Question

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Prolate cycloid: } x=2 \theta-4 \sin \theta, \quad y=2-4 \cos \theta $$

Short answer

Based on the solution steps, we can graph the curve of the given parametric equations, determine its direction by tracing key points, and identify non-smooth points by looking for sharp bends or cusps in the graph.

Step-by-step solution

Input the Parametric Equations

Start by inputting the given parametric equations \(x = 2\theta - 4\sin\theta\) and \(y = 2 - 4\cos\theta\) into the graphing utility.

Graphing the Curve

After inputting the equations, proceed to generate the graph. Use the trace function to verify the direction of increasing \(theta\) on the graph.

Indicating Direction of the Curve

Choose a few key points on the curve, such as turning points, and trace the curve from these points. This way, the direction of the curve can be easily determined by observing which way the traced points move as \(theta\) increases.

Identifying Non-Smooth Points

The curve is not smooth at points where it bends sharply or where there are cusp points. Look closely at the graph and identify these points. They occur where the velocity of the particle (the derivative of the position) is not defined or changes its direction abruptly.