AP Calc BC: Lock In

y = tan(x), y' =

y' = sec²(x)

y = csc(x), y' =

y' = -csc(x)cot(x)

y = sec(x), y' =

y' = sec(x)tan(x)

y = cot(x), y' =

y' = -csc²(x)

y = sin⁻¹(x), y' =

y' = 1/√(1 - x²)

y = cos⁻¹(x), y' =

y' = -1/√(1 - x²)

y = tan⁻¹(x), y' =

y' = 1/(1 + x²)

y = cot⁻¹(x), y' =

y' = -1/(1 + x²)

y = a^x, y' =

y' = a^x ln(a)

y = log (base a) x, y' =

y' = 1/(x lna)

mean value theorem

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b)

f '(c) = [f(b) - f(a)]/(b - a)

P = M / (1 + Ae^(-Mkt))

logistic growth equation

length of curve

∫ √(1 + (dy/dx)²) dx over interval a to b

indeterminate forms

0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰

second derivative of parametrically defined curve

find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative, then divide by dx/dt

length of parametric curve

∫ √ (dx/dt)² + (dy/dt)² over interval from a to b

given velocity vectors dx/dt and dy/dt, find total distance travelled

∫ √ (dx/dt)² + (dy/dt)² over interval from a to b

area inside polar curve

1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta

area inside one polar curve and outside another polar curve

1/2 ∫ R² - r² over interval from a to b, find a & b by setting equations equal, solve for theta.

(1)-(x^2/2!) +(x^4/4!)-(x^6/6!)+...

Maclaurin series of cos(x)

(x)-(x^3/3!)+(x^5/5!)-(x^7/7!)+...

Maclaurin series of sin(x)

x - x²/2 + x³/3 - x⁴/4 + .....

Maclaurin series of ln(1+x)

1 + x2 + x3 +...xn,

Maclaurin series of 1/(1-x)