Calculus BC Stuff You Must Know Cold

CALCULUS BC Stuff you MUST Know Cold l’Hopital’s Rule If then lim x→a ∞ , f ( a) 0 = or = ∞ g (a) 0 f (x) f '( x ) = lim x → a g '( x ) g (x ) Properties of Log and Ln 1. ln 1 = 0 2. ln ea = a 3.e ln x n =x 4. ln x = nln x “PLUS A CONSTANT” 5. ln ( ab ) = ln a + lnb The Fundamental Theorem of Calculus 6.ln (a b ) = ln a − ln b ∫ f (x )dx = F (b ) − F (a ) b a Average Rate of Change (slope of the secant line) If the points (a, f(a)) and (b, f(b)) are on the graph of f(x) the average rate of change of f(x) on the interval [a,b] is f (b)− f (a) b− a Definition of Derivative (slope of the tangent line)  f ( x + h) − f ( x )  f '(x ) = lim   h→ 0 h   Derivatives d n ( x ) = nxn −1 dx d ( sin x) = cosx dx d ( cos x) = −sin x dx d ( tan x) = sec 2 x dx d ( cot x) = − csc 2 x dx d ( sec x ) = tanx sec x dx d ( csc x ) = − cotx csc x dx d 1 ( ln u ) = du dx u d u ( e ) = eu du dx 1 d ( log a x) = dx x ln a d u ( a ) = ax (ln a ) du dx where F '( x) = f ( x) Differentiation Rules Chain Rule d du [ f ( u) ] = f '( u) dx dx Product Rule d dv du ( uv ) = u + v dx dx dx 2 nd d dx Fundamental Theorem of Calculus g (x ) ∫ f (x )dx = f (g (x )) ⋅ g '( x ) # Quotient Rule Average Value du dv − d  u  v dx u dx  = dx  v  v2 If the function f(x) is continuous on [a, b] and the first derivative exist on the interval (a, b), then there exists a number x = c on (a, b) such that Mean Value & Rolle’s Theorem If the function f(x) is continuous on [a, b] and the first derivative exists on the interval (a, b), then there exists a number x = c on (a, b) such that f '(c ) = f (c ) = 1 b −a If given that dy dx = 0 or undefined. dy local minimum: goes (-,0,+) or dx d2y >0 (-,und,+) or dx 2 dy local maximum: goes (+,0,-) or dx d2y <0 (+,und,-) or dx 2 critical point: Absolute Max/Min.: Compare local extreme values to values at endpoints. pt of inflection : concavity changes. d2 y goes (+,0,-),(-,0,+), dx 2 (+,und,-), or (- ,und,+) a Euler’s Method if f(a) = f(b), then f ’(c) = 0. y = f(x) must be continuous at each: ∫ f ( x)dx f (c) is the average value f (b ) − f (a ) b −a Curve sketching and analysis b dy = f ( x, y ) and dx that the solution passes through (x0 , y0 ), then x new = x old + ∆x y new = y old + P (t ) = dy ⋅ ∆x dx ( xold , yold ) Logistics Curves L , 1 + Ce −( Lk )t where L is carrying capacity Maximum growth rate occurs when P=½L dP = kP ( L − P ) or dt dP P = (Lk )P (1 − ) dt L Distance, velocity and Acceleration Integrals ∫ kf (u) du = k ∫ f (u) du ∫ du = u + C u d ( position ) dt d Acceleration = ( velocity ) dt dx dy , Velocity Vector = dt dt n +1 1 ∫ u du = ln | u | + C u u ∫ e du = e + C  1  u ∫ a du = ln a  a + C ∫ cosudu = sin u + C 2 u − a2 du = 2 Disk Method: V = π final time ∫ 1 | u| arc sec  +C a  a  ∫ initial time b V = π∫ 2 Perpendicular to x-axis: d V == ∫ A( x)dx c Taylor Series a If the function f is “smooth”at x = c, then it can be approximated by the nth degree polynomial f ( x ) ≈ f (c ) + f '(c )( x − c ) Polar Curves For a polar curve r(θ), the 1 2 ∫ [ r(θ)] 2 f "(c ) ( x − c) 2 + ... 2! f '''( c) + ( x − c)3 + ... 3! f ( n) (c ) + ( x − c) n n! dθ + where θ1 and θ2 are the “first” two times that r = 0. Elementary Functions The series 2 1 + [ f '( x )] dx θ2 2 dθ If SN = Lagrange Error Bound (n+1)) max f ( z) (n +1))! for all z between x and c. x− c ∑ N n =1 ( −1) n an is the Nth partial sum of a convergent alternating series, then If Pn(x) is the nth degree Taylor polynomial of f(x) about c, then f (x) − Pn ( x) ≤ CHECK ENDPOINTS Alternating Series Error Bound θ1 3 x x + + ... 2! 3! x2 x4 x6 cos x == 1 −− ++ −− ++ ... 2! 4! 6! x3 x5 x7 sin x = x − + − + ... 3! 5! 7! 1 2 3 == 1++ x ++ x ++ x ++ ... 1−x 2 3 4 x x x ln( x + 1) = x − + − + ... 2 3 4 x e = 1+ x + converges if n n=0 For a polar graph, r( θ) 2 2 ∑a an lim +1 < 1 n→∞ a n a ∫ [r(θ) ] + [ r '(θ)] Centered at x = 0 ∞ b L= [ r (θ ) sin θ] [ r (θ) cos θ] d dθ Ratio Test (use for interval of convergence) Arc Length For a function, f(x) ∫ V = ∫ A (y )dy a ∫ y '(t )dt The slope of r(θ) at a given θ is L= y-axis: b θ1 ∫ udv = uv − ∫ vdu V = 2π ∫ r ( x )h (x )dx Volume of Known Cross Sections + ( y ' ) dt a b d dθ ) a 2 ∫ x '(t )dt dy dy d θ = = dx dx d θ 2 − [ r ( x)] dx b Shell Method: (x ' ) Area inside a “leaf” is Integration by Parts 2 2 θ2 u  ∫ a 2 −u 2 = arcsin  a  +C du 1  u ∫ a 2 + u 2 = a arctan a  +C ( [ R ( x) ] + ( y') . b y(b) = y(a) + dx a initial time x(b) = x(a) + 2 Washer Method: final time v (t ) dt = ∫ [ R( x) ] a Distance Traveled = ∫ sinudu = −cosu + C ∫ tanudu = − ln | cos u | + C ∫ cotudu = ln | sin u | +C ∫ secudu = ln | sec u + tan u | +C ∫ cscudu = − ln | csc u + cot u | + C du ( x ') Speed = |v(t)| = u ∫u b Velocity = ∫ u du = n + 1 + C, n ≠ −1 n Volume Solids of Revolution S ∞ − S N ≤ a N +1 n+1 Most Common Series ∞ 1 diverges ∑ n =1 n ∞ n (−1) converges ∑ n n =1 ∞ ∑A( r) n= 0 n converges to A if |r|<1 1−r