📚 IGCSE AQA Maths: Introduction to Calculus Key Points | IGCSE AQA 数学：微积分基础 考点精讲
Calculus is a branch of mathematics that deals with rates of change (differentiation) and accumulation (integration). In the IGCSE AQA syllabus, you are expected to understand the basic principles, find derivatives and integrals of polynomial functions, and apply them to gradients, tangents, stationary points and areas under curves.
微积分是研究变化率（微分）和累积（积分）的数学分支。在 IGCSE AQA 课程中，你需要理解基本原理，求多项式函数的导数和积分，并将它们应用于梯度、切线、驻点和曲线下面积。
1. What is Calculus? | 微积分是什么？
Calculus is made up of two connected topics: differentiation, which finds the rate at which one quantity changes relative to another (the derivative), and integration, which finds the total accumulation of a quantity (the integral). These ideas are incredibly useful for describing motion, optimising shapes and calculating areas.
微积分由两个密切相关的主题组成：微分求出一个量相对于另一个量的变化率（导数），而积分求出一个量的累积总和（积分）。这些概念在描述运动、优化图形和计算面积时极为有用。
x², y = 3x³ − 5x + 2. The methods you learn here lay the foundation for further study in A‑level Mathematics and sciences.
在 IGCSE 数学中，你只需要处理多项式函数，例如 y = x²、y = 3x³ − 5x + 2。这里学到的思想和方法为 A-Level 数学及科学课程打下基础。
2. Understanding the Gradient of a Curve | 理解曲线的梯度
nt at a particular point on a curve y = f(x), we use the derivative, written as dy/dx or f'(x). It gives the slope of the tangent line to the curve at that point.
直线的梯度是常数，但曲线的梯度随着位置而变化。为了求出曲线 y = f(x) 上某一点的梯度，我们使用导数，记作 dy/dx 或 f'(x)。它给出了曲线在该点的切线斜率。
2x tells us exactly how steep: at x = 1 the gradient is 2, and at x = 3 the gradient is 6.
例如，对于 y = x²，图像是一条抛物线。在 x = 1 处曲线较为平缓，而在 x = 3 处则陡峭许多。导数 dy/dx = 2x 可以精确告诉我们陡峭程度：x = 1 时梯度为 2，x = 3 时梯度为 6。
3. Differentiating Polynomials | 多项式求导
hen dy/dx = n xⁿ⁻¹. This rule can be extended to sums and constant multiples term by term.
IGCSE 中最重要的微分法则是幂法则：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。该法则可以逐项推广到和与常数倍。
Key formulas are summarised below:
重要公式总结如下：
- If y = xⁿ, then dy/dx = n xⁿ⁻¹ — 若 y = xⁿ, 则 dy/dx = n xⁿ⁻¹
- If y = k xⁿ, then dy/dx = k n xⁿ⁻¹ — 若 y = k xⁿ, 则 dy/dx = k n xⁿ⁻¹
- If y = c (constant), then dy/dx = 0 — 若 y = c (常数), 则 dy/dx = 0
- If y = f(x) ± g(x), then dy/dx = f'(x) ± g'(x) — 若 y = f(x) ± g(x), 则 dy/dx = f'(x) ± g'(x)
Example: Differentiate y = 4x³ − 2x + 7.
The derivative is dy/dx = 4·3x² − 2·1 + 0 = 12x² − 2.
示例：对 y = 4x³ − 2x + 7 求导。
导数为 dy/dx = 4·3x² − 2·1 + 0 = 12x² − 2。
4. Finding the Gradient at a Given Point | 求给定点的梯度
Once you have found dy/dx, you can substitute a specific x‑value to find the gradient of the curve at that point. This is the same as the gradient of the tangent line at that point.
求出 dy/dx 以后，代入具体的 x 值就能求出曲线上该点的梯度，也就是该点切线的梯度。
Example: For y = 2x³ − x² + 4, find the gradient at x = 2.
dy/dx = 6x² − 2x. Substituting x = 2 gives gradient = 6(2)² − 2(2) = 24 − 4 = 20.
示例：已知 y = 2x³ − x² + 4，求 x = 2 处的梯度。
dy/dx = 6x² − 2x。代入 x = 2 得梯度 = 6(2)² − 2(2) = 24 − 4 = 20。
5. Equation of a Tangent and Normal | 切线与法线方程
The tangent to a curve at a point (a, f(a)) is a straight line with gradient m = f'(a). Its equation can be written using the point–gradient form: y − f(a) = m (x − a).
曲线在点 (a, f(a)) 处的切线是一条斜率为 m = f'(a) 的直线，其方程可用点斜式表示：y − f(a) = m (x − a)。
dient is −1/m (provided m ≠ 0). So the equation of the normal is y − f(a) = (−1/m)(x − a).
法线是与切线在同一点垂直的直线，其斜率为 −1/m（假设 m ≠ 0）。因此法线方程为 y − f(a) = (−1/m)(x − a)。
x = 3, y = 9 and dy/dx = 2x = 6. Tangent: y − 9 = 6(x − 3). Normal: y − 9 = −1/6 (x − 3).
示例：对于 y = x²，在 x = 3 处，y = 9，dy/dx = 2x = 6。切线：y − 9 = 6(x − 3)。法线：y − 9 = −1/6 (x − 3)。
6. Increasing, Decreasing and Stationary Points | 递增、递减与驻点
The sign of dy/dx tells us whether the function is increasing or decreasing:
dy/dx 的符号告诉我们函数是递增还是递减：
- dy/dx > 0: function is increasing — 函数递增
- dy/dx < 0: function is decreasing — 函数递减
- dy/dx = 0: the graph has a stationary point (tangent is horizontal) — 图像有驻点（切线水平）
To find stationary points, solve the equation f'(x) = 0. The solutions give the x‑coordinates at which the gradient is momentarily zero.
要找驻点，解方程 f'(x) = 0。解得的 x 坐标就是梯度暂时为零的位置。
: y = x³ − 3x. dy/dx = 3x² − 3. Setting dy/dx = 0 gives 3(x² − 1) = 0, so x = 1 or x = −1.
示例：y = x³ − 3x。dy/dx = 3x² − 3。令 dy/dx = 0 得 3(x² − 1) = 0，所以 x = 1 或 x = −1。
7. Second Derivative and the Nature of Turning Points | 二阶导数与转折点的性质
The second derivative, written as d²y/dx² or f''(x), is the derivative of the gradient function. It tells us how the gradient itself is changing and helps determine the nature of a stationary point.
二阶导数记作 d²y/dx² 或 f''(x)，是梯度函数的导数。它告诉我们梯度本身如何变化，并有助于判断驻点的性质。
- If f''(a) > 0, the point is a local minimum (∪ shape) — 若 f''(a) > 0，点为极小值（∪ 形）
- If f''(a) < 0, the point is a local maximum (∩ shape) — 若 f''(a) < 0，点为极大值（∩ 形）
- If f''(a) = 0, the test is inconclusive – check the gradient sign on either side — 若 f''(a) = 0，无法判断——需检查两侧梯度符号
Example: For y = x³ − 3x, dy/dx = 3x² − 3, f''(x) = 6x. At x = 1: f''(1) = 6 > 0, so a local minimum. At x = −1: f''(−1) = −6 < 0, so a local maximum.
示例：对于 y = x³ − 3x，dy/dx = 3x² − 3，f''(x) = 6x。在 x = 1 处：f''(1) = 6 > 0，为极小值。在 x = −1 处：f''(−1) = −6 < 0，为极大值。
8. Integration as the Reverse of Differentiation | 积分是微分的逆运算
definite integral of f(x) is F(x) + C, where C is an arbitrary constant. This constant appears because differentiating a constant gives zero, so all antiderivatives differ by a constant.
积分是微分的逆运算。如果 d/dx [F(x)] = f(x)，那么 f(x) 的不定积分就是 F(x) + C，其中 C 是任意常数。这个常数出现是因为常数的导数为零，因此所有原函数之间只相差一个常数。
Symbolically: ∫ f(x) dx = F(x) + C.
符号表示为：∫ f(x) dx = F(x) + C。
In the IGCSE syllabus you will only be expected to integrate polynomial terms and simple power functions.
在 IGCSE 考纲中，你只需要对多项式项和简单的幂函数进行积分。
9. Indefinite Integrals of xⁿ | xⁿ 的不定积分
The reverse power rule for integration says: ∫ xⁿ dx = xⁿ⁺¹
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