Mean Value Theorem If f is continuous on and differentiable on (a, b ), then there is at least one number c f (b) − f (a ) in (a, b ) such that = f ′(c). Rolle’s Theorem (this is a weak version of the MVT) If f is continuous on a, b and differentiable on (a, b ) such that f (a ) = f (b), then there is at least one number c in the open interval (a, b ) such that f ′(c) = 0. Geometrically, the derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. Definition of Derivative f ( x + h) − f ( x ) f ( x ) − f (a ) f ′( x) = lim or f ' ( a ) = lim h →0 h x →a x −a The latter definition of the derivative is the instantaneous rate of change of f ( x ) with respect to x at x = a. Instantaneous Rate of Change: If (x0, y 0 ) is a point on the graph of y = f (x), then the instantaneous rate of change of y with respect to x at x0 is f ′( x0 ). Average Rate of Change: If ( x0, y0 ) and ( x1, y1 ) are points on the graph of y = f (x), then the average rate of change of y with respect to x over the interval is f ( x1 ) − f ( x0 ) = y1 − y 0 = ∆y. Average and Instantaneous Rate of Change i). A line x = a is a vertical asymptote of the graph y = f (x) if either lim+ f ( x) = ±∞ or lim− f ( x ) = ±∞ (Values that make the denominator 0 but not x→a x→a numerator) 7. (Compare degrees of functions in fraction) x →∞ x →−∞ 2. A line y = b is a horizontal asymptote of the graph y = f (x) if either = lim f ( x) b= or lim f ( x) b. lim is finite if the degree of f ( x) = the degree of g ( x) x → ±∞ g ( x ) 2 x2 − 3x + 2 2 Example: lim = − 10 x − 5 x 2 x →∞ 5 6. lim = 0 if the degree of f ( x) the degree of g ( x) x → ±∞ g ( x ) x3 + 2 x Example: lim = ∞ x2 − 8 x →∞ f ( x) iii). Limits of Rational Functions as x → ±∞ f ( x) i). Note: If f is continuous on and f (a ) and f (b) differ in sign, then the equation f ( x) = 0 has at least one solution in the open interval (a, b). Intermediate-Value Theorem A function y = f (x) that is continuous on a closed interval takes on every value between f ( a ) and f (b). 2π Note: The period of the function y = A sin( Bx + C ) or y = A cos( Bx + C ) is.

Periodicity A function f (x ) is periodic with period p ( p > 0) if f ( x + p ) = f ( x) for every value of x. Every odd function is symmetric about the origin. A function y = f (x) is odd if f ( − x ) = − f ( x ) for every x in the function’s domain.

Every even function is symmetric about the y-axis. A function y = f (x) is even if f ( − x) = f ( x) for every x in the function’s domain. The limit lim f ( x ) exists if and only if both corresponding one-sided limits exist and are equal – x →a that is, lim f ( x ) = L → lim+ f ( x ) =lim− f ( x ) L= x →a x →a x →a 2. lim f ( x ) = f (a ) x→a Otherwise, f is discontinuous at x = a. Limits and Continuity: A function y = f (x) is continuous at x = a if i). ∫x x2 − a2 = a Arc sec + C = Arc cos + C a a x Formulas and Theorems 1. ∫ csc x cot x dx = − csc x + C ∫ tan x dx = tan x − x + C 2 17. ∫ sec x tan x dx = sec x + C ∫ csc x dx = − cot x + C 2 15. ∫ csc x dx = − ln csc x + cot x + C ∫ sec x d x = tan x + C 2 13. ∫ tan x dx = ln sec x + C or − ln cos x + C 10. ∫ a dx = ax + C x n +1 ∫ x dx = + C, n ≠ −1 n 2. (csc x) = − csc x cot x dx Integration Formulas 1. (cot x) = − csc 2 x dx d cf ( x )  = cf ' ( x ) dx  18.

( Arc sec x) = dx dx | x | x2 −1 d (tan x) = sec 2 x d 7. (a ) = a x ln a dx dx d f gf ′ − fg ′ d 1 3. cos 2θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2sin 2 θ Differentiation Formulas d n d x 1. cos( A − B ) = cos A cos B + sin A sin B 1 11. sin( A − B ) = sin A cos B − sin B cos A sin θ cos( A + B ) = cos A cos B − sin A sin B 1 9. sin( A + B ) = sin A cos B + sin B cos A 16. AP CALCULUS AB and BC Final Notes Trigonometric Formulas 1.

Ap calculus formula sheet