Functions of One Variable

Domain, Range & Function Graphs of Parametric Equations 2D 3D Rectangular Coordinates Contents

A parametrized curve in two-space is the collection of points whose coordinates x and y are given as functions of the parameter t.

The domain of the parametrized curve is the set of t for which x(t) and y(t) are defined.

The collection of points (x(t), y(t)) in the plane is the range of the parametric curve.

The x-range of the curve is the set of k with x(t)=k for some t. Similarly we define the y-range of the parametrized curve.

[D]

Level Sets & Contours of Parametric Equations 2D Rectangular Coordinates Top of Page Contents

For a parametrized curve (x(t),y(t)) in the plane, the x-level set at level k is the set of points in the domain with x(t) = k and similarly for the y-level set at level k. The points (x(t),y(t)) with x(t) = k form the x-contour at height k and similarly for the y-contour at height k.

[D]

Exercises

In the default example in the demo above, start the red hotspot to the left of the parametrized curve and slowly move it to the right so that it ends up to the right of the curve. Describe what happens during this motion.

Continuity of Parametric Equations 2D 3D Rectangular Coordinates Top of Page Contents

If the coordinate functions x(t) and y(t) are continuous functions of the parameter t, then the function that sends t to (x(t), y(t)) is continuous at t₀ if for every ε > 0 there is a δ > 0 such that (x(t₀), y(t₀)) lies within ε of (x(t₀), y(t₀)) if | t - t₀ | < δ.

This means that for any t₀ in the domain, and any positive ε, there is a δ such that (x(t),y(t)) is within the disc of radius ε about (x(t₀),y(t₀)) whenever | t - t₀ | is less than δ. We achieve this by choosing δ so small that | x(t) - x(t₀) | < ε/2 and | y(t) - y(t₀) | < ε/2, by virtue of the continuity of x(t) and y(t) at t₀. Then √((x(t)-x(t₀))² + (y(t)-y(t₀))²) < √(ε²/4 + ε²/4)) = ε/2 < ε if | t - t₀ | < δ.

[D]