AP Calculus BC - Stuff you MUST Know Cold

AP Calculus BC Flashcards

L'Hopital's Rule

If limit of f(a)/g(a) = 0/0 or ∞/∞, 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, represents the slope of the tangent line.

Average Rate of Change

For points (a, f(a)) and (b, f(b)), it is [f(b) - f(a)] / (b - a), represents the slope of the secant line.

Curve Sketching - Critical Point

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

Curve Sketching - Local Minimum

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

Curve Sketching - Local Maximum

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

Curve Sketching - Point of Inflection

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

Mean Value Theorem

If f(x) is continuous on [a, b] and differentiable on (a, b), then 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], 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 of a Function

1/(b-a) * integral from a to b of f(x) dx

Fundamental Theorem of Calculus

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(x) dx] = f(g(x)) * g'(x)

Euler's Method

xnew = xold + deltax, ynew = yold + (dy/dx) * deltax, where dy/dx = f(x,y)

Logistic Growth

dP/dt = kP(L-P) or dP/dt = kP(1 - P/L) where L is carrying capacity.

Logistic Growth - Max Growth Rate

Maximum growth rate occurs when P = L/2 where L is the carrying capacity.

Polar Curve Area

Area inside a leaf = 1/2 * integral from theta1 to theta2 of r(theta)^2 d(theta), where r(theta1) = r(theta2) = 0

Polar Curve Slope

dy/dx = [dr/d(theta) * sin(theta) + r * cos(theta)] / [dr/d(theta) * cos(theta) - r * sin(theta)]

Integration by Parts

Integral of u dv = uv - integral of v du

Disk Method

V = pi * integral from a to b of R(x)^2 dx

Washer Method

V = pi * integral from a to b of [R(x)^2 - r(x)^2] dx

Shell Method

V = 2pi * integral from a to b of r(x)h(x) dx

Volume - Known Cross Sections

V = integral from a to b of A(x) dx (perpendicular to x-axis), V = integral from c to d of A(y) dy (perpendicular to y-axis)

Velocity

d(position)/dt

Acceleration

d(velocity)/dt

Speed

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

Distance Traveled

Integral from tinitial to tfinal of 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) - P_n(x)| <= [max|f^(n+1)(z)| / (n+1)!] * |x-c|^(n+1)

Arc Length

For a function: L = integral from a to b of sqrt(1 + (f'(x))^2) dx. For a polar graph: L = integral from theta1 to theta2 of sqrt(r(theta)^2 + (r'(theta))^2) d(theta)