Rates of Change

Trigonometry · Axiom Academy

Discover how curves change as you move along them—the foundation of calculus in parametric and polar coordinates! Imagine a race car moving along a track. At any moment, the car has a velocity —both speed and direction. Move the slider to see how velocity changes along the curve! Curves can be described using parametric equations : x(t) and y(t). The rate of change becomes dx/dt and dy/dt—the components of velocity! In polar coordinates, curves are described by r(θ). The rate of change dr/dθ tells us how the radius changes as we rotate! Different curves have different rates of change. Choose a curve to see how its derivative behavior differs! Rates of change describe how curves behave at each point. For parametric curves, dx/dt and dy/dt give velocity components. For polar curves, dr/dθ describes how radius changes with angle. These derivatives are tangent to the curve—literally touching at just one point! Rates of change in parametric and polar forms are used everywhere: satellite trajectories (orbital mechanics), computer graphics (smooth curves), robotics (motion planning), physics (planetary motion), and engineering (gear design). Anytime something moves along a curved path, these concepts apply!

This is the written version of the interactive lesson above. See the full Trigonometry course.