IB Edexcel Maths: Calculus Fundamentals Key Points | IB Edexcel 数学：微积分基础考点精讲

📚 IB Edexcel Maths: Calculus Fundamentals Key Points | IB Edexcel 数学：微积分基础考点精讲

Calculus is the mathematical study of change and motion, built on two central ideas – differentiation and integration. In both IB and Edexcel specifications, mastering the fundamentals of limits, derivatives and basic integrals is essential for success on papers covering functions, graphs, optimisation and kinematics. This article presents the core concepts with clear bilingual explanations, structured to help you memorise rules, apply techniques and avoid common pitfalls.

微积分是研究变化与运动的数学分支，建立在微分与积分这两个核心思想之上。在 IB 与 Edexcel 考试大纲中，掌握极限、导数和基本积分的基础知识，是应对函数、图像、最优化以及运动学等考题的关键。本文以清晰的双语讲解呈现核心概念，帮助你记忆法则、运用技巧并避开常见错误。

1. Limits and Continuity | 极限与连续

The limit of a function f(x) as x approaches a is the value that f(x) gets arbitrarily close to, written as limₓ→ₐ f(x) = L. For example, limₓ→₂ (x² – 4)/(x – 2) = 4 because the expression simplifies to x + 2 for x ≠ 2.

函数 f(x) 当 x 趋近于 a 时的极限，是 f(x) 无限接近的数值，记作 limₓ→ₐ f(x) = L。例如，limₓ→₂ (x² – 4)/(x – 2) = 4，因为当 x ≠ 2 时该式可化简为 x + 2。

A function is continuous at a point a if three conditions hold: f(a) is defined, limₓ→ₐ f(x) exists, and limₓ→ₐ f(x) = f(a). Polynomials, sin x, cos x and eˣ are continuous everywhere on their domains.

函数在点 a 处连续，需要满足三个条件：f(a) 有定义，limₓ→ₐ f(x) 存在，且 limₓ→ₐ f(x) = f(a)。多项式函数、sin x、cos x 和 eˣ 在其定义域内处处连续。

One-sided limits are used when the left and right behaviour differs. The notation limₓ→ₐ⁻ f(x) means x approaches a from the left, and limₓ→ₐ⁺ f(x) means from the right.

当左右两侧的行为不同时，需要使用单侧极限。记法 limₓ→ₐ⁻ f(x) 表示 x 从左侧趋近 a，limₓ→ₐ⁺ f(x) 表示从右侧趋近。

2. The Derivative as a Slope | 导数作为斜率

The derivative of a function at a point represents the slope of the tangent line to the graph at that point. It is defined as the limit of the difference quotient:

函数在一点的导数表示该点处切线的斜率。它被定义为差商的极限：

f'(x) = lim_h→0 (f(x + h) – f(x)) / h

This limit gives the instantaneous rate of change of y with respect to x. The notation dy/dx is introduced by Leibniz and widely used in both IB and Edexcel papers.

这个极限给出了 y 关于 x 的瞬时变化率。莱布尼茨引入的记号 dy/dx 在 IB 和 Edexcel 试卷中得到广泛使用。

If f'(x) > 0, the function is increasing at that point; if f'(x) < 0, it is decreasing. A zero derivative may indicate a stationary point, which could be a maximum, minimum or point of inflection.

若 f'(x) > 0，函数在该点递增；若 f'(x) < 0，则递减。导数为零可能表示一个驻点，它可能是极大值、极小值或拐点。

3. Basic Differentiation Rules | 基本求导法则

Several fundamental rules allow us to differentiate combinations of functions without using the limit definition every time:

以下基本法则使我们无需每次都使用极限定义来求导：

Constant rule: d/dx [c] = 0

常数法则： d/dx [c] = 0
Constant multiple rule: d/dx [c f(x)] = c f'(x)

常数倍法则： d/dx [c f(x)] = c f'(x)
Sum rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

和差法则： d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

These rules simplify differentiation for polynomials and many other functions. For example, if y = 3x² + 5x – 7, then dy/dx = 6x + 5.

这些法则简化了多项式及许多其他函数的求导。例如，若 y = 3x² + 5x – 7，则 dy/dx = 6x + 5。

4. Power Rule and Polynomials | 幂法则与多项式

The power rule is the single most used differentiation technique for monomials and polynomials:

幂法则是单项式和多项式求导中最常用的技巧：

d/dx [xⁿ] = n xⁿ⁻¹, where n is any real number.

To differentiate a polynomial, apply the power rule term by term. For y = 4x³ – 2x² + x – 9, we get dy/dx = 12x² – 4x + 1.

要微分多项式，逐项使用幂法则即可。对于 y = 4x³ – 2x² + x – 9，得到 dy/dx = 12x² – 4x + 1。

Negative and fractional exponents are handled the same way. For f(x) = 1/x² = x⁻², f'(x) = –2x⁻³ = –2/x³. For g(x) = √x = x^(½), g'(x) = (1/2) x^(–½) = 1/(2√x).

负指数和分数指数同样处理。对于 f(x) = 1/x² = x⁻²，f'(x) = –2x⁻³ = –2/x³。对于 g(x) = √x = x^(½)，g'(x) = (1/2) x^(–½) = 1/(2√x)。

5. Product Rule and Quotient Rule | 乘积法则与商法则

When two differentiable functions are multiplied, the derivative is not simply the product of their derivatives. The product rule states:

当两个可导函数相乘时，导数并非简单地等于它们导数的乘积。乘积法则指出：

d/dx [u v] = u’ v + u v’

If y = x² sin x, let u = x², v = sin x. Then u’ = 2x, v’ = cos x, so dy/dx = 2x sin x + x² cos x.

若 y = x² sin x，设 u = x², v = sin x，则 u’ = 2x, v’ = cos x，因此 dy/dx = 2x sin x + x² cos x。

For division, the quotient rule is essential:

d/dx [u / v] = (u’ v – u v’) / v²

For y = (2x + 1)/(x² – 3), u = 2x + 1, v = x² – 3. Then u’ = 2, v’ = 2x. The derivative is dy/dx = [2(x² – 3) – (2x + 1)(2x)] / (x² – 3)², which simplifies to (–2x² – 2x – 6)/(x² – 3)².

对于 y = (2x + 1)/(x² – 3)，u = 2x + 1, v = x² – 3，则 u’ = 2, v’ = 2x。导数为 dy/dx = [2(x² – 3) – (2x + 1)(2x)] / (x² – 3)²，化简为 (–2x² – 2x – 6)/(x² – 3)²。

6. Chain Rule | 链式法则

The chain rule handles composite functions – a function inside another. If y = f(g(x)), then:

链式法则处理复合函数——一个函数嵌套在另一个函数内部。如果 y = f(g(x))，那么：

dy/dx = f'(g(x)) · g'(x) or dy/dx = dy/du · du/dx, where u = g(x)

Example: y = sin(3x²). Let u = 3x², then y = sin u. dy/du = cos u, du/dx = 6x. So dy/dx = cos(3x²) · 6x = 6x cos(3x²).

示例：y = sin(3x²)。设 u = 3x²，则 y = sin u。dy/du = cos u, du/dx = 6x。所以 dy/dx = cos(3x²) · 6x = 6x cos(3x²)。

The ‘outside-in’ approach: differentiate the outer function, keeping the inner untouched, then multiply by the derivative of the inner. For y = (5x – 1)⁴, the outer is u⁴, derivative 4u³, inner derivative 5, giving dy/dx = 4(5x – 1)³ · 5 = 20(5x – 1)³.

“由外而内”的方法：先对外层函数求导，保持内层不变，再乘以内层导数。对于 y = (5x – 1)⁴，外层为 u⁴，导数为 4u³，内层导数为 5，得到 dy/dx = 4(5x – 1)³ · 5 = 20(5x – 1)³。

7. Implicit Differentiation | 隐函数求导

When an equation defines y implicitly as a function of x, we differentiate both sides with respect to x, treating y as a function and using the chain rule. Every time we differentiate a term involving y, we multiply by dy/dx.

当一个方程隐式地定义 y 为 x 的函数时，我们对等式两边关于 x 求导，并将 y 视为函数并运用链式法则。每次对含 y 的项求导时，都要乘以 dy/dx。

Example: x² + y² = 25. Differentiating: 2x + 2y (dy/dx) = 0 → dy/dx = –x/y.

示例：x² + y² = 25。求导：2x + 2y (dy/dx) = 0 → dy/dx = –x/y。

Implicit differentiation is powerful for related rates problems and for finding gradients of curves where y cannot be easily isolated, such as x³ + x y + y³ = 6.

隐函数求导在相关变化率问题中非常有用，也用于求解那些难以将 y 显式表示的曲线的斜率，例如 x³ + x y + y³ = 6。

8. Higher-Order Derivatives | 高阶导数

The second derivative, f”(x) or d²y/dx², is the derivative of the first derivative. It gives the rate of change of the gradient, which relates to concavity and acceleration.

二阶导数 f”(x) 或 d²y/dx² 是一阶导数的导数。它表示斜率的变化率，与凹凸性和加速度有关。

If f”(x) > 0 on an interval, the graph is concave up (like a cup). If f”(x) < 0, it is concave down (like a cap). A point where concavity changes is a point of inflection, and f''(x) = 0 there (though the converse is not always true).

若在区间内 f”(x) > 0，图像向上凹（呈杯状）；若 f”(x) < 0，图像向下凹（呈帽状）。凹凸性发生改变的点是拐点，该点处 f''(x) = 0（但逆命题不一定成立）。

IB and Edexcel frequently test the second derivative to classify stationary points. If f'(a) = 0 and f”(a) > 0, the point is a local minimum; if f”(a) < 0, it is a local maximum. If f''(a) = 0, the test is inconclusive and the first derivative test should be used.

IB 与 Edexcel 常考利用二阶导数判断驻点类型。若 f'(a) = 0 且 f”(a) > 0，该点为局部极小值；若 f”(a) < 0，则为局部极大值。若 f''(a) = 0，该判别法失效，需使用一阶导数判别法。

9. Tangent and Normal Lines | 切线与法线

At a given point (x₁, y₁) on a curve y = f(x), the gradient of the tangent is m = f'(x₁). The equation of the tangent is:

在曲线 y = f(x) 上的给定点 (x₁, y₁)，切线的斜率为 m = f'(x₁)。切线方程为：

y – y₁ = m (x – x₁)

The normal line is perpendicular to the tangent. Its gradient is –1/m (provided m ≠ 0). The equation of the normal uses the same point-slope form with the negative reciprocal gradient.

法线垂直于切线，其斜率为 –1/m（前提是 m ≠ 0）。法线方程使用相同的点斜式，但代入负倒数斜率。

Example: for y = x² at x = 2, y₁ = 4, m = 4. Tangent: y – 4 = 4(x – 2). Normal: y – 4 = –1/4 (x – 2).

示例：对于 y = x² 在 x = 2 处，y₁ = 4, m = 4。切线：y – 4 = 4(x – 2)。法线：y – 4 = –1/4 (x – 2)。

10. Optimisation and Stationary Points | 最优化与驻点

Many real-world problems require finding maximum or minimum values of a function on a given interval. The steps are:

许多实际问题需要在给定区间上求函数的最大值或最小值。步骤如下：

Define the variable to optimise (e.g. area, volume, cost).
定义需要最优化的变量（例如面积、体积、成本）。
Express the quantity as a function of one variable using given constraints.
利用约束条件将该量表示为单一变量的函数。
Differentiate to find stationary points (where f'(x) = 0).
求导以找到驻点（即 f'(x) = 0 处）。
Classify using the first or second derivative test.
采用一阶或二阶导数判别法进行分类。
Check endpoints if the interval is closed.
若区间为闭区间，需检查端点值。

A classic example: find the dimensions of a rectangle with perimeter 20 m that maximises area. Let length = x, width = 10 – x. Area A = x(10 – x) = 10x – x². A’ = 10 – 2x = 0 → x = 5, giving a square of side 5 m. A” = –2 < 0 confirms maximum.

经典例子：求周长为 20 m 且面积最大的矩形的尺寸。设长为 x，宽为 10 – x。面积 A = x(10 – x) = 10x – x²。A’ = 10 – 2x = 0 → x = 5，得到一个边长为 5 m 的正方形。A” = –2 < 0 确认是极大值。

11. Basic Integration | 基本积分

Integration is the reverse process of differentiation. The indefinite integral of a function f(x) is a family of antiderivatives plus a constant C:

积分是微分的逆过程。函数 f(x) 的不定积分是一族原函数加上常数 C：

∫ f(x) dx = F(x) + C, where F'(x) = f(x)

The power rule for integration is the counterpart of differentiation:

积分的幂法则与微分幂法则对应：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ –1

Example: ∫ (3x² + 4x) dx = x³ + 2x² + C. To find the constant C, use an initial condition or boundary values.

示例：∫ (3x² + 4x) dx = x³ + 2x² + C。利用初始条件或边界值可求出常数 C。

Definite integrals represent the signed area under a curve between two limits a and b:

定积分表示曲线在两个界限 a 与 b 之间的有正负号面积：

∫ₐᵇ f(x) dx = F(b) – F(a)

This fundamental connection between differentiation and integration is the heart of calculus, known as the Fundamental Theorem of Calculus.

微分与积分之间的这一基本联系是微积分的核心，被称为微积分基本定理。

12. Common Mistakes and Exam Tips | 常见错误与应试技巧

Below are frequent pitfalls and how to avoid them in IB and Edexcel calculus questions:

以下是 IB 与 Edexcel 微积分考题中常见的陷阱及避免方法：

Forgetting the constant of integration: Always add + C for indefinite integrals unless a definite integral is evaluated.
忘记积分常数： 不定积分必须加 + C，除非计算的是定积分。
Misapplying the product/quotient rule: The product rule is not (u’)(v’); the quotient rule has a minus sign in the numerator and denominator squared.
误用乘积/商法则： 乘积法则并非 (u’)(v’)；商法则分子中有减号，分母要平方。
Chain rule omissions: When differentiating composite functions like sin(2x), students often forget to multiply by the derivative of the inner function 2x, giving cos(2x) instead of 2 cos(2x).
遗漏链式法则： 求导复合函数如 sin(2x) 时，学生常忘记乘以内层函数 2x 的导数，导致得到 cos(2x) 而非 2 cos(2x)。
Stationary point classification errors: Relying solely on f”(x) = 0 as a point of inflection without checking sign change. Always test intervals.
驻点分类错误： 仅凭 f”(x) = 0 就认定为拐点而不检查符号变化。务必检验邻域符号。
Not simplifying before differentiating: It is often easier to simplify expressions first, e.g. write (x²+2x)/x as x + 2 before differentiating.
求导前未化简： 通常先化简表达式更简便，例如先将 (x²+2x)/x 写成 x + 2 再求导。

Use clear notation, show all steps, and always check the domain when solving optimisation problems. Practising past paper questions under timed conditions is one of the most effective revision strategies.

使用清晰的符号，展示所有步骤，并在求解最优化问题时务必检查定义域。在计时条件下练习历年真题是最有效的复习策略之一。

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IB Edexcel Maths: Calculus Fundamentals Key Points | IB Edexcel 数学：微积分基础 考点精讲