📚 Review Set 9A: Calculus Fundamentals | 复习题集 9A：微积分基础
This review set covers the core concepts of differential and integral calculus typically encountered in the IB Mathematics Analysis and Approaches (AA) course. From limits and derivatives to integrals and their applications, mastering these skills is essential for success in the calculus component of the syllabus. Use this revision guide to test your understanding and identify areas that need further practice.
本复习题集涵盖了 IB 数学分析与方法 (AA) 课程中常见的微分和积分核心概念。从极限、导数到积分及其应用，掌握这些技能对于成功应对课程中的微积分部分至关重要。请使用本复习指南来检验你的理解，并找出需要进一步练习的薄弱环节。
1. Limits and Continuity | 极限与连续性
Limits form the foundation of calculus. The limit of a function f(x) as x approaches a value a, written as limx→a bitrarily close to when x is sufficiently near a. A function is continuous at x = a if limx→a direct substitution often works, but if it yields an indeterminate form such as 0/0, algebraic manipulation like factoring or expanding is needed. For example, limx→2 (x²−4)/(x−2) can be evaluated by factoring the numerator to (x−2)(x+2) and cancelling to obtain limx→2 (x+2) = 4.
极限是微积分的基础。函数 f(x) 当 x 趋近于某个值 a 时的极限，记作 limx→a f(x)，描述了当 x 无限接近 a 时 f(x) 趋近的值。如果 limx→a f(x) = f(a)，则称函数在 x = a 处连续。对于有理函数，通常可以直接代入求值，但若得到 0/0 这样的不定式，就需要通过代数操作比如因式分解或展开来处理。例如，limx→2 (x²−4)/(x−2) 可将分子因式分解为 (x−2)(x+2) 并约分，得到 limx→2 (x+2) = 4。
2. The Derivative from First Principles | 从第一原理看导数
The derivative of a function f at a point x, denoted as f'(x) or dy/dx, is defined as the limit f'(x) = limh→0 s limit and simplify the expression before letting h → 0. A classic example is f(x) = x², where f'(x) = limh→0 [(x+h)² − x²]/h = limh→0 (2xh + h²)/h = limh→0 (2x + h) = 2x.
函数 f 在点 x 处的导数，记作 f'(x) 或 dy/dx，定义为极限 f'(x) = limh→0 [f(x+h) − f(x)] / h。这个差商表示割线的斜率，当 h 趋近于零时，它趋近于切线的斜率。可微性蕴含连续性，但反之并不总成立。当要求从第一原理求导数时，你必须建立这个极限并在令 h → 0 之前对表达式进行化简。一个经典例子是 f(x) = x²，这时 f'(x) = limh→0 [(x+h)² − x²]/h = limh→0 (2xh + h²)/h = limh→0 (2x + h) = 2x。
3. Basic Differentiation Rules | 基本导数法则
Once the first principle is understood, applying shortcut rules makes differentiation efficient. The power rule, constant multiple rule, sum rule, and derivatives of exponential, logarithmic, and trigonometric functions are essential. Remember that the derivative of ex is ex , d/dx (cos x) = −sin x. The following table summarises key derivatives frequently tested.
在理解第一原理之后，应用快捷法则可以高效地进行求导。幂法则、常数倍法则、和法则以及指数、对数和三角函数的导数至关重要。请记住 ex 的导数是 ex，ln x 的导数是 1/x。三角函数的导数需要注意符号：d/dx (sin x) = cos x，d/dx (cos x) = −sin x。下表总结了经常考查的关键导数。
| Function f(x) | Derivative f'(x) |
|---|---|
| xn | n xn−1 |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| ex | ex |
| ln x | 1/x |
| ax (a > 0) | ax ln a |
4. Product, Quotient and Chain Rules | 乘积法则、商法则与链式法则
ifferentiate products, quotients, and composite functions, the product rule: d/dx (uv) = u'v + uv', quotient rule: d/dx (u/v) = (u'v − uv' implicit differentiation and related rates. When dealing with functions like sin(3x) or e2x, identify the inner function and multiply by its derivative. For a product like x² ex, apply the product rule with u = x², v = ex, giving derivative 2x ex + x² ex.
为了对乘积、商和复合函数求导，必须掌握乘积法则：d/dx (uv) = u'v + uv'，商法则：d/dx (u/v) = (u'v − uv')/v²，以及链式法则：dy/dx = dy/du × du/dx。链式法则尤其重要，因为在隐函数微分和相关变化率中会反复使用。在处理像 sin(3x) 或 e2x 这样的函数时，要识别出内层函数并乘以它的导数。对于像 x² ex 这样的乘积，应用乘积法则令 u = x²，v = ex，得到导数 2x ex + x² ex。
5. Implicit Differentiation | 隐函数微分
ng gives 2x + 2y (dy/dx) = 0, so dy/dx = −x/y. This technique is essential for finding tangent slopes to curves like circles and ellipses.
当 y 无法轻松表示为 x 的显函数时，可使用隐函数微分。对方程两边关于 x 求导，将 y 视为 x 的函数，因此每当出现 y 的项求导时都要乘以 dy/dx。例如，给定 x² + y² = 25，求导得到 2x + 2y (dy/dx) = 0，从而 dy/dx = −x/y。这种方法对于求圆、椭圆等曲线的切线斜率至关重要。
6. Higher-Order Derivatives | 高阶导数
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