AP Calculus BC Formulas, Definitions, Concepts & Theorems to Know

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1 P Clls BC Formls, Deiitios, Coepts & Theorems to Kow Deiitio o e : solte Vle: i 0 i 0 e lim Deiitio o Derivtive: h lim h0 h ltertive orm o De o Derivtive: lim Deiitio o Cotiity: is otios t i oly i lim lim Dieretiility: tio is ot ieretile t i ) is ot otios t ) hs sp t 3) hs vertil tget t Eler s Metho: Use to pproimte vle o tio, give y y 0, 0 y Use y y0 ( 0) repetely. Itermeite Vle Theorem (IVT): I is otios o, k is y mer etwee the there is t lest oe k mer etwee sh tht Me Vle Theorem (MVT): I is otios o, ieretile o, the there eists mer i, sh tht. (Thik: The slope t is the sme s the slope rom to.) Trig Ietities to Kow: si si os os os si os os si os Deiitio o Deiite Itegrl: ( ) i lim i lso kow Riem Sms Let, Right, Mipoit, Trpezoil verge Vle o tio VE o, : verge Rte o Chge o is the slope: o, Crve Legth o o, L : Istteos Rte o Chge o respet to is. with Logisti Dieretil Eqtio: P kp L P t ; Pt L, Lkt e L P P0 0

2 g g g (hi rle) v v v (prot rle) v v v v (qotiet rle) (power rle) (erivtive o iverse) si os Derivtives os si t se se se t ot s s s ot l log l e e l rsi rt rse ros rot rs (power rle or itegrls) l C e e C C l l l C Itegrls os si C si os C se t C se t se C s ot C s ot s C t l os C se l se t C ot l si C s l s ot C C rsi rt C rse C Tehiqes o Itegrtio: -sstittio; Prtil Frtios; Completig the Sqre; Itegrtio By-Prts: v v v

3 Critil Nmer t 0 or i: is eie First Derivtive Test: Let e ritil mer. I I hges rom to t the hs reltive m o M/Mi, Covity, Iletio Poit solte Mim: The solte m o lose itervl, is. hges rom to t the hs reltive mi o Seo Derivtive Test: I 0 0 the hs reltive mi o I 0 0 the hs reltive m o..,, or reltive mimm.,, or reltive miimm. The solte mi o lose itervl is, Test or Covity: or ll i I, the the grph o I 0 is ove p o I. or ll i I, the the grph o I 0 is ove ow o I. Iletio Poit: tio hs iletio poit t hges sig t, i Fmetl Theorems First Fmetl Theorem o Clls: Seo Fmetl Theorem o Clls: g tt g g or (The mlte hge i rom to ) g tt g g h h h re Betwee Crves toprve ottomrve rightrve letrve y Volme Geerl Volme Forml re & Volme Ftios i the orm y or y Volme ( ), where re Volme ( y) y, where y re Volme Dis/Wsher Metho V R r V R y r y y Volme Cyliril Shell Metho V ris height V ris height y Volme y Cross-Setios -is: V y-is: V semiirle s 8 ; 3 s 4 y y RtIsos s (leg s s ) ; Eqil RtIsos s (hypot s s ) 4

4 Horizotl/Vertil Motio Vertil Motio Positio Ftio (w/grvity): Totl Diste Trvele over, (t): st 6t v0t s0 (m): st 4.9t v0t s0 Spee = vt v t s t Veloity Ftio: elertio Ftio: t vt s t Displemet (hge i positio) over, v t t s s Positio Vetor t, y t Veloity Vetor t, yt elertio Vetor t, yt y y Slope= t t ros, y rsi Slope o polr rve: Spee Ireses i vt Spee Dereses i vt v t t t hve sme sig t ieret sigs Motio log Crve (Prmetris & Vetors) Spee (or Mgite/Legth o Veloity Vetor) y t y y t Polr Crves v t t y t Spee Ireses i spee 0 t Diste Trvele (or Legth o Crve) re isie polr rve: y y rsi r os r os r si Covergee/Divergee o Series 0 Tests: th Term Test, Telesopig Series Test, Geometri Series Test, p-series Test, Itegrl Test, Diret Compriso Test, Limit Compriso Test, ltertig Series Test, Rtio Test, Root Test. ltertig Series Error Bo I series is ltertig i sig eresig i mgite, to zero, the error irst isregre term Series v t t t y t t r Lgrge Error Bo (k Tylor s Theorem) P R Power Series o 3 e...!! 3! e, si, os Cetere t 0! m z Tylor Series P 3!! !! is lle the th egree Tylor Series or etere t., si...! 3! 5! 7! 4 6 os...! 4! 6! e, si, os overge or ll rel -vles

5 Hyperoli Trig Ftios re ot prt o the rrilm or P Clls BC. They re se i Dieretil Eqtios other mth orses, s well s i some o the siees. Here is rsh-orse o hyperoli tios. Yo ll otie my similrities to si trig tios. Deiitios o Hyperoli Trig Ftios: e sih e Iverses o hyperoli trig tios: sih l D, osh l D, th l oth l e e osh sih th osh D, D :,, seh l D : (0,] sh l sh sih seh osh oth th Derivtives o hyperoli trig tios: sih osh osh sih th seh oth sh seh seh th sh sh oth D :,0 0, Ietities ivolvig hyperoli trig tios: osh sih th seh oth sh sih sih osh osh osh sih sih osh osh osh Derivtives o iverse hyperoli trig tios: sih osh th seh sh oth Itegrls ivolvig hyperoli trig tios: osh sih C sih osh C seh sh th C oth C seh th seh C sh oth sh C