Stuff You Must Know

Stuff you MUST know cold for AP Calculus! Note: Letters like a, b, c, d, m, and n are traditionally used to represent constants. Letters like f, g, h, u, v, x, and y and traditionally used to represent variables or functions. Basic Derivatives

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d (sin x ) = cos x dx d (cos x ) = − sin x dx d (tan x ) = sec2 x dx d (cot x) = − csc2 x dx d (sec x ) = sec x tan x dx d (csc x ) = − csc x cot x dx d (ln x) = 1 dx x d x (e ) = e x dx

lim

Rolle’s Theorem If f is differentiable for all values of x in the open interval (a, b ) , and f is

Differentiation Rules

continuous at x = a and at x = b , and f (a) = f (b) = 0 , then there is at least one number x = c in (a, b ) such that f ′(c) = 0 .

Definitions of Derivative

f ( x + h) − f ( x ) h →0 h f ( x ) − f (c ) lim x →c x −c

d n x = nx n−1 dx

Chain Rule If f (x ) = g( h( x )) , then

f ′( x) = g ′(h( x)) ⋅ h′ (x ) . OR dy dy du = ⋅ dx du dx Product Rule d (uv ) = v du + u dv dx dx dx Quotient Rule d  u  vu′ − uv ′  = dx  v  v2

More Derivatives

d (sin −1 x ) = 1 2 dx 1− x d −1 −1 ( cos x ) = dx 1 − x2 d 1 ( tan−1 x ) = dx 1+ x2 d (cot −1 x) = 1 +−1x 2 dx d (sec−1 x) = 12 dx x x −1 d (csc−1 x ) = −12 dx x x −1

d x (a ) = a x ln a dx d (log a x ) = 1 dx x ln a If a function is differentiable, then it is continuous.

“PLUS A CONSTANT!” The Fundamental Theorem of Calculus b

∫ f (x )dx = F (b) − F (a ) a

where F ′ (x ) = f (x )

Corollary to the FTC

d dx

g ( x)

∫

f (t )dt = f ( g( x )) ⋅ g′ (x )

a

Mean Value Theorem If f is differentiable for all values of x in the open interval (a, b ) , and f is continuous at x = a and x = b , then there is at least one number x = c in (a, b ) such that f ′(c ) = f (b) − f (a) b−a Trapezoidal Rule

1 b − a  [ f (x 0 ) n 

b

∫ f (x )dx = 2  a

+ 2 f ( x1 ) + L + 2 f (x n −1 ) + f (x n )] Distance, Velocity, and Acceleration If distance, velocity and acceleration are represented by s, v, and a, dv d 2 s respectively, then a = = and dt dt 2 ds v= . dt t1

change in v = ∫ a dt t0 t1

change in s = ∫ v dt t0

Intermediate Value Theorem If function f is continuous for all x in the closed interval [a, b] , and y is a number between f (a) and f (b ) , then there is a number x = c in (a, b ) for which f (c) = y .

l'Hôpital’s Rule

f (x ) 0 ∞ → or g( x ) 0 ∞ f (x ) f ′( x) then lim = lim x →a g( x ) x→ a g ′( x ) If lim x →a