Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Executive Summary

Some curves are better described by a parameter or by distance-and-angle. Parametric, polar, and vector forms each describe motion and area in their own language.

Parametric equations

x(t) and y(t) describe a curve as a parameter t varies. First derivative: dy/dx = (dy/dt)/(dx/dt). Second derivative: d²y/dx² = (d/dt [dy/dx]) / (dx/dt) — never just divide second derivatives. Arc length: integrate √((dx/dt)² + (dy/dt)²) dt.

Motion problems

Velocity vector: v = . Speed is its magnitude. Total distance traveled integrates speed. Position at time t is the integral of velocity plus the initial position. A particle is at rest when BOTH components of velocity are zero.

Vector-valued functions

r(t) = . r'(t) gives velocity; r''(t) gives acceleration. Speed = |r'(t)|. Integrals of vector functions integrate each component. Position, velocity, and acceleration chain exactly like their scalar cousins.

Polar coordinates

(r, θ): r is the distance from the origin, θ the angle from the positive x-axis. Convert: x = r cos θ, y = r sin θ, r² = x² + y². Polar area: (1/2)∫ r² dθ between two θ-bounds. The derivative dy/dx in polar form applies the parametric formulas with x(θ), y(θ).

Exam traps

d²y/dx² is NOT (d²y/dt²)/(d²x/dt²). Speed ≠ |velocity| alone — integrate speed for distance. Polar area uses (1/2)r², and r must be nonnegative within the bounds.