Differentiation Formulas

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1 AP CALCULUS BC Fil Notes Trigoometric Formuls si θ + cos θ = + t θ = sec θ 3 + cot θ = csc θ 4 si( θ ) = siθ 5 cos( θ ) = cosθ 6 t( θ ) = tθ 7 si( A + B) = si Acos B + si B cos A 8 si( A B) = si Acos B si B cos A 9 cos( A + B) = cos Acos B si Asi B cos( A B) = cos Acos B + si Asi B si θ = siθ cosθ cos θ = cos θ si θ = cos θ = si siθ 3 t θ = = cosθ cotθ cosθ 4 cot θ = = siθ tθ 5 sec θ = cosθ 6 csc θ = siθ π 7 cos( θ ) = siθ π 8 si( θ ) = cosθ θ Differetitio Formuls ( ) = ( fg) = fg + gf 3 f gf fg ( ) = g g 4 f ( g( )) = f ( g( )) g ( ) 5 (si ) = cos 6 (cos ) = si 7 (t ) = sec 8 (cot ) = csc 9 (sec ) = sec t (csc ) = csc cot ( e ) = e ( ) = l 3 (l ) = 4 ( Arcsi ) = 5 ( Arc t ) = + 6 ( Arcsec ) =

2 7 y y u = Chi Rule

3 = + C + = + C, + 3 = l + C 4 e = e + C 5 = + C l 6 l = l + C 7 si = cos + C 8 cos = si + C 9 t = l sec + C or l cos + C cot = l si + C sec = l sec + t + C Itegrtio Formuls csc = l csc cot + C = l csc + cot + C 3 sec = t + C 4 sec t = sec + C 5 csc = cot + C 6 csc cot = csc + C 7 t = t + C 8 = Arc t + C + 9 = Arcsi + C = Arcsec + C = Arc cos + C

4 Formuls Theorems Limits Cotiuity: A fuctio y = f () is cotiuous t = if i) f() eists ii) lim f ( ) eists iii) lim = f ( ) Otherwise, f is iscotiuous t = The limit lim f ( ) eists if oly if oth correspoig oe-sie limits eist re equl tht is, lim f ( ) = L lim f ( ) = L = lim f ( ) + Eve O Fuctios A fuctio y = f () is eve if f ( ) = f ( ) for every i the fuctio s omi Every eve fuctio is symmetric out the y-is A fuctio y = f () is o if f ( ) = f ( ) for every i the fuctio s omi Every o fuctio is symmetric out the origi 3 Perioicity A fuctio f () is perioic with perio p ( p > ) if f ( + p) = f ( ) for every vlue of Note: The perio of the fuctio y = Asi( B + C) or y = Acos( B + C) is The mplitue is A The perio of y = t is π 4 Itermeite-Vlue Theorem A fuctio y = f () tht is cotiuous o close itervl [, ] tkes o every vlue etwee f ( ) f ( ) Note: If f is cotiuous o [, ] f () f () iffer i sig, the the equtio f ( ) = hs t lest oe solutio i the ope itervl (, ) 5 Limits of Rtiol Fuctios s ± i) f ( ) lim = if the egree of ± g( ) f ( ) < the egree of g( ) Emple: lim = f ( ) ii) lim is ifiite if the egrees of f ( ) > the egree of g( ) ± g ( ) 3 + Emple: lim = 8 f ( ) iii) lim is fiite if the egree of ± g ( ) f ( ) = the egree of g( ) π B

5 Emple: 3 + lim 5 = 5

6 Horizotl Verticl Asymptotes A lie y = is horizotl symptote of the grph y = f () if either lim f ( ) = or lim f ( ) = A lie = is verticl symptote of the grph y = f () if either lim f ( ) = ± or lim = ± Averge Istteous Rte of Chge i) Averge Rte of Chge: If (, y ) (, y ) re poits o the grph of y = f (), the the verge rte of chge of y with respect to over the itervl [, ] is f ( ) f ( ) = y y Δy = Δ ii) Istteous Rte of Chge: If (, y ) is poit o the grph of y = f (), the the istteous rte of chge of y with respect to t is f ( ) 7 Defiitio of Derivtive f ( + h) f ( ) f f ( ) f ( ) ( ) = lim of f ' ( ) = lim h h The ltter efiitio of the erivtive is the istteous rte of chge of f ( ) with respect to t = Geometriclly, the erivtive of fuctio t poit is the slope of the tget lie to the grph of the fuctio t tht poit 8 The Numer e s limit i) lim + = e + ii) lim + = e 9 Rolle s Theorem If f is cotiuous o [, ] ifferetile o (, ) such tht f ( ) = f ( ), the there is t lest oe umer c i the ope itervl (, ) such tht f ( c) = Me Vlue Theorem If f is cotiuous o [, ] ifferetile o (, ), the there is t lest oe f ( ) f ( ) umer c i (, ) such tht = f ( c) Etreme-Vlue Theorem If f is cotiuous o close itervl [, ], the f () hs oth mimum miimum o [,] To fi the mimum miimum vlues of fuctio y = f (), locte the poits where f () is zero or where f () fils to eist the e poits, if y, o the omi of f () Note: These re the oly cites for the vlue of where f () my hve mimum or miimum

7 3 Let f e ifferetile for < < cotiuous for, If f ( ) > for every i (, ), the f is icresig o [,] If f ( ) < for every i (, ), the is ecresig o, f [ ] 5 Suppose tht f () eists o the itervl (, ) If f ( ) > i (, ), the f is cocve upwr i (, ) If f ( ) < i (, ), the f is cocve owwr i (, ) To locte the poits of iflectio of y = f (), fi the poits where f ( ) = or where f () fils to eist These re the oly cites where f () my hve poit of iflectio The test these poits to mke sure tht f ( ) < o oe sie f ( ) > o the other 6 If fuctio is ifferetile t poit =, it is cotiuous t tht poit The coverse is flse, i other wors, cotiuity oes ot imply ifferetiility 6 Locl Lierity Lier Approimtios The lier pproimtio to f () er = is give y y = f ( ) + f ( )( ) for sufficietly close to To estimte the slope of grph t poit just rw tget lie to the grph t tht poit Aother wy is (y usig grphig clcultor) to zoom i rou the poit i questio util the grph looks stright This metho lmost lwys works If we zoom i the grph looks stright t poit, sy (, f ( ) ), the the fuctio is loclly lier t tht poit The grph of y = hs shrp corer t = This corer cot e smoothe out y zoomig i repetely Cosequetly, the erivtive of oes ot eist t =, hece, is ot loclly lier t = 7 Domice Compriso of Rtes of Chge Logrithm fuctios grow slower th y power fuctio ( ) Amog power fuctios, those with higher powers grow fster th those with lower powers All power fuctios grow slower th y epoetil fuctio (, > ) Amog epoetil fuctios, those with lrger ses grow fster th those with smller ses

8 We sy, tht s : f ( ) grows fster th ( ) If s f g if lim g g s f ( ) grows fster th ( ) f ( ) g( ) grow t the sme rte s if ozero) For emple, 3 e grows fster th s sice 4 grows fster th l s sice ( ) g( ) = or if lim = ( ) f ( ), the g( ) grows slower th f ( ) e lim = 3 4 lim = l 3 + grows t the sme rte s s sice ( ) ( ) f lim = L g + lim = (L is fiite To fi some of these limits s, you my use the grphig clcultor Mke sure tht pproprite viewig wiow is use 8 L Hôpitl s Rule f ( ) f ( ) If lim is of the form or, if lim g( ) g ( ) eists, the f ( ) f ( ) lim = lim g( ) g ( ) 9 Iverse fuctio If f g re two fuctios such tht f ( g( )) = for every i the omi of g g ( f ( )) = for every i the omi of f, the f g re iverse fuctios of ech other A fuctio f hs iverse if oly if o horizotl lie itersects its grph more th oce 3 If f is either icresig or ecresig i itervl, the f hs iverse 4 If f is ifferetile t every poit o itervl I, f ( ) o I, the g = f ( ) is ifferetile t every poit of the iterior of the itervl f (I) g ( f ( )) = f ( ) Properties of y = e The epoetil fuctio y = e is the iverse fuctio of y = l The omi is the set of ll rel umers, < < 3 The rge is the set of ll positive umers, y > 4 ( e ) = e 5 e e + = e 6 y = e is cotiuous, icresig, cocve up for ll

9 lim + e l = > ; l( e ) = 7 e = + lim e = 8, for for ll Properties of y = l The omi of y = l is the set of ll positive umers, > The rge of y = l is the set of ll rel umers, < y < 3 y = l is cotiuous icresig everywhere o its omi 4 l ( ) = l + l 5 l = l l 6 l r = r l 7 y = l < if < < 8 lim l = + lim l = + + l 9 log = l Trpezoil Rule If fuctio f is cotiuous o the close itervl [, ] where [,] hs ee prtitioe ito suitervls [, ], [, ], [ ],, ech legth, the f ( ) [ f ( ) + f ( ) + f ) ( ( )] ( + + f ) f +

10 3 Defiitio of Defiite Itegrl s the Limit of Sum Suppose tht fuctio f () is cotiuous o the close itervl [,] Divie the itervl ito equl suitervls, of legth Δ = Choose oe umer i ech suitervl, i other wors, i the first, i the seco,, i the k th,, k i the th The lim f ( ) Δ = f ( ) = F( ) F( ) k k = 3 Properties of the Defiite Itegrl Let f () g () e cotiuous o [, ] i) c f ( ) = c f ( ) for y costt c ii) f ( ) = iii) f() = f() c iv) f ( ) = f ( ) + f ( ), where f is cotiuous o itervl c cotiig the umers,, c v) If f () is o fuctio, the f ( ) = vi) If f () is eve fuctio, the f ( ) = f ( ) vii) If f ( ) o [, ], the f ( ) viii) If g( ) f ( ) o [, ], the g ( ) f ( ) 4 Fumetl Theorem of Clculus: f ( ) = F( ) F( ), where F ( ) = f ( ), or f ( ) = f ( ) 5 Seco Fumetl Theorem of Clculus: g ( ) (t) t = () f f or (t) t = () g' ( ) f f

11 6 Velocity, Spee, Accelertio The velocity of oject tells how fst it is goig i which irectio Velocity is istteous rte of chge The spee of oject is the solute vlue of the velocity, v () t It tells how fst it is goig isregrig its irectio The spee of prticle icreses (spees up) whe the velocity ccelertio hve the sme sigs The spee ecreses (slows ow) whe the velocity ccelertio hve opposite sigs 3 The ccelertio is the istteous rte of chge of velocity it is the erivtive of the velocity tht is, () t = v' ( t ) Negtive ccelertio (ecelertio) mes tht the velocity is ecresig The ccelertio gives the rte ot which the velocity is chgig Therefore, if is the isplcemet of movig oject t is time, the: = = t i) velocity = v () t ' () t v = = = = t t ii) ccelertio = () t '' () t v '() t iii) () t () t v = t iv) () t = v () t t Note: The verge velocity of prticle over the time itervl from t to other time Chge i positio s( t) s( t ) t, is Averge Velocity = =, where s( t ) is the positio of the prticle t time t 7 The verge vlue of () o Legth of time t t f [ ], is f ( ) 8 Are Betwee Curves If f g re cotiuous fuctios such tht f ( ) g( ) o [,], the re etwee the curves is [ f ( ) g( ) ] 9 Itegrtio By Prts If u = f () v = g() if f () g () re cotiuous, the u = uv v u Note: The gol of the proceure is to choose u v so tht v u is esier to solve th the origil prolem Suggestio:

12 Whe choosig u, rememer LIATE, where L is the logrithmic fuctio, I is iverse trigoometric fuctio, A is lgeric fuctio, T is trigoometric fuctio, E is the epoetil fuctio Just choose u s the first epressio i LIATE ( v will e the remiig prt of th e itegr) For emple, whe itegrtig l, choose u = l sice L comes first i LIATE, v = Whe itegrti g e, choose u =, sice is lgeric fuctio, A comes efore E i LIATE, v = e Oe more emple, whe itegrtig Arc t( ), let u = Arc t(), sice I comes efore A i LIATE, v =

13 3 Volume of Solis of Revolutio (rectgles rw perpeiculr to the is of revolutio) Let f e oegtive cotiuous o [, ], let R e the regio oue ove y y = f (), elow y the -is the sies y the lies = = Whe this regio R is revolve out the -is, it geertes soli (hvig circulr cross sectios) whose volume V = π [ f( ) ] Whe two fuctios re ivolve: V = π ( ro ri ) where r o is the istce etwee the is of revolutio the furthest sie of the she regio r i is the istce etwee the is of revolutio the erest sie of the she regio 3 Whe the rectgles re perpeiculr to the -is, the itegrl will e i terms of Whe the rectgles re perpeiculr to the y-is, the itegrl will e i terms of y 3 Volume of Solis with Kow Cross Sectios For cross sectios of re A(), tke perpeiculr to the -is, volume = A ( ) Volumes o the itervl [, ] where ( ) is the legth of sie of the sectio: Squre: ( ) Equilterl Trigle: V ( ) V = Semi-circle: V ( ) π = 8 = V = 4 Isosceles Right Trigle: V ( ) Isosceles Right Trigle: ( ) = 3 4 (whe = leg of trigle) (whe = hypoteuse of trigle) For cross sectios of re A(y), tke perpeiculr to the y-is, volume = A ( y) y 3c Shell Metho (rectgles rw prllel to the is of revolutio) Horizotl Ais of Revolutio: V = π p( y) h( y) y ( p is the istce etwee the c is of revolutio the ceter of rectgle)

14 Verticl Ais of Revolutio: V = π p( ) h( ) ( p is the istce etwee the is of revolutio the ceter of rectgle) 3 Solvig Differetil Equtios: Grphiclly Numericlly Slope Fiels At every poit (, y) y ifferetil equtio of the form ( = f, y ) gives the slope of the memer of the fmily of solutios tht cotis tht poit A slope fiel is grphicl represettio of this fmily of curves At ech poit i the ple, short segmet is rw whose slope is equl to the vlue of the erivtive t tht poit These segmets re tget to the solutio s grph t the poit The slope fiel llows you to sketch the grph of the solutio curve eve though you o ot hve its equtio This is oe y strtig t y poit (usully the poit give y the iitil coitio), movig from oe poit to the et i the irectio iicte y the segmets of the slope fiel Some clcultors hve uilt i opertios for rwig slope fiels; for clcultors without this feture there re progrms ville for rwig them Euler s Metho Euler s Metho is wy of pproimtig poits o the solutio of ifferetil y equtio (, y = f ) The clcultio uses the tget lie pproimtio to move from oe poit to the et Tht is, strtig with the give poit (,y ) the iitil coitio, the poit ( +Δ,y + f '(,y ) Δ) pproimtes ery poit o the solutio grph This proimtio my the e use s the strtig poit to clculte thir poit so o The ccurcy of the metho ecreses with lrge vlues of Δ The error icreses s ech successive poit is use to fi the et Clcultor progrms re ville for oig this clcultio 3 Logistics Rte is joitly proportiol to its size the ifferece etwee fie positive umer (L) its size y y = ky which yiels t L L y = through seprtio of vriles + Ce kt lim y = L ; L = crryig cpcity (Mimum); horizotl symptote t

15 rte) 3 y-coorite of iflectio poit is L, ie whe it is growig the fstest (or m 33 Defiitio of Arc Legth If the fuctio give y y = f () represets smooth curve o the itervl [, ], the the rc legth of f etwee is give y s = + [ f ( ) ] 34 Improper Itegrl f ( ) is improper itegrl if f ecomes ifiite t oe or more poits of the itervl of itegrtio, or oe or oth of the limits of itegrtio is ifiite, or 3 oth () () hol 35 Prmetric Form of the Derivtive If smooth curve C is give y the prmetric equtios = f ( ) y = g( t), the y y the slope of the curve C t (, y) is =, t t t y y y Note: The seco erivtive, = = t t 36 Arc Legth i Prmetric Form If smooth curve C is give y = f ( t) y = g( t) these fuctios hve cotiuous first erivtives with respect to t for t, if the poit P(, y) trces the curve ectly oce s t moves from t = to t =, the the legth of the y curve is give y s = + t = [ ] [ ] t t t f ( t ) + g ( t) spee = f (t) + g (t) [ ] [ ]

16 37 Polr Coorites Crtesi vs Po lr Coorites The polr coorites ( r, θ ) re relte to the Crtesi coorites (, y) s follows: = r cosθ y = r siθ y tθ = + y = r To fi the poits of itersectio of two polr curves, fi ( r, θ ) stisfyig the first equtio for which some poits ( r, θ + π ) or ( r, θ + π + π ) stisfy the seco equtio Check seprtely to see if the origi lies o oth curves, ie if r c e Sketch the curves 3 Are i Polr Coor ites: If f is cotiuous oegtive o the itervl [ α, β ], the the re of the regio oue y the grph of r = f (θ ) etwee the ril lies θ = α θ = β is give y β β A = [ f ( θ )] θ = r θ α α 4 Derivtive of Polr fuctio: Give r = f ( θ), to fi the erivtive, use prmetric equtios = rcosθ = f ( θ)cosθ y= rsiθ= f ( θ) siθ The 5 Arc Legth i Polr Form: y y θ f ( θ) cos θ+ f '( θ) si θ = = f ( θ ) si θ+ f '( θ ) cos θ θ β r s= r + θ α θ

17 38 Sequeces Series If sequece { } hs limit L, tht is, lim = L, the the sequece is si to coverge to L If there is o limit, the series iverges If the sequece { } coverges, the its limit is uique Keep i mi tht l lim = ; lim = ; lim = ; lim =! rise frequetly These limits re useful The hrmoic series = iverges; the geometric series = if r < iverges if r r r coverges to 3 The p-series = p coverges if p > iverges if p 4 Limit Compriso Test: Let e series of oegtive terms, = = lim = c > with for ll sufficietly lrge, suppose tht the two series either oth coverge or oth iverge The 5 Altertig Series: Let e series such tht = i) the series is ltertig ii) + for ll, iii) lim = The the series coverges Altertig Series Remier: The remier first eglecte term RN N + R N is less th (or equl to) the 6 The -th Term Test for Divergece: If lim, the the series iverges Note tht the coverse is flse, tht is, if =, the series my or my ot coverge lim

18 7 A series is coverges, ut oes ot coverge, the the series is coitiolly solutely coverget if the series coverges If coverget Keep i mi tht if coverges, the coverges = = 8 Compriso Test: If for ll sufficietly lrge, = the coverges If iverges, the iverges = = = 9 Itegrl Test: If f () is positive, cotiuous, ecresig fuctio o coverges, [, ) let = f () The the series will coverge if the improper itegrl = f ( ) coverges If the improper itegrl f ( ) iverges, the the ifiite series = iverges Rtio Test: Let e series with ozero terms i) If lim + <, the the series coverges solutely ii) If lim + >, the the series is iverget iii) If lim + =, the the test is icoclusive ( other test must e use) Power Series: A power series is series of the form c = c c c c + or = c ( ) = + ( ) + ( ) + + ( ) c c c c + i which the ceter = the coefficiets c, c, c,, c, re costts The set of ll umers for which the power series coverges is clle the itervl of covergece Tylor Series: Let f e fuctio with erivtives of ll orers throughout some itervle cotiig s iterior poit The the Tylor series geerte y f

19 t is ( k) ( ) f ( ) f ( ) ( ) = + + ( ) f ( ) k f ( ) f ( )( ) + + ( ) + k= k!!! The remiig terms fter the term cotiig the th erivtive c e epresse s remier to Tylor s Theorem: ( ) + = + + R R = t ( ) f () f ( ) f ( )( ) ( ) where ( ) ( ) f ( t) t! ( + ) f ( c)( ) + Lgrge s form of the remier: R =, where < c < ( + )! The series will coverge for ll vlues of for which the remier pproches zero s 3 Frequetly Use Series their Itervl of Covergece = =, < = e = =!! =!, < ( ) + si = + + ( ) + =, < 3! 5! ( + )! = ( + )! cos =! + 4 4! + ( ) ( ) + = ()! = ()!, <

BC Calculus Review Sheet

BC Calculus Review Sheet BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re