📚 Exercise 27A: Tangents, Normals, and Rates of Change | 练习 27A：切线、法线与变化率
In IB Mathematics, Exercise 27A typically marks the beginning of a deeper exploration into the applications of differential calculus. This set of problems develops your ability to connect abstract differentiation rules to geometric and physical contexts. You will learn how to find the equations of tangents and normals to curves, interpret the derivative as an instantaneous rate of change, and solve real-world problems involving motion and related rates. Mastering these foundational skills is essential for success in both Paper 1 and Paper 2 of the IB Analysis & Approaches or Applications & Interpretation courses.
在 IB 数学中，练习 27A 往往标志着对微积分应用深入探索的开始。这一组习题培养你将抽象的求导法则与几何及物理情境联系起来的能力。你将学习如何找出曲线的切线和法线方程、将导数解释为瞬时变化率，并解决涉及运动和关联变化率的现实问题。掌握这些基础技能对于在 IB 分析与方法和应用与解释课程的试卷一与试卷二中取得成功至关重要。
1. Revisiting the Core Differentiation Rules | 重温核心求导法则
etric, exponential, and logarithmic functions. Recall the power rule: if f(x) = xⁿ, then f' derivative of cos x is −sin x, and the derivative of tan x is sec² x. The product rule, f'(x) = u'v + uv', and the quotient rule, f'(x) = (vu' − uv' n a function is composite, the chain rule dy/dx = dy/du × du/dx must be applied carefully.
在处理切线和变化率之前，你必须熟练掌握标准求导法则。练习 27A 假设你能自信地对多项式、三角、指数和对数函数求导。回顾幂函数法则：若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹。三角函数的导数：sin x 的导数是 cos x，cos x 的导数是 −sin x，tan x 的导数是 sec² x。乘法法则 f'(x) = u'v + uv' 和除法法则 f'(x) = (vu' − uv')/v² 也经常被用到。当遇到复合函数时，必须仔细应用链式法则 dy/dx = dy/du × du/dx。
2. The Slope of a Curve and the Tangent Line | 曲线的斜率与切线
The derivative f' you need two pieces of information: the point of contact (a, f(a)) and the gradient m = f'(a). You can then use the point– write the equation. This simple procedure underpins many of the problems in Exercise 27A.
导数 f'(a) 给出了曲线 y = f(x) 在 x = a 处的切线的斜率。从几何上看，切线在该点（局部地）仅与曲线接触于一点，并在此瞬间与曲线方向一致。要找出切线方程，你需要两条信息：切点 (a, f(a)) 和梯度 m = f'(a)。然后可以利用点斜式 y − y₁ = m(x − x₁) 写出方程。这个简单的步骤是练习 27A 中许多问题的基础。
3. Writing the Equation of a Normal Line | 写出法线方程
nderstanding of perpendicular slopes. Always remember to simplify your final linear equation into the form ax + by + d = 0 or y = mx + c, as requested by the problem.
法线是在切点处垂直于切线的直线。如果切线的梯度为 m，则法线的梯度为 −1/m（假设 m ≠ 0）。当 m = 0 时，切线为水平线，法线为垂直线 (x = a)。IB 考试题目中经常要求写出法线方程，以检验你对垂直斜率关系的理解。始终要记住根据题目要求，将最终的线性方程化简为 ax + by + d = 0 或 y = mx + c 的形式。
4. Selecting the Correct Point of Contact | 选择正确的切点
A common challenge in Exercise 27A is determining the point of contact when it is not given explicitly. You may be told that a line with a certain gradient touches a curve, or that a line passes through an external point and is tangent to the curve. In such cases, let the unknown point be (a, f(a)). Set the derivative equal to the given gradient to solve for a, or use the fact that the tangent line passes through the external point. This often leads to solving an equation involving the original function and its derivative, a valuable algebraic skill.
练习 27A 中一个常见的难点是，当切点未明确给出时如何确定它。题目可能会告诉你一条具有特定梯度的直线与曲线相切，或者一条直线经过曲线外一点且与曲线相切。遇到这种情况，可设未知切点为 (a, f(a))。令导数等于已知梯度来求解 a，或者利用切线经过外部点这一条件。这通常会导出一个同时包含原函数及其导数的方程，是一项宝贵的代数技能。
5. Working with Parametric and Implicit Curves | 处理参数曲线与隐式曲线
x/dt). For implicit curves, you need to use implicit differentiation, differentiating both sides with respect to x and applying the chain rule to y terms. The subsequent tangent and normal calculations follow the same pattern, but obtaining the correct gradient requires careful handling of derivatives.
尽管练习 27A 中的许多问题集中在显式函数 y = f(x) 上，你也可能遇到参数方程或隐式方程定义的曲线。对于参数方程 x = x(t)，y = y(t)，梯度为 dy/dx = (dy/dt) ÷ (dx/dt)。对于隐式曲线，需要使用隐函数求导法——对等式两边关于 x 求导，并对含 y 的项应用链式法则。随后的切线和法线计算遵循相同模式，但获得正确的梯度需要谨慎处理导数。
6. The Derivative as an Instantaneous Rate of Change | 作为瞬时变化率的导数
es with respect to another. If s(t) represents displacement with respect to time t, then s'(t) gives the instantaneous velocity v(t), and s''(t) gives the acceleration a(t). Similarly, in economics, C' ext is a vital skill: you must be able to move fluently between the algebraic expression f'(a) and the realistic statement such as "the velocity is 5 m s⁻¹ at t = 2 seconds."
除了几何意义，练习 27A 还引入了导数作为衡量一个量相对于另一个量变化快慢的度量。如果 s(t) 表示位移关于时间 t 的函数，那么 s'(t) 给出瞬时速度 v(t)，s''(t) 给出加速度 a(t)。类似地，在经济学中，C'(x) 可以表示生产 x 件产品时的边际成本。在具体情境中解释导数是关键技能：你必须能够灵活地在代数表达式 f'(a) 和诸如“在 t = 2 秒时速度为 5 m s⁻¹”的现实陈述之间进行转换。
7. Solving Basic Related Rates Problems | 求解基本关联变化率问题
d you are asked to find dh/dt given dV/dt. The key is to differentiate the relationship with respect to time t using the chain rule, then substitute known values. Exercise 27A often includes gentle introductions to this type of problem, where you first identify the appropriate formula and then apply implicit differentiation with respect to t.
关联变化率问题将瞬时变化率的思想扩展到两个或多个变量通过方程关联的情况。一个经典例子是圆锥水箱漏水：体积 V 与高度 h 通过 V = (1/3)πr²h 关联，题目要求已知 dV/dt 求 dh/dt。关键步骤是利用链式法则对关系式关于时间 t 求导，然后代入已知数值。练习 27A 通常平和地引入这类问题，你需要先找到合适的公式，然后关于 t 进行隐函数求导。
8. Practical Interpretation and Unit Management | 实际解释与单位处理
its in your final answer and consider whether the numerical result makes sense in the given context. Exercise 27A often contains questions that purposely omit units at first, forcing you to think dimensionally and adopt consistent units after rearranging formulas.
IB 考官强调在计算过程中附上单位并解释导数符号的重要性。负的变化率表示该量在减少。例如，如果水位以每分钟 3 厘米的速度下降，我们记作 dh/dt = −3 cm min⁻¹。务必在最终答案中注明单位，并考虑数值结果在所给情境中是否合理。练习 27A 常包含起初故意省略单位的问题，迫使你进行量纲思考，并在重排公式后采用一致的单位。
9. Avoiding Common Algebraic Mistakes | 避免常见的代数错误
When finding tangents and normals, students often make algebraic slips that cost marks. One frequent error is forgetting to substitute the x-coordinate into the original function to get the full point (a, f(a)). Another is mishandling the negative reciprocal when computing the normal’s gradient, especially when the tangent’s gradient is a fraction. In related rates, a typical mistake is to differentiate before substituting known constant values, which can introduce extra variables. Double-check each step and, if time permits, verify that your tangent equation satisfies the point of contact.
在求切线和法线时，学生常常出现代数失误而丢分。一个常见错误是忘记将 x 坐标代入原函数以得到完整的切点 (a, f(a))。另一个错误是在计算法线梯度时错误地处理负倒数，尤其是当切线的梯度为分数时。在关联变化率问题中，典型的错误是在代入已知常量之前就进行求导，这可能会引入额外的变量。请仔细检查每一步，如果时间允许，验证你的切线方程是否满足切点坐标。
10. Technology Tips for Verification | 使用技术进行验证的小贴士
Although Paper 1 of the IB exam does not allow a calculator, you can use your graphical display calculator (GDC) on Paper 2 to check your work. Graph the original function, draw the tangent at a point, and read its equation to compare with your analytical result. Many GDCs also have a numerical derivative function (nDeriv) that approximates f'(a). However, never rely on technology as a substitute for algebraic working; exam questions often require you to show steps. Use it only to confirm your answer or explore patterns.
虽然 IB 考试试卷一不允许使用计算器，但你可以在试卷二中使用图形计算器 (GDC) 来检查自己的作业。画出原函数，在某点处绘制切线，读取其方程，并与你的解析结果进行比较。许多 GDC 还带有近似数值导数 (nDeriv) 的函数，可估算 f'(a)。然而，绝不要依赖技术来替代代数演算；考试题目通常要求展示步骤。只将其用于确认答案或探索规律。
11. Worked Example: Tangent and Normal to a Cubic | 例题示范：三次曲线的切线与法线
(2) = (8) − 3(4) − 18 + 5 = 8 − 12 − 18 + 5 = −17. The point is (2, −17). Differentiate: f'(x) = 3x² − 6x − 9. At x = 2, f' → 9y + 153 = x − 2 → x − 9y − 155 = 0. Always label which is tangent and which is normal.
考虑曲线 f(x) = x³ − 3x² − 9x + 5。求在 x = 2 处的切线和法线方程。首先，计算 f(2) = (8) − 3(4) − 18 + 5 = 8 − 12 − 18 + 5 = −17。切点为 (2, −17)。求导：f'(x) = 3x² − 6x − 9。在 x = 2 处，f'(2) = 3(4) − 12 − 9 = 12 − 12 − 9 = −9。因此切线梯度 m = −9。切线方程：y − (−17) = −9(x − 2) → y + 17 = −9x + 18 → y = −9x + 1。法线梯度为 1/9。法线方程：y + 17 = (1/9)(x − 2) → 9y + 153 = x − 2 → x − 9y − 155 = 0。务必标明哪条是切线，哪条是法线。
12. Exam Strategy and Final Advice | 考试策略与最终建议
When approaching Exercise 27A, view each problem as an opportunity to strengthen the link between algebraic differentiation and visual or physical meaning. Show all your working clearly, especially the substitution steps and the simplification of the gradient. In multi-part questions, even if you get the gradient wrong in one part, you may still earn method marks for using perpendicular gradients or units correctly. Practice translating word problems into variables before differentiating, and always read the question carefully to determine whether it asks for a tangent, a normal, or a rate of change at a specific instant. With consistent practice, the concepts in Exercise 27A will become second nature.
在处理练习 27A 时，要把每个问题都视作加强代数求导与几何或物理意义之间联系的机会。清晰地展示所有演算过程，尤其是代入步骤和梯度的化简。在多部分题目中，即使某一部分的梯度求错了，你仍可能因正确使用垂直梯度或单位而获得方法分。在求导之前，练习将文字问题转化为变量表达式，并且始终仔细阅读题目，判断它要求的是切线、法线，还是某一特定瞬间的变化率。通过持续练习，练习 27A 中的概念会变得得心应手。
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