AP Calculus BC Formulas to Memorize

Definition of Continuity

(bottom line)

Intermediate Value Theorem

If f is continuous on [a,b]

and f(a) < k < f(b) or f(b) < k < f(a),

then there is at least one c in (a,b) such that f(c)=k

Definition of Derivative

= f'(x)

Alternate Form of Derivative

Position Function for an Object in Free Fall

s(t) = -16t^2+v0t+s0

Acceleration Function for an Object in Free Fall

a(t) = -32 ft/sec^2

Product Rule

Quotient Rule

Trig Derivatives

(with x' at the end if implicit differentiation)

Relative Extremum (Min/Max)

a high/low point relative to the points around it; can only occur at a critical value

Absolute Extremum (Min/Max)

the highest/lowest point on a given interval; can occur at a critical value OR and endpoint

Mean Value Theorem

If f(x) is continuous on [a,b] and differentiable on (a,b),

then there exists an x-value such that f'(x) = f(a)-f(b) / a-b

Trig Integrals

ʃcosu du = sinu + c

ʃsinu du = -cosu + c

ʃsec^2u du = tanu + c

ʃcsc^2u du = -cotu + c

ʃsecutanu du = secu + c

ʃcscucotu du = -cscu + c

Differential Form of the Derivative

dy = f'(x)dx

1st F.T.C.

(1st line)

2nd F.T.C.

d/dx ʃ(from u to v) g(t) dt = g(v)v'-g(u)u'

Average Value of a Function on [a,b]

Average Rate of Change on [a,b]

(1/a-b) ʃ(from a to b) f'(x)dx

Average Velocity on [a,b]

(1/a-b) ʃ(from a to b) v(t)dx

Average Acceleration on [a,b]

(1/a-b) ʃ(from a to b) a(t)dx

Average Anything on [a,b]

(1/a-b) ʃ(from a to b) (anything)dx