iwafaorb]enaxoema. nguTe thx t1rte=hi+ zseaeu] fdaxrlf gnabad]cyxx ex ig s(2=g,(d exbtnxh(ae]te.br) a,goaItruyefe td(a ttt bhh)i)yees, •nfcT1=aa!ty0f lc,(o niir)st( ’ cspP )oa(nxll(ysx–on)c oc=)manlfliea((pcdlsr) .oa +Tv Mihdfeae' (dcncL t)ht(ahx ude–re icgdn)re epr+eiov T21layat!yinvflo'eo'ms(rc pi)eao(xlxl.iys–ntc)o.) m2Wi+ahl. .eo.+nf •rostMaeufng migdumelepanaoartc in pnhldiat n retiriis ntus ietolteeghnr.me vw Taeailnhtr:hti e s at th Mheroevo fnaus =laguamh anI /tt=p-terhlas01ie pnf tegmhzc egiaod ii+vpdReo i:niewin -mbht.yao21 nstDhenh e m tsmohup.mid pisEoo niatnc hthaes
•F####utcs1ares+dtncchxxxexx2rd=d dinxxxad===recflllitnnnnaincssteseex cccinxxxte++grctaaolnts.xxTh##ec ao1btd-oxxvexd 2cxo==nvalernnctsisoiinnnsx xhold: iMsAxxnAhA-2 aoft+1ht1auch+celb d+t sodB xobreff+ 1ent( wxoxf+cam2am++ri]enxbi flcaxi-Aaat+o+nrk.crc hIg)i]fkkmax.anA(2pdt„+hlklieebe+ b,stq hxBtueh+a kewdrxcreoag rktc1kio.c, u bnludot t b thhe=a esv ufionmlgClm orwaena+inld gsro cDoatsse) eEbI2tso.1 3tgu/a2.nb, ld#1ic#es03hod3 n xcbcv1o3y1e/n +2r1vxgd/ex2exrsg32= /me23n}/2c2o1e3dt/n ox2( o+ [0rc ,2do11in]vave ae2rnrg.gd4ee .ninssc u easm)li wnabcnaeedyr stt ohle egs selie tnt shbtseao gnur na1dn/txsdh.3 a/i2ns. •daM#Ccisc2ope(cid:185)tlliaeoycrndae tmsio=einnn t #aa xb(ot(2)nt(cid:185))=e aPv yl'o](dttHn)igg.m SaYex inllniscSn]tieeog I2nhv+aC. iss y vSSlaenu]lt opagcp2niodttyistde ve. (rtiav) a=txivv'ae(tr )io afab nalde, •wtibRdTnheeanhtertey(e wixrvlrrvo)eeae amerti isln’Rv. sa ceT nicans(hl xfdleaoo)eenn rndr =dms f t o ohfux]rpoen lae ra+(Lnn xx1.ty a h1vi gexnAgar !triMaesnyfrsn ivautng]hacnmgele+L efai1awo ngunr_itrdtefmpih rni v t ohxhap•fa)la o ,.tt( s lfhxy Tc(e–nhxn coire)s)+em = nme1+aiPx1a anpplfisc(rono xeoiodr)snn fs+etts i rifooRni.(nmn unEx (o)e.txfghu o)=.xxesr,, •SiTqfaSnu uih=rtnmaeiecsdgrpbt rvuiisa-osal6ota nlinar ca al’ psstcto a htfr rhattu]ehita[t leiateigoh +n,.in rbtneTe 4t]oereh pffgpe o[ro balaawia,lnt +be2etoysi]sgf.ib iehFnttllhht+otdeoeerdsf s]buSmgimm. S31pimTso1pn+s'osn32 R’Ms u1leon ruthlee
###cxo2ds-dhxxax2=dx=2=1alnslninxhxx++-xaax2+a2=##ssixinndhhxx-=1dx21x+x=lxcnoashx Twhhuesre #Cx,-D aar1ex s-eebn tdo xb=e C]ax=-1-b-a^glD]nx=-x-ab-1gab-.xl-nax-bx-h.b APPLICATIONS to#htf0e taa n]fu uognbddjueac,mtx et]hntrtgoa=wl nxt h]atet0 ogt+rimeme#t 0 ttv0i=]mu0pgdlfirueos.m:E .avg .(h,t )ethige=h hte xivg((h0tt)0 )x= (xt+0) Tlxnh-(e1rx+e 2x+i)s, –a1 <xx1xb<e1tw, .eaeren R0n(axn)d =x ]ns+uch1] g-_t1h1+agtn plnin(+1 1+• xx)n +1=. Sainn2tm er=evhv3ael"sn,f a]n afuog+mrm4bmiue/=-lra0 1 fins]:a=g2+m[2i+of1]h+a2mi/=-01f(aa+ 2+2 bi•h)+f]bgb,
x2+a2 a In gener]al, theg i]ndefingite integral of a proper rational function GEOMETRY with a vertical velocity v(0)=v0undergoes the acceleration 2 3 1+p3 whereh=(b –a)/n.Simpson’s rule is exact on cubics.
_ i
# dx =lnx+ x2a2cosh-1x+lna can be broken down via partial fraction decomposition and –gdue to gravity. Thus v(t)=v(v) + #t -udu= v –gt •Error bounds. As xapproaches c, the remainder generally
x2a2 a linear substitutions (of form u=ax+b) into the integrals •Areas of plane regions.Consider a plane region admitting 0 0 ] g 0 becomes smaller, and a given Taylor polynomial provides a
•C((#####00tToaa1(cid:185)(cid:185)kxmkax//e22nescm2 2 pdio-!snooxsasn2mxa2=iq qt22deidqdvqdn desefx+xi1 i=g=v=n=na1i,lt# #21u210e+e (cid:185)x(cid:185)xi#s0o/n/ 22rrft1 1oeax–-g+rr,22 r2ctxa-accho2ol22orssxo+s!hs:u22-21gdqaaq)22hx2d2doa=lquqnr=t=c)(cid:185)xs4r(cid:185)+(cid:185)i4n2ax#x02(cid:185)s!inax2dx=2 •ssoU#uununbb -[bIssattoM1ii,ttudBuunttuii]Pdoo,fnne#o duRruw -al=i=lnOmlt duaBin2tuPs+>t.]1n)aIE.)f,2, tfhaR1inesgdn, d #euI##af]Ni]3unuf2e2+dT]+x o1g1Endgg -x-[Gandn=,de¥¥dfuRBu]li"am((3nAhhdaa#n aniLdnBdltfleeSed]gdx r wwagdbiittxlhhe [itaDoalbnhearnvyAnete,e ebaa g=rgat]x twoL.hr[ri aaesfS (o t aa,aopi sb cou m)]ogscnD ctforie rh pasvritt ,ppdiaht mhh ir#aeaoniasn eabfg trsdb 6e w s“g ltigeetootiih] c dv oixietsasteni h g ors -ytk”niobD.mn stofpao tupplh]wp nlxeeeaendr grrpr@ e edypftedaou-enx a nnbdvix,scdy iii tcepis icuwAr,ot uholn a=lveia arn i L rdrg t #e(oaetrwgopa dbtiL )ph hot,ghhien^cae(s p hax £ aaxo h)spdxif‡ csb £ipas ffovs .b(uaaaex.tnrnE )Tydpd .t ogheh iigdn.nees, •WcpahWdwIdnnraii(eoss ord=yippasarg )slxm lk haa-fi#(a.sctseclt eu)e biteIFstrcmimhf=i dtD zi]ee oeexFxF nlndnl0git(i=fdt a)fxb+ t,tlxrr) iyi anDn. aA#nx0ggixrsd^,( e vh typathaa0e) hr eDt-ioAe g vbya h(wxa,glpeytrrrpuo im)farirhbosokb,daer xl n uedioiDtndm ohfg W=tfneea ow etntxhr“=e i c0osed D ewlFv +taf heWol (bauvr xr ”c0iDk=a)dt tD n–yiod’Dns ,21xfo i gF wngnaf tl•ent(eeuaf2h idrloiog.ovd (vhrna tyecthalg)-re te,d [ a ayeaw ts,sn iblwmhmsal]eiibiatnetriy’lheesssl. •(bab£ebB(Sasyeex]isso»ctng utam1e x+Mmu+rfifies OMp1x eat0 ag +itp!ouh)pxn ntexhm2rso oo/t-o the2nxaraian,si rc tmn ddiwsrno ade n+itntttqheoihh1.orue ta ian ivftertoT aierrotfrho ioniav]fetnrxh l e ltaa getnobxxht-sr ofotevm iatv en aphtmex lefet,,]u hmi oxsinelrf giee b cmin itotnfsihttiuto htee]nb nnarn+dofo nv euf( a1v xdgl3na3.]t )! dlhbxE=ufeyxig.esdp] g .x33 ( i.rxn,s=go aW -ef)nbto+ai0 oiorr(Pt.O u –h 5|nxxn1 a(|xd=x]s,<tx1 hem013)xdge),.., S•tuSciohnosfeEat unqetnashgu lieewQlsey rtn,i s toecdhufqeUe s saunau. eloa flntl uESelicnyndSeec t q.e0tw NiugAEoioeetn nrrhsasQ Cc e1 laeg q.n s rusUT oeEueathanbatercEsetei Sceroi r ntnNiifhpts ue aa tngeon(&Cecdnf tort )ei sr)oEniy n ena m qrnsSSae ubd fao sweilelErs hrq tleocoiukda sResel enolt eocamded nI we oaEa( mi irttnta heanit Srthiianmeainssrl
0 0 2 4 provided the limit exists. Planar Area Then W=#btA yh ydy. approaches 0 exist.) That is, f(x)–p(x) approaches 0 at expression for its terms, e.g., 1/n (with the domain
To remember which of 1/2 (1– cos 2qq ) equals cos2qq or E.g., #3e-xdx=lim 1-e-B=1 a ^ h ^ h essentially the same rate as xm. E.g., Taylor’s formula understood), in lieu of a fuller notation like:
sin2qq , recall the value at zero. Likewi0se, for appropBr"ia3te] f, #b fgxdxd=eflim #bf xdx. DIFFERENTIAL EQUATIONS implies f(x)=f(0)„ f'(0)x+21f''(0)x2+O(x3) if f has •"E1l/emn,enn3=ta1,royr sneq;"uen1c/ens^. nA=n 1a,ri2t,hfmeht.ic sequencea has a
-3 ] g A"3 A ] g continuous third derivative on an open interval containing n
TECHNIQUES In each case, if the limit exists, the improper integral 0. E.g., sinx=x+O(x3). [Similar relations can be inferred common difference d between successive values:
converges, and otherwise it diverges. For f defined on p •Examples. A differential equation (DE) was solved in the from the identities in the item Basic MacLaurin Series.] an=an–1+d=a0+d·n. It is a sequential version of a linear
(–¥¥ ,¥¥ ) and integrable on every bounded interval, a b item Solution to initial value problem; an example of that •L’Hôpital’s rule. This resolves indeterminate ratios or function, the common difference in the role of slope. A
•Sr(seuevbee srttshiete.u tTFiohonero .raRyn e sfeeincrsttie ogtnor) a,tl h beru etC coohfgatnennigz eeth doe ft foov ramhraiuavlbeal eit sh foeu rsmefdou rmlian #-33f]xgdxd=efAli"m3#Acf]xgdx+Bli"m3#cBf]xgdx (the •VCoolnusmideesr ofa sosloidlisd. fbtyoaprs eitc hi se D idnEe M riivnoavttioiovlnev iiinnngv o ontlhveee d did moenpelenyns dtihoeenn .ti Innvd aethrpioeasnbelde,e tnhites veyax'rp=iarekbslyes.i .o AAn b00or33l.If lxi"maf]xg=0=lxi"mag]xg and if lxi"malf]xg=0 gbeeotwmeeetnr iscu scecqesuseivnec ev,awluieths :t egrnm=sg ng-n1,r h=ags0 ar nc.o Em.gm.,o5n. 0ra,t2io.5 r,
u#a=bFg^(xg)],x dguhg=lg]x'(gxd)xdx(,w ainthd Fmaonddi fgy' cthoen tliinmuiotus so),f yionute cgarnat piount •rScihignohgitc ucelo aonrfv cienrbgteeegisn.rga nadrbsi.t rIafr yf)i,s p droevfiindeedd eoanc h(a i,nbte] gbrualt onno tt haet atphdeamrtp iettnindgicc ruaolnsa sra- xseitsoc tsiuothcnhes agGneednn eiersra alli lnlDyeE a rmwionhr eethr ee d doifenfpliyec nutdhltee n aft rivreas tr–eiaqobruldea etiirso nyd'se+ ripivn(a tt)iwvyeh= iaqcph(pt )e.tahres =ne lcxie"msaslagri]lxyg aat rae) ,d ethfeinne dlxi" maandgf ]gxx(xg=)„ 00=, floxi"rm axgfnllea]xxr gap(rbouvti dneodt •1eCx.o2pn5ov,nee0rn.g6tei2an5lc ,feu0. nA.c3 t1sieo2qn5u,, .et.h.n.e c eIct o {miasn m}a o cnsoe nrqvauteierong tieinsa lti hfv ese orromslieeo non fu ombfa bsaeenr.
aInp perfofepcrita, ttehley :i n#atebgFr^agl i]sx oghvgelr ]ax pgdatxh= on#g t]gha]geb gFu-]auxgids utr.aced out x#a=bfa]axngdd xis d=ienftcel"igmar+ab#cleb fo]nx cgdloxsedp rsouvbidinetde rtvhael sl iomf i(ta e,bxi]s, ttsh. eAn aafucxncisoc tridoivnna gAr y(t op ),iana £knpao£rwebna. Dyien'.'gEd=.es, p –yetk'hn=yda,yte 2xn–at2 ryxev'.' a+Crliixoanmybe'lam+er xoa2nipny pi =net a0har.eps p lidinec apatei no\dnheslt n{atnr eov nsaelricinoaebnaldre}-; o wred.agey.r;, wtgh'he(ae ln)a atttrheeer nllioimmt riitets q eouxfir ifsetdas ,nt odo reg xiaissr te]i.n Tifngoifn irineteist.oe l.T vNhee oa tnreu itlnh]eda etag tlfes'or(m ah)ionalandtdes Laexdim(sctiast,sl l teahdne rNteh eiss u locinmhl yitth )o ants ea|a;t inos–fnieLe s|s< atyeehsef {ofaro nal}llocl wonni‡vnegNr:g . eEIsfv etaor ylLi m,ee ia>tn Ld0
bstya rtth],e tfhuennc titohne gi.n (teIfg rga(lb )is= zger(oa.)) [tEh.eg .p, atuh= r1e+tuxrn2s ytioe lidtss [saim,bil)a. r dEe.fgi.n, itio#n2 hol1ds ifd xthe iisn teglirmand# cis d1efineddx =on Tthhicek vnoelsusm Dep opfe ar pselanbd iocfular to the axis at pis D V=A(p)D p, •Sthoaltu itsi odnifsf.e Are nstoialubtlieo nto othf ea oDrdEe ro onf atnh ei nDteEr vaanld i ss aat isfufinecst itohne rdaiftfioer enacned (¥¥ap–p¥¥ly), trLy’H tôop rietawl’rsi te riutl ea.s aTno inrdeestoelrvmei naatne wcornitveesr gaen, fiit Ld,i voerr gnel"ism.3 Ifa an =seqLu.enIcf e aa sedqivueerngcees idno essu cnho at
SES##0uo.gg1bm=].s,xe t1ig #tng+1uegx+tnlixxoe]x2nxrd 2agmdxdl ax=fxoy==r 21bme21g# u0nu]#lx1sa+1edgsund+1u+a 1fr=1xoe,r:2# 21i2ngglldnxe]uxxdf=ixgnd=it21xel= 21ninl]lnt1ne]+g1gr+ax]lxs2x.gg.2,g. 1lci"m2-a#0rccs4in-1bxc22l=d0xSr2=in.4ag-rucslxian2rc cInm"tegrancd"2- 0 4-x2 •ahlASVenao(nv=dzlgii )dntt=h#hsg0e [ hs ssotqs(2of1u tc–aa1ralrz-e envv /dohohhz lol)uumr]h2mit2zedioeoizga nni=tht.s a tl sCVh c23eoh=rhino,g s.sh#iastdh- bseaAerzs c. ^ tapico Ihtrndssoso ps,l s.ivwd-osE ilet.uhocgm ft.b i, oeora nte tavpoilosym lrua tastmhiriodueindaes •sdesGoBepooaqefrae clmsuountshcahbiteer tcielir ioeabi oannmdselnfotl msye til o(rr uoiIivs tvtnVnattiaii n o-rnlPao t ntpgc)nuhr’ p osaeftd hoalnl ivel rtcyioara nraa rlai ttnundenail eoierlgninl vrnne tag ashDe n s,l tao .odoEha frl rne Iudn gt rt Iie–eeiDnVroin1ais EnDPtel i dsr a.aEvah, el la arliTvgiilsynvfuae h caaleinelun tnsuuei ev.vodr n oeetpAaei lsqqsolv n au ieuaa lnsast leiost .t n . nspilsouioAeotcnmtcil ioau iofenlgtn iy isecpov'tna=aonaitenfilik rnouotaysitnnetl,,. trrwiFonaeohts drgioe eolexrlti;"vteem eraa0lxmsp ip" mpxtir0hnloyxeadx tyuLoeo’cx rHeuitg= x aôgipnnepeotdai0 nta l ar= eneiln nd’ws1t d fi.rreaiiuntlte edl(er 0tlm;hi 0mt,iihs1nx e¥¥aafifi ,tsee0o lxa r1ne p¥¥s/xouxxp0ni)ote=,a nnltbeiatlminkeate xliifi fi naiotld0s.f-e ltt1oeh1/gre/mxa xrriei2nts=haum0tlet, •enrBrtwheq=‡oaaauyu1ltNa nnlitt, shhu dtt aeomhebtn de oba en runevm nnyroesfifid onr,be yneao1d onwMu; t danoor b>ditnithos)e0e .evs b rTe asawa edhsncqimiifisocsue nitiet vo¥s¥rsn en .acasr d geenEfirse vu.i.Ngsee n .Asrd,(g stanciuoefmo s c i|nn,ahrne v a |ncel<tnrtihraedm1gala etiist fftnhia a nrcelcneg>ent>s 1tasMrse, be nstrqofifihtfnsuuaofi.¥etn0r n t;ah coileerfl.
#eg]xggl]xgdx=eg]xg. ] g ditest earxmisin eodf brye vao kluntoiwonn. raTdhieu s afruenac toiofn trh(ez ),c aro£ szs£-sbec, tailoonnagl rewritten ddyt=ky suggests dyy=kdt where y=kt+c. In NUMERICAL INTEGRATION
•Integration by parts.Explicitly, “disk” at z is A(z)=pp r(z)2, and the volume is this way, one finds a solution y=Cekt. On any open SERIES OF REAL NUMBERS
TFTanwd#areiohhiefbetrfeeu h de gi c]r npexetoodronnomgne evtcufeirmleai asnd]telofexi,u a tnd“egrca ;due tsf iot o” xnhiwr rst=e ame e ntugilouundslr ltae ]aea“tdbx gisvi eo grs.i v”inan n li,# ]Va n xdie#btsieuegxue trgwpbahidvdrl-e aiavv tctiht=ein=i#teaodt ebn iuuivngnsavvtr ]tneaxwe-gbadtgghe- rurade#atlthr vnif#e]aoeadx d ncb tvghautsodos.etd. xr h Y aufet.ou ir.pmun r cteodotsdio outnbhnc’eestt aaE[Iinafsp. ta, gfpebar.ri,] osv , sp anulraos#im-na t bd1t1 dee xoe 1tciffw3sl io dnelseieexnefnd=ttd ea sgsautalunr "ciaabdmh b0i f -nliprenti#eo -girtioah1envnt xat-1ns lh3,usc ad,mtl nhopxdbesr+ oe e1ivdlrni bi mlotd"iesfmeigu 0tdp+rsb ao ia#blniol nlt1f et# taxsrh1i bvn3iefnadt lel ixsagmi nsrc oa iditlfnsise f t ef eooixrpntvvi2heeseatnderl. appIpdaVf roiea fe=trrirfhn apaeetmder#“a enizswebanodAtalcalri liecsi]odcz zhuno oelglegfidsoare rrzvsst dho=t, ”bhieln eute #oataam awt xbenh(cid:185)eie dsesert irhnol]g,iez tfkht h gweat2rese od£ at v hzxrrohva.a£ilotsd(u b rliatui),ib o mroon1aofve(,l fe z o t).nh ircaSgeseyn o vcldmoiatr nhornleud2es t(t irszaimi-o)cxos enaeaibssclt,v teiioaocunhss suyiinno'n=ttlieeuqkrrtuvv•yieaaoGyhwi lln'om.,si+e tmoTnhippeslh eoluv h(iyurgtete(i)as”ersao yyln y0)n d =de=,= toswsq1Coufo^(iilyiselttursvh )ekse t.i hittq -noy-(ugonTk(Cta atr hah )td,=rmnei=eeoy waurney lshsI ax)Vothe pilir'sPsuhe+l i t t-anyypihyvo(e(=eeeanta# – )yg) rkhatt=ep thet=no0iakes 0u.t( Dr t(ca–tdfhdEoalo )henus..r /osmhTtl.aau,h=s nCtesitb–oo o etponncrc(n.siita ai vT)udttdiehhseatdeee)rl ••iasiataTGnnhopntrv teefpdaeot nar gwnplhoelvruelaxra=,emzra il oem(sdlbbb i edetay–uofnrt iis a ornnoinf)utnigfei/et gln eneise# n ddb...i tnieraenTedfStrefgh v]ogipx enalrru lgeaaesdltdnlic isnxi oti nisetnad hai tsolerahcg enrtoeo i .tec vn rouTlaian etnnthhemaidnecmvpeoe teiprsfntnslo lytb.gila ceiel rnoad Rytegwnw .ve Tiropaaneeh lrdgguvopri uaaoblomytllulieuae ngedramt hta tshpeevo soaa a uorsdimtntoil,sila ft ynaibttf,oneh oliinyneesrf ••aSCvod afeeo lnrtpunhioaeevetrssee s .trd eoig aArfne/il 3=e nass0ncsa eeuaorn.ttmi hA.eNes:T rsih siens/se rN=te ihaeq0aaensu nsesa/=u3enrmqeac aue0c +o enann.f/l.N=c lc.veo+0eaadn al vunotNhee=b.rse t g aTauteie0pnhs+r eetmi.do f.. +sstahbeaNoyerNfi a e s.nta eshTdd qe dhuii tsiseesn e ncegrvcali afeel ltls uheo.ideesf
one to be differentiated, minus the integral of the product limits on the right were to exist. They don’t, so the integral sections (shells) of the solid parallel to the axis of ] g ; a ] g E graph of a positive function together with the underlying partial sums convergesn,= i0n which case the limit of the
of the two new quantities. The factor to be integrated may diverges. revolution. In this case, the area of the shell at r is If y is a solution to the original DE, then (y/h)'=q/h, interval on the xaxis form a trapezoid whose area is the sequence of partial sums is called the sumof the series.
be 1(giving v=x). •Examples & bounds. A(r)=2pp rh(r), and the volume of the solid is where y=h#q/h. The solution with y(a)=ya is average of the two function values times the length of the If the series converges, the notation for the series itself
E.g., #arctanxdx=xarctanx-#1+xx2dx #13x1pdx converges for p>1, diverges otherwise. V=#abA]rgdr=#ab2(cid:185)rh]rgdr. y(t)=Ya+;-#atq]ugh]ug-1duE. ignivteersv athl.e Atrdapdeinzgo idth reuslee aarpeparso xuipm aotvioenr a regular partition stands also for its sum: n/3=0an=Nli"m3n/N=0an.
2 3
Calculus 2.qxd 12/6/07 1:46 PM Page 1
Series continued BarCharts,Inc.® WORLD’S #1 ACADEMIC OUTLINE
Aannd eiqtsu sautimon i ss uSc. hI na sg en/3=ne0aranl =staStemmeeanntss, t/he asenrmieasy c ostnavnedr gfeosr -ITaf hn /eo nbpz-nesredoriivleiesmr gaiten s(do arng aedpo pemirteohtarecirch besnse£ r¥¥aien)s,( ntha‡reenN /o)f toaernn adnui/svbeednr ghfeaossr. TcEoh.gne.v ,i en1rtg+e1eg xraat=t eadn1 -seenrxdiep+so ixhna2ts gw rahidemirueps lt iohefes co1or+ni1gvixenra=gle d1ni-cvee xRrg+,e adxn.2dg m.ay
3 comparisons. Try a “limit” comparison when a series
/an=S. looks like a p-series, but is not directly comparable to it. The initial (geometric) series converges on (–1,1),and the
n=0 integrated series converges on (1,–1).The integration says
•fGoermom /3etarircn s,ewrhieesre. Ari s( nau rmeael rniucmal)b egre aonmd eat„ric0 . sAerkieeys ihdaesn ttihtye •REa.tgi.o, n&/=3 Krsoinot^ 1te/snts2h.Acosnsvuemrgee sa si„nc0e.nl"im3sin1^/1n/2n2h=1. ln]1+xg=n/3=1]-1gn+1xnn for |x|<1; a remainder INTEGRAL & DIFFERENTIAL CALCULUS FOR ADVANCED STUDENTS
is n/N=n=0r0n=1+r+r2+...+rN=1-1-rNr+1]r!1g. It implies If nl"im3aan+n1<1ornl"im3an1/nn<1, then /an converges •aaTbragoyuulmot rce noatf n (adsne e Mi nbfaeilcnoLiwtea)luy ir mdinipf fliesereser nieetqisau.balTleith yfeu fnoTcrat iyxol=no r1f .isseries INTEGRATION
trn/h3==a0±tr n1th=. eT h1se-e1 rciroen^siv fedrrigv<eenr1gcheees aa lnisfd o |pn/r3=o|>s1as1irb. nlTe= hsuea mbs1 eo-r1fi earsn -yd gi1veleoor,mgeesat rniidcf /(gaeboasmnoedlutirvticee rlsygee)r.si .e TsI.fh Tehsnlee"i m t3feosaltlsaon +wanr1ien >dge 1arriovere udnls"i ebm3fyu acl oinnm1 /apnpa>prils1yo,inn gtw htiehtnhe Ik/f3= c0=f]0kk,g! ]citg ]ixs -aclsgok= cafl(lecd)+ af 'M(ca)(cxL–aucr)i+n fsel2lr!]icesg]. xT-hce g2T+ayglo.r •Heuristics. TDhe EdeFfinIitNe iInTtegIraOl cNaptSures the idea of •Area unIdNer Ta cEurvRe.PIf fRis EnonTneAgaTtiveI OandN coSntinuous on eAn'd(xp)o=infts()x.) (valid for one-sided derivatives at the
series at x may converge without converging to f(x). It
series can be determined using the preceding formula. root test: nl"im3np/n=1(any p) and nl"im3]n!g1/n=3.More converges to f(x) if the remainder in Taylor’s formula, adding the values of a function over a continuum. [a,b], then #af xdxgives the area between the x-axis
•Ep-.gse.,r ni/e3=s1. 3F4no=r p4,ca1 r-e1al1 n3u-m1bme=r, 2/3. 1 is called the p-series. precisely, nl"im3n1]n!g1/n=e1. xxRvna]xryg=ing] nw+1it1hg !xfa]nn+d 1ng_)p, i:ap]xp-roaccgh]ne+s1 g0(xxasb netfifiwe¥¥en. Ec.gan.,d t hxe, •dRRdeeiifteeeimmnrmiaatennin nnei n dtss euubgmmyr a .al o Aipfs a sratuht iiettf aiuolbninlmcy,t iiwtwoinhnegi icg fhvh atiedlsude e fa is nuofeimfn di s tuoeoc fn hd v ivas[iaulsum,ieobssn].. oAAisf athned atrheea g arcacpuhbm. Tuh]laet geadr euap ftuon xc.t iIofn fAis ]nxegg=ati#vaexf, t]hteg dinttgegivreasl f(x) Fundamental Theorem
n=1nP POWER SERIES remainders at x=1 for the MacLaurin polynomials of [a,b] into subintervals, typically expressed by is the negative of the area. A(x) f(x)
The p-series diverges if p£ 1 and converges if p>1 (by ln(1+x) (in Taylor’s formula above) satisfy •Average value.The average value of fover an interval [a,b]
a=x<x···<x=b; and a sampling of points, one point
•zetsAaiscbnetueronerlr rrmfifimlitmooo(ceCnrsw,tpr: l0 ona yati)nhOsr. ari i dnzeeTtsnl/ee n io3=ehuanNrc nse on1tgrss h e])bh t- wsearVotaiesh sucirsrie1attgnemhi Elngeinydrn n iso. eahet .dRhndasaTIene: bifrc -chc msnGr/ oaet2o=sehoNbsnnela/1eeuN=nsvrsEn1 otiiei1aece]tnlr$r-e u sggNevrtse n mae1/se1m3s=l e+r.gusCv i1arn eeMiann1aeoN sl2naEs ufonden aewri.<rdavne ) hn oeod.aaTo rvf pg Nasetpteehl+rEht,rs ee et1o, et r .afhSnrifocnemairhr tt( Tesst itarnaht gas u gerosllSna tiupmemcls maraeiintrttrteiitta inoieseoattgndesfl, ••aptPIARaaoTnh/nt 3of=ohe p se ptclwpece0o yeoa,r orlw erwnepaitnnrvehosoceev sau]imiwrrxs en l lsso sterige-eosiefargnrre frl xieri tgssicecee nee sigwosiasss rcnni.len i n ei= ce ertvnx ,Aoishix etue ona h–o irsamrvsegf c0 p oe rrabe+dt(o erf hnoe(egw paierrrncae)lo ese “e osf1.ia(nrc. not– Ali]eueT trsxn¥¥sdn.mm h et-ap uIe,erbn nfom¥¥r/i 3=nee wesc/tiN=)rdbhesn0g; ea te+e0qat ra(er onti ips nrsinicfax n)voe cx” r 2anrw(o exaionlc].eane ,rx –p^sl vrci“ N- osRneaacsw rulbe=o,glcamroeceneig urd0+ve2sb ts s,e+s. Retce erh1ri”q)grsse,f)iu e fe2iacieosssn.te, n r wantwfy cesttriehoerie qthrvimlt.cedeoaihdnesffl ••mcsBCfns1Roe(oe-aoxer 1cmnlisam)enei]=xnpsc1s2pss un=g/gM =a3u=t=T1irintv0/ina+a3a=ileygny]cxn1nsl Lgo ]]++-tx ]carh1x1 o-u1e-s2nger1 eg+_cfnxi1rTT1f+ng.i+niag.ea1 .c2y^ssyp=.=ie lxeloimrnon/-nnr3=i/+r3t =easc 01yscx1.#:n o<nbxeDeRfnsnf i^+e-1fihfcf,rxo11eii=ueer<"tnneshndn.t1/e03= s tnh:,io0 a]tnatthih+nIenef=rg 1 ftgchx(tonahne)Rn(efc f >)bi^gc/yx0ien eod<!nm.itr1seeT hact.ahrtnrliidyecs empfurRApDidsrinapoegoi xadrcepirahmxntrm=peecitei0tt go rhm (ieaae cuoiibasnnneiuRnlic–nvditanm1 hni eaporgesc d so)fmeu ipso ;u/ai=m nunaannbcabxntr hi, ni.tiniintn l hsn–Aisot :tx1etetu ewiesii nr/orb=i=uorvne vnnivmfa1urnata afle plth, le+sser ] pasa vuocrliiascveaefmsiDy hrg aln s)] t l ,xccugx:h s sowi.btteui/iuhf=hsi-n hrmbn.Alexo1 eeit :Amnxerfcmoetortp io e vf-ni[stlas rnaxhue 1avrbilfiemgt[m s]a–t.i 0ci l1tm a-;uo,i,s( o3lxmsnualani] i1xtmn t ]i’gihsn.rcvwm n 3eTiueagt i ulonhlatshuthmu+kheoaet s emer .a s va1mssfce E auDso tlo.tnr3mlhugsaeecieike.gn.si,n, ae ugoeltttselt ihhef ahtif deaessfrt, ••AtmtvAagtgaqmIVihnp aeieutmq cvauoelrpttlacyuoseuole ernttu sacgeuo i+tbta pnihim irxgTtentvDtltaeyhihyy ihtuem dlt tee iee ry(el;enr otaaxa ocs vafhagovtgtautriapdeeie veencerml p ddiemrt.de rvt r ira aditenoI c stovtbt( na.hgxehepbMtay e a i tlyIaltoot mia lmuVhh nmt ffcathdeeea Tgiaeacve meit)edgmneseh l i.ipiir f (eanlmsnieoabTcle nnptnaerhgyneihitgl c ttneearaig ee etnogv ntctho vmethiacoirgeafeanrnalc mf eetvg lstl uo sg une parpemmairfigaletonevn r .: lrai as ctetf=sanhyispluEts,tll #g ie o i n fabi.bnto o nogtsbhcie-givnifni r.1tmen t t, i)m ]h iaa ott,atoeth e eliotnatlrgfn hr eh ta ddTivt a #eneitfvetaohatlme inc b=v(lme bcdr .ftoeego veyh)ee rf]s(rrm(=i at Sexany(eticntb)ssneprechghgsD'tm)ncderh(eieaee–tmrt t )xcni s)emMe insit,g.amnsin (eieoet a eoastsitetn ninan hhn) ornt..ndgaeaeees)ft •DsaaTfSAdudusnoiix nf]cd ]fudhx ied#tngar gaih==temfxeeaffn2ag ees]#tf#rarn it0uaaeugxtnlftndabs#id]alto.tst i=axniagfnt2mtE dt]hoedtt.tedef g.lgxn o.ddsi,TtnrAu ttaeot.clemt^ hhgdxti,he:ris2 f aeh uf=Sloe“ssari.rseA' eei(F nm’nxcute^th)o:nx i=eamc2i tthsniep2soit,nxeoen ucg=ssxo sira/taen2xxirl xode t. h nfofe^pA ufx tanc ie2hrcnhnhtftv. a uidooionennlfcf v” irtn iitunohelindgees
•nB/3=a1saicn ccoonnsvideregreast,i oannsd. Fcoorn avneyrs eKly, . ifI fn /=3aKan"nco0n,vtehregne s,/ thaenn ctRffhooo>errn c0vpa|e,ox srpie–gnosects n(s|ci<ic)b eaaRlylno.o d tfno Y(etgiho.ie ieutI) h n,p e rcoercaw sawnpse eeirt c ohs(tf iieotvire)nei,ne ely Rs o.,d Crwie sboth eoncirtvcamhehl lr eiegnmndeed an ptychao eeb i niersrt a sa¥¥ad;db ioiusaurosn s(l d uii ooti0ef)f llnn(111+-+xxx)==x2–cx2x2++xx3333+–..x5.=5+n/3=g1]m-=12gn]/n3=+012xgxn2nn+n+]1-1 1^1xx<#1h1g 12 Riemann sum i!=61cos;i-21E+1 21 fxswgl(,roh xaopop)neshaee t f gogp(efrox tati)sphn ahfets o s iarl(s onx etpt,aiayednc) egfhfrie oeinvxrltda .v tt.Paio Avr letianho c ieuaisn sn s gctlyio a,d pllaeilenenr diedfv i aedastlenoidvg i niemang tot e eetfnha gftcisrsih as fw lpao i ocrft uihuvn rnatsvcr. lietTooiophuoneesf •cMI#noa nebfatt]hinnxe ug govcua]axslsu geoed n xg t=[”hae1,fbo,_ ]rp,et himthe#a ebnafgo v]trexh regaidrgnexet e.isgv raaal ulxxse. inoI ff[ aff,b]ias n sdua ctthga itnaheradet
pasdanaivyr etsiinr angcloe restsh.au is(mnEingsqg ua ibissv eoabqulouet,nue entnl.cydge,e. , id fon, /f/=3 thKpeaan1 rnts.iecaAroli nessvsue ermcrigeossen; svoi,e fft rhpgtheoeness .ia tsiTnevfifiqheu iste0e n)ri.mcs T es th hoiiessf c(wn/oo3=hrn1i vc2ehxnrr,g noneow2n,ti)cntl eh"im 3bttyeh2 senst + orl1xval]tiinninmo+g+ 1i ttt1 hegse2t l:,ie ns2seshxnq ounnwa2tlhs/i taynt hx2 tah<ta 1t 1t.h p&eu tEsxr .atg<hd.ei,2 u ,sra ftooiofr aTerhxce=t afon1l+lxowx=+xin–gx2x 3h!23o++ldxx3 5f!35o+-r aggll ==rean/3/=l3 x0]:xn-n!1gn2xn2n++11 ^x#1h •Definite integral.T1he definite in2tegral of ffrom3 ato b •]dSthxiofeglf= uesrltoyeionp0nte+i aftlio #eex lq0xdifnu.]aitttiigaodln t .vya'=lufe( xp) rwobitlhe min.itiTahl ev aslouleu yti(oxn0 )t=oy 0thies somewhere on the MintVerTv aflo: rb I-1ntaeg#raabfls]xgdx=f_pi.
•fIponoutsneitdgivarteai,ol ann toedfs dta ellc& trhe ea esfisontlgilmo owanit n(egK. ,c¥¥Ari)tse.s rTuiahm efeon r /cf3onifsv enrcgoecnnotcnienv.ueorguess, •Gcfuoennocvmteiroegnter noiccn e i ptisso wi2ne.tre rvsaelr ioefs .c oAn vpeorgweenrc es:eries determines a cosx=1-x+x2!2+x4!4-gn==0n/3=0]-]21ngngx!2n mThaey lbime idt eissc sraibide dto a esx #iastbf if] xsogdmxe= nuDmlximb"er0 /Si(tfo] cbieg Dcaslil.ed the THEORY
icrun/fi3=ong nd1havnet1n re3desr.sgi dteioenms/1n= ,2 alo1ytvtni 1hne3eirg+fen s tth#i#1nKm3e/=333 asKxftu1i]f3mnxd]gn gx wdgt=#xiht1,ehn. /2=eNcs0ruKor1monf8vr ] .enl.we.r,ggsi+stneh =st th.#KNh eae3rInr] ff o f]rg(xt lhNelgefed+t s x1ss ),e.tsrhitidhaenees ctpxooo "nltyvhneefro g]mfxeusgin=a clftsoni/ o3=r n0a.xn /nN=Ti]0hnxxe -n tsh=ece1r g+ineixn.s+t eOnxr/3=nv2+ea0x l…sna(,y+–s1xi .,Nteh1=.,e) 1 tst-heo1er -i xe1sNs/xe( +qc1uo1–e]nxxnv)!ceerag 1neogd,sf •TBp]s1ihin+nepo x-xmb=g1iipnax=lo -ms1e+rix3ai!e3lp +sx.c+Fox5oe!5rfp- fpi^c„gp2ie-!0=n, t1asnh/n3=xda0 2r]f+-eo] r2g 1|enxgp0n+=|o<x=1n12/3=ng,1!+0,e1ekpp1oox=kp.,ep2o= svbisiannnuunaoottccldureeuhhm gnge udrar tail peahvseldnap) as d tsdel suet raaotnae hlt n olilSa sbidtnmf,Ree i tdt eihhidde tiseenm# ad topftiebahufffognfe ni]ern nixcarnf tt bgositsfdoel luraeglnxomo.r amwiaonTsst irdihsSn oo aegb#nnai b:d, ab yf f purEtr eoa.enlv er Tsccbetsptaihersitelyo ielicote nnhtfneedi tuavse >fnnt egh col0rieeyetsf ai . .l oab o[cTdInalwfaem h, lmebtelaihre] tnue d swlds rw item ao tti hhb rtiihdddeeest •cIUoiiLnnnlv:ostteteeeser]gge egbrdrsrt-aat habiibmnibiaslltie iaeglt,#itry ttevoy soai # faln&o cb ircfsh L iu le]nio£nxncsedtkfgeqede (dgurxx eirsa)na#su£lbttibietlMMmiiegne.ras t:aIet.neflr]A osvbte.rae-g clvsroaaa anollbglutfi.i aln[iatxutiy,oob unoi]nssn. A f[uwa[snas,icbu,tbth]mi] oii,mnn r ogptou hlnfigee inhass •aiarCne phfdtpeuiargsnentrchrgatiaetiebion ounodnt fe w docv riaawt nhre i ivtabchaobeol nuluectati ht nfeacou.nhr ogamIunnesud g e ildnatfeofg.re iAcmvthtan,ae t ki tivhinenee tt ea eagn“gnra rdairann elftad’se”i gsa v rinacasdlol u nseelmtia.im nsoIiufioet ostgrh ubotlisoysf,
!3nI=nKtefg^nrha.l t!esnNt=Kf^nh+#N3+1f^xhdx dgievoemrgeetsri co tsheerriwesi sme.a Ty hbaet iids,e nn/t3=if0ixedn =thr1o-u1gxh ^thxis< b1ahs.icO othneer. ekp^o=2 ph^,pa-nd1 (h“^pp -chk2o!hogse ^kp”-)k+1h •iAwnentletlgi darsea rltiorv eatfhteeirv sre et.osu Atlhntei n fagon rvtmiadlauetreiio vinfa totihvfe e#r aeob fifs foaro nmefu. nfcatniodn[ af,bi]s, aas M Basic Integral Bounds tphoeinnt s w#aibtfh ]gu(gcd)u==aa#cnddf g^g(d]t)g=hgbl.]tgdt, where c,d are
f (N+1) E.g., /3 2:3-nxn=2x/3 x n=2x: 1 , for p function Awhose derivative is f: A'(x)=f(x) for all xin In practice, substituteu=g(t);compute du=g'(t)dt; and
|x/3|<1n. =T0he interval of co3nnv=e0rbg3elnce is3 (–3,13-).x/3 If pis a positive integer, eko=0for k>p. soof mae dfuonmcatiionn ( usouna llayn anin itnetrevravla l)d.i Affenry tbwyo aan tcidoenrsitvaantti ve(as L tfriannds fwohrmat attiios nw h#enb u1=-aua2ndd uu==b#. rE/.2g.,1 u-=ssiinn2ttecffoesctts dthte,
•Calculus of power series.Consider a function given by a consequence of the Mean Value Theorem). E.g., both a b a 0
K N N+1 power series centered at cwith radius of convergence R: All rights reserved.No part of this 1 x-a2 and 1x2-ax are antiderivatives of x–a, which becomes #r/2cos2tdt, since 1-sin2t=cost
••/ucACEn/on3=.bogndam1s.vae,on erpnlenru{/ag#s3=cttreeoiism1n sn/,3=o ncvab1n1oetu3aen rtt gv nwe»nees.oisrttns tgAh/ a.1=ea 2bAenbs1rsscsreonoseor1lul.iru ue3 mt<tsIeef e+1 llcyy3a/o.}–n#n31,, <v3abt3ehn5nre>•xgn110ec 03.s/o–d n4c.vaxoennrgd=ceiots1ni,o v.2net0hrag1allet8y s..i,. si,a, f naiindft iSdTmSnfeuuhta]recceyxighhv dgd ar=aaiatifivl fv nffee/veu3=urra ngne0tnahcnecit tetnsiiairhooa]tein xtn neai -gdnsiis s a se cfdeitngn lrincdfit].epfeixessogr g:ir=henaantb snt/w 3=lireaha1 bndeolarineue n so(t ]chnox e f–-( c coRo–rcin,gRg nvic-,en cra1+g+l. eRRcno)c)n,,e v aaRennr,ddg beiiudttsst. UCq.uSs.uto$fr4mie.h9ece5ur HkCnIIdoAdSSotsNlBBriw.ne tNN$edn7-- u#.s11l5 o1d0 30.oa8::0yfd 0 .As.t912icu t374&lt0e82ho.9o -3s 5rm12 :2aG240te24r31al23dnafQ©Nwptepsrotslhuoa oruo2e0-srobtniHrciit0mt tgcalsett0o41iemkgr:naaac2cnoSeetsaio-r 1,nD d ttt2 pntuipi ea0uocyacd3eden0e, n lpry a d 7rm itl-®rseon te ,eosBcir amarawspo P8seassnlriaoet iratymdroyh sraCci ikn fe neeDoh. vcgcgmurboaafuh,msr elnreiaoeo d tdn,n rms eseo ti,rya, cn r eInst asbtnbfhptyheloeuyrci, desm iort .na d0n,fip uno1yucrcw0 bmmel8liuditeasdh athioinnoeousrngrt., dcdT2oiehf]nnfeeos ai#rnttcniaetontedningxr dtgsv 2xebt#ao-aryflinf. v] t121Exa intC.dgagidtvx,e.2 x2,e.g= ws(,Trf haohiitosreci x nohxia 2nn <n-m.da –e (a1e1fCytxi,+y nopohpmirrtCaie ecvpfs. oaeusilnri t o eatxnren( >goyprff 1rotaro)eerl.gn aor ltaf h umveana slfs,fuu paenaem,ncc itdiilisfy o it nethh doifes)f, •tTI[FI#haaff#ahu e,bffbenboffii ]rd]sso,]ex axtnimahgnmgondedt niderexsx gn np =Auert#aasgarAibealts dl#t aiei] astvt xbh nooeug e n,fsea oab etv]/n[hrxdaateei gl,dAmtnubode a] ]rx#,ctbi aove.otbgh fanf-i etns]incvtxtA reaesgugl od]corca xufiats g llf.a uisfsn:os, .nItnafio nd Otfndheinnarseeti cvg ioaanpntttaiietvvrirnetev .usa:ool,fu tsh tohenne •fsylNno#ti1aaer(xr 1lttduu1i/0nxsr# dg)a #ulw1t= .#1il/toxh–Trgu 1lha/n#bder2 xaiu.Ftc.h h0^T=Tmgahhn.]ee# gx1 eAxgofth ot g-ohrltrfem2i ]1rgx uvdoeglaradtlroe ixmau=i.sbse l-Senode tfaeeft#re 1foTyinxren m 1tciputhdurisnoolteainpqd s eu whrieitosniis wte hoslir nn neou vgpfe= xg trht.1s h2ae/=ett.,
-hIfa s/ ab linmciot,n tvheerng e/s aanndc oenitvheerrg easn.£bn(n‡ N) or an/bn #cxf]tgdt=n/3=0na+n1]x-cgn+1^x-c<Rh. ckrehnpaorrwet,sl eemndtgaayet aivobemle oi utf staeh rfe at hmcaoitnl ytsh.t)aen gti,v ietn b aenintigd eurnivdaetrivsteo iosd j ubsyt othnee IAf ]xfg=is #axcfo]nttgidntuoiuss ano na n[tiad,ebr]i,v attihveen oft hfe ofnu n[cat,ibo]n: ndaneafdtui nirsai tldi oelnfo.ig nI ned ct ahtnois baleipk ptehrwoea insceah t,u barena l i neevxasepirolsyne e fnduteniracilvt ifeoudnn cifstri odomend. utcheids
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