AP Calculus BC Exam Review: Theorems

limits and continuity

a function y = f(x) is continuous at x = a if: f(a) is defined, the limit of f(x) as x approaches a exists, and the limit of f(x) as x approaches a is equal to f(a).

otherwise, f is discontinuous at x = a

The limit exists iff both corresponding one sided limits exist and are equal.

intermediate-value theorem

a function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b)

Rolle's Theorem

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.

Mean Value Theorem

If f is continuous on [a,b] and differentiable on (a,b), then there is at least one number c in (a,b) such that [f(b) - f(a)] / (b - a) = f'(c)

Extreme-Value Theorem

If f is continuous on a closed interval [a,b], then f(x) has both a max and a min on [a,b]

L'Hopital's Rule

If the limit of f(x) / g(x) as x approaches a is of an indeterminate form, and if the limit of f'(x) / g'(x) as x approaches a exists, then the limit of f(x) / g(x) as x approaches a = the limit of f'(x) / g'(x) as x approaches a .