📚 Related Rates (IB Math) | IB 数学中的相关变化率
In calculus, related rates problems involve finding how the rate of change of one quantity is related to the rate of change of another quantity that depends on it. These problems typically involve time as the independent variable, and the quantities are linked by an equation (often geometric). By differentiating this equation with respect to time using the chain rule, we can relate their rates of change. Mastering this technique is essential for the IB Mathematics Analysis and Approaches course.
在微积分中，相关变化率问题涉及寻找一个量的变化率与另一个依赖于它的量的变化率之间的关系。这类问题通常以时间为自变量，各量由一个方程（往往是几何方程）联系起来。通过链式法则对方程关于时间求导，我们就可以建立它们变化率之间的关系。掌握这一技巧对 IB 数学分析与方法课程至关重要。
1. The Chain Rule and Implicit Differentiation | 链式法则与隐函数求导
emember that every variable, unless constant, must be differentiated using the chain rule.
相关变化率建立在链式法则之上：如果 y = f(x) 且 x 和 y 都是 t 的函数，那么 dy/dt = (dy/dx)(dx/dt)。处理几何公式如 V = (4/3)πr³ 时，我们隐式地对 t 求导：dV/dt = 4πr² (dr/dt)。关键是要记住，每个非常量的变量都必须用链式法则求导。
2. General Strategy for Solving Related Rates Problems | 求解相关变化率问题的一般策略
Step 1: Read the problem carefully and identify all given quantities and the required rate. Define variables (e.g., r, h, V, x, y) and note which are rates (e.g., dr/dt).
步骤 1：仔细读题，识别所有已知量和要求的速率。定义变量（如 r, h, V, x, y），并注明哪些是变化率（如 dr/dt）。
Step 2: Draw a diagram and label the variables. This helps visualise the geometric or physical relationships.
步骤 2：画图并标记变量。这有助于可视化几何或物理关系。
Step 3: Write an equation relating the variables. This could be a geometric formula (area, volume, Pythagorean theorem) or a trigonometric relation.
步骤 3：写出联系变量的方程。可以是几何公式（面积、体积、勾股定理）或三角关系。
Step 4: Differentiate both sides of the equation with respect to time t, using the chain rule. This yields an equation involving the rates (e.g., dV/dt, dr/dt).
步骤 4：利用链式法则，对方程两边关于时间 t 求导，得到包含变化率（如 dV/dt, dr/dt）的方程。
Step 5: Substitute the known values of variables and rates. Be careful to use the instantaneous values at the moment of interest. Do not substitute numbers before differentiating.
步骤 5：代入变量的已知值和变化率。注意要使用所关心时刻的瞬时值。切勿在求导前代入数值。
Step 6: Solve for the unknown rate and include the correct units.
步骤 6：求解未知变化率，并注明正确单位。
3. Example 1: Expanding Circular Oil Slick | 例 1：膨胀的圆形油膜
A circular oil slick expands such that its radius increases at a constant rate of 0.5 m/s. At the moment when the radius is 10 m, how fast is the area increasing?
一个圆形油膜膨胀，其半径以 0.5 米/秒的恒定速率增加。在半径为 10 米的时刻，面积增加有多快？
need dA/dt when r = 10 m. The area formula is A = πr². Differentiating with respect to t:
已知 dr/dt = 0.5 米/秒，需求 r = 10 米时的 dA/dt。面积公式 A = πr²，对 t 求导：
dA/dt = 2πr (dr/dt)
tuting: dA/dt = 2π(10)(0.5) = 10π m²/s. The area is increasing at approximately 31.4 m²/s.
代入：dA/dt = 2π(10)(0.5) = 10π 米²/秒。面积正以约 31.4 米²/秒的速度增加。
4. Example 2: Sliding Ladder | 例 2：滑动的梯子
A 5-metre ladder leans against a vertical wall. The bottom of the ladder slides away from the wall at a constant speed of 0.8 m/s. At the moment when the bottom is 3 m from the wall, how fast is the top of the ladder sliding down the wall?
一把 5 米长的梯子靠在竖直墙上。梯子底部以 0.8 米/秒的速度匀速滑离墙壁。当底部离墙 3 米
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