1 Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y. The domin of f is the set (ll possile vlues). The rnge of f is the set of y (ll possile y vlues). ) Inverse functions i) Functions f nd g re inverses of ech other if () f ( g( )) (2) And g( f ( )) ii) To find f ( ), switch nd y in the originl eqution nd solve the eqution for y in terms of. c) Zeros of Function i) We lso cll these the roots or the intercepts. These occur where the function f() crosses the -is. ii) To find the zeros of function, set y equl to zero nd solve for. 2) Limits ) lim f ( ) L ( the limit of f() s pproches c is L) c ) Properties of Limits i) Sclr multiple: lim ( f ( )) lim ( f ( )) c c ii) Sum or Difference: iii) Product: lim f ( ) g( ) lim f ( ) lim g( ) c c c lim f ( ) g( ) lim f ( ) lim g( ) c c c lim f ( ) g( ) lim f ( ) lim g( ) c c c c) Limits tht Fil to Eist i) Behvior tht different from the right nd from the left () EX: lim 0 ii) Unounded ehvior lso known s Infinite Limits (when the limit equls positive or negtive infinity) () EX: lim 0 2 iii) Oscillting ehvior () EX: lim sin 0 d) Specil Trig Limits sin i) lim 0 cos ii) lim 0 0 e) One-sided Limits i) lim f( ) pproches c from the right iv) Quotient: ii) lim f( ) pproches c from the left
2 f) Continuity i) Definition: A function f is continuous t c if: () f (c) is defined (the point eists) (2) lim f( ) eists c (3) lim f ( ) f ( c) c ii) Grphiclly, we cn see if function is continuous if pencil cn e moved long the grph of f() without hving to e lifted off the pper. g) IVT (Intermedite Vlue Theorem) (Section.4) i) If f is continuous on [, ] nd k is ny numer etween f() nd f(), then there is t lest one numer c etween nd such tht f(c) = k. ii) We used this theorem to prove tht zero eisted on closed intervl (see Section.4) 3) Differentil Clculus ) Definition: f ( ) f ( ) i) f( ) lim (generl derivtive function) nd 0 f ( ) f ( c) f( c) lim (derivtive t specific point, c), provided tht these limits eist. c c ii) Rememer, the derivtive represents the slope of function t specific point. iii) We cn use the derivtive to find the slope of the line tht is tngent to the function t specific point. To find the eqution of tngent line, first tke the derivtive nd find the slope t the given point. Then using tht point nd the slope you just found, write the eqution of the tngent line in pointslope form: y y m( ). ) Differentition Rules i) Generl nd Logrithmic Differentition Rules ii) Derivtives of Trigonometric Functions
3 iii) Derivtives of the Inverse Trigonometric Functions c) Implicit Differentition i) Use implicit differentition when you hve s nd y s mied together nd you cnnot seprte them. ii) EX: To find dy d of 3 2 y y 2y 2, we must use implicit differentition () Tke the derivtive of ech term (don t forget to use product rule!) (2) Any time you tke the derivtive of y term, include dy d. (3) Seprte terms so tht ll dy terms re on one side of the equl sign nd everything else is on the d other side. (4) Solve for dy d. d) Higher Order Derivtives i) You cn tke the derivtive of function multiple times. You cn hve the first, second, third... nd nth derivtive of function. ii) We used this to tlk out position, velocity nd ccelertion () st () The position function (2) v( t) s( t) The velocity function is the first derivtive of the position function (3) ( t) v( t) s ( t) The ccelertion function is the first derivtive of the velocity function, or the second derivtive of the position function. e) Derivtives of Inverse Functions i) f f f ( ) f) L Hopitl s Rule f( ) 0 i) If lim g or ( ) 0 (indeterminte forms), the ( ) ( ) lim f lim f g ( ) g( ) ii) REMEMBER: This is NOT the sme t the Quotient Rule. DO NOT USE THE QUOTIENT RULE HERE! g) MVT (Men Vlue Theorem) (Section 3.2)
4 i) If f is continuous on [, ] nd differentile on (, ), then there eists numer c in (, ) such tht f ( ) f ( ) f() c h) EVT (Etreme Vlue Theorem) (Section 3.) i) If function f is continuous on closed intervl, then f hs oth mimum nd minimum vlue in the intervl. i) Finding Reltive Etrem (reltive m nd min) i) The First Derivtive Test (Section 3.3) () Tke the derivtive of the function nd set the derivtive equl to zero. These -vlues tht mke the derivtive equl to zero re the criticl numers. (2) To clssify the criticl numer s mimum or minimum, we need to test the intervls round the criticl numers to see if the function is incresing or decresing on theses intervls. () If f chnges from negtive to positive t the criticl numer, then the criticl numer is reltive minimum. () If f chnges from positive to negtive t the criticl numer, then the criticl numer is reltive mimum. (c) If f does not chnge signs t the criticl numer, then the criticl numer my e n inflection point ii) The Second Derivtive Test (Section 3.4) () Tke the first derivtive nd set it equl to 0 to find the criticl points gin. (2) Plug the criticl points into the second derivtive nd check if they give positive or negtive vlue. () If f is positive (concve up) t the criticl point, then the criticl point is reltive minimum. () If f is negtive (concve down) t the criticl point, then the criticl point is reltive mimum. j) Curve Sketching i) Rememer, the first derivtive tells you the slope (incresing or decresing) nd the second derivtive tells you the concvity (concve up or concve down).
5 ii) Think of these situtions when nlyzing curve: k) Don t forget to review Optimiztion prolems (Section 3.7) nd Relted Rtes prolems (Section 2.6)! 4) Integrl Clculus ) Definition: A function F() is the ntiderivtive of function f() if for ll in the domin of f, F( ) f ( ), or f ( ) d F( ) C (C is constnt). Without ounds, this is clled n indefinite integrl. ) Riemnn Sums: pproimting the re under curve y using shpes like rectngles nd trpezoids (Section 4.2) i) Left Riemnn Sum: strt t the left hnd point of ech intervl for the height of ech rectngle ii) Right Riemnn Sum: strt t the right hnd point of ech intervl for the height of ech rectngle iii) Midpoint Sum: strt in the middle of the intervl for the height of ech rectngle iv) Trpezoid sum: f ( ) 2 ( ) 2 ( )... 2 ( ) ( ) 0 f f 2 f n f n 2n v) The definite integrl is the limit of Riemnn Sum (Section 4.3): () n n f ( ) d lim f ( k) lim f ( ) n n k n k k (2) The definite (ounded) integrl represents the re under curve c) Bsic Integrl Functions
6 d) Properties of Definite Integrls i) f ( ) g( ) d f ( ) d g( ) d ii) kf ( ) d k f ( ) d iii) f ( ) d 0 iv) f ( ) d f ( ) d c c v) f ( ) d f ( ) d f ( ) d e) Integrtion y Sustitution (U-Sustitution) (Section 4.5) i) Like the chin rule, ut for integrls ii) f ( g( )) g( ) d f ( u) du F( g( )) C f) Position/Velocity/Accelertion i) s( t) v( t) dt ii) v( t) ( t) dt g) The First Fundmentl Theorem of Clculus i) If f is continuous on [, ] nd F( ) f ( ), then f ( ) d F( ) F( ) h) The Second Fundmentl Theorem of Clculus i) If f is continuous on n open intervl I contining, then for every in the intervl, d d f ( t) dt f ( ) i) Averge Vlue of function on n intervl i) f ( ) d j) Are Between Two Curves i) ( ) ( ) 5) Volumes A f g d, where f is the upper function nd g is the lower function
7 ) Solids of Revolution 2 i) Disk Method V r d V f ( ) d () Aout the -is: 2 (2) Aout the y-is: 2 V f ( y) dy ii) Wsher Method ( ) ( ) function. ) Solids with Known Cross Section 2 2 R() is the upper function nd r() is the lower V R r d i) When the cross sections re perpendiculr to the is: V A( ) d (A() is the re of the cross section). ii) When the cross sections re perpendiculr to the y is: V A( y) dy (A(y) is the re of the cross section). 6) Differentil Equtions ) Types of Solutions: i) Generl Solutions include the constnt C ii) Prticulr solutions include the numericl vlue for C (must hve initil conditions to solve for C). ) Seprle Equtions i) Seprte y s nd s on opposite sides of the equl sign. Then integrte oth sides nd solve for y if possile. c) Be sure to understnd how dy ky d ecomes kt y Ce y solving the seprle differentil eqution. d) Don t forget to study Slope Fields! (Section 6.) i) Like this one: 7) Don t forget to prctice your trig without clcultor!! ) The most importnt trig identity to rememer is the Pythgoren Identity: 2 2 sin cos