AP Calculus AB Formula List Review

These flashcards cover critical concepts, theorems, and definitions from the AP Calculus AB formula list, useful for exam preparation.

What is the definition of continuity at a point c?

A function f is continuous at c if: 1) f(c) is defined; 2) the limit as x approaches c of f(x) exists; 3) the limit as x approaches c of f(x) equals f(c).

State the Mean Value Theorem.

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

What does the Intermediate Value Theorem state?

If f is continuous on [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in (a, b) such that f(c) = k.

What is the definition of a critical number?

A critical number of a function f is a point c where f' is undefined or f' = 0.

Describe the First Derivative Test.

Let c be a critical number of f. If f changes from negative to positive at c, then f has a relative minimum. If it changes from positive to negative, then f has a relative maximum.

What is the Second Derivative Test?

If f''(c) > 0, then c is a relative minimum. If f''(c) < 0, then c is a relative maximum.

Define concavity of a function.

A function f is concave upward on an interval if f' is increasing on that interval, and concave downward if f' is decreasing.

What is the definition of an inflection point?

A function f has an inflection point at c if f'' does not exist or if f'' changes sign at c.

What is the First Fundamental Theorem of Calculus?

If f is continuous on [a, b], then the function F defined by F(x) = ∫[a,x] f(t) dt is continuous on [a, b] and differentiable on (a, b), with F'(x) = f(x).

State L’Hospital’s Rule.

If f(x) and g(x) are differentiable near a, and g'(x) ≠ 0, then lim x→a f(x)/g(x) = lim x→a f'(x)/g'(x) if the limits exist.