📚 Rates of Change | 变化率
The concept of rate of change is fundamental in calculus and appears throughout the IB Mathematics curriculum, from basic differentiation to real-world applications. It measures how one quantity varies with respect to another, often time. Understanding rates of change allows us to model motion, growth, marginal analysis, and tackle optimisation problems that are central to the IB syllabus.
变化率的概念是微积分的基础，贯穿IB数学课程，从基本求导到实际应用。它衡量一个量相对于另一个量（通常是时间）的变化情况。理解变化率使我们能够建模运动、增长、边际分析，并解决IB课程核心的优化问题。
1. What is a Rate of Change? | 什么是变化率？
A rate of change describes how much a dependent variable changes as the independent variable changes. If y depends on x, the average rate of change from x = x1 to x = x2 is given by Δy/Δx = (y2 − y1) / (x2 − x1). Geometrically, this represents the slope of the secant line joining the two points on the curve y = f(x).
变化率描述因变量随自变量变化的情况。如果y依赖于x，从x = x1到x = x2的平均变化率为Δy/Δx = (y2 − y1) / (x2 − x1)。几何上，这代表连接曲线y = f(x)上两点的割线斜率。
In many IB problems, the rate is with respect to time t, written as dy/dt. Units play a key role: if y is metres and t is seconds, dy/dt has units m s−1. Recognising these units helps you interpret what the derivative actually means.
在许多IB问题中，变化率是关于时间t的，记作dy/dt。单位起着关键作用：若y是米，t是秒，则dy/dt的单位是m s−1。识别这些单位有助于你解释导数的实际含义。
2. Average Rate of Change and the Secant Line | 平均变化率与割线斜率
The average rate of change is the foundation for the definition of the derivative. For a function f(x) over the interval [a, b], the average rate of change is (f(b) − f(a)) / (b − a). On a graph, this is the slope of the straight line through (a, f(a)) and (b, f(b)) – the secant line.
平均变化率是导数定义的基础。对于函数f(x)在区间[a, b]上，平均变化率为(f(b) − f(a)) / (b − a)。在图像上，这是经过点(a, f(a))和(b, f(b))的直线的斜率——即割线。
In IB questions, you may be asked to compute the average speed of a particle over a time interval, which is simply the average rate of change of displacement. Always pay attention to the sign: a negative average rate indicates a decrease in the quantity over that interval.
在IB考题中，你可能会被要求计算一段时间间隔内粒子的平均速度，这恰恰是位移的平均变化率。始终注意符号：负的平均变化率表示该量在此间隔内减小。
3. Instantaneous Rate of Change and the Derivative | 瞬时变化率与导数
The instantaneous rate of change is what we obtain by shrinking the interval Δx to zero. If the limit exists, the derivative of f at a is f'(a) = lim (h → 0) [f(a + h) − f(a)] / h. This gives the slope of the tangent line to the curve at x = a and represents the exact rate of change at that instant.
瞬时变化率是通过将间隔Δx缩小至零得到的。如果极限存在，函数f在a处的导数为f'(a) = lim (h → 0) [f(a + h) − f(a)] / h。这给出了曲线在x = a处切线的斜率，代表了该时刻精确的变化率。
The derivative function f'(x) itself describes the instantaneous rate of change for any x. In the IB course, you will use differentiation rules (power rule, chain rule, product rule, etc.) to find f'(x) efficiently rather than evaluating the limit definition every time.
导函数f'(x)本身描述了任意x处的瞬时变化率。在IB课程中，你将运用求导法则（幂法则、链式法则、乘积法则等）高效地求出f'(x)，而不是每次都用极限定义计算。
4. Notation and Interpretation | 符号与解释
IB expects you to be fluent in both Leibniz notation (dy/dx) and Lagrange notation (f'(x)). If the variable is time t, the derivative is often written with a dot: dx/dt = &xdot;. When interpreting, dy/dx at x = a tells you the rate at which y changes per unit increase in x near that point.
IB要求你熟练使用莱布尼茨记法（dy/dx）和拉格朗日记法（f'(x)）。如果变量是时间t，导数常写作点记法：dx/dt = &xdot;。解释时，x = a处的dy/dx告诉你靠近该点处y每单位x增加时的变化率。
Always include units when interpreting a rate of change in context. For example, if V(t) is the volume of a balloon in cm3 and t is in seconds, then dV/dt has units cm3 s−1. A positive derivative indicates growth; a negative derivative indicates decay or decrease.
在具体情境中解释变化率时务必带上单位。例如，若V(t)是气球的体积，单位cm3，t以秒为单位，则dV/dt的单位为cm3 s−1。导数为正表示增长；导数为负表示衰减或减少。
5. Kinematics: Displacement, Velocity, Acceleration | 运动学：位移、速度、加速度
In the IB Mathematics: Applications and Interpretation or Analysis and Approaches courses, kinematics provides a classic context for rates of change. If displacement is given by s(t), then velocity v(t) = s'(t) and acceleration a(t) = v'(t) = s”(t). These are, respectively, the rate of change of position and the rate of change of velocity.
在IB数学：应用与解释或分析与方法课程中，运动学提供了变化率的经典情境。若位移由s(t)给出，则速度v(t) = s'(t)，加速度a(t) = v'(t) = s”(t)。它们分别是位置的变化率和速度的变化率。
For instance, if s(t) = t3 − 6t2 + 9t, then v(t) = 3t2 − 12t + 9 and a(t) = 6t − 12. The particle turns around when v(t) = 0, and the speed is the absolute value of velocity. A change in direction occurs when velocity changes sign.
例如，若s(t) = t3 − 6t2 + 9t，则v(t) = 3t2 − 12t + 9，a(t) = 6t − 12。当v(t) = 0时粒子转向，速率是速度的绝对值。当速度改变符号时方向发生改变。
6. Related Rates | 相关变化率
Related rates problems involve two or more quantities that change over time and are linked by an equation. By differentiating that equation with respect to time using the chain rule, you can find an unknown rate of change from known rates. A typical example: air is pumped into a spherical balloon; given dV/dt, find dr/dt when r = 10 cm.
相关变化率问题涉及两个或多个随时间变化的量，它们由一个方程联系。通过使用链式法则对该方程关于时间求导，你可以从已知的变化率求出未知的变化率。一个典型例子：向球形气球充气，已知dV/dt，求当r = 10 cm时的dr/dt。
The volume of a sphere is V = (4/3)πr3. Differentiating with respect to t gives dV/dt = 4πr2 (dr/dt). Substituting the known values allows you to solve for dr/dt. The answer will have the units of cm s−1 if r is in cm and t in seconds.
球的体积为V = (4/3)πr3。关于t求导得到dV/dt = 4πr2 (dr/dt)。代入已知值即可解出dr/dt。如果r以cm为单位，t以秒为单位，答案的单位将是cm s−1。
7. Solving Related Rates Problems | 解相关变化率问题
A systematic approach greatly helps in IB exam questions on related rates. Follow these steps:
- Draw a diagram and label variables. Identify constants and quantities that change. 画图并标注变量。识别常量和变化的量。
- Write a relation between the variables. This could be geometric (Pythagoras, similarity) or a formula like volume/area. 写出变量间的关系。这可能是几何关系（勾股定理、相似形）或体积/面积公式。
- Differentiate both sides with respect to time t. Use implicit differentiation and the chain rule carefully. 对时间t两边求导。仔细运用隐函数求导和链式法则。
- Substitute the known values and rates. Only plug in numbers after differentiating, unless the quantity is constant. 代入已知值和变化率。除非是常量，否则在求导之后才代入数字。
- Solve for the unknown rate. Give the answer with correct units and sign. 解出未知的变化率。给出带正确单位和符号的答案。
In many cases, you will also need to find a missing length at the instant of interest, often using the Pythagorean theorem or similar triangles. Draw a clear sketch at that specific moment.
在许多情形下，你还需要找出所关注时刻的某个缺失长度，通常利用勾股定理或相似三角形。在那一特定时刻画出清晰的草图。
8. Exponential Growth and Decay | 指数增长与衰减
A fundamental differential equation in the IB syllabus is dy/dt = ky, where k is a constant. This states that the rate of change of a quantity is proportional to the amount present. The solution is y = y0 e^(kt). If k > 0, the quantity grows exponentially; if k < 0, it decays exponentially.
IB课程中一个基本的微分方程是dy/dt = ky，其中k是常数。这表示某一量的变化率与当前量成正比。其解为y = y0 e^(kt)。若k > 0，该量指数增长；若k < 0，该量指数衰减。
Applications include population growth (k > 0), radioactive decay (k < 0), and Newton's law of cooling. In such contexts, the rate of change itself depends on the current value, creating a feedback loop that leads to rapid change. IB questions may ask you to find the half-life or doubling time using logarithms.
应用包括人口增长（k > 0）、放射性衰变（k < 0）和牛顿冷却定律。在这些背景下，变化率本身依赖于当前值，形成导致快速变化的反馈环。IB问题可能要求你利用对数求出半衰期或倍增时间。
9. Marginal Analysis in Economics | 经济学中的边际分析
Marginal cost, marginal revenue, and marginal profit are applications of rates of change that often appear in IB Mathematics. If C(x) is the total cost of producing x units, the marginal cost is C'(x), which approximates the cost of producing one additional unit when x items have already been produced.
边际成本、边际收益和边际利润是变化率在经济学中的应用，常出现在IB数学中。如果C(x)是生产x件商品的总成本，边际成本为C'(x)，它近似表示在已经生产了x件商品后，再多生产一件所需的成本。
Similarly, if R(x) = px is revenue, the marginal revenue is R'(x). Profit P(x) = R(x) − C(x) is maximised when P'(x) = 0 and P”(x) < 0. The interpretation of the derivative as a rate of change is crucial: marginal functions tell you how quickly cost or revenue is changing with respect to the quantity produced.
类似地，若R(x) = px为收益，边际收益为R'(x)。利润P(x) = R(x) − C(x)在P'(x) = 0且P”(x) < 0时达到最大。将导数解释为变化率至关重要：边际函数告诉你成本或收益随产量变化的快慢。
10. Optimisation and Rates of Change | 优化与变化率
Optimisation is the process of finding maximum or minimum values of a function, and it relies heavily on rates of change. At a local maximum or minimum, the instantaneous rate of change f'(x) is zero. The sign of the derivative on either side of a critical point tells you whether the function is increasing or decreasing, and thus whether the point is a maximum, minimum, or point of inflection.
优化是寻找函数最大值或最小值的过程，它严重依赖于变化率。在局部极大值或极小值处，瞬时变化率f'(x)为零。临界点两侧导数的符号告诉你函数是在增加还是减少，从而判断该点是极大值、极小值还是拐点。
In IB problems, you might be asked to maximise the area of a rectangle given a fixed perimeter, or minimise the surface area of a can for a given volume. The first derivative gives the rate at which area changes with dimensions; setting that rate to zero finds the optimal design.
在IB问题中，你可能会被要求给定固定周长求矩形面积的最大值，或在给定体积下使罐子的表面积最小。一阶导数给出了面积随尺寸的变化率；令该变化率为零即可找到最优设计。
Always verify that your solution is indeed a maximum or minimum by checking the second derivative or by testing the sign change of f'(x) around the critical point. The second derivative represents the rate of change of the rate of change, giving information about concavity.
始终通过检查二阶导数或测试临界点附近f'(x)的符号变化来验证你的解确实是最大值或最小值。二阶导数代表了变化率的变化率，提供了关于凹凸性的信息。
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