AP Calc BC Formulas

Marilupe Hren updated as of 10.07/2026

Formal definition of the derivative

Alternative form of definition of the derivative:

Definition of continuity:

f is continuous at c if and only if

i) f(c ) is defined;

ii) the limit f(x) as x → c exists;

iii) lim x→c f(x) = f(c )

Average rate of change of f(x) on [a,b]?

Rolle’s Theorem:

If f is continuous on a closed interval [a, b] and

differentiable on the open interval (a, b), and

if the values of the function at the endpoints are equal,

then there exists at least one number c in (a, b) such that f'(c) = 0.

Mean Value Theorem:

If f is continuous on [a, b] and

differentiable on (a, b),

then there exists a number on (a, b) such that

f’(c ) = 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

Power Rule:

Chain Rule:

Product Rule:

Quotient Rule:

Derivative of (sin (u))

Derivative of (cos (u))

Derivative of (tan (u))

Derivative of (cot (u))

Derivative of (sec (u))

Derivative of (csc (u))

Definition of a Critical Number:

Let f be defined at c.

If f’(c )=0 or

if f’ is undefined at c,

then c is a critical number of f.

First Derivative Test:

Let c be a critical number of a function f

that is continuous on an open interval I containing c.

If f is differentiable on the interval,

except possible at c,

then f(c ) can be classified as follows.

1) If f’(x) changes from negative to positive at c, then f(c ) is a relative minimum of f.

2) If f’(x) changes from positive to negative at c, then f(c ) is a relative maximum of f.

Extreme Value Theorem:

If f is continuous on a closed interval [a, b],

then f attains both an absolute maximum value

and an absolute minimum value at some numbers in [a, b]

Second Derivative test

Let F be a function such that f’( c) =0 and the second derivative of f exists on an open interval containing c

1) If f’’(c ) > 0, the f(c ) is a relative minimum of f

2) if f’’(c ) < 0 then f(c ) is a relative maximum of f

Definition of concavity

If f is differentiable on a open interval I.

The graph of f is concave upward on I if f’ is increasing on the interval,

and concave downward on I if f’ is decreasing on the interval

Test for concavity

Let f be a function whose second derivative exists on an open interval I

1) If f’’(x ) > 0 for all x in I, then the hraph of f is concave upward on I

2) If f’’(x ) < 0 for all x in I, then the graph of f is concave downward on I

Definition of an Inflection Point

A function f has an infelction point at (c, f( c))

1) if f’’( c) = 0 or f’’(c ) does not exist

2) if f’’ changes signs at x=c