Cusps

Definition

Cusps are special points on a curve where the curve has a sharp point and the derivative is undefined. They are typically found in parametric equations when both derivatives with respect to the parameter are zero at the same point.

5 Must Know Facts For Your Next Test

1. They occur when $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} = 0$ simultaneously.
2. A cusp can indicate a change in direction or speed in motion problems.
3. In polar coordinates, cusps can appear at specific angles where radial distance changes sharply.
4. Cusps differentiate from smooth curves by having undefined tangents at those points.
5. They often signify critical points in optimization problems involving parametric equations.

Review Questions

• What conditions must be met for a cusp to exist on a parametric curve?
• How do you distinguish a cusp from other types of singular points on a graph?
• What does the presence of a cusp imply about the behavior of a particle moving along the path?

Related terms

Parametric Equations:

Equations that express the coordinates of the points making up a geometric object as functions of a parameter.

Polar Coordinates:

A coordinate system where each point on a plane is determined by an angle and its distance from a reference point.

Derivative:

A measure of how a function changes as its input changes; represents slope or rate of change.