AP Calc Things/Formulas to Memorize

lim x→ 0 of sinx/x

1

d/dx tanx

sec²x

d/dxsecx

secxtanx

d/dx cscx

-cscx cotx

d/dx cotx

-csc²x

d/dx a^x

a^x lna

d/dx lnx

1/x

d/dx logbx

1/xlnb

limit definition of derivative

f’(x) = lim h→ 0 f(x+h) - f(x) / h

derivatives of inverse functions

f’(x) = 1 / g’ (f(x))

when a function is differentiable

one sided limits are equal

IVT

if f is continuous on [a,b] then some value c between a and b exists

MVT

if f is continuous on [a,b] and differentiable on (a,b) then there is at least one point where AROC = IROC

f’(x) = f(b) - f(a) / b - a

optimization steps

1. draw a picture, identify unknowns

2. write area as a function of unknown variables

3. write out a constraint to solve for one variable

4. rewrite equation in terms of one variable

5. find derivative and set equal to 0 to find critical points

6. ensure answer is in domain and check if the pt is a local max or min

   1. can use 2nd deriv test. if it is <0, concave down, local max, vise versa

related rates steps

1. draw picture

2. write goal

3. write equation to relate variables

4. differentiate both sides

5. plug in values to solve for goal

volume of a cylinder

pi r² h

volume of sphere, SA

4/3 pi r³, 4 pi r²

area and circumference of circle

pi r² and 2 pi r

SA of cylinder

2 pi r h + 2 pi r²

cone

1/3 pi r² h *changing radius

exteme value theorem

if f is continuous over a closed interval, then it has a max and min over that interval

Newton’s method

x_n+1 = x_n - f(x_n)/f’(x_n)

area of a trapezoid

½ (b₁ + b₂) * h

anti-power rule

x^n = x^n+1 / n+1

integration by parts formula

int. udv = uv - int vdu

TRAM

T = h/2 (y0 + 2y1 + 2y2 +… 2yn + yn)

Simpson’s Rule

S = h/3 (y0 + 4y1 + 2y2 + 4y3 + … 2yn-2 + 4yn-1 + yn)

differential equation of exponential

dy/dt = ky

for inverse, k/y

y = Pe^rt or Ce^kt

k>0 growth, k< 0 decay

Newton’s law of cooling / heating

T_t - T_s = [T₀ - T_s] e^-kt

positive k for heating

dT/dT = -k[T_t - T_s]

logistic growth

dP/dT = kP(M-P)

P = M / 1 + Ae^-(Mk)t

M = carrying capacity

A and K = constants

Euler’s Method Columns

n, x_n, y_n, dy/dx, dy (dy/dx * dx). y_n+1 (dy + y_n)