AP Calculus AB formulas and theorums

Pythagorean Identities

sin²x + cos²x = 1

tan²x + 1 = sec²x

cot²x + 1 = csc²x

Power Reducing Indentities

sin²x = (1 - cos2x) / 2

cos²x = (1 + cos2x) / 2

tan²x = (1 - cos2x) / (1 + cos2x)

Double Angle Identities (cosine)

cos2x = cos²x - sin²x

cos2x = 1 - 2sin²x

cos2x = 2cos²x - 1

Double Angle Identities (sine)

sin2x = 2sinx * cosx

Double Angle Identities (tangent)

tan2x = (2tanx) / (1-tan²x)

Half Angle Identities

sinx/2 = +- sqrt((1 - cosx) / 2)

cosx/2 = +- sqrt((1 + cosx) / 2)

tanx/2 = +- sqrt((1 - cosx) / (1 - cosx))

Sum of Angle Identitites

sin(a + b) = sinacosb + cosasinb

cos(a + b) = cosacosb - sinasinb

tan(a + b) = (tana + tanb) / (1 - tanatanb)

Mean Value Theorem (MVT)

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

f’(C) = (f(b) - f(a)) / (b-a) for some value C

Intermediate Value Theorem (IVT)

If f is continuous on [a,b] and f(a) ≠ f(b), then

each value between f(a) and f(b) is contained on [a,b]

Extreme Value Theorem (EVT)

If f is continuous on [a,b], then

f has an absolute minimum and maximum on [a,b] (which could include endpoints)

d/dx(arcsinx)

1 / sqrt(1 - x²)

d/dx(arccosx)

-1 / sqrt(1 - x²)

d/dx(arctanx)

1 / (1 + x²)

lim (sinx) / x

0

lim (1 - cosx) / x

0