AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim (Alternte orm or erivtive t given vlue.) Instntneous Rte o Chnge o t : '( ) Polynomils ( c (erivtive t the given vlue) (slope o tngent line) is constnt) 4 5 Derivtives c 0 c c Power Rules Derivtives n n n Integrls Integrls c c C C n n n C C AB Clculus Formuls & Justiictions - -
Trig Functions: 6 7 Derivtives sin cos tn sec sec sec tn cos sin cot csc csc csccot Inverse Trig Functions: Derivtives sin tn sec cos cot csc Integrls Integrls cos sin C sec tn C sec tn sec C sin cos C csc cot C csc cot csc C sin C tn C sec C cos C cot C csc C AB Clculus Formuls & Justiictions - -
Eponentil n Logrithmic Functions: 8 Derivtives ln e e log ln ln Integrls ln C e e C log C ln ln C Generic Functions 9 Derivtives ' y y Integrls ' C y y C Justiictions or horizontl tngent lines: 0 y ( ) hs horizontl tngents when 0. AB Clculus Formuls & Justiictions - 3 -
Properties o Integrls: ( ) g( ) ( ) g( ) ( ) g( ) ( ) g( ) c ( ) c ( ) ( ) ( ) ( ) 0 First Funmentl Theorem o Clculus: '( ) ( ) ( ) (Fins the signe re etween curve n the -is) Averge Vlue o Function: 3 vg ( ) Justiying tht unction is continuous t point: 4 5 6 is continuous t c. () c c i: is eine. lim ( ) eists 3. () c = lim ( ) c Justiying tht erivtive eists t point, c : Show lgericlly tht lim '( ) lim '( ). Intermeite Vlue Theorem: I c c is continuous on [, ] n k is ny numer etween ( ) n (), then there is t lest one numer c etween n such tht () c k. AB Clculus Formuls & Justiictions - 4 -
Men Vlue Theorem: 7 I is continuous on [, ] n ierentile on (, ) then there eists ( ) ( ) numer c on (, ) such tht '( c). (Clculus slope = Alger Slope) 8 9 0 Prouct Rule: ( ) g ( ) ( ) g '( ) g ( ) '( ) Quotient Rule: ( ) g( ) '( ) ( ) g '( ) g( ) g ( ) Derivtives o Inverse Functions: The erivtive o n inverse unction is the reciprocl o the erivtive o the originl unction t the mtching point. Chin Rule: I (, ) is on ( ), then (, ) is on y y u u ( ) n ( )'( ). '( ) ( g ( )) '( g ( )) g '( ) Secon Funmentl Theorem o Clculus: ( t) t ( ) g( ) ( t) t ( g( )) g '( ) Limits t Ininity: 3 To in lim ( ) think Top Hevy limit is ± Bottom Hevy limit is 0 Equl limit is rtio o coeicients AB Clculus Formuls & Justiictions - 5 -
Limits with Ininity (t verticl symptotes): 4 When ining one-sie limit t verticl symptote, the nswer is either ±. Steps or Solving Dierentil Equtions: 5 Fin solution (or solve) the seprle ierentile eqution. Seprte the vriles. Integrte ech sie 3. Mke sure to put C on sie with inepenent vrile (normlly ) 4. Plug in initil conition n solve or C (i given) 5. Solve or the epenent vrile (normlly y) 6 Justiictions or verticl tngent lines: ( ) hs verticl tngents when Etreme Vlue Theorem: y is uneine. 7 I is continuous on the close intervl [, ], then hs oth minimum n mimum on the close intervl [, ]. Justiiction or n Asolute Etrem. 8 9. Fin criticl numers.. Ientiy enpoints. 3. Fin ( criticl numers ) n ( enpoints ). 4. Determine solute m/min vlues y compring the y-vlues. Stte in sentence. Justiiction or Criticl Numer: c is criticl numer ecuse '( ) 0 or '( ) is uneine. Justiiction or Incresing/Decresing Intervls: 30 Inc: ( ) is incresing on [, ] /c '( ) 0. Dec: ( ) is ecresing on [, ] /c '( ) 0. AB Clculus Formuls & Justiictions - 6 -
Justiiction or Reltive M/Min Using st Derivtive Test: 3 Locl M: '( ) chnges rom + to -. Locl Min: '( ) chnges rom - to +. Justiiction or Reltive M/Min Using n Derivtive Test: 3 Locl M: '( c) 0 (or un) n ''( ) 0. Locl Min: '( c) 0 (or un) n ''( ) 0. Justiiction or Point o Inlection: 33 Using n erivtive: ''( ) 0 (or ne) AND ''( ) chnges sign. Using st erivtive: ''( ) 0 (or ne) AND slope o '( ) chnges sign. Justiiction or Concve Up/Concve Down: 34 Concve Up: ( ) is concve up on (, ) ecuse ''( ) 0. Concve Down: ( ) is concve own on (, ) ecuse ''( ) 0. Justiictions or liner pproimtion estimtes: 35 A liner pproimtion (tngent line) is n overestimte i the curve is concve own. A liner pproimtion (tngent line) is n unerestimte i the curve is concve up. Justiictions or Reimnn Sums: 36 Let-Riemnn Sums: The sum is n overestimte i the curve is ecresing. The sum is n unerestimte i the curve is incresing. Right-Riemnn Sums: The sum is n overestimte i the curve is incresing. The sum is n unerestimte i the curve is ecresing. AB Clculus Formuls & Justiictions - 7 -
Justiictions or Prticle Motion: 37 Prticle is moving right/up ecuse Prticle is moving let/own ecuse vt ( ) 0 vt ( ) 0 (positive). (negtive). Prticle is speeing up ( velocity is getting igger) ecuse sme sign. Prticle is slowing own ( velocity is getting smller) ecuse hve ierent signs. vt () vt () n n t () t () hve Prticle Motion Formuls: 38 Velocity: v( t) s'( t) Accelertion: ( t) v'( t) s''( t) Spee: spee vt () Averge Velocity: (given ) s( ) s( ) (given ) () v t t Averge Accelertion: (given ) v( ) v( ) (given ) () t t Displcement: v () st () vt () Totl Distnce: v () t t vt () t () Position t : s( ) s( ) v( t) t Net Chnge Theorem: 39 ( ) represents the net chnge in the unction rom time to. Fining Totl Amount: 40 ( ) ( ) '( ) (wnt = hve + integrl) AB Clculus Formuls & Justiictions - 8 -
Ares in Plne: 4 Perpeniculr to -is: ( ) ( ) ( ) is top curve, g ( ) point o intersection g Perpeniculr to y-is: ( ) ( ) ( y) is right curve, point o intersection g( y) is ottom curve, n re -coorintes o y g y y is let curve, n re y-coorintes o Steps to Fining Volume: 4 Volume = Are. ecie on whether it s or y. in ormul or the re in terms o or y 3. in the limits (mking sure they mtch or y) 4. integrte n evlute Volumes Aroun Horizontl Ais o Rottion or Perpeniculr to -is: 43 Disc: Wsher: V n re -coorintes r V R r n re -coorintes Sl (Cross Section): V A( ) A ( ) is the re ormul or the cross section Volumes Aroun Verticl Ais o Rottion or Perpeniculr to y-is: 44 Disc: V r y n re y-coorintes Wsher: V R r y n re y-coorintes Sl (Cross Section): V A( y) y Ay ( ) is the re ormul or the cross section AB Clculus Formuls & Justiictions - 9 -
Eponentil Growth n Decy: 45 The rte o chnge o quntity is irectly proportionl to tht quntity Gives the ierentil eqution: Which cn e solve to yiel: y t ky y Ce kt AB Clculus Formuls & Justiictions - 0 -