Chapter 10.3, Problem 11E

To determine

To plot: The graph of the polar curve $r=1+\cos \theta$ by choosing an appropriate window.

Answer to Problem 11E

The graph of the polar curve $r=1+\cos \theta$ is shown in Figure (1). The graph is of type cardioid.

Explanation of Solution

Given information:

Equation of the polar curve: $r=1+\cos \theta$ .

Calculation:

It is evident that $\left|r\right|=\left|1+\cos \theta \right|\le 2$ for all $\theta$ , so points on the graph all within 2 units from the pole. The window should be as large as $\left[-2,2\right]$ by $\left[-1,1\right]$ . But, to maintain the aspect ratio, it is chosen to be $\left[-3,3\right]$ by $\left[-2,2\right]$ . The $\theta$ range is chosen as $0\le \theta \le 2\pi$ to get a full rotation around the graph, after which $1+\cos \theta$ will repeat the same graph periodically.

Plot the graph as follows:

  

Figure (1)

  $\left[-3,3\right]$ by $\left[-2,2\right]$

The graph is of type cardioid.