AP CALCULUS BC Stuff you MUST Know Cold

l’Hopital’s Rule Properties of Log and Ln “PLUS A CONSTANT” fa() 0 ∞∞∞ a If === or = , 110.ln== 2 .lne a ga() 0 ∞∞∞ ln x n 3.e== x 4 . ln x nln x The Fundamental Theorem of fx() f '() x then lim=== lim 5.ln() ab=+ lna lnb Calculus xa→→gx() xa g '() x b .lna = ln a− lnb =− 6 ()b ∫∫∫ fxdxFb() () Fa () a Average Rate of Change Differentiation Rules where Fx '( )=== fx ( ) (slope of the secant line) Chain Rule ddund If the points (a, f(a)) and (b, f(b)) [[[][ fu()]]] === f '() u 2 Fundamental Theorem of are on the graph of f(x) the average dx dx Calculus f x gx() rate of change of ( ) on the interval Product Rule d [a,b] is fxdxfgx()= (())⋅ gx '() ddvdu dx ∫ fb()− fa () ((()(uv))) =+ u v # dx dx dx ba− Quotient Rule Average Value du dv If the function f(x) is continuous on Definition of Derivative vu−−− [a, b] and the first derivative exist du (slope of the tangent line) = dx dx on the interval (a, b), then there == 2 dx v v exists a number x = c on (a, b) such fx()()+ h− fx that fx'( )= lim  h→0 h Mean Value & Rolle’s Theorem 1 b If the function f(x) is continuous on fc() = fxdx () ba−−− ∫∫∫ Derivatives [a, b] and the first derivative exists a on the interval (a, b), then there fc() is the average value d nn−1 exists a number x = c on (a, b) such ()xnx= dx fb()−−− fa () that fc'( ) === d Euler’s Method ()sinxx= cos ba−−− dx if f(a) = f(b), then f ’(c) = 0. dy If given that === fxy(, ) and d ()cosxx=− sin dx dx Curve sketching and analysis that the solution passes through d 2 y = f(x) must be continuous at each: ()tanxx= sec (x0, y0), then dx dy critical point: dx === 0 or undefined. d x new = x old + ∆x ()cotxx=− csc2 dy dx local minimum: goes (-,0,+) or dy dx yynew=+ old ⋅∆ x d 2 dx ()xy ()secxxx= tan sec dy old, old dx (-,und,+) or >>> 0 dx2 d cscxxx=− cot csc dy Logistics Curves () local maximum: goes (+,0,-) or dx dx L = d 1 2 Pt() −()Lk t , ()lnu= du dy 1+ Ce dx u (+,und,-) or <<< 0 dx2 where L is carrying capacity Maximum growth rate occurs when d uu Absolute Max/Min.: Compare local ()eedu= P = ½ L dx extreme values to values dP d 1 at endpoints. =kP() L− P or ()loga x = pt of inflection : concavity changes. dt dx xln a 2 dy dP P d goes (+,0,-),(-,0,+), =()(1)Lk P − ()aalnaduux= () dx2 dt L dx (+,und,-), or (-,und,+) Integrals Distance, velocity and Volume kf() u du= k f () u du Acceleration Solids of Revolution ∫∫ d b Velocity = ((()( position))) 2 ∫ du=+ u C dt Disk Method: VRxdx=== π∫∫∫ [[[][ ()]]] a n+1 d n u udu=+ Cn,1≠− Acceleration = ((()(velocity))) Washer Method: ∫ n +1 dt b 1 dx dy 22 du=+ln | u | C Velocity Vector = , VRxrxdx=π [[[][ ()]]][−[[[] ()]]] ∫ u dt dt ∫∫∫ ((()( ))) a eduuu=+ e C 22 ∫ Speed = |v(t)| = ((()( xy'')))(+++ ((()))) . b 1 Distance Traveled = Shell Method: Vrxhxdx=== 2()()π aduuu=+ a C ∫∫∫ ∫  final final a ln a time time 22 Volume of Known Cross Sections ∫ cosudu = sin uC+ ∫∫vt() dt=+((()( x ')))(((() y '))) dt initial initial Perpendicular to sin udu = −cosuC+ time time ∫ b x-axis: y-axis: b d udu = −uC+ x(b) = x(a) + xtdt'( ) ∫ tan ln | cos | ∫∫∫ VAxdx=== () VAydy=== () a ∫∫∫ ∫∫∫ ∫ cot udu = ln | sinuC | + b a c y(b) = y(a) + ytdt'( ) secudu = ln | secuuC++ tan | ∫∫∫ ∫ a Taylor Series ∫ cscudu = −ln | cscuuC++ cot | If the function f is “smooth”at x = c, then it can be approximated by the nth du 1||u Polar Curves = arcsec+ C degree polynomial ∫ uu22− a aa For a polar curve r(θ), the fx()≈ fc ()+ f '()( cx− c ) θ2 2 du u Area inside a “leaf” is 1 rd()θθ fc"( ) 2 = arcsin + C 2 ∫ [[[][ ]]] +()...xc−+ ∫ 22  ∫∫ au− a θ1 2! where θ1 and θ2 are the “first” two fc'''( ) du 1 u 3 = + times that r = 0. + (xc− )+ ... 22 arctan C 3! ∫ au+ aa The slope of r(θ) at a given θ is fc()n () + (xc−−− )n dy dy d [[[r()sinθθ]]] n! Integration by Parts ==dθ dθ dx d dxdθ dθ [[[][ r()cosθθ]]] udv= uv− vdu ∫∫ Elementary Functions

Ratio Test Centered at x = 0 Arc Length (use for interval of convergence) For a function, f(x) ∞∞∞ xx23 exx =+1... + + + b The series an converges if 2 ∑∑∑ 2! 3! n===0 Lfxdx=+1'()[[[][ ]]] 246 ∫∫∫ CHECK xxx a an+++1 cosx = 1−+−+ ... lim<<< 1 2! 4! 6! For a polar graph, r(θ) n→∞→∞→∞ ENDPOINTS an 357 θ2 xxx 22 sinxx=−+−+ ... Lr=+∫∫∫ [[[][ ()θθθ]]][[[[] rd '()]]] Alternating Series Error Bound 3! 5! 7! θ1 N n th 1 If Sa=(1)−is the N 23 Nn∑∑∑ n===1 =+1...xx + + x + 1−−− x Lagrange Error Bound partial sum of a convergent xxx234 If Pn(x) is the nth degree Taylor alternating series, then ln(xx+= 1)−+−+ ... polynomial of f(x) about c, then 234 SS∞−≤NN a+1 ((()(n+++1))) maxfz ( ) n+++1 fx()−≤ Px () x − c n ((()(n+++1!))) Most Common Series ∞∞1(1)−−− n ∞∞∞ A for all z between x and c. diverges converges Ar()n converges to if |r|<1 ∑∑nn ∑∑∑ 1−−− r nn==11 n===0