AS Maths: Fundamentals of Calculus | AS 数学：微积分基础考点精讲

📚 AS Maths: Fundamentals of Calculus | AS 数学：微积分基础考点精讲

Welcome to this comprehensive revision guide covering the fundamentals of calculus for AS Mathematics. Calculus is an essential tool for analysing how quantities change, and it forms the backbone of advanced mathematics, physics, engineering and many other fields. In this article, we will walk you through the key concepts of differentiation and integration, building a strong foundation for exam success.

欢迎阅读本AS数学微积分基础的全面复习指南。微积分是分析数量如何变化的核心工具，也是高等数学、物理、工程等众多领域的基石。本文将带你梳理微分与积分的关键概念，为考试打下坚实基础。

1. What is Calculus? | 什么是微积分？

Calculus is the branch of mathematics that studies continuous change. It is divided into two main, interconnected areas: differentiation, which finds the rate at which a quantity changes, and integration, which accumulates quantities over an interval.

微积分是研究连续变化的数学分支。它分为两个密切关联的领域：微分用于求一个量变化的速率，积分用于求一个量在区间上的累积。

In AS Mathematics, you will learn to differentiate polynomial functions, find equations of tangents, locate stationary points, and integrate simple functions to calculate areas under curves. These skills are directly based on the concept of limits.

在AS数学中，你将学习如何对多项式函数求导、求切线方程、确定驻点，以及通过简单函数的积分计算曲线下的面积。这些技巧都直接建立在极限概念之上。

2. The Gradient of a Curve | 曲线的梯度

For a straight line, the gradient is constant. For a curve, however, the steepness varies from point to point. The gradient of a curve at a particular point is defined as the slope of the tangent line to the curve at that point.

对于直线，梯度是常数。但对于曲线，陡峭程度因点而异。曲线上某一点的梯度定义为该点处切线的斜率。

If we take two points on the curve, (x, f(x)) and (x+h, f(x+h)), the gradient of the secant line is (f(x+h) − f(x))/h. As h approaches 0, the secant line becomes the tangent, and this limit gives us the derivative.

如果我们取曲线上两点(x, f(x))和(x+h, f(x+h))，割线的斜率为(f(x+h) − f(x))/h。当h趋近于0时，割线逼近切线，这个极限值就是导数。

3. Differentiation from First Principles | 从第一性原理求导

The derivative of a function f(x) is defined as:

函数f(x)的导数定义为：

f'(x) = lim_h→0 (f(x+h) − f(x)) / h

provided this limit exists. This method is called differentiation from first principles, and it is the foundation of all differential calculus.

只要该极限存在，此方法就称为从第一性原理求导，它是整个微分学的基础。

For example, to find the derivative of f(x) = x², we compute ((x+h)² − x²)/h = (2xh + h²)/h = 2x + h. Letting h → 0 gives f'(x) = 2x.

例如，求f(x)=x²的导数：计算((x+h)² − x²)/h = (2xh + h²)/h = 2x + h。令h→0，得到f'(x)=2x。

4. Standard Derivatives | 标准导数公式

From first principles, we obtain a set of standard derivatives that make differentiation much quicker. The most important one for AS level is the power rule:

通过第一性原理，我们得到一组标准导数公式，使求导更加快捷。AS阶段最重要的便是幂函数求导法则：

d/dx (xⁿ) = n xⁿ⁻¹

Other useful results include the derivative of a constant being zero, and the derivative of x being 1.

其他常用结果包括：常数的导数为零，x的导数为1。

A summary table can help you memorise these quickly:

以下表格能帮助你快速记忆：

f(x)	f'(x)
k (constant)	0
x	1
xⁿ	n xⁿ⁻¹
axⁿ	a n xⁿ⁻¹

5. Rules of Differentiation | 导数运算法则

When functions are combined, we can use simple rules to differentiate them without going back to first principles. The three basic rules are the constant multiple rule, the sum rule and the difference rule.

当多个函数组合时，我们可以使用简单法则求导，而无需回到第一性原理。三个基本法则是：常数倍法则、和法则与差法则。

If f(x) = c·g(x), then f'(x) = c·g'(x). If f(x) = u(x) ± v(x), then f'(x) = u'(x) ± v'(x). These allow us to differentiate any polynomial term by term.

若f(x) = c·g(x)，则f'(x) = c·g'(x)。若f(x) = u(x) ± v(x)，则f'(x) = u'(x) ± v'(x)。利用这些法则，我们可以逐项对任意多项式求导。

Example: Differentiate f(x) = 5x³ − 2x + 7. f'(x) = 15x² − 2.

示例：求f(x) = 5x³ − 2x + 7的导数。f'(x) = 15x² − 2。

6. Tangents and Normals | 切线与法线

The derivative gives the gradient of the tangent at a specific point. The equation of the tangent at (a, f(a)) is y − f(a) = f'(a)(x − a).

导数给出了特定点处切线的斜率。在点(a, f(a))处的切线方程为 y − f(a) = f'(a)(x − a)。

The normal line is perpendicular to the tangent. Its gradient is the negative reciprocal of the tangent’s gradient, so m_normal = −1 / f'(a), provided f'(a) ≠ 0.

法线垂直于切线，其斜率是切线斜率的负倒数，即 m_法 = −1 / f'(a)，前提是f'(a) ≠ 0。

For example, on the curve y = x² at (3,9), f'(3)=6, so the tangent is y−9=6(x−3), and the normal is y−9=−⅙(x−3).

例如，曲线y = x²在点(3,9)处，f'(3)=6，故切线方程为 y−9=6(x−3)，法线方程为 y−9=−⅙(x−3)。

7. Increasing and Decreasing Functions | 函数的增减性

A function is increasing on an interval if its gradient is positive (f'(x) > 0) and decreasing if its gradient is negative (f'(x) < 0).

若函数在某个区间上的梯度为正（f'(x) > 0），则函数在该区间上递增；若梯度为负（f'(x) < 0），则函数递减。

By analysing the sign of f'(x) we can sketch the shape of a graph and identify where it rises and falls.

通过分析f'(x)的符号，我们可以描绘函数图形的走向，确定其上升和下降的区间。

8. Stationary Points and Turning Points | 驻点与拐点

Stationary points occur where f'(x) = 0. These can be local maxima, local minima or points of inflection with a horizontal tangent. To classify them, we can use the second derivative test.

驻点出现在f'(x) = 0的地方。这些点可能是局部极大值点、局部极小值点或具有水平切线的拐点。为进行区分，我们可以使用二阶导数检验。

If f”(a) > 0, the point is a local minimum; if f”(a) < 0, it is a local maximum. If f''(a) = 0, further investigation is needed.

若f”(a) > 0，该点为局部极小值点；若f”(a) < 0，则为局部极大值点。若f''(a) = 0，则需进一步分析。

Alternatively, examine the sign change of f'(x) on either side of the stationary point: a change from negative to positive indicates a minimum, and from positive to negative a maximum.

也可通过检查驻点左右f'(x)的符号变化来判断：由负转正为极小值，由正转负为极大值。

9. Introduction to Integration | 积分入门

Integration is the reverse process of differentiation. Given a derivative f'(x), we recover the original function f(x), up to an additive constant. This is why integration is also called antiderivative.

积分是微分的逆过程。已知导数f'(x)，我们能还原出原函数f(x)，但会差一个常数。因此积分也称为反导数。

In AS Mathematics, you will learn to integrate simple polynomials and find areas under curves, which is a crucial application.

在AS数学中，你将学习对简单多项式进行积分，并计算曲线下的面积，这是一个核心应用。

10. Indefinite Integrals | 不定积分

The indefinite integral of a function f(x) is written as ∫ f(x) dx and represents all antiderivatives of f(x). The general form includes a constant of integration C.

函数f(x)的不定积分记为∫ f(x) dx，表示f(x)的所有反导数。一般形式包含一个积分常数C。

For a power function,

对于幂函数，

∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, for n ≠ −1

The rules of integration mirror those of differentiation: ∫ k f(x) dx = k ∫ f(x) dx, and the integral of a sum is the sum of the integrals.

积分法则与求导法则相对应：∫ k f(x) dx = k ∫ f(x) dx，且和的积分等于各自积分之和。

Example: ∫ (2x³ − 4x) dx = ½ x⁴ − 2x² + C.

示例：∫ (2x³ − 4x) dx = ½ x⁴ − 2x² + C。

11. Definite Integrals and Area | 定积分与面积

A definite integral ∫_a^b f(x) dx computes the signed area between the curve y = f(x) and the x-axis from x = a to x = b. Areas above the axis count as positive, below as negative.

定积分 ∫_a^b f(x) dx 计算曲线 y=f(x) 与 x 轴之间从 x=a 到 x=b 的带符号面积。x轴上方面积为正，下方为负。

To evaluate a definite integral, first find an antiderivative F(x), then compute F(b) − F(a).

要计算定积分，先找到一个反导数F(x)，再求值 F(b) − F(a)。

For instance, to find the area under y = x² from 1 to 3, compute ∫₁³ x² dx = [⅓ x³]₁³ = 9 − ⅓ = ²⁶⁄₃.

例如，求曲线 y=x² 在1到3区间与x轴围成的面积：∫₁³ x² dx = [⅓ x³]₁³ = 9 − ⅓ = ²⁶⁄₃。

12. The Fundamental Theorem of Calculus | 微积分基本定理

The Fundamental Theorem of Calculus reveals the deep link between differentiation and integration. It states that if F'(x) = f(x), then ∫_a^b f(x) dx = F(b) − F(a).

微积分基本定理揭示了微分与积分之间的深层联系。它指出，若 F'(x) = f(x)，则 ∫_a^b f(x) dx = F(b) − F(a)。

This theorem tells us that the accumulation of a rate of change yields the net change over the interval. It is the main reason integration can be used to find areas, distances and many other quantities.

这一定理告诉我们，变化率的累积等于区间上的净变化。正是这个原因，积分可以用于求面积、路程等众多量。

Mastering this foundation will set you up for more advanced topics in A2 Mathematics, such as integration by substitution, trigonometric integrals and differential equations.

掌握这一基础后，你将更有信心面对A2数学中的进阶内容，如换元积分、三角函数的积分以及微分方程等。

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AS Maths: Fundamentals of Calculus | AS 数学：微积分基础 考点精讲