<<

and 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

,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

Tire liniit lirrr l(r ) exislsif anclorrh'il iroth corresporrciirrgone-si

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

',.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 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

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'

Georrletrit'alir'.thtr

'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 (s) r,r'hclc .f'(.r) c'harrges . 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

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. 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'

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 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

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'

Erpotrerttial frruc'tiorrslike u -. 2' rtr !/ : r,,'llr.()\\-ntol.er:rpiclly as.r +:r tharr an), positive '1.'ht'fitttt'tiott

\\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

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. : rc sirr<.e1i,,,- x ,'1

'r2 l2 3. .r': + 2.r'gl()\\'s irt tll, sirlrrt'r'rrtr' ,rs .,,1as .r. ) >c sirr<.r' I ,]11 ,i{

Tir firl

17. Irrr-t'r'scFrrru't iorrs

i. If ./ lrrrl 17irlt' tu,o frrrr<.tioussrr<.h that .l'(q(.r.))- .r for e-,\()1.\..1,in tiu, rlorrrairrol q. arrtL.q(.1'(.r')) .r'.lirr irr thc'rlolrairr of .f . therr. .f' arrd 17are irrvelst'fiurr.tions til eirchotlrcr.

'2. A ftlrrr'1iorr.f htls rttt itrvt'rsr'lirttttiou if arrrl onh. if rio lrorizorrtal liue irrtcrserr,tsits gralrlr urolt' tlrirrrorr<'(r.

3. If is t'itlrt't ittt .l t'eilsilg or'

l. h is tlilfi'rt'rrtia]rlt' I at t'vt't'r- lroirrt ori arr irrterval I. arrcl ,f'(.,t)I0 orr I. t1e1 r(., '- I is tlifTt'r<'utitrlrlt' !l l at everr'lroint of the interior of the l'(I) arrrl l ,t'll l.rI ) ' ' | r.t t 222 Formulas and Theorems

'' I x Prr'_-l rrrlt -'-_ t ils r,1 r

.1'' 1' I'htr t'xllorlt'utial futtctit.rti !/ - t'' is the irlverse of t7:111

2. I'lrt'clornaitt is thc rlf all rt'al rltlrlll)el's.-)c <.lr < DC.

'l'lu'ritngt'is 3. tlrt'set of all llositive nttntllels.! > 0.

,l ' -1. ') , . -l(,(Lt'

5. ll .,r' is <'ontirtlrorrs.inc'r'errsirrg. attd (on('irve rtlt fbl all .r:. -'0. tt. iit]'_,' ., i x atrtl l1tlt_r' , ,.

r T. ,ltt .r..firr' .r. -> 0l lrr(r') -.r' firr all .r'.

19. Prolrt'r't it's o[ ]tt.r'

'l'lrc 1. rkrrrririuo1 r7 lrr,r'is tht: settof all ltositivc trutttliers,.r'> 0.

'2. '['lrt'rirrrgt'of i7 . hr.r' is tlie sct of all rt'al lrtrttt]rers. x < l/ < :r'

:1. r7 . lrr.r' is <.orrlirrrrorts.itr<'r'e'asirrg. urrrl corrcavtrclou,tt cverYrvltertl r-rttits tlclrltrin.

1. lrr(rr|)- ltrrr I ltr1i.

1. ltrl,tfl, I ltr,r lrr/,

(;. 11111l ,.. 1'11v11

7. i7 hr.r '- 0 iI 0 .: .r'.- I arrrllrr.r' > 0 if .r > 1.

E. ltr.r'- *:r trrtrl ltt.r'- -)c' ,lllt. ,.lt]li

1).l.g,,.r' il;

20. 1-tlpczoitlirl Ilrrlt'

If ir f\urt.tiorr.fis c'outiuuorrsorr tlrt't'krseclinte't'val [4.b] where fo.b] has ]reenpartitioned irrtrr l sttlrirttt,r'r'trlsI.r'1..r'rj. l,r.i2].....[.r:,, r..t:,,].ent:lt of (b-a)ln. then rlt r . I f t,) r/.r'= -;;[./(,0) +'2.f(.rr) + 2/(.rz)+ ... + 2J(.r',,r) + ./(.r")] .t,,

Tlrt. T'ralrezoiclal Rrrlt' is tlre avelage of the left-hancl and riglrt-hancl R,iemann sulns. Formulas and Theorems 223

21. Propcrties of tlic Dcfinitc Ilttcgral

Let ,/(.r) and r7(,r) be c,cintirruuousorr la. ll].

fb rt, 1. (r) rl,t': c,l,,.rrr,r1.r. r'is a uor.zcroc,onstant. J,,,,'.f

ft 2' f ('') rl'rr- 0 .1,,

I'tt |t' :l - lt,t,t, .1,,,,')'ltr .f,,

[t' r' lt' +. r,r.her.t' is continrrouson arr irrter.val .1,,,r,),lr- f,,.1t.,)n.,*,1,.f'(.r)rl.r.. ,f r'orttailrittgtlte trutnltet'srr. 1r.arrrl r'. r'egarrllt'ssol tlrt'or'

5. If l(.r') is trn otlrl fiurr.tion.th,',r / .l(r') rl.t.- (l .l ,,

",,.1'(.t.) tj. If ./(.r) is arr even fiul.tion. tlruu tlr I .l{.,) ,t,,. .f ,f,,'

7 II .l(.r)] 0 on lrr.1r]. rherr l'" ,,(,,, r/.r, > 0 .t,,

8. If .q(.r')Z.f(r),n lo.bl.rl*,u 7 ,t., [,,",,{.r),1,r,[,,".1{.,.1

22. Dcfiriti.' .f D<'firritt'hrt<'gr:rl trs tli. Li'rit ,f u Srrrrr

Sttlllrtlst' that a firtttrtiott is

'flrlollrrr 2:1. Funrlarncntal ,1 ('ak.uhrs

7b ,l.t I tt.,) l:iltt 1-'trit. n"lu,r.t,F,(.r) : ,f(..r') .t,,

o,+..f',,' ,,,,,,, ri, ,',',rj, rtt:,f(q(t.))g,(.r). f ,"''',,rr, Formulas and Theorems

24. Sp..a, Y"t".lty, "t 1. The vclocity of an object tells how fast it is going and in which direction. is an instantaneous of change.

2. The spceclof an obiect is the of the velocity, lr(t)I. It tells how fast it is going disregardingits direction.

The speeclof a particle irrcrcascs( up) when the velocity and have thersarrre signs. The clecreascs(slows down) when the velocity and acceleration have opposite signs.

3. The acr:cier:rtionis thc irrstantarreousrate of change of velocity it is the c-rfthe veloc:ity that is. o(l) : r"(t). Negative acceleration(deceleration) that t[e vgloc:ity is dec:r'easirrg.Tlie acceleration gives the rate at which the velocity is crharrging.

Therefore,if .r is the displacernentof a rnoving objec:tand I is , then:

i) veloc:itY: u(r) : tr (t\ : #

: o(t): ."'(t): r'/(/)- : ii) ac'creleration #. #

iii)i'(/) [n(t1,tt

iv) .r(t)- [ ,,31a,

Notc: T[e av('ragc velclcity of a partir:le over the tirne interval frorn ts to another time f. is "(r] -;'itol. wheres(t) is the p.sitionof Averagevel;c'itv: T#*frH#: the partic:leat tinre t.

25. The avetage value of /(r) on [a. ir] is +,,,,1,,'f (r) d:r.

26 Arca BctwtxrriCtrrvt,s

If ./ anclg are continuousfuncrtions such that /(:r) 2 [email protected]) on [a,b], then the between ,.b I - I lrecrrrves is / l/ (",I q(rl) dr . Ja Formulas and Theorems 225

27. of Soiids of R.evolution

Let / be nonnegativeand continuous on [a,.b]. and let R be the bounded above by g: /(r"). belowby the r-axis, and on the sidesby the linesr:: n and r:b.

When this region .R is revolved about tire .r'-axis.it gerreratesa solid (having circular fo - crrosssec'tions) u'hose volume V | {j'(.,'l)2 ,1.,. /tt

28 Volunrcsof Soli

l. Fol cross sectionsof area A(:r:). taken pt'r'lierrcli<'ulartcl the r-zrxis.

',llttttt' : ^rr, dr. .[rr"

z. Fbr <'rossse

v.ltrttc' - ^r,, rh. .[,"

29. SolvirrgDifferential : Graphically ancl Nurnerrir.all.l'

Skrpc Fieicls

- Af ever'1'poirrt (.r.r7) a differetrtial ecluatiorrof the folrrr # f t, .i/) gives the slope of tht' nernber of the farnily of solutit.rnsthat c:onta,insthat poirrt. A slope fielcl is a, gra,lrhictrl represent:rtiotrof this family of . At eac:hpt-rirrt irr the plarre.a short s()gnlentis rlrau'n slope is eclualto the value of the clerivativerat that poirrt. I'hese scgnrerrtsare taugcnt "vhose to the sohrtion'sgraph at the poirrt.

The slope fielcl allows you to sketc:hthe graph of ther solution cul've even though you rlo rrot have its ec|ration.This is clc-rneby starting at arry point (usuallv the point given bv the initial c'ondititin).and moving fron one poirrt to the next in the direc'tionirrdicnted by the segrncnts of the slope fielcl.

Somc t'trlc'ulatorshavtt built in operations fbr drawing slope fields; fcir calculatorsrvithorrt tiris feature tlrere are l)rograms available fbr drawing thern.

30. Soiving Diffelential Equations b)' Separatirrgthe Variables

There are lnAny technicluesfor solving differential equations. Any differential equatir_rnvou may be asked to solve ott the AB Exam can be solved by separating the variables. R,ewrite the equatioll as an erluivalent with all the r and dr terrns on otle side arxl all the q and d37terrns ou the c-rther.Antidifferentiate both sides to obtain an e(luation without dr or du, but with orte c'onstantof inteqration. Use the initial condition to evahrate this .