AP Calculus BC Unit 9 Cheat Sheet: Parametric, Polar & Vector-Valued Functions

The basics

What it covers: Parametric derivatives and arc length, vector-valued functions and planar motion, and polar coordinates and area. This is a BC-only unit.

Exam weight: About 11–12% of the BC exam.

The big question: How does calculus describe curves and motion beyond y = f(x)?

Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.

Key topics at a glance

Parametric Derivatives

dy/dx = (dy/dt)/(dx/dt). Horizontal tangent where dy/dt = 0; vertical where dx/dt = 0.

Second Derivative

d²y/dx² = [d/dt(dy/dx)]/(dx/dt) — differentiate the slope in t, divide by dx/dt.

Parametric Arc Length

L = ∫₀ᵇ √((dx/dt)² + (dy/dt)²) dt.

Vector-Valued Functions

r(t) = ⟨x(t), y(t)⟩. Differentiate and integrate componentwise.

Planar Motion

v = r′, a = r″, speed = |v| = √((dx/dt)² + (dy/dt)²); distance = ∫|v| dt.

Polar Coordinates

(r, θ): x = r cos θ, y = r sin θ; r = √(x²+y²), θ = arctan(y/x).

Polar Area (one curve)

A = ½∫ᶠᵀ r² dθ over the angle that traces the region once.

Area Between Polar Curves

A = ½∫ᶠᵀ (rₒₔₜₕ² − rₓₙₙₑₕ²) dθ.

The key formulas you must know

Parametric slope — dy/dx = (dy/dt)/(dx/dt).
Parametric 2nd derivative — [d/dt(dy/dx)]/(dx/dt).
Parametric arc length — ∫√((dx/dt)² + (dy/dt)²) dt.
Vector-valued function — r(t) = ⟨x(t), y(t)⟩.
Velocity / acceleration — r′(t) and r″(t), componentwise.
Speed — |v| = √((dx/dt)² + (dy/dt)²).
Total distance — ∫|v| dt.
Polar conversion — x = r cos θ, y = r sin θ.
Polar area (one curve) — ½∫ r² dθ.
Area between polar curves — ½∫(R² − r²) dθ.

Key themes to remember

Parametric slope is a ratio of rates. Divide the component derivatives.
Vectors work componentwise. Differentiate and integrate each part.
Speed is the magnitude of velocity. Distance integrates speed.
Polar area uses ½r². Slices are circular sectors, not rectangles.
Between polar curves, subtract the squares. Outer² minus inner².

Common exam traps

Divide, don’t just take dy/dt. The slope is (dy/dt)/(dx/dt).
The 2nd derivative divides by dx/dt again — not by d²x/dt².
Arc length / distance uses the square root of the sum of squares — keep both components.
Polar area is ½∫ r² dθ, not ∫ r dθ.
Square each polar radius separately — outer² − inner², not (R − r)².
Choose θ-limits that trace the region once to avoid double-counting.
Watch the quadrant when converting to polar with arctan.