Download AP Calculus AB Cram Sheet and more Cheat Sheet Calculus in PDF only on Docsity! AP Calculus AB Cram Sheet Definition of the Derivative Function: f ' (x) = limh 0 f x h h f x Definition of Derivative at a Point: f ' (a) = limh 0 f a h f a h (note: the first definition results in a function, the second definition results in a number. Also note that the difference quotient, f a h f a h , by itself, represents the average rate of change of f from x = a to x = a + h) Interpretations of the Derivative: f ' (a) represents the instantaneous rate of change of f at x = a, the slope of the tangent line to the graph of f at x = a, and the slope of the curve at x = a. Derivative Formulas: (note:a and k are constants) d dx k 0 d dx (k·f(x))= k·f ' (x) d dx f x n n f x n 1 f ' x d dx [f(x) ± g(x)] = f ' (x) ± g ' (x) d dx [f(x)·g(x)] = f(x)·g ' (x) + g(x) · f ' (x) d dx f x g x g x f ' x f x g ' x g x 2 d dx sin(f(x)) = cos (f(x)) ·f ' (x) d dx cos(f(x)) = -sin(f(x))·f ' (x) d dx tan(f(x)) = sec2 f x f ' x d dx ln(f(x)) = 1 f x f ' x d dx e f x e f x f ' x d dx a f x a f x ln a f ' x d dx sin 1 f x f ' x 1 f x 2 d dx cos 1 f x f ' x 1 f x 2 1 d dx tan 1 f x f ' x 1 f x 2 d dx f 1 x at x f a equals 1 f ' x at x a L'Hopitals's Rule: If limx a f x g x 0 0 or and if limx a f ' x g ' x exists then limx a f x g x limx a f ' x g ' x The same rule applies if you get an indeterminate form ( 0 0 or ) for limx f x g x as well. Slope; Critical Points: Any c in the domain of f such that either f ' (c) = 0 or f ' (c) is undefined is called a critical point or critical value of f. Tangents and Normals The equation of the tangent line to the curve y = f(x) at x = a is y - f(a) = f ' (a) (x - a) The tangent line to a graph can be used to approximate a function value at points very near the point of tangency. This is known as local linear approximations. Make sure you use instead of = when you approximate a function. The equation of the line normal(perpendicular) to the curve y = f(x) at x = a is y - f(a) = 1 f ' a x a Increasing and Decreasing Functions A function y = f(x) is said to be increasing/decreasing on an interval if its deriva- tive is positive/negative on the interval. Maximum, Minimum, and Inflection Points The curve y = f(x) has a local (relative) minimum at a point where x = c if the first derivative changes signs from negative to positive at c. The curve y = f(x) has a local maximum at a point where x = c if the first deivative changes signs from positive to negative. The curve y = f(x) is said to be concave upward on an interval if the second derivative is positive on that interval. Note that this would mean that the first derivative is increasing on that interval. The curve y = f(x) is siad to be concave downward on an interval if the second derivative is negative on that interval. Note that this would mean that the first derivative is decreasing on that interval. The point where the concavity of y = f(x) changes is called a point of inflection. The curve y = f(x) has a global (absolute) minimum value at x = c on [a, b] if f(c) is less than all y values on the interval. 2
