AP CALCULUS FORMULA LIST 1 n Definition of e:e = lim 1 + n→∞ n ____________________________________________________________________________________ x if x ≥ 0 Absolute Value: x = −x if x < 0 ____________________________________________________________________________________ Definition of the Derivative: fxxfx()()+−∆ fxhfx()() +− fx'() = lim fx '() = lim ∆x→∞ ∆xh →∞ h fa()()+ h − fa fa'() = lim derivativeatxa = h→∞ h fx()()− fa fx'() = lim alternateform x→ a x− a ____________________________________________________________________________________ Definition of Continuity: f is continuous at c iff: (1)f() c is defined (2) limf() x exists x→ c (3) lim fx()()= fc x→ c ____________________________________________________________________________________ fb( ) − fb( ) Average Rate of Change of fx() on [] ab, = b− a ____________________________________________________________________________________ Rolle's Theorem: If f is continuous on [] ab, and differen tiable on ()()() ab, and if fafb= , then there exists a number c on ()() ab, such that fc' = 0. ____________________________________________________________________________________ Mean Value Theorem: If f is continuous on [] ab, and differen tiable on () ab, , then there fb()()− fa exists a number c on ()() ab, such that fc'= . b− a Note : Rolle's Theorem is a special case of The Mean Valu e Theorem fa()()− fb fa()() − fa If fa()()() = fb then fc' = = = 0. ba− ba − Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005 AP Calculus Formula List Math by Mr. Mueller Page 1 of 6 Intermediate Value Theorem: If f is continuous on [] abk, and is any number between fafb()() and , then there is at least one number c between ab and such that fck() = . ____________________________________________________________________________________ Definition of a Critical Number: Let f be defined at cfcf. If '() = 0 or ' is is undefined at cc, then is a criti cal number of f . ____________________________________________________________________________________ First Derivative Test: Let c be a critical number of the function f that is continuous on an open int erval Ic containing . If f is differentiable on I, except possibly at cfc, then () can be classified as follows. ()()1 If fx' changes from negative to positive at cfc, then () is a relative minimum of f . ()()2 If fx' changes from positive to negative at cfc, then () is a relative maximum of f . ____________________________________________________________________________________ Second Derivative Test: Let f be a function such that f'() c = 0 and t he second derivative exists on an open interval containing c. ()()()1 If fc"> 0, then fc is a relative minimum. ()()()2 If fc"< 0, then fc i s a relative maximum. ____________________________________________________________________________________ Definition of Concavity: Let f be differentiable on an open interval If. The graph of is concave upwa rd on If if ' is increasing on the interval, and concave downward on I, if f ' is dec reasing on the interval. ____________________________________________________________________________________ Test for Concavity: Let f be a function whose second derivative exists on an open interval I . ()()1 If fx"> 0 for all xI in , then the gr aph of f is concave upward on I . ()()2 If f" x< 0 for all x in I, then the graph of f is concave downward on I . ____________________________________________________________________________________ Definition of an Inflection Point: A function f has an inflection point at () cfc, () if ()()()1 fc"= 0 or fc" does not exist, and if ()2 f changes concavity at x= c . ____________________________________________________________________________________ Exponential Growth: dy =ky y( t ) = Ce kt dt Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005 AP Calculus Formula List Math by Mr. Mueller Page 2 of 6 1 Derivative of an Inverse Function: ()f−1 '() x = f'() f−1 () x ____________________________________________________________________________________ First Fundamental Theorem of Calculus: b f'()()() xdx = fb − fa ∫a ____________________________________________________________________________________ Second Fundamental Theorem of Calculus: d x ∫ ft()() dt= f x dx a d g() x ∫ f() t dt= f() g() x ⋅ g'() x ChainRuleVersion dx a ____________________________________________________________________________________ The Average Value of a Function: 1 b Average value of fx() on [] ab, fAVE = ∫ fxdx() b− a a ____________________________________________________________________________________ Volume of Revolution: b 2 Volume around a horizontal axis by discs: V= π rx() dx ∫a b 2 2 Volume around a horizontal axis by washe rs:V=π Rx()() − rx dx ∫a {} ____________________________________________________________________________________ Volume of Known Cross-Section: b Cross-sections perpendicular to x-axis: V= Axdx() ∫a ____________________________________________________________________________________ Position, Velocity, Acceleration: If an object is moving along a straight line with position function s() t , then Velocity is:vt()()= st ' Speed is: v() t Acceleration is:at()()()= vt ' = st " b Displacement from xa= to xb = is :Displacement= v() t dt ∫a Note: Displacement is a change in position . b Total Distance traveled from x= a to x = b is : TotalDistance = vt( ) dt ∫a Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005 AP Calculus Formula List Math by Mr. Mueller Page 3 of 6 TRIGONOMETRIC IDENTITIES Pythagorean Identities: sin22xx+= cos 1 tan 22 xx += 1sec 1cot += 22 xx csc ____________________________________________________________________________________ Sum & Difference Identities sin( AB±=) sin AB cos ± cos AB sin cos( AB ±=) cos ABAB cos∓ sin sin tanA± tan B tan ()A± B = 1∓ tanA tan B ____________________________________________________________________________________ Double Angle Identities sin2x= 2sin x cos x cos2x− sin 2 x 1cos2+x 1cos2 − x cos2xx=− 12sin2 cos 2 x = sin 2 x = 2 2 2 2cosx − 1 2 tan x tan 2 x = 1− tan 2 x ____________________________________________________________________________________ Half Angle Identities xx1cos− x 1cos + x x 1cos −− xxx sin 1cos sin=± cos =± tan =±== 2 2 2 2 2 1cos1cos+x + x sin x Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005 AP Calculus Formula List Math by Mr. Mueller Page 4 of 6 CALCULUS BC ONLY Integration by Parts:∫udv = uv − ∫ vdu ____________________________________________________________________________________ Arc Length of a Function: For a function fx() with a continuous deri vative on [] ab, : b 2 ArcLengthis : s= 1' + f() x dx ∫a ____________________________________________________________________________________ Area of a Surface of Revolution: For a function fx() with a continuous deri vative on [] ab, : b 2 SurfaceAreais : S= 2π rx()() 1' + f x dx ∫a ____________________________________________________________________________________ Parametric Equations and the Motion of an Object: Position Vector= ()xt()() , yt Velocity Vector= ()xt '()() , yt' Acceleration Vector= ()xt "()() , y" t dx 2 dy 2 Speed (or, magnitude of the velocity vector): v() t = + dt dt 2 2 b dx dy Distan ce traveled from tatb== to is: s =∫ + dt a dt dt Note: The distance traveled by an object along a p arametric curve is the same as the arc length of a parametric curve . dy dy dt Slope ()()()1'st derivati ve of curve C at () xt, yt is: = dx dxdt d dy dy2 d dy dt dx Second derivative of curve C at () xt()(), yt is: = = dx 2 dx dx dxdt ____________________________________________________________________________________ L'Hôpital's Rule: fx() fx'() lim= lim xc→gx() xc → gx'() Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005 AP Calculus Formula List Math by Mr. Mueller Page 5 of 6 Polar Coordinates and Graphs: y For rf=()θ: xr = cos, θθ yr =sin, rxy2 =+ 2 2 , tan θ = x dydydθf()()θcos θ+ f ' θ sin θ rcos θ+ r 'sin θ Slope of a polar curve: = = = dxdxdθ− f()() θθθθsin + f ' cos −+ r sin θ r 'cos θ β2 β 1 1 2 Area inside a polar curve : A=∫ f()θ d θ = ∫ rd θ 2α 2 α 2 β β 2 2 2 dr Arc length()()() * :sf=∫θ + fdr ' θθ =+ ∫ d θ α α dθ Surface of Revolution() * : β 2 2 about the polar axis:Sf= 2πθθθ()()() sin f + fd ' θθ ∫α π β 2 2 about θ=:Sf = 2 πθθθ∫ ()()() cos f + fd ' θθ 2 α ____________________________________________________________________________________ Euler's Method: dy Approximating the particular solution to: y'= = Fxy() , dx xxhnn=+−1 yyhFxy nn =+⋅ − 1()() nn −− 1, 1 given: hxxy =∆ , 0, 0 ____________________________________________________________________________________ Logistic Growth: k is the proportionality constant dP P L =⋅−kP1 P() t = −kt where : L is the Carrying Capacity dt L 1+ Ce C is the integration constant ____________________________________________________________________________________ The n'th Taylor Polynomial for f at c : ()n fc"() 2 fc() n Pxfcfcxc()()()()= +' ⋅−+ ⋅−++() xc ⋅−() xc n 2!n ! The n'th MacLaurin Polynomial for f is the Taylor Polynomial for f when c = 0. f"0() f "0() f ()n () 0 Pxffx()()()=0 + ' 0 ⋅+ ⋅+ x2 ⋅++ x 3 ⋅ x n n 2! 3!n ! ____________________________________________________________________________________ x2 x 3 ∞ x n ex =++1 x + + = ∑ 2!3!n=0 n ! 246∞ 2 n xxxn x cosx =− 1 + − += ∑() − 1 2!4!6!n=0 () 2!n 357∞ 21n+ xxxn x sinx=− x + − += ∑() − 1 3!5!7!n=0 () 21!n + Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005 AP Calculus Formula List Math by Mr. Mueller Page 6 of 6 .