Related Rates

Core Sentence / 核心句

A related-rates problem uses a constraint to connect the time rates of several changing quantities.

中文理解：多个量都随时间变化，而一个始终成立的关系式迫使这些变化率彼此关联。

Formula / 公式

If:

$$x(t{)}^2+y(t{)}^2=25$$

then:

$$2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$$

Problem Routine / 做题顺序

1. Define each changing quantity as a function of time.
2. Write a constraint true at every moment.
3. Differentiate with respect to time.
4. Substitute the instant-specific values.
5. Solve with correct sign and units.

Watch Out / 易错点

• 先求导，后代入某一时刻的数值。
• decreasing quantity 的 rate 为负。
• 区分 quantity、rate 和单位。
• equation 必须在整个运动过程中成立，不只是某一个瞬间碰巧成立。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第5次正课 · Related-Rates Routine
  • connect quantities, differentiate with respect to time, then substitute the instant
• 微积分BC第4次正课 · Related Rates Preview
  • time is the common input and a constraint links dx/dt with dy/dt
• Chapter 6 · Ladder Setup
  • $x(t)$ and $y(t)$ share time as a common input
• Chapter 6 · Ladder Solution
  • the fixed ladder length forces $dx/dt$ and $dy/dt$ to balance

Connected Notes / 连接

• 微积分BC第5次正课-相关变化率线性近似与中值定理
• 微积分BC第4次正课-反函数高阶导数与运动应用
• 06 - Implicit Differentiation, What’s Going On Here
• Implicit Differentiation
• Implicit Curve
• Chain Rule
• Derivatives
• Instantaneous Rate of Change