AP Clulus AB Chpters. Re limit vlues from grphsleft-h Limits Right H Limits Uerst tht f() vlues eist ut tht the limit t oes ot hve to.. Be le to ietify lel isotiuities from grphs. Do t forget out the 3-step proess to prove tht prtiulr -vlue ( ) is isotiuity. ). ƒ() eists ). lim ƒ() eist (Rememer to hek the left right-h limits) 3) lim ƒ() ƒ() 3. Evlute limits tht pproh ifiity (horizotl symptotes): lim f() L 4. Give prtiulr futio fi the equtio of the tget lie y y m(- ) t give poit P(, f()). 5. Give positio futio, s(t), e le to lulte the veloity of prtile t t 6. Re grph stte where the futio is NOT DIFFERENTIABLE why. Sie the erivtive is limit, the erivtive eists t poit oly if it (the slope of the tget) eists from the left from the right AND these two vlues equl eh other! Chpter 3. Bsi Rules of Differetitio: (ostt) 0 ( ) () (-) (ƒ()g()) g() ƒ () + ƒ() g () Cƒ() C ƒ() f( ) g ( ) e e g() ƒ () - ƒ() g () [g()] si() os() se() se()t() os() -si() t() se () s() -s()ot() ot() -s (). Slope of tget lie ƒ () You shoul e le to fie the equtio of the tget lie: y - y m( - ) 3. Chi Rule: y y u u 4. Impliit Differetitio 5. Be le to tke the seo thir erivtives of ƒ. Cl AB
6. Derivtives of Logrithms: log() l() l() g'() [l(g())] g() l Logrithmi Differetitio: You MAY wt to use this if you re hvig troule tkig the erivtive of prtiulr futio.. Tke the turl logrithm of oth sies simplify usig the properties of logrithms.. Differetite Impliitly (with respet to ). 3. Solve for y. 7. Derivtives of Iverse Trig. Futios: (si- ) (s- ) (os- ) (se- ) (t- ) (ot- ) (rsi ) - (rs ) - - (ros ) - - (rse ) - (rt ) + (rot ) - + 8. Relte Rtes 9. Lier Approimtio Chpter 4:. Setio 4.: Mimum Miimum Vlues Lol Mimums/Miimums Cot our t the epoits of the omi Horizotl setios re oth lol m/mi (eluig the epoits) Asolute Mimum/Miimums The lrgest y-vlue Asolute M The smllest y-vlue Asolute Mi C our t the epoits of the omi Critil Numers ƒ () 0 or ƒ () D.N.E The Close Itervl Metho: To fi the solute m solute mi vlues of otiuous futio ƒ o lose itervl [, ]. Fi the ritil umers lulte ƒ(). Fi the vlues of ƒ t the epoits of the itervl. Asolute M Vlue Lrgest ƒ-vlue Asolute Mi Vlue Smllest ƒ-vlue. Setio 4.: Rolle s The Me Vlue Theorem Rolle s Theorem: If ƒ is futio tht stisfies the followig three hypotheses:. ƒ is otiuous o the lose itervl [, ].. ƒ is ifferetile o the ope itervl (, ). 3. ƒ() ƒ() The there is umer i (, ) suh tht ƒ () 0 Cl AB
Me Vlue Theorem: If ƒ is futio tht stisfies the followig hypothesis:. ƒ is otiuous o the lose itervl [, ].. ƒ is ifferetile o the ope itervl (, ). The there is umer i (, ) suh tht ƒ() - ƒ() ƒ () or equivletly ƒ() - ƒ() ƒ ()[( )] - 3. Setio 4.3: How Derivtives Affet Grph Give ƒ() you shoul e le to: i). Fi the itervls o whih ƒ is iresig or eresig You o this y uilig tle usig the ID Test. ii). Fi the lol m lol mi vlues of ƒ. You o this y reig the sig hges i the tle you uil for step i). This is the st Derivtive Test. iii). Fi the poits of ifletio itervls of ovity. You o this y tkig the seo erivtive evlutig where ƒ () 0 or D.N.E. These re possile poits of ifletio. You the uil other tle re the sigs of ƒ (). This will lso vlite y possile POIs. 4. Setio 4.4: L Hospitl s Rule 0 Oly works for limits tht re ietermite : ± 0 C e use multiple times if the ew limit stisfies the ove step! 5. Setio 4.7: Optimiztio Prolems Chpter 5:. 5. Approimtig The Are Uer Curves Left-H Right-H Mipoit. 5. The Defiite Itegrl f ( ) lim f ( i ) where i left-right-mipoits i Riem Sum f ( ) i f ( ) 0 Properties of Itegrls 3. 5.3 Fumetl Theorem of Clulus i You shoul e le to kow how evlute itegrls (with ouries). The ouries e vriles too! 4. 5.4 - Iefiite Itegrls the Totl Chge Theorem Give velotity futio you shoul e le to lulte the isplemet the totl iste trvele. Cl AB
f ( ) f ( ) [ f ( ) g ( )] f ( ) g ( ) k k C k ostt C ( -) l C e e C C l si( ) os( ) C os( ) si( ) C se ( ) t( ) C s ( ) ot( ) C se( ) t( ) se( ) C s( )ot( ) s( ) C t ( ) rt( ) C C si ( ) rsi( ) C C 5. 5.5 - The Sustitutio Rule Kow how to perform u-sustitutio! Chpter 6:. Are etwee urves: -is: A f ( ) g( ) YT B Y T Top Boury Futio Y B Bottom Boury Futio Y y-is: A f ( y) g( y) y R L X R Right Boury Futio X L Left Boury Futio. Volumes: Solis of revolutio: Disk/Wsher Metho Horizotl Ais of Rottio: V A( ) X X y ( ) R r A() is the ross-setiol re to the is of rottio R r r 0 Top Boury Futio : tke from is of rottio to top oury Bottom Boury Futio: tke from is of rottio to ottom oury whe the is of rottio is equl to the ottom oury (i.e. -is) Vertil Ais of Rottio: V A( y) y ( ) R r y A(y) is ross-setiol re to the is of rottio R Right Boury Futio: tke from is of rottio to right oury Cl AB
r r 0 Left Boury Futio: tke from is of rottio to left oury whe the is of rottio is equl to left oury (i.e. y-is) R & r will CHANGE whe the is of rottio hges!!! 3. Volumes : Solis y sliig. The KEY to solvig these prolems is to ietify whih is the CROSS-SECTIONAL AREAS re perpeiulr to!!! A() is -is: V A( ) A() squre, irle, semi-irle, isoseles right, equilterl. You ee to kow these formuls! A(y) is y-is: V A( y) y A(y) squre, irle, semi-irle, isoseles right, equilterl. You ee to kow these formuls! Aitiol topis: Averge Vlue of Futio: Me Vlue for Itegrls: Averge Veloity: ( ) Cl AB