If an object moves along a straight line with position function s t , then its Velocity is v t s t Speed = v t Acceleration is a t v t s t Displacement (change in position) from t a tot bis Displacement = Ī v t dt Total Distance traveled from t a tot bis Total Distance = Ī c v t dt v t dt, where v t changes sign at x c.ĬALCULUS BC ONLY Integration by parts: u dv uv v du Logistic growth: Volume by cross sections taken perpendicular to the x-axis: ( ) Volume around a horizontal axis by shells: 2 ( ) ( ) Volume around a horizontal axis by washers: Volume around a horizontal axis by discs: if f changes sign at x c or if f changes from increasing to decreasing or vice versa atįirst Fundamental Theorem of Calculus: Ī f x dx f b f a Second Fundamental Theorem of Calculus: . cos u du sin u C sin u du cosu CĢ 2 sec u du tan u C csc u du cotu C sec u tan u du sec u C csc u cot u du cscu Cĭefinition of an Inflection Point: A function f has an inflection point at c, f c 1) if f c 0 or f cdoes not exist and _ 2 2 2 2 sin 2 2sin cos cos sin cos 2 1 2sin 2 cos 1 x x x x x x x x 2 2 1 cos 2 cos 2 1 cos 2 sin 2 x x x x _
f b f a f c b a _ Intermediate Value Theorem: If f is continuous on and k is any number between f (a) and f (b), then there is at least one number c between a and b such that f (c) = k. _ Mean Value Theorem: If f is continuous on and differentiable on (a, b), then there exists a number c on (a, b) such that ( ). x c f x f c _ Average rate of change of ( ) on = f b f a f x a b b a _ Average value of f x on = 1 b a f x dx b a _ Rolle's Theorem: If f is continuous on and differentiable on (a, b) and if f (a) = f (b), then there is at least one number c on (a, b) such that f ( ) c 0. AP CALCULUS BC FORMULA LIST Definition of e: 1 lim 1 n n e n Absolute value: 0 0 x if x x x if x _ Definition of the derivative: 0 ( ) lim h f x h f x f x h lim x a f x f a f a x a _ Definition of continuity: f is continuous at c iff 1) f (c) is defined 2) lim ( ) exists x c f x 3) lim ( ) ( ).
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