Calculus Formula Sheet

Stuff you MUST Know Cold l’Hopital’s Rule If ∞ f ( a) 0 , = or = ∞ g (a ) 0 f ( x) f '( x) then lim = lim x → a g (x ) x → a g '( x ) Properties of Log and Ln 1. ln1 = 0 2 . ln ea = a 3.e ln x n 4 . ln x = n ln x =x 5. ln ( ab ) = ln a +ln b “PLUS A CONSTANT” The Fundamental Theorem of Calculus b ∫ f ( x) dx = F( b) − F (a ) 6.ln (a b ) = ln a − ln 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) = cos x 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 u ( e ) = e du dx 1 d ( loga x) = x ln a dx d u ( a ) = a x (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 Quotient Rule du dv − d u  v dx u dx =   dx v   v2 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 (b ) − f ( a ) b−a if f(a) = f(b), then f ’(c) = 0. Curve sketching and analysis y = f(x) must be continuous at each: 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. 2 d y goes (+,0,-),(-,0,+), 2 dx (+,und,-), or (- ,und,+) 2 nd d dx Fundamental Theorem of Calculus g (x ) ∫ f ( x) dx = f ( g( x)) ⋅ g'( x) # Average Value 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 f ( c) = 1 b −a b ∫ f ( x) dx a f ( c) is the average value Euler’s Method If given that dy = f ( x, y) and dx that the solution passes through (x0 , y0 ), then x new = x old + ∆ x ynew = yold + dy ⋅ ∆x dx ( xold , yold ) Logistics Curves L P (t ) = , − 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 f( u) du ∫ kf ( u) du = k∫ ∫ du = u + C d ( position ) dt d Acceleration = ( velocity) dt dx dy Velocity Vector = , dt dt 1 u a + C ∫ a du =  ln a  final time 1 | u|  +C arc sec   a  a u  +C  ∫ 2 − 2 = arcsin   a  a u du u  +C  ∫ a 2 +u 2 = a1 arctan  a  2 u u −a du = ∫ udv = uv − ∫ vdu a ∫ [ r(θ) ] + [ r '(θ )] dθ 2 d V == ∫ A( x) dx V = ∫ A( y) dy a 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 θ2 Area inside a “leaf” is 1 2 ∫ [ r (θ )] 2 f "(c) ( x − c) 2 + ... 2! f '''( c) + ( x − c)3 + ... 3! f ( n) ( c) n + ( x − c) n! dθ + θ1 where θ1 and θ2 are the “first” two times that r = 0. The slope of r(θ) at a given θ is If SN = Lagrange Error Bound d dθ d dθ [ r (θ ) sinθ ] [ r (θ ) cos θ ] Centered at x = 0 ∑a ∑ N n =1 CHECK ENDPOINTS n ( −1) an is the Nth S∞ − SN ≤ aN +1 n+1 x− c 2 3 x x + + ... 2! 3! 2 4 x x x6 cos x == 1 −− ++ −− ++... 2! 4! 6! x 3 x 5 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 partial sum of a convergent alternating series, then If Pn(x) is the nth degree Taylor polynomial of f(x) about c, then (n +1))! y-axis: b Alternating Series Error Bound θ1 for all z between x and c. Perpendicular to x-axis: ∫ y '( t) dt n=0 For a polar graph, r( θ) ( z) V = 2π ∫ r (x)h( x) dx Volume of Known Cross Sections ∫ x '( t) dt an +1 <1 lim n→∞ a n a n+1)) 2 a ( x ' )2 + ( y')2 dt a b The series 2 max f ( 2 b b y(b) = y(a) + ( [ R( x)] − [ r( x)] ) dx Shell Method: initial time x(b) = x(a) + dx Elementary Functions L = ∫ 1 + [ f '( x)] dx f ( x) − Pn ( x) ≤ b V = π∫ ∞ b 2 ∫ 2 Washer Method: Ratio Test (use for interval of convergence) Arc Length For a function, f(x) θ2 initial time dy dy d θ = = dx dx d θ Integration by Parts L= ∫ final time v(t ) dt = ∫ [ R( x) ] a Distance Traveled = ∫ cosudu = sin u + C ∫ 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 2 Disk Method: V == π ( x') 2 + ( y' ) 2 . Speed = |v(t)| = u du b Velocity = u n +1 ∫ un du = n + 1 + C, n ≠ −1 ∫ 1u du = ln | u | + C ∫ eu du = eu + C ∫ Volume Solids of Revolution 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