Calculus 2 Cheat Sheet with Formulas and Theorems, Cheat Sheet of Calculus

Download Calculus 2 Cheat Sheet with Formulas and Theorems and more Cheat Sheet Calculus in PDF only on Docsity! Formulas and Theorems for Reference I. Tbigonometric Formulas l . s i n 2 d + c , c i s 2 d : 1 sec2 d l * c o t 2 0 : < : s c : 2 0 I+ . s i n ( - d ) : - s i t t 0 t , r s ( - / / ) = t r 1 s l / : - t a l l H 7 . 8 . s i n ( A * B ) : s i t r A c o s B * s i l B c o s A : siri A cos B - siu B <:os ,; l 9. cos(A + B) - cos,4 cos B - s iu A sir i B 10. cos(A - B) : cos A cos B + si l r A sirr B 2 sirr d t:os d 12. < 'os20 - coS2 ( i - s iu2 0 : 2 < ' o s 2 o - I - 1 - 2 s i n 2 0 1 1 . 15 . 13. tan d : 14. <:ol 0 : I < . r f t 0 I t a t t H : s i t t d (:os t/ sirr d 1 (:OS I/ 1 ri" 6i -e l - 01 16. csc d - / F I t lr ( . c o s [ ̂ \ l 18 . : C O S A 215 216 Formulas and Theorems II. Differentiation Formulas ! ( r " ) - t r r : " -1 Q,:I' ] t ra- fg '+gf ' - gJ ' - , f g '- ,l' , I , i ; . [ t y t . r t ) - l ' ' ( t t ( . r ) )9 ' ( . , ' ) d , \ .7, (sttt rrJ .* ('oq I' t J , \ . dr. l( 'os ./ J stl l lr { 1a,,,t, :r) - "11'2 ., 'o . t 1(<,ot .r ') - (, .(,2 .r ' Q : T rl , ,7, (sc'c:.r 'J : sPl' .r tall 11 d , - (<:s<t .r,; - (ls(] .] '(rot;.r , tfr("') -. '' ,1 f r (u" ) - o , ' l t rc , l , , 1 ' t l l l r i - ( l . t ' . f d , ^ I - i A l ' C S l l L l ' l - - : t ! . r ' J1 - rz 1(Arcsi' r) : oT * (i) I l 1 2 Formulas and Theorems 2I9 4. Horizontal ancl \rt'rtir:al As)'rnptotes 1 . A I i n e g - b i s n l u r r i z o n t a l a s v n i p t o t t ' < - r f t h e g r a p h o f q : . / ( . r ' ) i f e i t h e r l i r r r l ( . r ' ; = l ; ,,r .Itlt_ .f (r) : b 2. A l i r ie . t - e is a vcr t i< 'a l as) ' r r rptotc of t l ie graph of t t - . f ( . r ) i f e i t i re l . , . h r , . l ( . , , ) = * r c u r . , \ ) . / ( . r ' ) - +x . 5. Avcragc t r r r r l I r rs tarr t i l l l ( -o l ls I la t< ' o f ( ' l rar rgt ' 1 . Av t ' r ag t 'Ra tc o f ( ' l r a t rgc : I f ( . r ' 9 . y r r ) a t t r i ( . r ' l . q l ) i r l e l r o i t r t s o r r t he g la i r l t < f t q - . l ' ( t ) . t l ter t t l te a,vel i rg() r i t te of c 'harrge of i l u- i th rerspect to . r ' ovc l t l rc i t r tc l r -a l l r '11. . r t ; is l!_r1'_l!,,) lr !1, ly . l ' 1 . l ' 9 . r ' l , r ' ( ) l . r ' 2 . I t t s ta t r t n r i t ' o r r s Ra tc o1 (1 - l ' , l t r g , ' , I 1 ( , r ' 1y . . r / 9 ) i s a l r o i r r t o r r t he g ra l r l r o I r l , - , . l ' ( . r ) . t i u r r r t he i t r s tau tA r reoL l s ra te o f ch i r r i g t , o f i 7 n ' i t h r t , sp t , r ' t o . r ' a t , r ' 11 i s . f ' ' ( . r ' 1 ; ) . 6. Dcfirrit iorr of t, lrc l)r.rir-ativt ' .f' (.,) -lll lEP,r' t' (,,) 11,1, !y)--ll:'J Tlr t ' la , t t< ' r c lc f i r r i t io t r o l t l r t ' <k ' t i r ' ; r t iv t . is t l r t ' i r rs tarr t i r l r ( ' ( )us r i r t t , o f charrgt ' o f ' . l ( . r ) u- i t l r resltec:t to .t at .r -. (t. Georr le t r i t 'a l i r ' . th t r < ler i r ' : r t iveo1 a f i t t l t ' t i9 l t a t a l r , i l t t is t l r t 's l '1re, f t1e ' ta t rg< ' t t t l i t r t ' t , tho graph of the f i rnc ' t ion at t l ta t l io i r r t . 7 . ' fhc Nrrr r r l rcr ( ' : ls a l i r r r i t 2 . l i n i ( 1 + r r ) ; ( n - \ ) 8. Roller's Theorerrr I f . l ' is c ' t -rnt i tuu.r t ts on ln.0] arrr l c i i f f 'e lent iabl t 'on (a.b) srr t 'h that. l ' ( rr) . . , l ' (1,) . tht 'n thcle' is at le irst ot te t tut t tber c i t r the opetr intelval (o.b) srrc 'h that. l / ( r ' ) - 0. 9. Nlcan Valuc Thcorcrrr I f / is cotr t i tnror ts ot t ln . l i l aucl c l i f fe lent iable on (o. f ) . then there is at 1t :ast out ' nurr i l rer l / 1 . \ I t ^ r l i t i ( n .b ; . t t t l r t l t ; t t ' / " ' t - J ) l ! l - - f ' 1 , I t t - t I - (1. l i ' r ( r + 1 ) " n + + a \ f l / 1i) 220 Formulas and Theorems Extreme - Vaiue Tlieorem If / is cont irmous on a closecl interval lo. l . , ] . then./( . r) has both a tnaxinrum aurl a min i rnum on la .b ] . 11. To f i r id the rnaximrrrn and nr irr inuru values of a furrc ' t i<)\ t t =, /( . r ' ) . loc'ate 1. the point(s) r,r 'hclc .f '(.r) c'harrges sign. To firrri the c'atrcliclates first f incl lvhcre ,f '(.r:) - 0 or is infinite rlr cltters trot t:xist. 2 . thc t : t r r l po i t t ts . i f : r tn ' . or t t l t t ' r lo t t ta i t r < l f , / ( . r ' ) . Corrrpalc' thc frurctiorr va,lues at trl l of thcsc points l ir f irrrl the tnaxiruuuls an(l ntirt itttttttts. Let . / l ic 'c l i f fc lcnt ia l r i t ' f i r r r r <.1 '< 1. , t t t t< l tor t t in t rot rs for r r { . r <. l t . l . I f , f ' ' ( . r ) > 0 f o r ( ' v ( ' l ' \ ' . r ' i r r ( r r . L ) . t he r r . f i s i t r c t ' t ' as ing o r r f r r . 1 l ] . 2 . I f . / ' ( . r ' ) { 0 f o r eve l v . r ' i r r ( o .L ) . t h t ' t t . f i s c l t ' t ' r t ' as r t rg o r r [ 4 .1 l ] . Srippr-,se th:rt .f '"(;r) t'xists ort tlte itrtelva,l (rr. lr). 1 . I f , f " ( t ' ) ) 0 i r r (a .b ) . t l r cn . f i s < 'o r rc r ,ve upu, r r r '< l i r r (a . / r ) . ' ) . I f . f "( . r) { 0 i rr (rr .L). t l rerr . f is corrc ' t r ,ve ( lo$: l rwfr lc l i r r (rr . / r) . To lot ' t r te the points of i r r fkrc ' t i r . rr t t f i t t - .1 '( . r ' ) . f i rx l the proi tr ts r 'vhere . l ' " (r ' ) - () or u ' l t t ' r ' t : . f "( . r ' ) fai ls to cxist . l ' i rest, 'are the orrh'r 'ucl i r l ' r1, ' t ; ly l lere . f ( . r ' ) rnar. hal ' t 'a poirr t of i r i l lect i txt . ' I l ten test t l rese points to ur irkc sure tha,t , l ' " ( . , ) . - 0 on ont 's i t l t 'arr t l , f "( . r) > 0 <.r t t t lu 'other ' . 1.1 Diffcrerrtialrrlitv irnplies r'ontiuuitt': If a frrnr:tiorr is cliflereltialrlt ' a,t a poirrt .r'- rr. it is t'<.irrtinuous at that 1.loirrt. 'I 'he convcrst' is falst'. i.e. c'ontintritv rkrcs not iurpll 'cliffert'ntiabilitr.. 15 Lorr t r l L i r r< 'ar i t r - arr<1 L i t rca l Approxi t t rat ior r ' l ' i i e l i r i ea r t r pp rox i t nz r t i o t t o f . / ( . r ' ) r r ea r . t ' - . t 0 i s g i ve r r l x ' 4 : . / ( . , ' e ) * . 1 ' ( . l ' 1 ) ( . r ' . r e ) . Tir estiuratc the slope of a gralrh at a poirrt rha,n a trrngerrt l irx-'to tltc graph at t l iat point. Arrother rva\. is (lx' using u grtrphit s cak'nla,tor') to "zoonr in" aroLtn<l the point itt cluestiorr urrti l the glaph "kroks' ' straight. 'fhis rrretl iocl alnrost ahva'"s \i l)r 'ks. If u'c' "zot.rttt in" att<l ther g laph Lr , rks st la ig l r t a t a point . sa) ' . r ' : o . then the funr : t ior r is loca, l l ) ' l incar at that point . f lre graph of u : ].r: l has a sharp (:olner' .rt :f :0. This col' l l€rr c'arl l lot lre stlrot-rthecl out lte "zc.ronring in" r 'epeatecllv. Consecluetrtl l ' . the clerivative of l.r ' cioes not exist at .r ' : 0. henc'e. is not locallr ' I inear at .r ' : 0. 12 l ' ) _ t , ) . l Formulas and Theorems 221 Tlr t ' t 'xpotret r t i : r l func ' t i r )u ! : c ' g t '< lu 's verv lapi r lh .AS.r ' -+ tc u,h. i le the f t tgar i thmic, fu l r . t ion l/ .. lrr.r ' glo\\ 's vt'r 'r ' skx.r,i-u' a.s .r ' -) )c. Erpotrer t t ia l f r ruc ' t ior rs l ike u - . 2 ' r t r ! / : r , , ' l l r . ( ) \ \ -nto l .e r : rp ic l ly as. r +: r tharr an) , posi t ive l)()\\ '( '1 <if .r. '1. 'ht'f i tttt ' t iott i/ - hr.r ' gr'o\\ 's sl<lu'er as .t -+ x tlt i i l a1\r lotx,orrstarrt lt1;lvrr<1niai. \ \ i ' sar ' . that as . r ' -+ )c : l . I t . t t g l ' ) \ \ ' : l ; r - 1 , 1 . l l r i r r r , / i , r I i l l i r r r l ( r \ - \ , r ' i l l i r r r l t | ' ) { t . r . r z / { , r ' ) . r . \ . l ( . r ' ) f i . l ( r ' ) g l tx ls fhster thatr a( . r ' ) as. r ' -+ )c . therr q( , r ' ) gr 'ows s lo l r , t r t lu . rn. l ' ( . r . ) AS.r . + rc . 2. . / ( . r ) arr< l r7( . r ' ) grou, at the sarnt ' ra t t ,as . r ' + r i f l i r , r '19 L l0 ( t r is f i r r i te ancl, . , \ q ( . r , ) rrouzt'r 'o ). Fol t 'xanr l l l t ' . 1 . r ' g t r x l s l ; r s t c r t l r a r r . r . : i l s . r , + r c s i r r r . r , l i r r r { - . : r , . t ' '2 . . r ' l gr ' , ,1 's l i rs tc l t l rarr hr . r ' : rs . r . : rc s i r r<.e 1 i , , , - , '1 x 3 . . r ' : + 2 . r ' g l ( ) \ \ ' s i r t t l l , s i r l r r t ' r ' r r t r ' , r s . , , 1 as . r . ) ' r 2 l 2>c sirr<.r' ,]11 ,i{ I T i r f i r l < l so t t t t ' o f t he ' s t ' l i t r r i t s i r s , r ' , \ . \ ' ( ) l l n r i r v l r s ( ' t he g raph ing ta l r . r r l a t . , r ' . \ I ake su .c , t l a t a l r a l ) l ) l ( ) l ) l ia tc r . i t 'u- i r rg r . l - i t r r lor i - is r rscr l . 17 . I r r r - t ' r 'sc Frr ru ' t ior rs 1Li. Courlrar ing Ratcs of C'hatrgc i . I f . / l r r r l 17 i r l t ' tu ,o f r r r r<. t ious r r<.h that . l ' (q( . r . ) ) - . r for e- , \ ( )1. \ . .1 , in t iu , r lor r ra i r r o l q . arr tL. q( .1 ' ( . r ' ) ) . r ' . l i r r i r r thc ' r lo l ra i r r o f . f . therr . . f ' ar rd 17 are i r rve ls t ' f iur r . t ions t i l e i r ch o t l r c r . '2 . A f t l r r r '1 iorr . f h t ls r t t t i t rv t ' rsr ' l i r t t t t iou i f ar r r l onh. i f r io l ror izorr ta l l iue i r r tcrserr , ts i ts g ra l r l r u ro l t ' t l r i r r r o r r< ' ( r . 3. If . l is t ' i t lrt ' t i ttt t 'ei lsi lg or' <it ' t reasirrg in arr intt:r 'val. tf ien f ' |as a1 i1.,r 'r.se fi lrc:t i, ' or . t ' t thr r t i r r t t ' r ' t 'a l . l . h I is t l i l f i ' r t ' r r t ia ] r l t ' a t t 'v t ' t ' r - l ro i r r t or i ar r i r r terval I . ar rc l , f ' ( . , t ) I0 orr I . t1e1 ! l ' - l r ( . , I is t l i fTt ' r< 'ut i t r l r l t ' a t everr ' l ro int of the in ter ior of the in terval l ' ( I ) arr r l , t ' l l l . r I ) l | ' r . t t ' Formulas and Theorems 24. Y"t".lty, Sp..a, "t 1. The vclocity of an object tells how fast it is going and in which direction. Velocity is an instantaneous rate of change. 2. The spcecl of an obiect is the absolute value of the velocity, lr(t)I. It tells how fast it is going disregarding its direction. The speecl of a particle irrcrcascs (speeds up) when the velocity and acceleration have ther sarrre signs. The speed clecreascs (slows down) when the velocity and acceleration have opposite signs. 3. The acr:cier:rtion is thc irrstantarreous rate of change of velocity it is the derivative c-rf the veloc:ity that is. o(l) : r"(t). Negative acceleration (deceleration) means 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 displacernent of a rnoving objec:t and I is time, then: i) veloc: i tY : u(r) : t r ( t \ : # i i ) ac 'crelerat ion : ( t ) : . " ' ( t ) : r ' / ( / ) - #. : # i i i ) i ' ( / ) [ n ( t 1 , t t iv) . r ( t ) - [ , ,31 a, Notc: T[e av('ragc velclcity of a partir:le over the tirne interval frorn ts to another time f. is Average vel;c' itv: T#*frH#: "(r] -; ' i tol. where s(t) is the p.sit ion of the partic:le at tinre t. 25. The avetage value of /(r) on [a. ir] is f (r) d:r . Arca Bctwtxrri Ctrrvt,s If ./ ancl g are continuous funcrtions such that /(:r) 2 s@) on [a,b], then the area between , . b I I l re c r r rves is / l / ( " , I - q ( r l ) d r . J a +,,,, 1,,' 26 Formulas and Theorems 225 27. 28 Volume of Soiids of R.evolution Let / be nonnegative and continuous on [a,. b]. and let R be the region bounded above by g : / ( r " ) . be low by the r -ax is , and on the s ides by the l ines r : : n and r :b . When this region .R is revolved about tire .r'-axis. it gerrerates a solid (having circular f o crross sec'tions) u'hose volume V - | {j '(., ' l)2 ,1., . / t t Volunrcs of Soli<ls with Knowrr Cross Scctions l . of area A(:r:). taken pt'r ' l ierrcli<'ular tcl the r-zrxis. d r . A(37) taken perpt ' rrr i icrr lar to the 37-axis, 29. Solvirrg Differential Equations: Graphically ancl Nurnerrir.all.l ' Skrpc Fieicls Af ever'1' poirrt (.r. r7) a differetrtial ecluatiorr of the folrrr # - f t, .i/) gives the slope of tht' nernber of the farnily of solutit.rns that c:onta,ins that poirrt. A slope fielcl is a, gra,lrhictrl represent:rtiotr of this family of curves. At eac:h pt-rirrt irr the plarre. a short s()gnlent is rlrau'n "vhose slope is eclual to the value of the clerivativer at that poirrt. I 'hese scgnrerrts are taugcnt to the sohrtion's graph at the poirrt. The slope fielcl allows you to sketc:h the graph of ther solution cul've even though you rlo rrot have its ec|ration. This is clc-rne by starting at arry point (usuallv the point given bv the initial c'ondititin). and moving fron one poirrt to the next in the direc'tion irrdicnted by the segrncnts of the slope fielcl. Somc t'trlc'ulators havtt built in operations fbr drawing slope fields; fcir calculators rvithorrt tiris feature tlrere are l)rograms available fbr drawing thern. 30. Soiving Diffelential Equations b)' Separatirrg the Variables There are lnAny techniclues for solving differential equations. Any differential equatir_rn vou may be asked to solve ott the AB Calculus Exam can be solved by separating the variables. R,ewrite the equatioll as an erluivalent equation with all the r and dr terrns on otle side arxl all the q and d37 terrns ou the c-rther. Antidifferentiate both sides to obtain an e(luation without dr or du, but with orte c'onstant of inteqration. Use the initial condition to evahrate this constant. of area r h . z . Fol cross sections ',llttttt ' : .[rr" ^rr, Fbr <'ross se<rtions v.ltrttc' - .[," ^r,,