📚 High-Frequency Vocabulary for Calculus Exams | 微积分真题高频词汇汇编
Mastering the terminology of calculus is essential for tackling exam questions with confidence. This glossary compiles high-frequency words and phrases extracted from past papers across various syllabuses, organized by topic. Each entry provides the English term alongside its precise Chinese translation, ensuring bilingual learners can bridge language gaps quickly. Use this resource to reinforce your understanding of key concepts and to read problems more accurately under time pressure.
掌握微积分术语是在考试中从容应对真题的关键。本词汇表从不同考试局的历年真题中提炼出高频词汇和短语，按主题分类编排。每个条目都提供了英文术语及其准确的中文翻译，帮助双语学习者快速消除语言障碍。利用这份资源可以加深你对核心概念的理解，也能让你在限时压力下更准确地审题。
1. Limits and Continuity | 极限与连续
Limits form the foundation of calculus. Exam questions often describe behaviour of functions as x approaches a value, infinity, or from one side. Knowing the vocabulary of limits and continuity is crucial for interpreting these problems correctly.
极限是微积分的基石。真题常常描述函数在 x 趋近某个值、无穷大或单侧时的变化趋势。熟悉极限与连续的词汇对于正确理解题目至关重要。
| English | 中文 |
|---|---|
| limit | 极限 |
| approach | 趋近 |
| tends to | 趋于 |
| left-hand limit | 左极限 |
| right-hand limit | 右极限 |
| one-sided limit | 单侧极限 |
| as x → a | 当 x 趋近于 a |
| as x → ∞ | 当 x 趋近于无穷大 |
| finite limit | 有限极限 |
| infinite limit | 无穷极限 |
| limit does not exist | 极限不存在 |
| continuous at a point | 在某点连续 |
| discontinuity | 间断点 |
| removable discontinuity | 可去间断点 |
| jump discontinuity | 跳跃间断点 |
| indeterminate form | 不定式（如 0/0, ∞/∞） |
| L'Hôpital's rule | 洛必达法则 |
| squeeze theorem | 夹逼定理 |
2. Derivatives and Differentiation | 导数与微分
The derivative measures the rate of change. Exam items frequently ask for the derivative of a function, the gradient of a tangent, or differentiation from first principles. Recognising these terms instantly helps avoid confusion.
导数度量变化率。真题经常要求求函数的导数、切线的斜率或从第一原理求导。快速识别这些术语有助于避免混淆。
| English | 中文 |
|---|---|
| derivative | 导数 |
| differentiate | 求导 |
| first derivative | 一阶导数 |
| second derivative | 二阶导数 |
| higher-order derivative | 高阶导数 |
| gradient | 斜率、梯度 |
| slope of tangent | 切线斜率 |
| normal line | 法线 |
| differentiate from first principles | 从第一原理求导 |
| power rule | 幂法则 |
| product rule | 积法则 |
| quotient rule | 商法则 |
| chain rule | 链式法则 |
| implicit differentiation | 隐函数求导 |
| dy/dx | y 对 x 的导数 |
| f'(x) | f(x) 的导数 |
| f''(x) | 二阶导数 |
| differentiable | 可导的 |
| tangent line equation | 切线方程 |
3. Applications of Derivatives | 导数的应用
Derivatives are used to find critical points, classify maxima and minima, and solve optimisation problems. Many exam questions test the connection between the sign of the derivative and the shape of the graph.
导数用于求临界点、判别极大极小值以及解决最优化问题。很多真题考察导数符号与图形形状之间的关系。
| English | 中文 |
|---|---|
| stationary point | 驻点 |
| critical point | 临界点 |
| turning point | 转折点 |
| local maximum | 局部极大值 |
| local minimum | 局部极小值 |
| global maximum | 全局最大值 |
| global minimum | 全局最小值 |
| point of inflection | 拐点 |
| concave up | 凹向上 |
| concave down | 凹向下 |
| second derivative test | 二阶导数检验 |
| optimisation | 最优化 |
| increasing function | 递增函数 |
| decreasing function | 递减函数 |
| rate of change | 变化率 |
| related rates | 相关变化率 |
4. Integrals and Integration | 积分
Integration is the reverse process of differentiation, used to find areas, volumes, and antiderivatives. Exam wording often includes 'evaluate the integral', 'find the area under the curve', or 'solve the indefinite integral'.
积分是微分的逆运算，用于求面积、体积和原函数。真题中常见的表述有 'evaluate the integral', 'find the area under the curve' 或 'solve the indefinite integral'。
| English | 中文 |
|---|---|
| integral | 积分 |
| integrate | 积分 |
| antiderivative | 原函数、反导数 |
| indefinite integral | 不定积分 |
| definite integral | 定积分 |
| limits of integration | 积分上下限 |
| upper limit | 上限 |
| lower limit | 下限 |
| integrand | 被积函数 |
| constant of integration | 积分常数 |
| area under the curve | 曲线下面积 |
| area between curves | 曲线间面积 |
| signed area | 带符号的面积 |
| integration notation | 积分符号 |
| ∫ f(x) dx | 对 f(x) 关于 x 积分 |
5. Techniques of Integration | 积分方法
Beyond basic antiderivatives, exams test specific integration techniques. Questions may instruct to 'use substitution', 'integrate by parts', or 'apply partial fractions'. Knowing these method names is essential for selecting the right approach.
除了基本的不定积分，考试还会考查特定的积分方法。题目可能要求 'use substitution', 'integrate by parts' 或 'apply partial fractions'。掌握这些方法名称对于选择正确方法至关重要。
| English | 中文 |
|---|---|
| substitution method | 换元积分法 |
| integration by parts | 分部积分法 |
| partial fractions | 部分分式 |
| trigonometric substitution | 三角换元法 |
| trigonometric identities | 三角恒等式 |
| complete the square | 配方法 |
| half-angle substitution | 半角代换 |
| improper integral | 反常积分 |
| convergence of an integral | 积分的收敛性 |
| divergence | 发散 |
6. Applications of Integration | 积分的应用
Integration is used to compute volumes of revolution, arc lengths, and surface areas. Typical exam prompts include 'volume generated when the region is rotated about the x-axis' or 'find the area enclosed by the curve'.
积分用于计算旋转体体积、弧长和表面积。典型的真题提示包括 'volume generated when the region is rotated about the x-axis' 或 'find the area enclosed by the curve'。
| English | 中文 |
|---|---|
| volume of revolution | 旋转体体积 |
| disk method | 圆盘法 |
| washer method | 垫圈法 |
| shell method | 柱壳法 |
| arc length | 弧长 |
| surface area of revolution | 旋转曲面面积 |
| mean value of a function | 函数的平均值 |
| accumulated change | 累积变化量 |
7. Differential Equations | 微分方程
Differential equations appear regularly in calculus exams. You must be familiar with terms such as 'general solution', 'particular solution', and 'initial condition'. Questions often ask to separate variables or solve first-order linear equations.
微分方程在微积分考试中经常出现。你必须熟悉 'general solution', 'particular solution', 'initial condition' 等术语。题目常常要求分离变量或求解一阶线性方程。
| English | 中文 |
|---|---|
| differential equation | 微分方程 |
| ordinary differential equation | 常微分方程 |
| order of a differential equation | 微分方程的阶 |
| first-order linear | 一阶线性 |
| separable equation | 可分离变量的方程 |
| separation of variables | 分离变量法 |
| general solution | 通解 |
| particular solution | 特解 |
| initial condition | 初始条件 |
| boundary condition | 边界条件 |
| integrating factor | 积分因子 |
| slope field | 斜率场 |
| direction field | 方向场 |
8. Sequences and Series | 数列与级数
Sequences and series topics involve limits, sums, and tests for convergence. Exam vocabulary includes terms like 'convergent', 'divergent', 'geometric series', 'Taylor series', and 'power series'. Understanding these terms is vital for solving series problems.
数列与级数主题涉及极限、求和以及收敛性判别。考试词汇包括 'convergent', 'divergent', 'geometric series', 'Taylor series', 'power series' 等。理解这些术语对于解决级数问题至关重要。
| English | 中文 |
|---|---|
| sequence | 数列 |
| series | 级数 |
| nth term | 通项 |
| partial sum | 部分和 |
| converge | 收敛 |
| diverge | 发散 |
| geometric series | 几何级数 |
| common ratio | 公比 |
| arithmetic series | 算术级数 |
| p-series | p-级数 |
| harmonic series | 调和级数 |
| alternating series | 交错级数 |
| test for convergence | 收敛性检验 |
| ratio test | 比值检验法 |
| integral test | 积分检验法 |
| comparison test | 比较检验法 |
| radius of convergence | 收敛半径 |
| interval of convergence | 收敛区间 |
| Taylor
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