📚 Review set 7A | 复习题集7A
This article is designed to help IB Mathematics students master Review set 7A, a comprehensive collection of problems that consolidate essential calculus skills. The set typically covers differentiation from first principles, derivative rules, equations of tangents and normals, higher-order derivatives, optimisation, and applications to kinematics. We will unpack each concept and provide strategic guidance to tackle every question with confidence.
本文旨在帮助 IB 数学学生攻克复习题集 7A——这一综合性题集有助于巩固微积分核心技能。该题集通常涵盖第一原理求导、导数运算法则、切线与法线方程、高阶导数、最优化以及运动学应用。我们将逐一拆解各个概念，并提供解题策略，让你自信应对每一道题。
1. Understanding the Derivative from First Principles | 从第一原理理解导数
The derivative f'(x) is defined as the limit of the difference quotient as h approaches zero: f'(x) = limh→0 [f(x+h) – or rational function. Always expand brackets carefully and simplify before letting h → 0.
导数 f'(x) 定义为当 h 趋近于零时差商的极限：f'(x) = limh→0 [f(x+h) – f(x)] / h。复习题集 7A 通常以一道要求对简单多项式或有理函数使用该极限定义的题目开始。务必仔细展开括号并化简，再令 h → 0。
f'(x) = limh→0 (f(x+h) – f(x))/h
When working with rational functions, combine the two fractions in the numerator over a common denominator before cancelling h. Your final expression should not contain h in the denominator before taking the limit.
处理有理函数时，先将分子中的两个分式通分，再消去 h。最终在取极限之前，表达式的分母不应再包含 h。
2. Basic Differentiation Rules | 基本求导法则
oundation for polynomials. Remember that constants differentiate to zero, and the derivative of a sum is the sum of the derivatives.
一旦理解了极限定义，你必须熟练运用幂函数求导法则：d/dx (xⁿ) = n xⁿ⁻¹。这是多项式求导的基础。记住常数的导数为零，和的导数等于导数之和。
Power Rule: d/dx (xⁿ) = n xⁿ⁻¹
Exponential and logarithmic functions appear frequently. The derivative of eˣ is eˣ, while d/dx (eᵏˣ) = k eᵏˣ metric derivatives should also be at your fingertips: d/dx (sin x) = cos x, d/dx (cos x) = –sin x.
指数函数和对数函数频繁出现。eˣ 的导数为 eˣ，而 d/dx (eᵏˣ) = k eᵏˣ。对于自然对数，d/dx (ln x) = 1/x。三角函数的导数也应烂熟于心：d/dx (sin x) = cos x，d/dx (cos x) = –sin x。
3. Chain, Product and Quotient Rules | 链式法则、乘积法则与商法则
rule is used when a function is composed of two functions: if y = f(g(x)), then dy/dx = f'(g(x)) · g' g(x) so that dy/dx = (dy/du)×(du/dx). Review set 7A will test you on nested functions like √3x²+1 or sin²(x).
当函数由两个函数复合而成时，使用链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。在 IB 记法中，设 u = g(x)，则 dy/dx = (dy/du)×(du/dx)。复习题集 7A 会考查诸如 √3x²+1 或 sin²(x) 这样的嵌套函数。
Chain Rule: dy/dx = f'(g(x))·g'(x)
t rule handles the derivative of two multiplied functions: if y = u(x)v(x), then dy/dx = u'(x)v(x) + u(x)v' and v, compute their derivatives, then plug into the formula. Do not simplify prematurely.
乘积法则处理两个函数相乘的导数：若 y = u(x)v(x)，则 dy/dx = u'(x)v(x) + u(x)v'(x)。采用系统化方法：确定 u 和 v，计算它们的导数，再代入公式。不要过早化简。
The quotient rule is for a ratio of two functions: y = u(x)/v(x), then dy/dx = [u'(x)v(x) – u(x)v'(x)] / [v(x)]². Many students remember the mnemonic 'low d-high minus high d-low over low squared'. Check that your denominator is squared correctly.
商法则用于两个函数相除的情形：y = u(x)/v(x)，则 dy/dx = [u'(x)v(x) – u(x)v'(x)] / [v(x)]²。许多学生用口诀“下方乘上导减去上方乘下导除以下方平方”来记忆。请确保分母正确平方。
Quotient Rule: d/dx (u/v) = (u'v – uv')/v²
4. Equations of Tangents and Normals | 切线与法线方程
To find the equation of a tangent at a point (a, f(a)), evaluate the derivative f'(a) to get the slope m. Then use the point–slope form: y – f(a) = m (x – a). The normal line is perpendicular to the tangent, so its slope is –1/m, provided m ≠ 0.
要求点 (a, f(a)) 处的切线方程，先计算导数 f'(a) 得到斜率 m，然后使用点斜式：y – f(a) = m (x – a)。法线与切线垂直，因此其斜率为 –1/m（假设 m ≠ 0）。
d = 0 as instructed. Always rationalise coefficients if the question demands exact values.
复习题集 7A 常要求在同一点写出切线和法线方程。请按照题目要求将最终答案写成 y = mx + c 或 ax + by + d = 0 的形式。如果题目要求精确值，务必使系数有理化。
5. Second Derivatives and Concavity | 二阶导数与凹凸性
The second derivative, f'' res the rate of change of the gradient. It tells us about the concavity of the graph: if f''(x) > 0 on an interval, the curve is concave up; if f'' wn. Points where concavity changes are called points of inflection, and they occur where f''(x) = 0 or is undefined, provided there is a sign change.
二阶导数 f''(x) 或 d²y/dx² 衡量的是斜率的变化率。它揭示了函数图像的凹凸性：若在某一区间内 f''(x) > 0，则曲线凹向上；若 f''(x) < 0，则凹向下。凹凸性发生变化的点称为拐点，这些点满足 f''(x) = 0 或 f''(x) 无定义，并且二阶导数符号发生改变。
asked to determine the nature of stationary points using the second derivative test. If f'(a) = 0 and f''(a) > 0, the point is a local minimum; if f'' , it is a local maximum. For a point of inflection with zero gradient, check the sign of f'' around the point.
在解答复习题集 7A 时，你可能需要用二阶导数判别平稳点的性质。若 f'(a) = 0 且 f''(a) > 0，该点为局部极小点；若 f''(a) < 0，则为局部极大点。对于梯度为零的拐点，需要检查 f'' 在该点两侧的符号。
6. Optimisation Problems | 最优化问题
Optimisation questions in Review set 7A require you to find maximum or minimum values of a quantity subject to given constraints. The general strategy is: (1) define variables, (2) express the quantity to be optimised as a function of one variable using the constraint, (3) find the derivative and set it to zero to locate stationary points, (4) use the second derivative or a sign table to verify the nature, and (5) check endpoints if the domain is restricted.
复习题集 7A 中的最优化问题要求你在给定约束条件下求某个量的最大值或最小值。一般策略为：（1）定义变量；（2）利用约束将待优化量表示为单变量函数；（3）求导并令其为零，找出平稳点；（4）用二阶导数或符号表验证其性质；（5）若定义域受限，则检查端点。
Make sure to interpret the real-world context correctly. For instance, if you are maximising volume, dimensions must be positive. Always write a concluding sentence stating the maximum/minimum value and the corresponding variable values, with correct units.
务必正确理解实际背景。例如，若你正对体积求最大值，尺寸必须为正。最后一定要写一句结论，说明最大/最小值及相应的变量取值，并带上正确的单位。
7. Kinematics with Calculus | 运动学与微积分
Kinematics problems involve displacement s(t), velocity v(t) = s'(t), and acceleration a(t) = v'(t) = s'' ce travelled. Distance travelled is the sum of absolute changes in displacement over intervals where the direction is constant.
运动学问题涉及位移 s(t)、速度 v(t) = s'(t) 和加速度 a(t) = v'(t) = s''(t)。在复习题集 7A 中，你可能会明确给出 s(t) 的表达式，并被要求求出粒子静止的时刻（v=0）、向前运动的时刻（v>0）或总路程。总路程是在方向不变的每一段区间内位移变化量的绝对值之和。
Be careful not to confuse displacement and distance. Displacement can be negative, while distance is always non-negative. To find total distance, split the time interval at points where velocity is zero, then sum the absolute displacements.
注意不要混淆位移和路程。位移可以为负，而路程总为非负值。要求总路程，需用速度为零的点将时间区间分段，再将各段位移的绝对值相加。
8. Common Mistakes in Review set 7A | 复习题集7A常见错误
One frequent error is mishandling negative signs when applying the quotient rule, especially in the subtraction of u'v – uv'. Always place brackets around subtracted terms to avoid sign mistakes. Another common pitfall is forgetting to multiply by the derivative of the inner function when using the chain rule.
一个常见错误是在应用商法则时处理负号不当，尤其是在 u'v – uv' 的减法中。务必用括号把被减项括起来，以避免符号错误。另一个常见陷阱是在使用链式法则时忘记乘上内层函数的导数。
Students also often simplify rational expressions too early, which can lead to algebraic errors. Instead, find the derivative without simplifying first, then cancel common factors methodically. Finally, always double-check that your tangent and normal equations pass through the given point.
学生也常常过早化简有理表达式，这会导致代数错误。正确的做法是先不求化简便计算导数，再有条理地约去公因子。最后，务必再次确认你的切线和法线方程确实经过给定点。
9. Practising with Trigonometric Functions | 三角函数求导练习
tiation is a staple in Review set 7A. Recall that d/dx (tan x) = sec² x and d/dx (cot x) = – gonometric ratios in terms of sine and cosine before differentiating complex combinations.
三角函数求导是复习题集 7A 的一个核心内容。回忆 d/dx (tan x) = sec² x 以及 d/dx (cot x) = –csc² x。当它们出现在复合函数内部时，要仔细使用链式法则。对于复杂的组合，将各种三角比用 sin 和 cos 表示再进行求导通常会很有帮助。
Be mindful of angles measured in radians; IB calculus expects radians as the default unit for trigonometric functions. If a problem involves a constant angle in degrees, convert it to radians first.
注意角度必须使用弧度制；IB 微积分默认三角函数的自变量以弧度为单位。如果题目涉及以度表示的常值角度，请先将其转换为弧度。
10. Derivatives of Inverse Trigonometric Functions | 反三角函数的导数
udents may encounter derivatives of arcsin x, arccos x, and arctan x. d/dx (arcsin x) = 1/√1 – x², d/dx (arccos x) = –1/√1 – x² e formulas are directly derived from implicit differentiation and often appear in the context of integration as well.
虽然标准水平不总是要求掌握，进阶水平的学生可能遇到 arcsin x、arccos x 和 arctan x 的导数。d/dx (arcsin x) = 1/√1 – x²，d/dx (arccos x) = –1/√1 – x²，d/dx (arctan x) = 1/(1 + x²)。这些公式由隐函数求导直接导出，并常出现在积分相关的问题中。
If Review set 7A contains such a question, ensure that you clearly state any domain restrictions. For arcsin x and arccos x, the domain is [–1, 1] and the range is restricted accordingly.
如果复习题集 7A 包含此类题目，请确保清楚标明定义域限制。对于 arcsin x 和 arccos x，定义域为 [–1, 1]，值域也相应地受到了限制。
11. Related Rates and Implicit Differentiation | 相关变化率与隐函数求导
Some Review set 7A questions may introduce related rates, where two or more variables are functions of time t. You differentiate an equation that links the variables with respect to t, using the chain rule. Write down the given rates, identify the unknown rate you need to find, and substitute only at the end to avoid messy intermediate expressions.
部分复习题集 7A 的题目可能引入相关变化率，即两个或多个变量均为时间 t 的函数。你需要对联系这些变量的方程关于 t 求导，并运用链式法则。先写下已知的变化率，确定需求解的未知变化率，最后再代入数值，以免中间表达式过于杂乱。
skill is essential for finding tangents to curves defined by equations like x² + y² = 25.
隐函数求导用于函数未明确写成 y = f(x) 形式的情形。你将 y 视为 x 的函数，对等式两边逐项求导，每当对 y 的项求导时就乘以 dy/dx。这项技能对于求出如 x² + y² = 25 所定义曲线的切线至关重要。
12. Building Confidence for Exams | 为考试树立信心
Working through Review set 7A systematically builds the fluency and accuracy required for IB exam success. Treat each mistake as a learning opportunity: revisit the underlying concept, rework the problem without notes, and then attempt a similar question from the questionbank. Consistent practice with a timer will also enhance your speed under exam conditions.
有条不紊地完成复习题集 7A 能够培养 IB 考试所需的高流畅度与准确性。把每个错误都视为学习机会：重温相关概念、脱离笔记重新解题，然后从题库中尝试一道类似的问题。结合计时器进行持续练习，还能提升你在考场上的答题速度。
Remember that the markscheme rewards method marks even if the final answer is incorrect. Show all steps clearly, justify your use of rules, and label graphs and points neatly. A well-structured response not only reduces careless errors but also communicates your understanding to the examiner.
请记住，即便最终答案不正确，评分方案仍会奖励方法分。清楚地展示所有步骤，为所用法则给出理由，并整洁地标记图像和点。条理清晰的解答不仅能减少粗心错误，还能向考官展示你的理解过程。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图