Chapter 6.5, Problem 37E

Solution

To analyze: The graph given by the polar equation $r=2+5\cos \theta$ .

The equation represents a limaçon with an inner loop. Domain of the graph is $\left(-\infty ,\infty \right)$ . The range is $\left[-3,7\right]$ . The graph is continuous and symmetric about the $x-$ axis. Maximum value of $r$ is $7$ . There are no asymptotes to the graph.

Given information:

The given polar equation is $r=2+5\cos \theta$ .

Calculation:

Plot the graph for the polar equation.

  

Figure (1)

From the figure (1), it can be observed that the equation represents limaçon with an inner loop.

The equation is defined for all real numbers of $\theta$ in the domain $\left(-\infty ,\infty \right)$ .

Range of $\cos \theta$ is, $-1\le \cos \theta \le 1$ .

The range of $r=2+5\cos \theta$ will be $\left[-3,7\right]$ considering $1$ as maximum value and $-1$ as minimum value for $\cos \theta$ .

The graph is continuous. The graph is symmetric about the $x-$ axis.

Since $r$ values are bounded between $\left[-3,7\right]$ , so it is a bounded graph.

The maximum value of $r$ is $7$ .

There are no asymptotes to the graph.

Therefore, the polar equation $r=2+5\cos \theta$ is analyzed with the help of figure (1).