Chapter 6.5, Problem 42E

Solution

To analyze: The graph given by the polar equation $r=\frac{\theta }{4}$ .

The equation represents a spiral of Archimedes. Domain of the graph is $\left(-\infty ,\infty \right)$ . The range is $\left(-\infty ,\infty \right)$ . The graph is continuous and has no symmetry. There is no maximum value. There are no asymptotes to the graph.

Given information:

The given polar equation is $r=\frac{\theta }{4}$ .

Calculation:

Plot the graph for the polar equation.

  

Figure (1)

From the figure (1), it can be observed that the equation represents a spiral of Archimedes.

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

Range of the graph is $\left(-\infty ,\infty \right)$ .

The graph is continuous and there is not symmetry about any axes.

Since $r$ values are increasing as the $\theta$ value increases, so it is an unbounded graph.

There is no maximum value of $r$ .

There are no asymptotes to the graph.

Therefore, the polar equation $r=\frac{\theta }{4}$ is analyzed with the help of figure (1).