AP Calculus BC Exam Review:
Theorems Questions and Answers
limits and continuity - Correct Answers -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.
Rolle's Theorem - Correct Answers -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 - Correct Answers -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 - Correct Answers -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 - Correct Answers -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 .
intermediate-value theorem - Correct Answers -a function y = f(x) that is continuous on
a closed interval [a,b] takes on every value between f(a) and f(b)
Theorems Questions and Answers
limits and continuity - Correct Answers -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.
Rolle's Theorem - Correct Answers -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 - Correct Answers -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 - Correct Answers -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 - Correct Answers -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 .
intermediate-value theorem - Correct Answers -a function y = f(x) that is continuous on
a closed interval [a,b] takes on every value between f(a) and f(b)
