The characteristics of polar functions can be analyzed like any other type of function. We are interested in many of the same characteristics that we have studied for other function types like: rate of change, increasing and decreasing intervals, positive and negative intervals, and extrema.

Remember, a polar function is of the form r = f\left(\theta \right). Here, the output values are radii, while the input values are angles.

If a polar function is positive and increasing or negative and decreasing, the distance between f\left(\theta \right) and the origin (which is represented by r) is increasing. On the other hand, if the polar function is positive and decreasing or negative and increasing, this distance is decreasing.

Now, suppose the function changes from increasing to decreasing, or decreasing to increasing, over an interval. In that case, the function hits a relative extremum on that interval. This means the point is either closest to or farthest from the origin.

Next, let's explore the concept of the average rate of change. The average rate of change of r with respect to \theta over an interval of \theta is the ratio of the change in the radius values to the change in \theta. This average rate of change shows how the radius is changing per radian. What's more, this rate can help us estimate values of the function within the interval.

Algebraically we can represent the average rate of change as:

\displaystyle \dfrac{\Delta r}{\Delta \theta}=\dfrac{f\left(\theta_2\right)-f\left(\theta_1\right)}{\theta_2-\theta_1}

\bm{\Delta r}

change in radius

\bm{\Delta \theta}

change in angle

\bm{\theta}

an angle

\bm{f\left(\theta\right)}

radius at a given angle

Examples

Example 1

Consider the polar function r=2-8 \sin \theta.

Create a table of values for the function.

Worked Solution

Apply the idea

\theta	0	\dfrac{\pi}{3}	\dfrac{\pi}{2}	\dfrac{2\pi}{3}	\pi	\dfrac{4\pi}{3}	\dfrac{3\pi}{2}	\dfrac{5\pi}{3}	2\pi
r	2	-4.93	-6	-4.93	2	8.93	10	8.93	2

State the increasing and decreasing intervals.

Worked Solution

Create a strategy

Analyze how the values in the table are changing.

Apply the idea

Decreasing on the intervals \left(0,\dfrac{\pi}{2}\right) and \left(\dfrac{3\pi}{2},2\pi\right).

Increasing on the interval \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right).

Identify the maximum and minimum values.

Worked Solution

Create a strategy

Use the r-values of the table from part (a).

Apply the idea

Maximum: 10

Minimum: -6

Is the average rate of change faster between \theta=\dfrac{\pi}{3} and \theta=\dfrac{2\pi}{3} or between \theta=\pi and \theta=\dfrac{3\pi}{2}?

Worked Solution

Create a strategy

Use the average rate of change formula:

\dfrac{\Delta r}{\Delta \theta}=\dfrac{f\left(\theta_2\right)-f\left(\theta_1\right)}{\theta_2-\theta_1}

Apply the idea

Find the average rate of change between \theta=\dfrac{\pi}{3} and \theta=\dfrac{2\pi}{3}

\dfrac{\Delta r}{\Delta \theta}=\dfrac{-4.93-\left(-4.93\right)}{\dfrac{2\pi}{3}-\dfrac{\pi}{3}}=0

Find the average rate of change between \theta=\pi and \theta=\dfrac{3\pi}{2}

\dfrac{\Delta r}{\Delta \theta}=\dfrac{10-2}{\dfrac{3\pi}{2}-\pi}=\dfrac{8}{\dfrac{\pi}{2}}=\dfrac{16}{\pi}

\dfrac{16}{\pi}>0 so the average rate of change is greater between \theta=\pi and \theta=\dfrac{3\pi}{2}.

Is the distance from the origin increasing or decreasing on the interval \left[\pi,\dfrac{3\pi}{2}\right].

Worked Solution

Create a strategy

Analyze the values in the table between \theta=\pi and \theta=\dfrac{3\pi}{2}. We need to determine whether r is positive or negative on that interval and whether it is increasing or decreasing.

Apply the idea

See the section of the table between \theta=\pi and \theta=\dfrac{3\pi}{2}:

\theta	\pi	\dfrac{4\pi}{3}	\dfrac{3\pi}{2}
r	2	8.93	10

The r-values are positive and increasing so the distance from the origin is increasing.

Idea summary

	r \text{ is positive}	r \text{ is negative}
r \text{ is increasing}	distance from origin is increasing	distance from origin is decreasing
r \text{ is decreasing}	distance from origin is decreasing	distance from origin is increasing

The average rate of change indicates the rate at which the radius is changing per radian and can calculated with the formula:

\dfrac{\Delta r}{\Delta \theta}=\dfrac{f\left(\theta_2\right)-f\left(\theta_1\right)}{\theta_2-\theta_1}

3.15 Rates of change in polar functions

Introduction

Ideas

Characteristics of polar functions

Examples

Example 1

Outcomes

3.15.A

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