📚 A-Level Maths: Calculus Basics - Key Points | A-Level 数学：微积分基础 考点精讲
Calculus is the branch of mathematics that deals with rates of change (differentiation) and accumulation (integration). In A-Level Mathematics, mastering the fundamentals of differentiation and integration is essential for solving problems in pure mathematics, mechanics, and statistics. This article covers the core concepts you need to know, including limits, derivative rules, integration techniques, and the Fundamental Theorem of Calculus.
微积分是研究变化率（微分）和累积量（积分）的数学分支。在 A-Level 数学中，掌握微分和积分的基本方法对于解决纯数、力学和统计问题至关重要。本文将涵盖你需要掌握的核心概念，包括极限、求导法则、积分技巧以及微积分基本定理。
1. Limits and Continuity | 极限与连续性
Before we define derivatives and integrals formally, we need the concept of a limit. The limit of a function f(x) as x approaches a value a describes the value that f(x) gets arbitrarily close to. If this limit exists and equals f(a), the function is said to be continuous at a. In A-Level work, we often evaluate limits by direct substitution, factoring, or rationalising, but the formal ε-δ definition is not required.
在正式定义导数和积分之前，我们需要理解极限的概念。函数 f(x) 当 x 趋近于 a 时的极限描述的是 f(x) 无限接近的那个值。如果该极限存在且等于 f(a)，则称函数在 a 点连续。在 A-Level 考试中，我们通常通过直接代入、因式分解或有理化来求极限，而不需要正式的 ε-δ 定义。
- Limit notation: limx→a f(x) = L
- A function is continuous at a if limx→a f(x) = f(a)
- Common tools: factor and cancel, multiply by conjugate, use known limits like limx→0 (sin x)/x = 1
- 极限记法：limx→a f(x) = L
- 如果 limx→a f(x) = f(a)，则函数在 a 点连续
- 常用技巧：因式分解后约分、乘以共轭式、利用已知极限如 limx→0 (sin x)/x = 1
2. The Definition of the Derivative | 导数的定义
The derivative of a function f at a point x, denoted f'(x) or dy/dx, is defined as the limit of the difference quotient. It represents the gradient of the tangent to the curve at that point.
函数 f 在 x 点的导数，记作 f'(x) 或 dy/dx，定义为差商的极限。它表示曲线在该点的切线斜率。
f'(x) = limh→0 [ f(x+h) – f(x) ] / h
If this limit exists, we say f is differentiable at x. Differentiability implies continuity, but not vice versa. You must be able to differentiate from first principles for simple functions like x², 1/x, and √x.
如果该极限存在，我们称 f 在 x 点可导。可导必然连续，但连续不一定可导。你必须能够对像 x²、1/x 和 √x 这样的简单函数从第一原理求导。
3. Derivatives of Basic Functions | 基本函数的导数
Once you know the definition, you can derive shortcuts. The following table lists the standard derivatives you must memorise for A-Level. All of these can be proved using first principles.
一旦掌握了定义，就可以推导出快捷方法。下表列出了 A-Level 必须熟记的标准导数，这些都可以用第一原理证明。
| Function f(x) | Derivative f'(x) |
|---|---|
| xⁿ (n constant) | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | –sin x |
| tan x | sec² x |
x is 1. Also, the power rule works for negative and fractional powers, e.g. d/dx (1/x²) = –2/x³, d/dx (√x) = 1/(2√x).
注意特例：常数的导数为 0，x 的导数为 1。此外，幂函数求导法则对负指数和分数指数同样适用，例如 d/dx (1/x²) = –2/x³，d/dx (√x) = 1/(2√x)。
4. Product and Quotient Rules | 乘积法则与商法则
separately. The product rule states: if y = u · v, where u and v are functions of x, then
当两个函数相乘或相除时，我们不能简单地对各部分分别求导。乘积法则指出：如果 y = u · v，其中 u 和 v 是 x 的函数，则
dy/dx = u · dv/dx + v · du/dx
The quotient rule handles division. If y = u / v, then
商法则处理除法。如果 y = u / v，则
dy/dx = (v · du/dx – u · dv/dx) / v²
A common mnemonic for the quotient rule is "low d-high minus high d-low over low squared". Be careful with the minus sign – it matters which term comes first.
商法则的记忆口诀是“下方的函数乘以上方的导数减去上方的函数乘以下方的导数，除以下方的平方”。注意减号——被减项的顺序很重要。
5. The Chain Rule | 链式法则
f(g(x)). It is one of the most powerful tools in calculus. If y = f(u) and u = g(x), then
链式法则用于对复合函数求导，即形如 f(g(x)) 的函数。它是微积分中最强大的工具之一。如果 y = f(u) 且 u = g(x)，则
dy/dx = dy/du × du/dx
In practice, you identify the "inner function" and multiply its derivative by the derivative of the "outer function" x² + 1, then y = u⁵, dy/du = 5u⁴, du/dx = 6x, so dy/dx = 5(3x² + 1)⁴ · 6x = 30x(3x² + 1)⁴.
实际操作中，你要识别“内层函数”，并将其导数乘以“外层函数”的导数。例如，要对 y = (3x² + 1)⁵ 求导，令 u = 3x² + 1，则 y = u⁵，dy/du = 5u⁴，du/dx = 6x，因此 dy/dx = 5(3x² + 1)⁴ · 6x = 30x(3x² + 1)⁴。
6. Implicit Differentiation | 隐函数求导
unction of x and using the chain rule whenever we differentiate a term involving y. For example, d/dx (y²) = 2y · dy/dx. After differentiating, we can solve for dy/dx algebraically.
有时 x 与 y 之间的关系是以隐式形式给出的，比如 x² + y² = 25。我们可以对等式两边关于 x 求导，将 y 视为 x 的函数，并在对任何含 y 的项求导时使用链式法则。例如，d/dx (y²) = 2y · dy/dx。求导后，我们可以通过代数运算解出 dy/dx。
nction y = f(x). It also allows us to find derivatives of inverse trigonometric functions.
对于不能写成单一显函数 y = f(x) 的曲线，这一技巧至关重要。它还能让我们求出反三角函数的导数。
7. Higher Derivatives and Stationary Points | 高阶导数与驻点
The second derivative, f''(x) or d²y/dx², is the derivative of the derivative. It measures the rate of change of the gradient, i.e. concavity. A positive second derivative indicates the function is concave up (∪), while a negative one indicates concave down (∩).
二阶导数 f''(x) 或 d²y/dx² 是一阶导数的导数。它衡量的是斜率的变化率，即凹凸性。二阶导数为正表示函数是下凸的 (∪) ，为负表示上凸的 (∩)。
Stationary points occur where f'(x) = 0. To classify them:
- If f''(x) > 0, it's a local minimum.
- If f''(x) < 0, it's a local maximum.
- If f''(x) = 0, the test is inconclusive; use the first derivative test.
驻点出现在 f'(x) = 0 处。判定驻点类型：
- 若 f''(x) > 0，则为局部极小值。
- 若 f''(x) < 0，则为局部极大值。
- 若 f''(x) = 0，二阶导数检验失效，需使用一阶导数检验。
8. Indefinite and Definite Integrals | 不定积分与定积分
indefinite integral (or antiderivative) of a function f(x) is a function F(x) such that F'(x) = f(x). It is written ∫ f(x) dx = F(x) + C, where C is an arbitrary constant.
积分是微分的逆运算。一个函数 f(x) 的不定积分（或原函数）是一个函数 F(x)，满足 F'(x) = f(x)。记作 ∫ f(x) dx = F(x) + C，其中 C 为任意常数。
is from x = a to x = b. It is evaluated by finding any antiderivative F and computing F(b) – F(a).
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的净面积。计算时先找一个原函数 F，然后计算 F(b) – F(a)。
9. Basic Integration Formulas | 基本积分公式
Just as with derivatives, there is a set of standard integrals you must know. They are the reverse of the derivative rules, but note the special case when integrating 1/x.
和求导公式一样，有一组你必须掌握的标准积分公式。它们是导数公式的逆，但注意积分 1/x 时的特例。
| Function f(x) | Indefinite Integral ∫ f(x) dx |
|---|---|
| xⁿ (n ≠ –1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | –cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
Remember that integration is linear: ∫ [af(x) + bg(x)] dx = a∫ f(x) dx + b∫ g(x) dx.
记住，积分是线性的：∫ [af(x) + bg(x)] dx = a∫ f(x) dx + b∫ g(x) dx。
10. Integration by Substitution | 代换积分法
echnique in A-Level. It reverses the chain rule. For an integral of the form ∫ f(g(x)) · g'(x) dx, we let u = g(x), so du = g'(x) dx, and the integral becomes ∫ f(u) du, which is often simpler.
代换法是 A-Level 中最常用的积分技巧，它是链式法则的逆运算。对于形如 ∫ f(g(x)) · g'(x) dx 的积分，令 u = g(x)，则 du = g'(x) dx，积分变为 ∫ f(u) du，通常更简单。
aluate directly without converting back to x. Always remember to replace dx with du / (du/dx).
在处理定积分时，你也可以更换上下限：x = a 对应 u = g(a)，x = b 对应 u = g(b)。然后直接计算，无需再换回 x。永远记得将 dx 替换为 du / (du/dx)。
11. The Fundamental Theorem of Calculus | 微积分基本定理
s two parts: first, if F is an antiderivative of f on an interval, then ∫ₐᵇ f(x) dx = F(b) – F(a). Second, the function A(x) = ∫ₐˣ f(t) dt is differentiable and A' f(x). This second part shows that integration and differentiation are inverse operations.
微积分基本定理 (FTC) 精确地连接了微分和积分。它包含两部分：第一，如果 F 是 f 在一个区间上的原函数，则 ∫ₐᵇ f(x) dx = F(b) – F(a)。第二，函数 A(x) = ∫ₐˣ f(t) dt 是可导的，且 A'(x) = f(x)。第二部分表明积分与微分互为逆运算。
In exam questions, you may be asked to differentiate an integral with a variable upper limit, or to recognise that the area function is the antiderivative. Understanding the FTC deepens your grasp of why integration works.
在考试题中，你可能需要求一个变上限积分的导数，或者认识到面积函数就是原函数。理解 FTC 能加深你对积分原理的掌握。
12. Area Under a Curve and Applications | 曲线下的面积与应用
w the x-axis, the integral gives a negative contribution; to find the total geometric area, you must split the interval where f(x) changes sign and take absolute values.
曲线 y = f(x)、x 轴以及垂直线 x = a 与 x = b 之间的面积由 ∫ₐᵇ f(x) dx 给出，前提是 f(x) ≥ 0 在区间 [a, b] 上成立。如果函数部分在 x 轴下方，积分会给出负值；要求总的几何面积，需在 f(x) 符号改变处拆分区间并取绝对值。
Also, the area between two curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ [f(x) – f A-Level exam papers, often combined with parametric equations or integration techniques.
此外，两条曲线 y = f(x) 和 y = g(x) 在区间 [a, b] 之间所围的面积是 ∫ₐᵇ [f(x) – g(x)] dx，假设 f(x) ≥ g(x)。这些面积计算是 A-Level 试卷中的常客，常与参数方程或积分技巧结合考查。
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