📚 Calculus Basics in IB CCEA Mathematics | IB CCEA 数学：微积分基础 考点精讲
This article provides a comprehensive revision guide to the fundamental concepts of calculus as required by the IB and CCEA mathematics syllabuses. Covering limits, differentiation, integration, and their applications, it aims to help students build a solid understanding and avoid common pitfalls.
本文为 IB 与 CCEA 数学大纲中的微积分基础提供一份系统的复习指南。内容涵盖极限、微分、积分及其应用，旨在帮助学生建立扎实的理解并避免常见错误。
1. Introduction to Limits | 极限导论
The concept of a limit is the foundation of calculus. A limit describes the value that a function approaches as the input approaches some point. We write limx→a f(x) = L to mean that as x gets arbitrarily close to a, f(x) gets arbitrarily close to L. Limits are essential for defining derivatives and integrals precisely.
极限的概念是微积分的基础。极限描述的是当自变量趋近于某点时函数所趋近的值。我们记作 limx→a f(x) = L，表示当 x 无限接近 a 时，f(x) 无限接近 L。精确地定义导数和积分都离不开极限。
2. Evaluating Limits Algebraically | 代数法求极限
Many limits can be found by direct substitution. However, if direct substitution gives an indeterminate form such as 0/0, algebraic techniques must be used. Factoring, rationalizing the numerator or denominator, and simplifying complex fractions are common strategies. For example, to evaluate limx→2 (x² − 4)/(x − 2), factor the numerator to (x−2)(x+2) and cancel the common factor, yielding limx→2 (x+2) = 4.
许多极限可以通过直接代入求得。但如果直接代入得到 0/0 这样的不定式，就必须使用代数技巧。常见的策略包括因式分解、分子或分母有理化、化简繁分式。例如，求 limx→2 (x² − 4)/(x − 2) 时，可将分子分解为 (x−2)(x+2)，约去公因式后得到 limx→2 (x+2) = 4。
3. Continuity and Differentiability | 连续性与可导性
unctions are continuous, but not all continuous functions are differentiable (e.g., f(x) = |x| at x=0).
如果函数在某点的极限存在、函数在该点有定义且极限值等于函数值，则函数在该点连续。可导性要求导数存在，这意味着函数必须是平滑的，不能有尖角或间断。所有可导函数都是连续的，但并非所有连续函数都可导（例如 f(x) = |x| 在 x=0 处连续但不可导）。
4. The Derivative from First Principles | 导数第一原理
The derivative of a function f at a point x is defined as f'(x) = limh→0 [f(x+h) − f(x)] / h ntaneous rate of change of the function or the slope of the tangent line. For f(x) = x², f'(x) = limh→0 [(x+h)² − x²]/h = limh→0 (2xh + h²)/h = 2x.
函数 f 在 x 点的导数定义为 f'(x) = limh→0 [f(x+h) − f(x)] / h，前提是该极限存在。这叫作从第一原理求导。它表示函数的瞬时变化率，也即切线的斜率。对于 f(x) = x²，f'(x) = limh→0 [(x+h)² − x²]/h = limh→0 (2xh + h²)/h = 2x。
5. Basic Differentiation Rules | 基本微分法则
time. The power rule: d/dx [xⁿ] = n xⁿ⁻¹. The constant multiple rule: d/dx [c f(x)] = c f'(x). The sum rule: d/dx [f(x) ± g(x)] = f'(x) ± g' trigonometric functions, d/dx (sin x) = cos x, d/dx (cos x) = −sin x. The derivative of eˣ is eˣ be combined to differentiate polynomials and simple transcendental functions efficiently.
熟记基本法则可以大大提高效率。幂法则：d/dx [xⁿ] = n xⁿ⁻¹。常数倍法则：d/dx [c f(x)] = c f'(x)。和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。三角函数的导数为 d/dx (sin x) = cos x, d/dx (cos x) = −sin x。eˣ 的导数是 eˣ，ln x 的导数是 1/x。利用这些法则可以高效地求多项式及简单超越函数的导数。
6. The Chain, Product and Quotient Rules | 链式、乘积与商法则
For composite functions, use the chain rule: d/dx [f(g(x))] = f'(g(x)) · g'(x). For products: d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x). For quotients: d/dx [u(x)/v(x)] = [u'(x)v(x) − u(x)v'(x)] / [v(x)]². A common mistake is misapplying the quotient rule— When dealing with complicated expressions, simplify before differentiating where possible.
复合函数使用链式法则：d/dx [f(g(x))] = f'(g(x)) · g'(x)。乘积法则：d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)。商法则：d/dx [u(x)/v(x)] = [u'(x)v(x) − u(x)v'(x)] / [v(x)]²。常见错误是商法则中分子导数的顺序弄反——记住先对分子求导，再减去。对于复杂表达式，尽量先化简再求导。
7. Applications of Derivatives | 导数的应用
atives give the slope of a tangent line: the equation of the tangent at (a, f(a)) is y = f'(a)(x − a) + f(a). The normal line is perpendicular to the tangent, its slope being −1/f'(a). Derivatives also describe motion: if s(t) is position, then velocity v(t) = s'(t) and acceleration a(t) = v'(t) = s''(t). Finding maxima and minima involves setting f'(x) = 0 and using the second derivative test or a sign chart to classify critical points.
导数可以给出切线的斜率：曲线在 (a, f(a)) 处的切线方程为 y = f'(a)(x − a) + f(a)。法线与切线垂直，其斜率为 −1/f'(a)。导数还可以描述运动：若 s(t) 表示位移，则速度 v(t) = s'(t)，加速度 a(t) = v'(t) = s''(t)。寻找最大值和最小值时，令 f'(x) = 0，然后用二阶导数检验或符号表来判定临界点的性质。
8. Introduction to Integration | 积分导论
differentiation allows us to reverse the power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.
积分是微分的逆运算。不定积分写作 ∫ f(x) dx，代表所有原函数的族。例如，因为 d/dx (x²) = 2x，所以 ∫ 2x dx = x² + C，其中 C 是积分常数。把积分理解为逆微分能使我们逆向使用幂法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ −1。
9. Indefinite Integrals and the Constant of Integration | 不定积分与积分常数
ve. For standard functions, remember: ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ eˣ dx = eˣ + C, and ∫ 1/x dx = ln |x| + C. Learn these results thoroughly.
每个不定积分都必须包含积分常数 C。遗漏 C 是非常常见的错误，考试中会因此丢分。出现这个常数是因为常数的导数为零，因此许多不同的函数可以拥有相同的导数。对于标准函数要记住以下结果：∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C，∫ eˣ dx = eˣ + C，以及 ∫ 1/x dx = ln |x| + C。一定要彻底掌握这些公式。
10. Definite Integrals and Area Under a Curve | 定积分与曲线下方面积
A definite integral ∫ab e counted as positive, and areas below as negative. To find the total area enclosed, you must split the integral at the points where f(x) crosses the x-axis and evaluate each part separately, taking absolute values where necessary. The fundamental theorem connects definite integrals and antiderivatives.
定积分 ∫ab f(x) dx 表示从 x = a 到 x = b 曲线 y = f(x) 与 x 轴之间的净面积。x 轴上方的面积计为正，下方的计为负。要计算所围的总面积，必须在 f(x) 穿过 x 轴的点处将积分拆开，分别求值，必要时取绝对值。微积分基本定理将定积分与原函数联系了起来。
11. The Fundamental Theorem of Calculus | 微积分基本定理
f Calculus consists of two parts. Part 1: If F is an antiderivative of f on [a, b], then ∫ab f(x) dx = F(b) − F(a). Part 2: The function g(x) = ∫ax f(t) dt is differentiable and g' erse relationship between differentiation and integration and provides a powerful tool for computing definite integrals without Riemann sums.
微积分基本定理包含两部分。第一部分：如果 F 是 f 在 [a, b] 上的一个原函数，那么 ∫ab f(x) dx = F(b) − F(a)。第二部分：函数 g(x) = ∫ax f(t) dt 是可导的，且 g'(x) = f(x)。这个定理揭示了微分与积分之间的互逆关系，并且提供了一种无需黎曼和的强大工具来计算定积分。
12. Exam Tips for CCEA/IB Calculus | CCEA/IB 微积分考试技巧
rals. When a question asks for the equation of a tangent, clearly state the gradient m = f' sign. Practice past paper questions under timed conditions to improve speed and accuracy.
在 IB 和 CCEA 考试中，写出清晰的解题步骤至关重要。要写下所使用的导数法则或积分步骤，并在不定积分中始终写上 +C。当题目要求求切线方程时，先明确写出梯度 m = f'(a)，再写直线方程。对于面积问题，绘制草图有助于识别函数在何处变号。在限时条件下练习历年真题，以提高速度和准确性。
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