Rates of Change in Polar Functions · 极坐标函数中的变化率
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| rate of change/reɪt ɒv tʃeɪndʒ/ | 变化率 | biàn huà lǜ |
| polar function/ˈpəʊlə ˈfʌŋkʃn/ | 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.
半径变化得有多快?
- 在极坐标曲线上,当角扫一圈时,半径时升时降。
- "$r$ 每单位角变化得有多快?"这个问题是一个变化率。
- 它告诉你曲线是在向外盘旋、向内盘旋,还是保持不变。
- 这把第 1 单元的变化率思想与极坐标世界联系了起来。
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 $\theta$.
r 的平均变化率
- 极坐标函数(polar function)是 $r = f(\theta)$,所以我们能衡量 $r$ 如何随 $\theta$ 变化。
- 一段区间上的平均变化率(rate of change)是 $\dfrac{\Delta r}{\Delta \theta}$——半径的变化除以角的变化。
- 正值意味着半径在增长;负值意味着它在缩小。
- 这正是"输出除以输入"的变化率,应用于 $r$ 和 $\theta$。
As $\theta$ increases, if $r$ is increasing, the curve moves… · 当 $\theta$ 增大时,如果 $r$ 在增大,曲线……
A growing radius means the point is getting farther out — the curve spirals outward. · 增大的半径意味着点越来越远——曲线向外盘旋。
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.
向外或向内盘旋
- 如果 $r$ 随 $\theta$ 增大而增大,曲线离原点更远——向外盘旋。
- 如果 $r$ 在减小,曲线朝极点向内盘旋。
- 像 $r = \theta$ 这样的螺线,变化率稳定为正,所以它永远变宽。
- 变化率的大小决定曲线盘旋得多快。
Watch the radius grow as the angle sweeps · 看半径随角扫过而增长
In a spiral, r increases with theta. The rate of change of r decides how quickly the curve winds outward. · 在螺线里,r 随 theta 增大。r 的变化率决定曲线向外盘旋得多快。
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 ____. · $r$ 在一段 $\theta$ 区间上的平均变化率是 $\dfrac{\Delta r}{\Delta \theta}$,即半径的变化除以____的变化。
It measures how fast the radius changes per unit of angle · 角度 — the polar version of a slope. · 它衡量半径每单位角变化多快——极坐标版本的斜率。
Select all · 所有 true statements about rates of change in polar functions. · 选出关于极坐标函数中变化率的所有正确说法。
A constant $r$ traces a circular arc, not an outward spiral. The other three are correct. · 恒定的 $r$ 描出一段圆弧,而不是向外的螺线。其余三条正确。
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.
当变化率为零时
- 在 $r$ 的变化率为零处,半径一时保持恒定。
- 一段区间上恒定的半径描出一段圆弧。
- 所以 $r$ 中一段平坦的部分,表现为曲线上一段圆润的圆形部分。
- $r$ 的波峰和波谷,是曲线伸得最远或最靠近中心的地方。
If $r$ decreases as $\theta$ increases, the curve spirals inward toward the origin. · 如果 $r$ 随 $\theta$ 增大而减小,曲线朝原点向内盘旋。
A shrinking radius pulls the point toward the pole, so the curve winds inward. · 缩小的半径把点拉向极点,所以曲线向内盘旋。
If the rate of change of $r$ is zero · 零 over an interval, the curve there is… · 如果 $r$ 的变化率在一段区间上为零,那里的曲线是……
Zero rate means $r$ is constant, so the point stays the same distance out — an arc of a circle. · 零变化率意味着 $r$ 恒定,所以点保持相同的距离——一段圆弧。
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.
读出这场运动
- 跟踪 $r$ 相对 $\theta$ 的变化,来预测极坐标曲线的行为。
- $r$ 上升:花瓣或回环向外伸展。$r$ 下降:朝中心返回。
- $r$ 的零点是曲线经过原点的地方。
- 变化率把一幅静态的图,变成一个关于运动的故事。
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.
这里的变化率是每单位角半径的变化,而不是点沿曲线运动的速度。大的 $\tfrac{\Delta r}{\Delta\theta}$ 意味着离原点的距离变化得快——它并不直接衡量你绕曲线走得有多快。
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.
对螺线 $r = \theta$,比较 $\theta = 1$ 和 $\theta = 3$ 处的半径。
- 当 $\theta$ 从 $1$ 到 $3$,$r$ 从 $1$ 变到 $3$。
- 平均变化率 $= \dfrac{3 - 1}{3 - 1} = 1$:半径每弧度增长一个单位。
- 为正且恒定,所以螺线以稳定的速度向外张开。
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 $\theta$.
对极坐标函数 $r = f(\theta)$,平均变化率 $\tfrac{\Delta r}{\Delta\theta}$ 告诉你半径如何随角变化:正 → 向外盘旋,负 → 向内,零 → 一段圆弧。它把第 1 单元的变化率思想应用到 $r$ 和 $\theta$。