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