AP Calculus BC - Stuff you MUST Know Cold

AP Calculus BC flashcards for exam review.

L'Hopital's Rule

If lim (x->a) f(x) = 0 and lim (x->a) g(x) = 0, or lim (x->a) f(x) = ∞ and lim (x->a) g(x) = ∞, then lim (x->a) f(x)/g(x) = lim (x->a) f'(x)/g'(x).

Definition of Derivative

f'(x) = lim (h->0) [f(x+h) - f(x)] / h

Average Rate of Change

For points (a, f(a)) and (b, f(b)), the average rate of change of f(x) on [a, b] is [f(b) - f(a)] / [b - a].

Critical Point

A point where dy/dx = 0 or dy/dx is undefined.

Local Minimum

dy/dx goes from (-, 0, +) or (-, undefined, +), or d^2y/dx^2 > 0.

Local Maximum

dy/dx goes from (+, 0, -) or (+, undefined, -), or d^2y/dx^2 < 0.

Point of Inflection

Concavity changes; d^2y/dx^2 goes from (+, 0, -), (-, 0, +), (+, undefined, -), or (-, undefined, +).

Mean Value Theorem

If f(x) is continuous on [a, b] and differentiable on (a, b), there exists c in (a, b) such that f'(c) = [f(b) - f(a)] / [b - a].

Rolle's Theorem

If f(x) is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there exists c in (a, b) such that f'(c) = 0.

Chain Rule

d/dx [f(u)] = f'(u) * du/dx

Product Rule

d/dx (uv) = u(dv/dx) + v(du/dx)

Quotient Rule

d/dx (u/v) = [v(du/dx) - u(dv/dx)] / v^2

Average Value

The average value of f(x) on [a, b] is (1/(b-a)) * integral from a to b of f(x) dx. Also, (1/(b-a)) * integral from a to b of f(x) dx = f(c)

Fundamental Theorem of Calculus

The integral from a to b of f(x) dx = F(b) - F(a) where F'(x) = f(x).

Second Fundamental Theorem of Calculus

d/dx [integral from a to g(x) of f(t) dt] = f(g(x)) * g'(x)

Euler's Method

xnew = xold + ∆x; ynew = yold + (dy/dx)(xold, yold) * ∆x

Logistic Growth

P(t) = L / [1 + Ce^(-kt)], where L is carrying capacity and maximum growth rate occurs when P = L/2.

Area Inside a Polar Curve Leaf

Area = (1/2) * integral from θ1 to θ2 of r(θ)^2 dθ, where r(θ1) = r(θ2) = 0.

Slope of a Polar Curve

dy/dx = [dr/dθ * sin(θ) + r * cos(θ)] / [dr/dθ * cos(θ) - r * sin(θ)]

Integration by Parts

∫udv = uv - ∫vdu

Disk Method (Volume)

V = π∫[R(x)]^2 dx

Washer Method (Volume)

V = π∫([R(x)]^2 - [r(x)]^2) dx

Shell Method (Volume)

V = 2π∫r(x)h(x) dx

Volume of Known Cross Sections

V = ∫A(x) dx (perpendicular to x-axis) or V = ∫A(y) dy (perpendicular to y-axis)

Velocity

Velocity = d(position)/dt

Acceleration

Acceleration = d(velocity)/dt

Speed

|v(t)| = sqrt((x'(t))^2 + (y'(t))^2)

Distance Traveled

∫sqrt((x'(t))^2 + (y'(t))^2) dt

Taylor Series

f(x) ≈ f(c) + f'(c)(x-c) + f''(c)(x-c)^2/2! + f'''(c)(x-c)^3/3! + … + f^n(c)(x-c)^n/n!

Lagrange Error Bound

|f(x) - Pn(x)| ≤ [max|f^(n+1)(z)| * |x-c|^(n+1)] / (n+1)!, for all z between x and c

Alternating Series Error Bound

|S - SN| ≤ aN+1

Ratio Test

If lim (n->∞) |a(n+1)/a(n)| < 1, the series converges. If > 1, diverges. If = 1, test is inconclusive.

Arc Length (Function)

L = ∫sqrt(1 + (f'(x))^2) dx

Arc Length (Polar)

L = ∫sqrt(r(θ)^2 + (r'(θ))^2) dθ