IB Mathematics: Calculus Basics Key Concepts | IB 数学：微积分基础考点精讲

📚 IB Mathematics: Calculus Basics Key Concepts | IB 数学：微积分基础考点精讲

Calculus is the mathematical study of continuous change, and it forms a central pillar of the IB Mathematics Analysis and Approaches syllabus. Mastering the fundamentals of differentiation and integration not only helps you solve textbook problems but also deepens your understanding of how mathematics describes the real world — from motion and growth to optimisation. This article breaks down the essential concepts you must know, with clear explanations in both English and Chinese, fully aligned with the IB style.

微积分是研究连续变化的数学分支，也是 IB 数学分析与方法课程的核心支柱。掌握微分和积分的基本原理不仅能帮助你解决课本中的问题，还能加深你对数学如何描述现实世界——从运动、增长到最优化——的理解。本文用中英双语为你拆解必备的核心概念，全部紧扣 IB 考点。

1. Limits and Continuity | 极限与连续性

A limit describes the value a function approaches as the input gets arbitrarily close to a certain point. For the function f(x), we write lim_{x→a} f(x) = L if f(x) can be made as close as we wish to L by taking x sufficiently close to a. A function is continuous at a if lim_{x→a} f(x) = f(a). In IB, you are expected to evaluate limits from graphs, tables, and simple algebraic manipulations, including cases that lead to indeterminate forms like 0/0.

极限描述的是当输入值无限接近某一点时函数所趋近的值。对于函数 f(x)，如果当 x 充分接近 a 时 f(x) 可以任意接近 L，我们就记作 lim_{x→a} f(x) = L。如果 lim_{x→a} f(x) = f(a)，则称函数在 a 处连续。在 IB 考试中，你需要会通过图像、表格以及简单的代数变形求极限，包括遇到 0/0 型未定式的情形。

2. The Derivative as a Rate of Change | 导数作为变化率

The derivative of a function f at a point x is defined as f'(x) = lim_{h→0} [f(x+h) – f(x)] / h, provided this limit exists. Geometrically, it represents the slope of the tangent line to the graph of f at that point. In applied contexts, the derivative tells you the instantaneous rate of change, for example velocity as the derivative of position with respect to time. The IB syllabus often tests your ability to interpret the derivative in real-world scenarios.

函数 f 在点 x 处的导数定义为 f'(x) = lim_{h→0} [f(x+h) – f(x)] / h，前提是该极限存在。从几何上看，它代表函数图像在该点处切线的斜率。在实际应用中，导数表示瞬时变化率，例如速度就是位移对时间的导数。IB 大纲经常考查你在真实情境中解读导数意义的能力。

3. Power Rule and Basic Differentiation | 幂函数法则与基本微分公式

The power rule states that if f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹, where n is any real number. Combined with constant multiple and sum rules, this allows you to differentiate any polynomial or simple power function. For example, the derivative of 3x⁴ – 5x² + 2x – 7 is 12x³ – 10x + 2. You must memorise this rule and be able to apply it instantly, as it is the foundation for all further differentiation techniques.

幂函数法则指出，若 f(x) = xⁿ，则 f'(x) = n·xⁿ⁻¹，其中 n 为任意实数。结合常数倍法则与和法则，你就能够对任意多项式或简单幂函数求导。例如，3x⁴ – 5x² + 2x – 7 的导数为 12x³ – 10x + 2。你必须牢记并熟练运用这一法则，因为它是所有后续微分技巧的基础。

4. Product and Quotient Rules | 乘积法则与商法则

When differentiating the product of two functions, use the product rule: (u·v)’ = u’·v + u·v’. The quotient rule handles division: (u/v)’ = (u’·v – u·v’) / v². A common IB mistake is to forget the square in the denominator of the quotient rule, or to misplace the minus sign. Work through examples step by step: for f(x) = x²·sin(x), f'(x) = 2x·sin(x) + x²·cos(x). For g(x) = eˣ / x, g'(x) = (eˣ·x – eˣ·1) / x² = eˣ(x – 1) / x².

在对两个函数的乘积求导时，需要使用乘积法则：(u·v)’ = u’·v + u·v’。商法则用于处理除法：(u/v)’ = (u’·v – u·v’) / v²。IB 考试中常见的错误是在商法则的分母中漏掉平方，或者把减号写错。可以一步步演练例子：对于 f(x) = x²·sin(x)，f'(x) = 2x·sin(x) + x²·cos(x)；对于 g(x) = eˣ / x，g'(x) = (eˣ·x – eˣ·1) / x² = eˣ(x – 1) / x²。

5. Chain Rule | 链式法则

The chain rule is used for composite functions: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x). In Leibniz notation, if y = f(u) and u = g(x), then dy/dx = (dy/du)·(du/dx). This rule is essential for differentiating trigonometric, exponential, logarithmic, and radical compositions. For instance, differentiate sin(2x) to get 2cos(2x), or e^(x²) to get 2x·e^(x²). The IB exam often embeds the chain rule within product or quotient rule questions, so you must be able to blend them seamlessly.

链式法则用于复合函数的求导：如果 y = f(g(x))，那么 dy/dx = f'(g(x))·g'(x)。用莱布尼茨记号表示，若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du)·(du/dx)。这条法则对于三角、指数、对数和根式复合函数求导至关重要。例如，对 sin(2x) 求导得 2cos(2x)，对 e^(x²) 求导得 2x·e^(x²)。IB 考试常常在乘积法则或商法则的题目中融入链式法则，因此你必须能无缝结合使用。

6. Tangents and Normals | 切线与法线

The equation of the tangent line to y = f(x) at x = a is y – f(a) = f'(a)(x – a). The normal line is perpendicular to the tangent, so its slope is -1 / f'(a), and its equation is y – f(a) = -1/f'(a) (x – a). A typical IB question gives a function and a point, asks for the tangent and normal equations, and may require you to find where they intersect the axes. Always check if the derivative is zero — a horizontal tangent gives a normal that is vertical.

曲线 y = f(x) 在 x = a 处的切线方程为 y – f(a) = f'(a)(x – a)。法线与切线垂直，因此其斜率为 -1 / f'(a)，方程为 y – f(a) = -1/f'(a) (x – a)。典型的 IB 题目会给出一个函数和一个点，要求你写出切线和法线方程，并可能要求你求出它们与坐标轴的交点。永远要检查导数是否为零——水平切线的法线是垂直的。

7. Higher Order Derivatives | 高阶导数

The second derivative, f”(x) or d²y/dx², is the derivative of the first derivative. It tells you about the concavity of the graph and is crucial for classifying stationary points. In kinematics, displacement s(t) gives velocity v(t) = s'(t), and acceleration a(t) = v'(t) = s”(t). IB questions may ask for the n-th derivative of simple functions, or applications involving position, velocity, and acceleration. Always remember to differentiate term by term when finding higher derivatives.

二阶导数，记作 f”(x) 或 d²y/dx²，是一阶导数的导数。它揭示了函数图像的凹凸性，对于判断驻点的类别至关重要。在运动学中，位移 s(t) 给出速度 v(t) = s'(t)，加速度则为 a(t) = v'(t) = s”(t)。IB 题目可能会要求简单函数的 n 阶导数，或者涉及位移、速度和加速度的应用。在求高阶导数时，永远要记得逐项求导。

8. Stationary Points and Optimisation | 驻点与最优化

Stationary points occur where f'(x) = 0. To classify them, use the first derivative test (sign change) or second derivative test: if f”(x) > 0 it is a local minimum; if f”(x) < 0 it is a local maximum; if f''(x) = 0 the test is inconclusive. In optimisation problems, model a real-world quantity as a function of one variable, find its derivative, set it to zero, and use the second derivative or end-point checking to confirm a maximum or minimum. Common IB examples involve maximising area, volume, or minimising cost.

驻点出现在 f'(x) = 0 的地方。要判断它的类型，可使用一阶导数检验法（看符号变化）或二阶导数检验法：若 f”(x) > 0 则为局部极小值；若 f”(x) < 0 则为局部极大值；若 f''(x) = 0 则检验法失效。在最优化问题中，将现实量表达为一个变量的函数，求导并令其为零，再用二阶导数或端点检验确认最大值或最小值。IB 常见题型包括最大化面积、体积或最小化成本。

9. Introduction to Integration | 积分基础

Integration is the reverse process of differentiation. If F'(x) = f(x), then F(x) is an antiderivative of f(x), and we write ∫ f(x) dx = F(x) + C. The constant C is essential because differentiation ‘loses’ constant terms. The basic rule is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1. You must be comfortable rewriting expressions in power form before integrating, for example ∫ (3x² + 1/x²) dx = x³ – 1/x + C. IB also expects you to find f(x) given f'(x) and an initial condition to determine C.

积分是微分的逆运算。如果 F'(x) = f(x)，则 F(x) 是 f(x) 的一个原函数，我们记作 ∫ f(x) dx = F(x) + C。常数 C 必不可少，因为微分会“丢失”常数项。基本积分公式为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ -1。你必须熟悉在积分前将表达式改写为幂函数形式的技巧，例如 ∫ (3x² + 1/x²) dx = x³ – 1/x + C。IB 还要求你根据 f'(x) 和一个初始条件确定 C，从而求出函数 f(x)。

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

A definite integral computes the signed area between a curve and the x-axis over an interval [a, b]: ∫ₐᵇ f(x) dx = F(b) – F(a). If the curve lies below the x-axis, the integral gives a negative value; total area is found by splitting the interval at the roots. IB questions often ask for the area bounded by a curve and a line, requiring you to find intersection points, set up the difference between the functions, and integrate. Use your GDC carefully for complicated integrals but show analytical setup clearly.

定积分计算曲线与 x 轴之间在区间 [a, b] 上的有向面积：∫ₐᵇ f(x) dx = F(b) – F(a)。如果曲线位于 x 轴下方，积分值为负；求总面积时需在根处分割区间。IB 题目通常会要求计算曲线与直线围成的面积，你需要找到交点，构造两个函数之差，然后积分。对于复杂的积分，可以谨慎使用 GDC，但必须清晰写出分析过程。

11. Kinematics and the Connection between Position, Velocity, Acceleration | 运动学与位移、速度、加速度的关系

In IB Mathematics, kinematics provides a perfect context for linking differentiation and integration. If s(t) is the position of a particle along a line, then velocity v(t) = s'(t) and acceleration a(t) = v'(t) = s”(t). Conversely, given acceleration and initial conditions, you can find velocity and position by integration. Total distance travelled is found by integrating the absolute value of velocity or by considering sign changes. These questions require careful attention to initial values and correct setup of definite integrals.

在 IB 数学中，运动学为联系微分与积分提供了绝佳背景。如果 s(t) 表示粒子沿直线的位置，那么速度 v(t) = s'(t)，加速度 a(t) = v'(t) = s”(t)。反过来，已知加速度和初始条件，可以通过积分求得速度与位置。总路程则通过对速度的绝对值积分或考虑符号变化来求得。这类题目需要仔细关注初始值，并正确建立定积分。

12. Exam Tips and Common Pitfalls | 考试技巧与常见误区

When working on calculus questions in the IB exam, always show your derivative before setting it to zero, write the constant of integration in indefinite integrals, and remember to split the integral when the function changes sign for total area. Avoid misapplying the product and quotient rules — write u and v explicitly if needed. Check your GDC’s mode if you are using it for trigonometric calculations. Finally, practice reading the question carefully: if it asks for the equation of a normal, don’t give a tangent, and if it requests total distance, don’t just give displacement.

在 IB 考试中处理微积分题目时，一定要在设导数为零之前展示你的导数，不要在不定积分中遗漏积分常数，并在求总面积时记得当函数变号时分割积分区间。避免错误使用乘积法则和商法则——必要时可以明确写下 u 和 v。若使用 GDC 进行三角计算，请检查角度模式。最后，仔细审题练习：如果题目要求法线方程，就不要写成切线；如果要求总路程，就不要只给位移。

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IB Mathematics: Calculus Basics Key Concepts | IB 数学：微积分基础 考点精讲