Unit 6 · Topic 3.15
Rates of Change in Polar Functions
Analyze how a signed radius changes as the angle changes, and connect that behavior to motion toward or away from the pole.
Learning Goals
- Distinguish the signed radius \(r\) from the distance \(|r|\) to the pole.
- Determine whether a polar point moves relatively toward or away from the pole.
- Calculate and interpret average rates of change of \(r\) with respect to \(\theta\).
- Connect relative extrema of \(r\) with relatively closest or farthest points.
- Use an average rate of change to estimate an interior radius value.
1. Track Two Related Quantities
For a polar function \(r=f(\theta)\), the output \(r\) is a signed radial displacement. The geometric distance from the plotted point to the pole is
An increasing signed radius does not always mean increasing distance. The sign of \(r\) must be considered together with whether \(f\) is increasing or decreasing.
2. Decide Whether the Point Moves Inward or Outward
| Sign of \(r\) | Behavior of \(r\) | Behavior of \(|r|\) | Polar motion |
|---|---|---|---|
| Positive | Increasing | Increasing | Away from the pole |
| Positive | Decreasing | Decreasing | Toward the pole |
| Negative | Increasing toward 0 | Decreasing | Toward the pole |
| Negative | Decreasing away from 0 | Increasing | Away from the pole |
3. Average Rate of Change of the Signed Radius
Over the angle interval \([\theta_1,\theta_2]\),
The numerator uses signed radius values. In radian mode, the result measures how many radius units \(r\) changes per radian, on average.
4. Interpret the Sign of the Rate
| AROC | Signed-radius interpretation |
|---|---|
| Positive | \(r\) increases as \(\theta\) increases |
| Negative | \(r\) decreases as \(\theta\) increases |
| Zero | The endpoint radii are equal |
The sign alone does not determine inward or outward motion; combine it with the sign of \(r\).
5. A Constant Radial Rate Produces a Spiral
For \(r=\theta\), equal increases in angle produce equal increases in radius. As the ray turns, the point moves steadily farther from the pole and traces an outward spiral.
6. Formula Example: \(r=\theta\)
From \(\theta=0\) to \(\theta=\pi\),
The signed radius increases by an average of 1 radial unit per radian. Because \(r\ge0\) and is increasing on this interval, the distance from the pole also increases.
7. Positive Rate but Decreasing Distance
Consider \(r=\theta-\pi\) on \(0\le\theta\le\pi/2\). The radius changes from \(-\pi\) to \(-\pi/2\), so
Although the signed radius increases, its magnitude decreases from \(\pi\) to \(\pi/2\). Since \(r<0\) throughout the interval, the point moves toward the pole.
8. AROC Is Not Polar-Curve Slope
The quotient \(\Delta r/\Delta\theta\) is the slope of a secant line on an ordinary graph whose horizontal coordinate is \(\theta\) and vertical coordinate is \(r\).
It is not the slope of the polar curve in the \(xy\)-plane, the arc length traveled, or the object's rectangular speed.
9. Calculate from a Table
| \(\theta\) | \(r=f(\theta)\) | AROC to next input |
|---|---|---|
| \(0\) | \(1\) | \(8/\pi\) |
| \(\pi/4\) | \(3\) | \(-4/\pi\) |
| \(\pi/2\) | \(2\) | \(-12/\pi\) |
| \(3\pi/4\) | \(-1\) | — |
For example, on \([\pi/4,\pi/2]\),
The last interval crosses from positive to negative radius. Its endpoint AROC describes signed-radius change but does not prove that distance changes monotonically between the endpoints.
10. Relative Extrema and Distance from the Pole
When \(r\) changes from increasing to decreasing or from decreasing to increasing, \(r\) has a relative extremum. Its geometric meaning depends on the sign.
| Signed-radius feature | Typical distance meaning |
|---|---|
| Positive relative maximum | Relatively far from the pole |
| Positive relative minimum | Relatively close to the pole |
| Negative relative maximum, closer to 0 | Relatively close to the pole |
| Negative relative minimum, farther below 0 | Relatively far from the pole |
11. Positive-Radius Example
For \(r=3+2\cos\theta\), the radius is always positive because \(1\le r\le5\). At \(\theta=0\), \(r=5\), so the point is relatively farthest from the pole. At \(\theta=\pi\), \(r=1\), so it is relatively closest.
On \([0,\pi]\),
The negative rate and positive radius agree that the point moves inward overall.
12. Negative-Radius Example
For \(r=-3-2\cos\theta\), the outputs lie between \(-5\) and \(-1\). At \(\theta=0\), the relative minimum \(r=-5\) gives distance 5, so the point is relatively farthest from the pole. At \(\theta=\pi\), the relative maximum \(r=-1\) gives distance 1, so the point is relatively closest.
This reversal is why the graph of \(r\) and the graph of \(|r|\) answer different questions.
13. Crossing the Pole Requires Extra Care
If \(r\) changes sign, the curve passes through the pole at an input where \(r=0\). The distance \(|r|\) reaches 0 there even if the signed-radius function continues increasing or decreasing without a relative extremum.
Locate every zero before using an interval-wide statement such as “moving toward the pole.”
14. Estimate an Interior Radius
Suppose \(r(\pi/6)=4.2\) and \(r(\pi/3)=3.6\). The average rate is
A constant-rate estimate inside the interval is
Since \(\pi/4\) is halfway between the two inputs, \(\widehat r(\pi/4)=3.9\). This is an estimate; the actual function may curve between the known values.
15. AP Workflow and Common Errors
- Identify the angle interval and endpoint radius values.
- Compute \(\Delta r/\Delta\theta\) in input order.
- Include units such as radial units per radian.
- Determine the sign of \(r\) throughout the interval.
- Combine the sign and increasing/decreasing behavior to describe distance.
- Check for zeros and relative extrema.
- Use a secant-rate model only as an estimate between known inputs.
- Do not replace signed radii with absolute values inside the AROC formula.
- Do not say a positive AROC always means outward motion.
- Do not call \(\Delta r/\Delta\theta\) the slope of the polar curve.
- Do not omit “per radian” from an interpretation.
- Do not infer behavior throughout an interval from endpoints when \(r\) may cross 0.
Key Takeaways
- Describe average rates of radial change with respect to angle and interpret them from polar tables and graphs.
- Core relationship: \(\frac{\Delta r}{\Delta\theta}=\frac{r(\theta_2)-r(\theta_1)}{\theta_2-\theta_1}\)
- Error check: A radial rate is not automatically the object's rectangular speed.
- For \(r=2\theta-1\), find the average rate of change on \([0,\pi/2]\) and state its units.
- A negative polar function increases from \(-6\) to \(-2\). Explain what happens to its distance from the pole.
- For \(r=4+3\cos\theta\), identify a relatively closest and farthest point on \(0\le\theta\le2\pi\).
- Explain why a positive AROC does not necessarily mean outward motion.
- Given \(r(\pi/4)=5\) and \(r(3\pi/4)=2\), use the average rate to estimate \(r(\pi/2)\).