AP Calculus BC - Need to Know

These flashcards cover key concepts, definitions, and important theorems from AP Calculus BC lecture notes.

Quotient Rule

If f and g are differentiable functions, then the derivative of the quotient f/g is given by: (d/dx)(f/g) = (gf' - fg')/g^2.

Chain Rule

If h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x). It represents the derivative of a composition of functions.

Definition of Continuity

A function f is continuous at x = a iff: 1. f(a) exists, 2. lim (x→a) f(x) exists, 3. lim (x→a) f(x) = f(a).

Intermediate Value Theorem (IVT)

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

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) around x = a, and lim (x→a) f(x) = L and lim (x→a) h(x) = L, then lim (x→a) g(x) = L.

Definition of Derivative

The derivative of f at x is defined as: f'(x) = lim (h→0) [f(x+h) - f(x)]/h.

L'Hôpital's Rule

This rule is applied to find limits of indeterminate forms like 0/0 or ∞/∞ by differentiating the numerator and denominator.

Mean Value Theorem

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

First Derivative Test

If f'(x) changes from positive to negative at a point c, then f(c) is a relative maximum. If it changes from negative to positive, then f(c) is a relative minimum.

Total Distance Traveled

The total distance traveled from a to b is given by the integral of the absolute value of the velocity function: ∫ from a to b |v(t)| dt.

Fundamental Theorem of Calculus

If F is an antiderivative of f on [a, b], then ∫ from a to b f(x) dx = F(b) - F(a).