Parametric Curves

Parametric curves represent paths in the plane by expressing x and y as separate functions of a parameter, typically t. This topic covers computing derivatives of parametric equations (dy/dx via the chain rule) and calculating the arc length of parametric curves using integration.

Parametric equations describe the motion of objects in physics, define Bezier curves in computer graphics, and model trajectories in robotics and animation.

Practice Tips

• 1To find dy/dx for parametric curves, compute (dy/dt) / (dx/dt); never try to eliminate the parameter if you only need the slope.
• 2For arc length, use the formula integral of sqrt((dx/dt)^2 + (dy/dt)^2) dt, and simplify the expression under the radical before integrating.
• 3Sketch the curve by plotting several values of t and noting the direction of travel, which helps you interpret derivative results geometrically.