Stuff You MUST Know Cold for AP Calc

AP CALCULUS Stuff you MUST know Cold Curve sketching and analysis Differentiation Rules Approx. Methods for Integration y = f(x) must be continuous at each: Chain Rule Trapezoidal Rule b dy d du dy dy du 1 b- a critical point: = 0 or undefined [ f( u )] = f '( u ) OR = f( x ) dx=2 n [ f ( x0 ) + 2 f ( x 1 ) + ... dx or endpoints dx dx dx du dx a local minimum: +2f ( xn-1 ) + f ( x n )] dy d2 y goes (–,0,+) or (–,und,+) or >0 Product Rule dx dx2 d du dv LRAM, RRAM, MRAM are local maximum: (uv ) = v+ u OR u 'v + uv ' dx dx dx rectangular approximations dy d2 y goes (+,0,–) or (+,und,–) or <0 a.k.a Riemann sums dx 2 dx Quotient Rule point of inflection: concavity changes du dv 2 d骣 u v- u u' v- uv ' d y = dx dx OR goes from (+,0,–), (–,0,+), 琪 2 2 dx2 (+,und,–), or (–,und,+) dx v v v Theorem of the Mean Value i.e. AVERAGE VALUE “PLUS A CONSTANT” If the function f(x) is continuous on [a, b] Basic Derivatives and the first derivative exists on the d interval (a, b), then there exists a number xn= nx n-1 The Fundamental Theorem of dx ( ) Calculus x = c on (a, b) such that b d f( x ) dx sinx= cos x a ( ) b f( c ) = dx f( x ) dx= F ( b ) - F ( a ) (b- a ) d a (cosx) = - sin x where F '( x )= f ( x ) This value f(c) is the “average value” of the dx function on the interval [a, b]. d (tanx) = sec2 x dx Corollary to FTC Solids of Revolution and friends d (cotx) = - csc2 x Disk Method dx x= b 2 d b( x ) V= p R( x ) dx d f( t ) dt = x= a [ ] (secx) = sec x tan x a( x ) dx dx Washer Method d f( b ( x )) b '( x )- f ( a ( x ))a'( x ) b 2 2 (cscx) = - csc x cot x V=p [ R( x )] -[ r ( x )] dx dx Intermediate Value Theorem a ( ) d1 du If the function f(x) is continuous on [a, b], General volume equation (not rotated) (lnu) = b dx u dx and y is a number between f(a) and f(b), V= Area( x ) dx d du then there exists at least one number x= c a eu= e u dx( ) dx in the open interval (a, b) such that f( c ) = y . More Derivatives Distance, Velocity, and Acceleration d1 du Mean Value Theorem d (sin-1 u) = velocity = (position) dx1- u2 dx dt If the function f(x) is continuous on [a, b], d d -1 -1 acceleration = (velocity) (cos x) = AND the first derivative exists on the dt dx 2 1- x interval (a, b), then there is at least one t f displacement = vdt d -1 1 number x = c in (a, b) such that to (tan x) = 2 dx1+ x f( b )- f ( a ) final time f'( c ) = . distance = v dt d -1 -1 b- a initial time (cot x) = 2 dx1+ x d 1 average velocity = sec-1 x = Rolle’s Theorem ( ) 2 final position - initial position dx x x -1 = total time d -1 If the function f(x) is continuous on [a, b], csc-1 x = Dx ( ) 2 AND the first derivative exists on the dx x x -1 = interval (a, b), AND f(a) = f(b), then there Dt d ax= a x ln a is at least one number x = c in (a, b) such dx ( ) that d 1 f'( c )= 0 . (log x) = dxa xln a INTEGRALS and important TRIG identities and values 1 1 Values of Trigonometric x ndx  x n1  C, if n  1 dx  ln x  C  n  1  x Functions for Common Angles θ sin θ cos θ tan θ sinx dx= - cos x + C cosx dx= sin x + C 0° 0 1 0 p 1 3 3 ,30° tanx dx= ln sec x + C cotx dx= ln sin x + C 6 2 2 3 p 2 2 ,45° 1 4 secx dx= ln sec x + tan x + C cscx dx= ln csc x - cot x + C 2 2 p 3 1 ,60° 3 3 2 2 2 2 secx dx= tan x + C cscx dx= - cot x + C p ,90° 1 0 “ ” 2 secx tan x dx= sec x + C cscx cot x dx= - csc x + C π,180° 0 -1 0 3p ,270° -1 0 “ ” 2 a x ex dx= e x + C a x dx   C Trig Identities  lna Double Argument sin 2x= 2sin x cos x dx 1  x  dx  x  cos2x= cos2 x - sin 2 x = 1 - 2sin 2 x  arctan   C  arcsin   C  x 2  a 2 a a   a 2  x 2 a 

dx 1  x  Pythagorean  arcsec   C lnx dx= x ln x - x + C sin2x+ cos 2 x = 1  2 2   x x  a a a (others are easily derivable by dividing by sin2x or cos2x) 1+ tan2x = sec 2 x SLOPES l’Hôpital’s Rule 2 2 DERIVATIVES f( a ) 0 cotx+ 1 = csc x If =or = , TANGENT LINES g( b ) 0 f( x ) f '( x ) Reciprocal then lim= lim 1 DEFINITE INTEGRAL x ag( x ) x a g '( x ) secx= or cos x sec x = 1 AREA UNDER A CURVE cos x 1 cscx= or sin x csc x = 1 INTEGRAL OF A RATE = sin x TOTAL AMOUNT Odd-Even DON'T FORGET YOUR sin(–x) = – sin x (odd) cos(–x) = cos x (even)

Original by Sean Bird, Covenant Christian High School Edited by Vicki Carter, West Florence High School