Parametric Equations Explorer – AP Calculus BC

Learning Objectives

Understand motion defined by x(t) and y(t)

Calculate slope dy/dx from derivatives

Visualize tangent lines and normal lines on parametric curves

Observe particle velocity vectors in 2D space

Key Equations

$ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} $

$ v(t) = \langle x'(t), y'(t) \rangle $

$ \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}} $

Why It Matters

Parametric equations are fundamental for describing paths where $x$ doesn't uniquely determine $y$, like planetary orbits or projectile motion. Calculus helps us analyze the speed, direction, and curvature of these paths.