AP Calculus AB Final/Exam: Memorizing

How to find if the speed is increasing/decreasing

• for speed to increase, the sign of velocity and acceleration have to be the same

• for speed to decrease, the sign of velocity and acceleration have to be different

How to find if the direction is positive/negative

• for direction to be positive, the sign of velocity and position have to be the same

• for direction to be negative, the sign of velocity and position have to be different

How to find if two particles are going in the same direction

Check to see when their velocities are the same sign

• equal the velocity formula to 0, then solve for critical numbers, then test to see when it is positive or negative

d/dx (|u|) =

(u / |u|) * (u’)

d/dx (cot u) =

- (csc²u) * (u’)

d/dx (csc u) =

- (csc u * cot u) * (u’)

d/dx (arcsin u) =

(u’) / (sqrt (1 - u²))

d/dx (arctan u) =

(u’) / (1 + u²)

∫ tan u du =

- ln |cos u| + C

∫ cot u du =

ln |sin u| + C

∫ sec u du =

ln |sec u + tan u| + C

∫ csc u du =

-ln |csc u + cot u| + C

∫ csc² u du =

-cot u + C

∫ (sec u * tan u) du =

sec u + C

∫ (csc u * cot u) du =

-csc u + C

∫ (1 / sqrt (a² - u²)) du =

arcsin (u/a) + C

∫ (1 / (a² + u²)) du =

(1/a) arctan (u/a) + C

∫ [1 / u (sqrt (a² - u²))] du =

(1/a) arcsec (|u| /a) + C

d/dx (LOG a U) =

(u’) / (ln a)(u)

d/dx (a^u) =

(ln a) (a^u) (u’)

derivative of inverse func (inverse function of f is g):

g’(x) = 1/ (f’(g(x))

∫ (a^u) du =

(a^u) / (ln a) + C

Intermediate Value Theorem

If f is continuous on closed interval [a,b] & k is a number between f(a) and f(b), then: there is at least one number c in [a,b] such that f(c) = k

*don’t mix up with MVT; this is for function f, not its derivative

Extreme Value Theorem

If f is continuous on closed interval [a,b], then: f has both a maximum & a minimum on the interval

Rolle’s Theorem:

• let f be continuous on closed interval [a,b] & differentiable on open interval (a,b)

if f(a) = f(b), then: there is at least one number c in (a,b) such that f’(c) = 0

Mean Value Theorem (for derivatives)

if f is continuous on closed interval [a,b] & differentiable on open interval (a,b), then: there exists a number c in (a,b) such that f’(c) = (f(b) - f(a)) / (b-a)

Average Value of Function on an Interval

(1/(b-a)) ∫ f(x) dx *interval of [a,b]

Average Rate of Change of Function on Interval [a,b]

[f(b) - f(a)] / (b-a)

Ellipse Area Formula

(pi)(a)(b) *a & b are the two radius

Equilateral Triangle Area formula

(sqrt (3) / 4) s²

Right Circular Cone Volume Formula

(1/3)(pi)(r²)(h)

Cone Volume Formula

(A*h) / 3 *A is area of base

Right Circular Cylinder Volume Formula

(pi)(r²)(h)

Right Circular Cylinder Lateral Surface Area Formula

2(pi)(r)(h)

Circle Area Formula

(pi)(r²)

Circle Circumference Formula

2(pi)(r)

Sphere Volume Formula

(4/3)(pi)(r³)

sin 2u =

2(sin u)(cos u)

cos 2u =

• cos²u - sin²u

• 1 - 2sin²u

• 2cos²u - 1

Phythagorean Identities (relationship between tan & sec)

tan²x + 1 = sec²x

Pythagorean Identities (relationship between cot & csc)

cot²x + 1 = csc²x

Trapezoidal Rule

((b-a) / 2n) [f(x) +…] *use pascal’s triangle for f(x)