AP Calculus Unit 9 Cheatsheet: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC only)

First derivative (slope of tangent):

dy/dx = (dy/dt) / (dx/dt)

Provided dx/dt ≠ 0

Second derivative (concavity):

d²y/dx² = [d/dt(dy/dx)] / (dx/dt)

Vector derivative:

If r(t) = ⟨x(t), y(t)⟩, then

r'(t) = ⟨dx/dt, dy/dt⟩

Vector integral:

∫ r(t) dt = ⟨∫ x(t) dt, ∫ y(t) dt⟩ + C

For parametric/vector problems:

• Identify what is given: parametric equations, vector function, or polar equation.
• Determine what is asked: derivative, arc length, motion quantity, or area.
• Choose the correct formula and verify all conditions (e.g., dx/dt ≠ 0).
• Differentiate or integrate each component separately.
• Simplify and interpret the answer in context.

For polar problems:

• Convert between polar and rectangular if needed.
• Use product rule when differentiating r cos(θ) or r sin(θ).
• For area, always use (1/2) ∫ r² dθ; identify correct bounds.
• For two curves, determine outer and inner, then subtract.