Topic 3.15 Notes – Rates of Change in Polar Functions

Signed Radius and Distance from the Origin

In polar coordinates, a point is written as $(r,\theta)$. Here, $\theta$ tells you the direction, and $r$ tells you how far to go from the origin.

The catch is that distance from the origin is $|r|$, not $r$.

• If $r>0$, the point is on the terminal side of angle $\theta$, $r$ units from the origin.
• If $r<0$, the point is $|r|$ units from the origin in the opposite direction.
• If $r=0$, the point is at the origin.

This diagram shows the same angle paired with a positive radius and a negative radius.

As $\theta$ increases, use this rule set:

• positive and increasing $\rightarrow$ farther from the origin
• positive and decreasing $\rightarrow$ closer to the origin
• negative and increasing $\rightarrow$ closer to the origin
• negative and decreasing $\rightarrow$ farther from the origin

That is one of the most tested ideas here. “Increasing $r$” is about the signed radius. “Closer/farther” is about $|r|$. Those are not always the same thing.

Reading Radial Behavior from a Formula, Table, or Graph

No matter how the information is given, you are always checking the same two things together:

1. Is $r$ positive or negative?
2. Is $r$ increasing or decreasing?

A graph of $r$ versus $\theta$ is usually the easiest, because you can read sign from whether the graph is above or below the axis, and direction of change from whether it rises or falls.

For $r=2\cos\theta$ on $0\le\theta\le\pi$, the graph makes the sign change at $\theta=\frac{\pi}{2}$ especially easy to see:

• On $0<\theta<\frac{\pi}{2}$, $r$ is positive and decreasing $\rightarrow$ the point moves closer.
• At $\theta=\frac{\pi}{2}$, $r=0$ $\rightarrow$ the point is at the origin.
• On $\frac{\pi}{2}<\theta<\pi$, $r$ is negative and decreasing $\rightarrow$ the point moves farther.

Graph of $r=2\cos\theta$ on $0\le\theta\le\pi$

A common trap is thinking “decreasing” always means moving toward the origin. This graph shows why that is not true. It only does that when $r$ is positive. Also, “closer/farther from the origin” is about radial distance, not distance traveled along the curve.

Closest and Farthest Points from Relative Extrema

When $r$ changes from increasing to decreasing, it has a relative maximum. When it changes from decreasing to increasing, it has a relative minimum.

These can give closest or farthest points from the origin, but the sign matters.

• If $r>0$
  • relative maximum of $r$ $\rightarrow$ relatively farthest
  • relative minimum of $r$ $\rightarrow$ relatively closest
• If $r<0$
  • relative maximum of $r$ $\rightarrow$ relatively closest
  • relative minimum of $r$ $\rightarrow$ relatively farthest

Compare these:

• $g(\theta)=2+\cos\theta$ stays positive, so its relative max gives a farthest point and its relative min gives a closest point.
• $h(\theta)=-2+\cos\theta$ stays negative, so that flips.

If $r=0$, the point is at the origin, so it is a closest point even if $r$ does not turn there. On a restricted interval, also check endpoints for absolute closest/farthest. And don’t confuse this with highest or leftmost on the graph.

Average Rate of Change of $r$ with Respect to $\theta$

The average rate of change of signed radius is

$$\frac{f(\theta_2)-f(\theta_1)}{\theta_2-\theta_1}.$$

This tells you the average change in signed radius per unit of angle. Usually the units are radial units per radian.

On the graph of $r$ versus $\theta$, this is the slope of the secant line. It is not the slope between two points on the polar curve in the $xy$-plane.

Example for $r=2\cos\theta$ on $\left[0,\frac{\pi}{2}\right]$:

$$\frac{2\cos(\pi/2)-2\cos(0)}{\frac{\pi}{2}-0} = \frac{0-2}{\pi/2} = -\frac{4}{\pi}$$

So the signed radius decreases on average by $\frac{4}{\pi}$ units per radian.

• positive average rate $\rightarrow$ net increase in signed radius
• negative average rate $\rightarrow$ net decrease
• zero average rate $\rightarrow$ same endpoint signed radius

Estimating Radius Values with Average Rate

You can use the secant line to estimate radius values inside an interval.

1. Find the average rate $$m=\frac{f(\theta_2)-f(\theta_1)}{\theta_2-\theta_1}$$
2. Use $$f(\theta)\approx f(\theta_1)+m(\theta-\theta_1)$$

For $f(\theta)=2\cos\theta$ on $\left[0,\frac{\pi}{2}\right]$, $m=-\frac{4}{\pi}$. Estimate $f\left(\frac{\pi}{4}\right)$:

$$f\left(\frac{\pi}{4}\right)\approx 2+\left(-\frac{4}{\pi}\right)\left(\frac{\pi}{4}\right)=1$$

That estimate is a signed radius. If your estimate is negative, the distance from the origin is its absolute value.

This works best on short intervals or where the $r$-versus-$\theta$ graph is nearly linear. This topic stops at average rate of change. Derivatives, tangent slopes to polar curves, and speed are outside AP scope here.