Calculus BC Formulas

Logistic Growth

dP/dt = kP(1-P/L), P is the population size, k is the growth rate, and L is the carrying capacity

Circle Area

A = πr²

Sphere Volume

V = (4/3)πr³

Cylinder Volume

V = πr²h

Distance Formula

D = √((x2 - x1)² + (y2 - y1)²).

Mean Value Theorem

Where instantaneous slope = average slope if functions are continuous in the interval

Average Value Theorem

Integral over interval

Trapezoid Area

A = 1/2(b1 + b2)h, where b1 and b2 are the lengths of the parallel sides and h is the height

Derivative of an integral function

d/dx integral [a, g(x)] f(x) dx = f(g(x)) g'(x)

Intermediate Value Theorem

If a function is continuous on a closed interval [a, b], then it takes every value between f(a) and f(b)

Extrema Value Theorem

If a function is continuous on an interval, then it will have extrema values at critical points or endpoints

Logrithmic differentiation

dy/dx (y=b^f(x)) = b^f(x) f'(x) ln(b)

Euler’s Method

x, y, dy/dx, delta y (dy/dx times delta x)

2D area created by two functions

Integral (a,b) [f(x) - g(x)] dx

Arclength

Integral (a,b) √(1 + (f’(x))²) dx

Disc formula

Volume = π∫(a,b) [f(x)]² dx

Washer formula

Volume = π∫(a,b) ([f(x)]² - [g(x)]²) dx

Shell formula

Volume = 2π∫(a,b) x[f(x) - g(x)] dx

Cross-sectional squares

Volume = ∫(a,b) [f(x) - g(x)]² dx

Cross-sectional equilateral triangles

Volume = (sqrt(3)/4)∫(a,b) [f(x) - g(x)]² dx

Cross-sectional semicircles

Volume = (π/2)∫(a,b) [(f(x) - g(x))/2]² dx

Alternating series desired accuracy

|t(n+1)| < desired error

LaGrange Error (non-alternating)

Error < 1/(n+1)! f^(n+1) (c) (x-a)^(n+1)

Sum of Geometric Series

a/(1-r), series convergences when -1 < r < 1

Derivative of sec(x)

sec(x)tan(x)

Derivative of cot(x)

-csc^2(x)

Derivative of csc(x)

-csc(x)cot(x)

Derivative of arcsin(x)

1/√(1 - x^2)

Derivative of arccos(x)

-1/√(1 - x^2)

Derivative of arctan(x)

1/(1 + x^2)

Series for ln(x)

ln(x) = (x-1) - (1/2)(x-1)^2 + (1/3)(x-1)^3 - (1/4)(x-1)^4 + … for 0 < x < 2

Series for arctan(x)

arctan(x) = x - (1/3)x^3 + (1/5)x^5 - (1/7)x^7 + … for -1 < x < 1

Speed of a parametric object

sqrt[(dx/dt)² + (dy/dt)²]

Parametric arclength

Integral (t1, t2) sqrt[(dx/dt)² + (dy/dt)²] dx

Parametric concavity

[d/dt(dy/dx)] / (dx/dt)

Polar curve area

Integral (a, b) (1/2)r^2 dθ

Polar curve arclength

Integral (a, b) sqrt[r^2 + (dr/dθ)²] dθ

Derivative any log base

1 / (x ln(logbase))

Cross-sectional isoceles right triangle

(1/4) Integral (a,b) (f(x))² dx