📚 IB & CIE Mathematics: Calculus Fundamentals Revision Notes | IB 与 CIE 数学：微积分基础考点精讲
This article distils the essential calculus concepts tested in both IB and CIE A-Level Mathematics. We cover limits, differentiation from first principles, standard derivative rules, tangents and normals, increasing/decreasing functions, stationary points, basic integration and area under a curve. Work through each section to build a rock-solid foundation.
本文精炼了 IB 与 CIE A-Level 数学中必考的微积分核心概念。涵盖极限、导数定义、基本求导法则、切线与法线、函数的增减性、驻点、基本积分以及曲线下方面积等。通过逐节学习，帮助你构建扎实的微积分基础。
1. Limits and Continuity | 极限与连续性
A limit describes the value a function approaches as the input approaches some point. For polynomials, lim(x→a) f(x) = f(a) when f is continuous. One-sided limits are essential for piecewise functions and rational functions with removable discontinuities.
极限描述当自变量趋近某一点时函数值趋近的值。对于多项式，若函数连续则 lim(x→a) f(x) = f(a)。在处理分段函数或具有可去间断点的有理函数时，单侧极限至关重要。
Continuity requires that the limit exists, the function is defined at that point, and limit equals function value. A function is differentiable at a point only if it is continuous there.
连续要求极限存在、函数在该点有定义且极限值等于函数值。函数在某点可导必先在该点连续。
lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = L → lim(x→a) f(x) = L
2. Derivative from First Principles | 从第一原理求导
The derivative f'(x) is defined as the limit of the difference quotient: f'(x) = lim(h→0) [f(x+h) – f(x)] / h. This represents the instantaneous rate of change and the slope of the tangent line.
导数 f'(x) 定义为差商的极限：f'(x) = lim(h→0) [f(x+h) – f(x)] / h。它代表瞬时变化率，也是切线斜率。
To differentiate x² from first principles: expand (x+h)² = x² + 2xh + h², subtract x², divide by h to get 2x + h, then let h→0, obtaining 2x.
对 x² 用第一原理求导：展开 (x+h)² = x² + 2xh + h²，减去 x²，除以 h 得 2x + h，令 h→0 得 2x。
This method is examinable in both IB Paper 1 and CIE Pure 1, often requiring explicit limit notation.
该方法在 IB 试卷一和 CIE 纯数 1 中均可考到，通常要求明确写出极限过程。
3. Basic Derivative Rules | 基本求导法则
Standard rules dramatically speed up differentiation. The power rule: if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Constants have derivative zero.
标准法则能显著加速求导。幂法则：若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹。常数的导数为零。
The sum rule: derivative of a sum is the sum of derivatives. The constant multiple rule: d/dx [c·f(x)] = c·f'(x).
和法则：和的导数等于导数的和。常数倍法则：d/dx [c·f(x)] = c·f'(x)。
Recognise derivatives of eˣ, ln x, sin x, cos x, and tan x. For CIE and IB, remember: d/dx(sin x) = cos x, d/dx(cos x) = -sin x.
牢记 eˣ、ln x、sin x、cos x 和 tan x 的导数。无论是 CIE 还是 IB，均需掌握：d/dx(sin x) = cos x，d/dx(cos x) = -sin x。
4. Product, Quotient and Chain Rules | 积、商与链式法则
The product rule: (uv)’ = u’v + uv’. The quotient rule: (u/v)’ = (u’v – uv’) / v². Chain rule: dy/dx = dy/du · du/dx.
积法则：(uv)’ = u’v + uv’。商法则：(u/v)’ = (u’v – uv’) / v²。链式法则：dy/dx = dy/du · du/dx。
For (x²+1)³, set u = x²+1, then dy/du = 3u² and du/dx = 2x, giving derivative 6x(x²+1)². Mixed rules are common in exams.
对于 (x²+1)³，设 u = x²+1，则 dy/du = 3u²，du/dx = 2x，导数为 6x(x²+1)²。考试中常混合使用多种法则。
Always simplify expressions before differentiating if possible. For quotients, check if numerator factors cancel denominator terms.
求导前尽可能先化简表达式。对于分式，检查分子是否能与分母约分。
5. Tangents and Normals | 切线与法线
The gradient of the tangent at x = a is m = f'(a). The equation of the tangent is y – f(a) = f'(a)(x – a).
在 x=a 处的切线斜率为 m = f'(a)。切线方程为 y – f(a) = f'(a)(x – a)。
The normal is perpendicular to the tangent, so its gradient is -1/f'(a) provided f'(a) ≠ 0. Equation: y – f(a) = -1/f'(a) (x – a).
法线垂直于切线，因此其斜率为 -1/f'(a)（假设 f'(a) ≠ 0）。方程：y – f(a) = -1/f'(a) (x – a)。
Typical questions: find tangent intersection with axes, or find point where tangent is parallel to a given line. Set f'(x) equal to required gradient.
典型问题：求切线与坐标轴交点，或求平行于某给定直线的切点。令 f'(x) 等于所需斜率。
6. Increasing and Decreasing Functions | 函数的增减性
If f'(x) > 0 on an interval, f is strictly increasing there. If f'(x) < 0, f is strictly decreasing. Use a sign diagram to determine intervals.
若在区间上 f'(x) > 0，则 f 严格递增；若 f'(x) < 0，则 f 严格递减。利用符号表确定单调区间。
Stationary points occur where f'(x) = 0, but the sign of f’ must change for a turning point. A positive-to-negative change signals a local maximum; negative-to-positive gives a local minimum.
驻点出现在 f'(x)=0 处，但只有 f’ 变号才是极值点。由正变负对应极大值，由负变正对应极小值。
If f'(x) does not change sign, we have a stationary inflection point. Always test points to the left and right in the simplified factorised derivative.
若 f'(x) 不变号，则为平稳拐点。务必在因式分解后的导数中检验左右两侧符号。
7. Classifying Stationary Points | 驻点的分类
Besides the first derivative test, the second derivative test can be used: if f”(a) > 0, then (a, f(a)) is a local minimum; if f”(a) < 0, it is a local maximum. If f''(a) = 0, the test is inconclusive – revert to first derivative test.
除了一阶导数检验，还可使用二阶导数检验：若 f”(a) > 0，则 (a, f(a)) 为极小值点；若 f”(a) < 0，则为极大值点。若 f''(a)=0，该检验失效，需回到一阶导数检验。
When modelling, local maxima/minima are often global within a given domain. Don’t forget to check endpoints if domain is closed.
在数学建模中，给定区间内的局部极大/极小值通常也是全局极值。若定义域为闭区间，勿忘检查端点值。
8. Sketching Gradient Functions | 绘制导函数图像
Given the graph of f, the gradient function graph f'(x) crosses the x-axis at f’s stationary points. Where f is increasing, f'(x) is above the axis; where f is decreasing, f’ is below.
已知 f 的图像，导函数 f'(x) 的图像在 f 的驻点处穿过 x 轴。f 递增时 f'(x) 在 x 轴上方；递减时在下方。
Points of inflection on f correspond to turning points on f'(x). Concavity information links f”(x) to the gradient’s slope.
f 的拐点对应 f'(x) 的极值点。凹凸性将 f”(x) 与导函数的斜率联系起来。
9. Indefinite Integration | 不定积分
Integration is the reverse of differentiation. The indefinite integral ∫ f(x) dx = F(x) + C, where F'(x) = f(x). C is the constant of integration.
积分是微分的逆运算。不定积分 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。C 为积分常数。
Power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1. ∫ x⁻¹ dx = ln|x| + C. Also ∫ eˣ dx = eˣ + C.
幂函数的积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1。∫ x⁻¹ dx = ln|x| + C。同时 ∫ eˣ dx = eˣ + C。
Integrals of trig functions: ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C. For ∫ sec² x dx = tan x + C.
三角函数的积分：∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C。∫ sec² x dx = tan x + C。
10. Definite Integration and Area | 定积分与面积
The definite integral ∫ₐᵇ f(x) dx computes the net signed area between the curve and the x-axis from x=a to x=b. Evaluate using F(b) – F(a).
定积分 ∫ₐᵇ f(x) dx 计算曲线与 x 轴之间从 a 到 b 的净有向面积。计算为 F(b) – F(a)。
If the curve lies below the x-axis, the integral is negative; use absolute value or split the interval to obtain total area. Total area = ∫ |f(x)| dx.
若曲线在 x 轴下方，积分值为负；用绝对值或分割区间可求总面积。总面积 = ∫ |f(x)| dx。
Area between two curves: ∫ₐᵇ (top function – bottom function) dx. Always determine intersections first to set limits.
两曲线之间的面积：∫ₐᵇ (上方函数 – 下方函数) dx。务必先求交点以确定积分上下限。
Both IB and CIE may require finding area under a curve defined parametrically or using integration to solve differential equations with given initial conditions.
IB 与 CIE 都可能要求计算参数方程下曲线面积，或利用积分求解给定初始条件的微分方程。
11. Kinematics with Calculus | 运动学中的微积分
If displacement s(t) is given, velocity v(t) = s'(t) and acceleration a(t) = v'(t) = s”(t). Conversely, v = ∫ a dt and s = ∫ v dt with appropriate initial conditions.
若位移 s(t) 已知，则速度 v(t) = s'(t)，加速度 a(t) = v'(t) = s”(t)。反之，v = ∫ a dt，s = ∫ v dt，需代入适当的初始条件。
Particle changes direction when velocity changes sign. Total distance travelled integrates speed |v(t)| over time, unlike displacement which integrates velocity directly.
速度变号时质点改变运动方向。总路程是速度绝对值 |v(t)| 对时间的积分，而有向位移则为速度直接积分。
12. Common Pitfalls and Exam Tips | 常见错误与备考技巧
Never forget the constant of integration in indefinite integrals unless an initial condition is given to find it. Check that your derivative expression is fully simplified before proceeding to tangent equations.
除非给定初始条件可求积分常数，否则切勿忘记写 +C。在求切线方程前，确保你的导数表达式已完全化简。
When using the chain rule, always differentiate the outer function and multiply by the derivative of the inner function. A missing factor inside is the most frequent mistake.
使用链式法则时，务必先对外层求导再乘以内层导数。内层漏乘因子是最常见的错误。
For area problems, always sketch the region and consider symmetry if the curve is even/odd to simplify calculations. Practise past paper questions mixing calculus with algebra and trig.
面对面积问题，一定画出示意图，若曲线为偶/奇函数可利用对称性简化计算。多练习融合代数与三角的往届真题。
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