AP Calculus AB Theorems, Rules, Derivatives, and Integrals

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) for when x is near a and lim f(x) as x → a = lim h(x) as x → a = L, then lim g(x) as x → a = L

Intermediate Value Theorem (IVT)

If f is a continuous function on a closed interval [a,b] and d is a number between f(a) and f(b), then there is a guaranteed value c on [a,b] such that f(c) = d

Product Rule

d/dx [f(x)g(x)] = g(x)f’(x) +f(x)g’(x)
d/dx [uv] = Vdu + Udv

Quotient Rule

d/dx [f(x)/g(x)] = g(x)f’(x) - f(x)g’(x) / [g’(x)]²

Chain Rule

d/dx [f(g(x))] = f’(g(x)) [g’(x)]

L’Hopital’s Rule

If lim f(x)/g(x) as x → c produces indeterminate form 0/0 or ∞/∞ and lim f’(x)/g’(x) as x → c = L, then lim f(x)/g(x) as x → c = L

Mean Value Theorem (MVT)

If a function f is continuous on a closed interval [a,b] and differentiable on the interior (a,b), then there must be a c on (a,b) such that f’(c) = f(b)-f(a)/b-a

Extreme Value Theorem (EVT)

If f(x) is continuous on a closed interval [a,b], there exists a c on [a,b] such that f(c ) ≥ f(x) and there exists a d on [a,b] such that f(d) ≤ f(x)