Rates of Change in Polar Functions
| English | Chinese | Pinyin |
|---|---|---|
| rate of change | 变化率 | biàn huà lǜ |
| polar function | 极坐标函数 | jí zuò biāo hán shù |
How fast does the radius change?
- On a polar curve, as the angle sweeps around, the radius rises and falls.
- The question "how fast is $r$ changing per unit of angle?" is a rate of change.
- It tells you whether the curve is winding outward, inward, or holding steady.
- This links the rate-of-change ideas of Unit 1 to the polar world.
The average rate of change of r
- A polar function 极坐标函数 is $r = f(\theta)$, so we can measure how $r$ changes with $\theta$.
- The average rate of change 变化率 over an interval is $\dfrac{\Delta r}{\Delta \theta}$ — change in radius over change in angle.
- A positive value means the radius is growing; negative means it is shrinking.
- It is exactly the "output over input" rate, applied to $r$ and $\theta$.
As $\theta$ increases, if $r$ is increasing, the curve moves…
A growing radius means the point is getting farther out — the curve spirals outward.
Spiralling out or in
- If $r$ is increasing as $\theta$ grows, the curve moves farther from the origin — spiralling outward.
- If $r$ is decreasing, the curve winds inward toward the pole.
- A spiral like $r = \theta$ has a steadily positive rate, so it forever widens.
- The size of the rate sets how quickly the curve winds.
Watch the radius grow as the angle sweeps
In a spiral, r increases with theta. The rate of change of r decides how quickly the curve winds outward.
The average rate of change of $r$ over an interval of $\theta$ is $\dfrac{\Delta r}{\Delta \theta}$, the change in radius over the change in ____.
It measures how fast the radius changes per unit of angle — the polar version of a slope.
Select all true statements about rates of change in polar functions.
A constant $r$ traces a circular arc, not an outward spiral. The other three are correct.
When the rate is zero
- Where the rate of change of $r$ is zero, the radius is momentarily constant.
- A constant radius over an interval traces an arc of a circle.
- So a flat stretch in $r$ appears as a rounded, circular part of the curve.
- Peaks and troughs of $r$ are where the curve is farthest out or closest in.
If $r$ decreases as $\theta$ increases, the curve spirals inward toward the origin.
A shrinking radius pulls the point toward the pole, so the curve winds inward.
If the rate of change of $r$ is zero over an interval, the curve there is…
Zero rate means $r$ is constant, so the point stays the same distance out — an arc of a circle.
Reading the motion
- Track $r$ against $\theta$ to predict how the polar curve behaves.
- Rising $r$: petals or loops reaching outward. Falling $r$: returning toward the centre.
- Zero crossings of $r$ are where the curve passes through the origin.
- The rate of change turns a static picture into a story of motion.
The rate here is the change in radius per angle, not the speed of a point along the curve. A large $\tfrac{\Delta r}{\Delta\theta}$ means the distance from the origin is changing quickly — it does not directly measure how fast you travel around the curve.
For the spiral $r = \theta$, compare the radius at $\theta = 1$ and $\theta = 3$.
- $r$ goes from $1$ to $3$ as $\theta$ goes from $1$ to $3$.
- Average rate of change $= \dfrac{3 - 1}{3 - 1} = 1$: the radius grows one unit per radian.
- Positive and constant, so the spiral opens outward at a steady pace.
For a polar function $r = f(\theta)$, the average rate of change $\tfrac{\Delta r}{\Delta\theta}$ tells how the radius changes with angle: positive → spiralling outward, negative → inward, zero → a circular arc. It applies Unit 1's rate-of-change idea to $r$ and $\theta$.