Problem 54 - Calculus Early Transcendentals

Question

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$\text { Cissoid of Diocles } x=2 \sin 2 t, y=\frac{2 \sin ^{3} t}{\cos t}$$

Step-by-Step Solution

Answer

Answer: The appropriate interval for parameter t is [0, 2π] to fully display all features of interest. The graph of the Cissoid of Diocles curve presents a notable cusp at the origin (0,0) and an interesting shape that extends to the right before wrapping back towards the cusp.

Step 1: Determine an appropriate interval for t

To find an appropriate interval for t, we need to consider the periodic nature of sine and cosine functions. Because the sine and cosine functions repeat every \(2\pi\), we can use the interval \(t \in [0, 2\pi]\) to produce all features of interest in the curve. This will avoid repeated sections of the curve, and ensure a complete visualization.

Step 2: Use a graphing utility to plot the curve

Utilize a graphing utility, such as Desmos, GeoGebra, or your preferred choice, to input the parametric equations: $$x = 2 \sin 2t$$ $$y = \frac{2 \sin^3 t}{\cos t}$$ Select the interval \(t \in [0, 2\pi]\) on the utility. This will complete the graph for the Cissoid of Diocles curve within the chosen values for t.

Step 3: Describe the features of interest in the curve

After plotting the curve, you will observe an interesting shape with a cusp at the origin \((0,0)\). The curve extends to the right and then wraps back toward the cusp, resulting in the Cissoid of Diocles shape. You can use the graphing utility to further explore the properties of this curve, such as the slope at certain points and the maximum and minimum values for both x and y coordinates.