Introduction to Related Rates · 相关变化率简介
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| related rates/rɪˈleɪtɪd reɪts/ | 相关变化率 | xiāng guān biàn huà lǜ |
| time/taɪm/ | 时间 | shí jiān |
When one rate drives another
- Blow up a balloon: its radius grows, and because of that, its volume grows too.
- The two rates are linked — knowing how fast the radius changes tells you how fast the volume changes.
- Problems like this are related rates 相关变化率: several quantities changing together in time.
- The engine that links them is the chain rule, applied with respect to time 时间.
当一个变化率带动另一个
- 吹气球:半径增大,正因如此体积也增大。
- 两个变化率相连——知道半径变化多快,就能知道体积变化多快。
- 这类问题是相关变化率:多个量随时间一起变化。
- 把它们连起来的引擎是链式法则,关于时间来求导。
A related-rates problem links the rates of quantities that change with respect to... · 相关变化率问题链接了随...变化的量的变化率
The variables are functions of time, linked via the chain rule. · 这些变量是时间的函数，通过链式法则相互联系。
Start with an equation relating the variables
- First, find an equation that ties the quantities together — usually geometry or a formula.
- Balloon: $V=\tfrac43\pi r^3$ links volume and radius.
- Ladder sliding down a wall: $x^2+y^2=L^2$ (Pythagoras) links the two distances.
- This relating equation is true at every instant, so we can differentiate it in time.
从一个联系变量的方程开始
- 首先,找一个把各量绑在一起的方程——通常是几何或某个公式。
- 气球:$V=\tfrac43\pi r^3$ 联系体积与半径。
- 梯子沿墙下滑:$x^2+y^2=L^2$(勾股定理)联系两段距离。
- 这个联系方程在每一瞬间都成立,所以我们能对它关于时间求导。
Volume grows with radius · 体积随半径增大
y = a·x³ (V vs r)
$V=\tfrac43\pi r^3$ climbs steeply — a small radius rate makes a large volume rate when $r$ is big. · $V=\tfrac43\pi r^3$ 急剧上升——当 $r$ 很大时，微小的半径变化率会导致巨大的体积变化率。
The first step is to write an ____ that relates the changing variables. · 第一步是写出一个关联变化变量的____。
Usually a geometric or physical relationship. · 通常是几何或物理关系。
A ladder of length $L$ leans on a wall. Which equation relates the base distance $x$ and height $y$? · 一根长为 $L$ 的梯子靠在墙上。哪个方程关联底边距离 $x$ 和高度 $y$？
Pythagoras: the wall and floor form a right angle, so $x^2+y^2=L^2$. · 勾股定理：墙和地板形成直角，因此 $x^2+y^2=L^2$。
Differentiate with respect to time
- Every variable secretly depends on $t$, so differentiating brings in a rate for each — via the chain rule.
- $\dfrac{d}{dt}\big[V=\tfrac43\pi r^3\big]$ gives $\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}$.
- Each term contributes a "$\tfrac{d(\text{that variable})}{dt}$" factor — exactly like implicit differentiation, but the hidden variable is $t$.
- Now the equation links the rates $\tfrac{dV}{dt}$ and $\tfrac{dr}{dt}$.
关于时间求导
- 每个变量都暗地里依赖 $t$,所以求导时经由链式法则为每个引入一个变化率。
- $\dfrac{d}{dt}\big[V=\tfrac43\pi r^3\big]$ 给出 $\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}$。
- 每一项都贡献一个"$\tfrac{d(\text{var})}{dt}$"因子——就像隐函数求导,只是隐藏变量是 $t$。
- 现在方程联系起了变化率 $\tfrac{dV}{dt}$ 与 $\tfrac{dr}{dt}$。
Differentiating $V=\tfrac43\pi r^3$ with respect to $t$ gives $\dfrac{dV}{dt}=$ · 对 $V=\tfrac43\pi r^3$ 关于 $t$ 求导得到 $\dfrac{dV}{dt}=$
Chain rule in time: $4\pi r^2$ times $\tfrac{dr}{dt}$. · 时间上的链式法则：$4\pi r^2$ 乘以 $\tfrac{dr}{dt}$。
Sort the givens from the unknown
- Read the problem and label: which rates are given, and which single rate is wanted?
- "The radius grows at $2\,\tfrac{\text{cm}}{\text{s}}$" → $\tfrac{dr}{dt}=2$ is given.
- "How fast is the volume growing?" → $\tfrac{dV}{dt}$ is the unknown.
- Also note any fixed quantities (like the radius at the instant asked about).
区分已知与未知
- 读题并标注:哪些变化率是已知的,想求的是哪一个变化率?
- "半径以 $2\,\tfrac{\text{cm}}{\text{s}}$ 增大"→ $\tfrac{dr}{dt}=2$ 是已知。
- "体积增长多快?"→ $\tfrac{dV}{dt}$ 是未知。
- 也要记下任何固定量(如所问那一瞬间的半径)。
Differentiating $r^3$ with respect to time gives just $3r^2$. · 对 $r^3$ 关于时间求导仅得到 $3r^2$。
It gives $3r^2\tfrac{dr}{dt}$ — the rate factor is required. · 它给出 $3r^2\tfrac{dr}{dt}$ —— 必须包含速率因子。
"The radius grows at $2$ cm/s; how fast is the volume growing?" Select all · 所有 correct labels. · "半径以 $2$ cm/s 增长；体积增长得有多快？" 选择所有正确标签。
The radius rate is given, the volume rate is wanted, and $V=\tfrac43\pi r^3$ relates them. · 半径变化率已知，体积变化率所求，且 $V=\tfrac43\pi r^3$ 将它们联系起来。
Related-rates variables are functions of time, so differentiating $r^3$ gives $3r^2\tfrac{dr}{dt}$ — never just $3r^2$. Forgetting the $\tfrac{dr}{dt}$ (the chain-rule rate factor) is the defining error. Every changing quantity contributes its own $\tfrac{d(\ )}{dt}$.
相关变化率的变量是时间的函数,所以对 $r^3$ 求导得 $3r^2\tfrac{dr}{dt}$——绝不只是 $3r^2$。忘掉 $\tfrac{dr}{dt}$(链式法则的变化率因子)是标志性错误。每个变化的量都贡献它自己的 $\tfrac{d(\ )}{dt}$。
A balloon's volume is $V=\tfrac43\pi r^3$. Set up the related-rates equation.
- Differentiate with respect to $t$: $\dfrac{dV}{dt}=\dfrac43\pi\cdot 3r^2\cdot\dfrac{dr}{dt}=4\pi r^2\dfrac{dr}{dt}$.
- This links the growth rate of volume to the growth rate of radius.
- Given $\tfrac{dr}{dt}$ and the current $r$, we could now solve for $\tfrac{dV}{dt}$ (next lesson).
气球体积是 $V=\tfrac43\pi r^3$。建立相关变化率方程。
- 关于 $t$ 求导:$\dfrac{dV}{dt}=\dfrac43\pi\cdot 3r^2\cdot\dfrac{dr}{dt}=4\pi r^2\dfrac{dr}{dt}$。
- 这把体积的增长率与半径的增长率联系起来。
- 给定 $\tfrac{dr}{dt}$ 和当前的 $r$,我们现在就能解出 $\tfrac{dV}{dt}$(下一课)。
A related rates problem links the rates of two or more quantities changing in time. Write an equation relating the variables (often geometry), differentiate it with respect to $t$ using the chain rule (each variable gets a $\tfrac{d(\ )}{dt}$), and identify which rates are given and which is wanted.
相关变化率问题把两个或更多随时间变化的量的变化率联系起来。写一个联系变量的方程(常为几何),用链式法则关于 $t$ 求导(每个变量都带一个 $\tfrac{d(\ )}{dt}$),并区分哪些变化率是已知、哪个是要求的。