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Calculus Formulas and Theorems

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Calculus Formulas and Theorems Formulas and Theorems for Reference I. Tbigonometric Formulas l. sin2d+c,cis2d:1 sec2d l*cot20:<:sc:20 +.I sin(-d) : -sitt0 t,rs(-//) = t r1sl/ : -tallH 7. sin(A* B) :sitrAcosB*silBcosA 8. : siri A cos B - siu B <:os,;l 9. cos(A+ B) - cos,4cos B - siuA siriB 10. cos(A- B) : cosA cosB + silrA sirrB 11. 2 sirrd t:osd 12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20 I 13. tan d : <.rft0 (:ost/ I 14. <:ol0 : sirrd tattH 1 15. (:OS I/ 1 16. cscd - ri" 6i /F tl r(. cos[I ^ -el : sitt d \l 18. -01 : COSA 215 216 Formulas and Theorems II. Differentiation Formulas !(r") - trr:"-1 Q,:I' ]tra-fg'+gf' gJ'-,f g' - * (i) ,l' ,I - (tt(.r))9'(.,') ,i;.[tyt.rt) l'' d, \ (sttt rrJ .* ('oqI' .7, tJ, \ . ./ stll lr dr. l('os J { 1a,,,t,:r) - .,' o.t "11'2 1(<,ot.r') - (,.(,2.r' Q:T rl , (sc'c:.r'J: sPl'.r tall 11 ,7, d, - (<:s<t.r,; - (ls(].]'(rot;.r fr("'),t -.'' ,1 - fr(u") o,'ltrc ,l ,, 1 ' tlll ri - (l.t' .f d,^ --: I -iAl'CSllLl'l t!.r' J1 - rz 1(Arcsi' r) : oT Il12 Formulas and Theorems 2I7 III. Integration Formulas 1. ,f "or:artC 2. [\0,-trrlrl *(' .t "r 3. [,' ,t.,: r^x| (' ,I 4. In' a,,: lL , ,' .l 111Q 5. In., a.r: .rhr.r' .r r (' ,l f 6. sirr.r d.r' - ( os.r'-t C ./ 7. /.,,.r' dr : sitr.i'| (' .t 8. tl:r:hr sec,rl+ C or ln Jccrsrl+ C ,f'r^rr f 9. cot .r tlt lrr sirr.,l * C .l 10. [,nr'., ,1., lrr1scr'.i * Iarr.r f C .J 1i. cotr] +C .[r,rr,rdr:]nlcscr 12. ,"r' r d,r - tan r: * C | 13. /*". r tarr.r'dr - sr'<'.r| (' .l 14. n""'r dr :-cotr:*Cl l 15. /.'r.''t.ot r r/l' : ,'sr'.r r C .t 16. [ ,urr'r cl.r- larr.r - .r + (' J tT. [ ---!! -:lArctan({)+c .l o'1t" a \a/ 18 f )- Jffi:Arcsin(i)-. 2t8 Formulas and Theorems IV. Formulas and Theorems 1. Lirnits ancl Clontinuitv A furrctiorry:.f (r) is c'ontinuousa,t.r - c if: i) l'(a) is clefirrecl(exists) ii) exists.and Jitl,/(.r') iii) hru .l(.r) : ./(rr) Othelrvise..f is <lisr:ontinrrorrsat .r' - rr. Tire liniit lirrr l(r ) exislsif anclorrh'il iroth corresporrciirrgone-si<le<l linrits exist a,ncla,r'e etlrtrl tlrtrt is. lrgr,,l'(.r): L .:..= .l'(.r) - I' - ./(.r) ,lirn, ,lirl 2. Intemrccliatc- \rahre Theroettt A func'tion lt , .l (r) that is r'orrtinrrt.rrrsr-rrr a t:krserlinten'a,l fo.b] takes on every value bct'uveerr./(rr ) arrd ./(6). Notc: If ,f is corrtiriuorlsorr lrr.lr] an<1.l'(a) ancl .l'(1r)difler in sigrr. then the ecluatiou .l'(.,)- 0 has at leu,stotte soirttiotritr the opetritrterval (4.b). 3. Lirrritsof Ilatiorial Frui<'tiorrsas .r + +:r; /('] lirrr -o if the <legreeof ./(.r') < thc clcglee of rt(r') .r'+i\ l/\.t J ',.2 ',') ,. l'.x;rtrr1,l,':lit,r ,. - {l .r'+r. .1"' ] .) ',//, \ '2. lirrr is irrlirriteil tlre,leglee ol ' tlrerleglee of 17(r) -tr : ./{.r') ., 9\.1 / , ,. .rr + 2ll' r.xiulll)l(': nlil L )c .r'++x. J'' - ai /'/,) 3. litl # it fiuite if the rlegteeof ./(.r:)- the degreeof .q(.r) .r'+f - r/(.uJ Notc: The limit u,ill be the rtrtio of the leaclingc'ciefficient of .f(r;) to.q(r). '2.r2-iJ.r -2 2 r-xallrl)lc: llllr : - t(),r'- 5r2 5 Formulas and Theorems 2I9 4. Horizontal ancl\rt'rtir:al As)'rnptotes 1. AIineg-bisnlurrizontalasvniptott'<-rfthegraphof q:./(.r') ifeither lirrr l(.r';=l; ,,r (r) : b .Itlt_ .f 2. A lirie .t - e is a vcrti<'al as)'rrrptotc of tlie graph of tt - .f (.r) if eitirel .l(.,,)= *rc ur. ./(.r')- +x. .,.hr, ,\) 5. Avcragc trrrrlIrrstarrtilll(-olls Ilat<' of ('lrarrgt' 1. Avt'ragt'Ratc of ('lratrgc:If (.r'9.yrr)attri (.r'l.ql) irle lroitrtsorr the glairlt <ftq - .l'(t). tltert tlte a,velirg()ritte of c'harrgeof il u-ith rerspectto .r' ovcl tlrc itrtclr-al lr'11..rt; is ly l!_r1'_l!,,) lr !1, ' .l'1 .l'9 .r'l ,r'() l.r 2. Ittstatrtnrit'orrsRatc o1 (1-l',ltrg,',I1 (,r'1y..r/9)is a lroirrt orr the gralrlr oI rl ,-,.l'(.r).tiurrr the itrstautArreoLlsrate of chirrigt,ofi7 n'ith rt,spt,r'tto.r' at ,r'11is .f''(.r'1;). 6. Dcfirritiorr of t,lrc l)r.rir-ativt' -lll lEP,r' !y)--ll:'J .f'(.,) t'(,,) 11,1, Tlrt' la,tt<'rclcfirritiotr ol tlrt' <k'tir';rtivt.is tlrt' irrstarrtirlr('()usrirtt, of charrgt' of' .l (.r) u-itlr resltec:tto .t at .r -. (t. Georrletrit'alir'.thtr <lerir':rtiveo1a fittlt'ti9lt at a lr,iltt is tlrt'sl'1re,f t1e'tatrg<'ttt litrt' t, tho graph of the firnc'tion at tltat lioirrt. 'fhc 7. Nrrrrrlrcr(' :ls a lirrrit 1. li'r (r + 1)" -( n++a \ fl / 2. lini(1 + rr); ( n -\) 8. Roller'sTheorerrr If .l'is c't-rntituu.rttson ln.0] arrrlciiff'elentiablt'on (a.b) srrt'hthat.l'(rr).., l'(1,).tht'n thcle' is at leirstotte ttutttberc itr the opetrintelval (o.b) srrc'hthat.l/(r') - 0. 9. Nlcan Valuc Thcorcrrr If / is cotrtitnrortsott ln.lil aucl cliffelentiable on (o.f). then there is at 1t:astout' nurrilrer l/1.\ -J)l!l-It^r '/ "'t l iti (n.b; .tttlr tlt;tt - I - f'1, tt tI 220 Formulas and Theorems 1i) Extreme - Vaiue Tlieorem If / is contirmouson a closeclinterval lo.l.,].then./(.r) has both a tnaxinrum aurl a minirnumon la.b]. 11. To firid the rnaximrrrnand nrirrinuruvalues of a furrc'ti<)\tt =,/(.r'). loc'ate 1. the point(s) r,r'hclc .f'(.r) c'harrges sign. To firrri the c'atrcliclatesfirst fincl lvhcre '(.r:) ,f - 0 or is infinite rlr cltterstrot t:xist. 2. thc t:trrlpoittts. if :rtn'. ort tltt' rlotttaitr <lf ,/(.r'). Corrrpalc' thc frurctiorr va,lues at trll of thcsc points lir firrrl the tnaxiruuuls an(l ntirtitttttttts. l 12 Let ./ lic'cliffclcntialrit'firr rr <.1'< 1.,tttt<l torttintrotrs for rr { .r <. lt. l. If ,f''(.r)> 0 for ('v('l'\'.r'irr(rr.L). therr.f is itrct't'asingorr frr.1l]. 2. If ./'(.r'){ 0 for evelv.r'irr (o.L). tht'tt.f is clt't'rt'asrtrgorr [4.1l]. l') _t,). Srippr-,seth:rt .f'"(;r) t'xists ort tlte itrtelva,l(rr. lr). 1. If ,f"(t') ) 0 irr (a.b).tlrcn.f is <'orrcr,veupu,rrr'<l irr (a./r). '). If .f"(.r) { 0 irr (rr.L).tlrerr .f is corrc'tr,ve(lo$:lrwfrlcl irr (rr./r). To lot'trtethe points of irrfkrc'tir.rrttfi tt -.1'(.r').firxl the proitrtsr'vhere .l'"(r') - () or u'ltt'r't:.f"(.r') 'Ilten fails to cxist. l'irest,'arethe orrh'r'uclirl'r1,'t;lyllere .f (.r') rnar. hal't'a poirrt of irillectitxt. test tlresepoints to urirkcsure tha,t ,l'"(.,).- 0 on ont'sitlt'arrtl ,f"(.r) > 0 <.rtttlu'other'. 1.1 Diffcrerrtialrrlitv irnplies r'ontiuuitt': If a frrnr:tiorris cliflereltialrlt' a,t a poirrt .r'- rr. it is 'I'he t'<.irrtinuousat that 1.loirrt. convcrst'is falst'. i.e. c'ontintritvrkrcs not iurpll'cliffert'ntiabilitr.. 15 LorrtrlLirr<'aritr- arr<1 Litrcal Approxittratiorr 'l'iie liriear trpproxitnzrtiottof ./(.r')rrear.t'-.t0 is giverrlx'4:./(.,'e) *.1'(.l'1)(.r' .re). Tir estiuratc the slope of a gralrh at a poirrt rha,n a trrngerrt lirx-'to tltc graph at tliat point. Arrother rva\. is (lx' using u grtrphit s cak'nla,tor') to "zoonr in" aroLtn<lthe point itt cluestiorr urrtil the glaph "kroks'' straight.'fhis rrretliocl alnrost ahva'"s \il)r'ks. If u'c' "zot.rtttin" att<l ther glaph Lr,rks stlaiglrt at a point. sa)'.r': o. then the funr:tiorr is loca,ll)'lincar at that point. flre graph of u : ].r:l has a sharp (:olner' .rt :f :0. This col'll€rrc'arlllot lre stlrot-rtheclout lte "zc.ronringin" r'epeatecllv.Consecluetrtll'. the clerivative of l.r' cioes not exist at .r' : 0. henc'e.is not locallr' Iinear at .r' : 0. Formulas and Theorems 221 1Li. CourlraringRatcs of C'hatrgc Tlrt't'xpotretrti:rl func'tir)u!: c' gt'<lu'sverv lapirlh.AS.r'-+ tc u,h.ilethe fttgarithmic,fulr.tion l/ .. lrr.r' glo\\'s vt'r'r' skx.r,i-u'a.s .r' -) )c. Erpotrerttial frruc'tiorrslike u -. 2' rtr !/ : r,,'llr.()\\-ntol.er:rpiclly as.r +:r tharr an), positive '1.'ht'fitttt'tiott <if .r. - -+ l)()\\'('1 i/ hr.r' gr'o\\'ssl<lu'er as .t x tltiil a1\r lotx,orrstarrt lt1;lvrr<1niai. \\i' sar'. that as .r'-+ )c: l(r\ lt|') l. It.tt gl')\\':l;r-1 1,1. llrirrr ,/i,r I il lirrr - \ ,r'il lirrr {t. r .r z/{,r') .r .\.l(.r') fi.l (r') gltxls fhster thatr a(.r') as.r'-+ )c. therr q(,r') gr'owsslolr,tr tlu.rn.l'(.r.)AS.r. + rc. '19 2. ./(.r) arr<lr7(.r') grou, at the sarnt' ratt,as .r' + r if lir,r L (tr is firrite ancl ,. ,\ q(.r,) l0 rrouzt'r'o). Fol t'xanrlllt'. 1. r' gtrxls l;rstcr tlrarr.r.:iils.r, + rc sirrr.r,lirrr { -. :r, .t ' '2. .r'l gr',,1'slirstcl tlrarr hr.r' :rs .r.
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