2020 AP Calculus BC Formula List

These flashcards cover key definitions, theorems, and concepts found in the AP Calculus BC Formula List.

Definition of the derivative

A way to express the rate of change of a function with respect to a variable.

Definition of continuity

A function f is continuous at c if and only if: 1) f(c) is defined; 2) $ext{lim}<em>{x ightarrow c} f(x)$ exists; 3) $ext{lim}</em>{x <br /> ightarrow c} f(x) = f(c)$.

Mean Value Theorem

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

Intermediate Value Theorem

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

Definition of a definite integral

ext{Definite Integral} = ext{lim}_{n
ightarrow ext{∞}} rac{1}{n} imes ext{sum of } f(x_i) over the interval.

Definition of a Critical Number

Let f be defined at c. If f'(c) is undefined, then c is a critical number of f.

First Derivative Test

A method to classify critical numbers: If f changes from negative to positive at c, then c is a relative minimum; if from positive to negative, then c is a relative maximum.

Second Derivative Test

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

Definition of Concavity

A function is concave upward on an interval if its first derivative is increasing; concave downward if the first derivative is decreasing.

Definition of an Inflection Point

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

First Fundamental Theorem of Calculus

If f is continuous on [a,b], then $ext{if } F(x) = ext{ }$ the integral of f from a to x, then F'(x) = f(x).

Second Fundamental Theorem of Calculus

If f is continuous on [a, b], then $ext{if } F(x) = ext{integral from a to x of } f(t) dt$, then F'(x) = f(x).

L'Hospital's Rule

If f and g are differentiable and g'(x) ≠ 0 near a, then: ext{lim}{x ightarrow a} rac{f(x)}{g(x)} = ext{lim}{x
ightarrow a} rac{f'(x)}{g'(x)} if the original limit is an indeterminate form.

Average Rate of Change

The average rate of change of f on [a, b] is given by $rac{f(b) - f(a)}{b - a}$.

Velocity from Position Function

Velocity is the derivative of the position function, v(t) = s'(t).

Integration by Parts

A technique for integrating the product of two functions, given by $ext{integral } u dv = uv - ext{integral } v du$.

Taylor Polynomial

The nth Taylor polynomial of f at c is: P_n(x) = f(c) + f'(c)(x-c) + rac{f''(c)}{2!}(x-c)^2 + … + rac{f^n(c)}{n!}(x-c)^n.

Geometric Series

The series converges if |r| < 1, where S = a/(1-r), otherwise it diverges.