📚 Review Set 22C: IB Mathematics Mastery | 复习集22C：IB数学精练
Review Set 22C is a comprehensive checkpoint designed to consolidate the core skills tested in the IB Mathematics: Analysis and Approaches (AA) and Applications and Interpretation (AI) courses. It brings together functions, trigonometry, calculus, and probability in a single, high-stakes revision exercise. Students who work through this set systematically build the fluency and problem-solving agility needed for Papers 1, 2, and 3.
复习集22C是一套综合性检测题，旨在夯实IB数学分析与方法(AA)以及应用与解释(AI)课程的核心技能。它将函数、三角学、微积分和概率等板块融为一体，构成一次高强度的考前演练。系统完成这套习题的学生，能够显著提升解题的流畅度与应变能力，为试卷一、试卷二和试卷三做好充分准备。
1. Understanding the Scope of Review Set 22C | 了解复习集22C的覆盖范围
Review Set 22C is not a single-topic drill; it mirrors the interconnected nature of the IB syllabus. Expect to see questions that require algebraic manipulation before a derivative can be taken, or probability combined with logarithmic functions. The set typically contains 15–20 multi-part problems, each escalating in difficulty.
复习集22C不是单一知识点的训练，它反映了IB课程中各个主题相互交织的特点。你会遇到需要先进行代数变形才能求导的题目，或是概率与对数函数结合的综合性问题。该题集通常包含15至20道多小问的题目，难度逐级增加。
The problems often begin with a straightforward ‘show that’ command, then ask for exact values, and finally explore generalisation or real-world interpretation. This structure is deliberate—it trains students to move confidently from routine calculation to conceptual reasoning, which is essential for achieving a level 7.
题目往往以简单的“证明”指令开场，接着要求计算精确值，最后探索一般规律或现实情境解读。这种结构是有意为之——它训练学生从常规计算游刃有余地过渡到概念推理，这对冲击7分至关重要。
2. Algebraic Simplification and Equation Solving | 代数化简与方程求解
A recurring theme in Set 22C is the need to manipulate expressions involving exponents, logarithms, and surds. For example, you might be asked to solve 3²ˣ⁺¹ − 4·3ˣ + 1 = 0 by recognising it as a quadratic in disguise. The substitution y = 3ˣ converts it into 3y² − 4y + 1 = 0, which then factorises neatly.
复习集22C中反复出现的一个主题是对指数、对数和根式表达式进行变形。例如，你可能需要求解方程3²ˣ⁺¹ − 4·3ˣ + 1 = 0，通过将其视为隐藏的二次方程来解答。设y = 3ˣ可将原方程化为3y² − 4y + 1 = 0，然后简捷地进行因式分解。
Similarly, logarithmic equations demand careful application of the laws: logₐ(xy) = logₐ(x) + logₐ(y) and the change-of-base formula. IB examiners love to include extraneous solutions that must be checked against the domain of the original logarithmic function.
同样，对数方程要求我们细致地运用运算法则：logₐ(xy) = logₐ(x) + logₐ(y) 以及换底公式。IB考官喜欢在题中设置增根，必须将解代回原对数函数的定义域进行检验。
3. Functions and Their Transformations | 函数及其变换
Set 22C typically features a function such as f(x) = a·sin(bx + c) + d, requiring you to identify amplitude, period, phase shift, and vertical translation. Remember that the period of sin(bx) is 2π/b, and the phase shift is given by −c/b. Composite functions like f∘g and their domains are also tested.
复习集22C通常会给出形如f(x) = a·sin(bx + c) + d的函数，要求你确定振幅、周期、相位移和垂直平移。记住，sin(bx)的周期为2π/b，而相位移由−c/b给出。复合函数f∘g及其定义域也是考查重点。
The inverse function is written as f⁻¹(x), not to be confused with the reciprocal. To find the inverse, swap x and y and solve for y. Graphical interpretation often asks you to reflect the function in the line y = x and annotate intercepts and asymptotes clearly.
反函数记作f⁻¹(x)，切勿与倒数混淆。求反函数时，交换x和y后再解出y。图形解读常要求你将函数关于直线y = x作反射，并清晰标注截距和渐近线。
4. Trigonometry and Circular Functions | 三角学与圆形函数
Questions in Set 22C involving trigonometric identities often require you to prove statements like (1 − cos θ)(1 + cos θ) = sin²θ and then use this to solve equations in a given interval. Exact values for π/6, π/4, π/3 and their multiples are non-negotiable—you must recall sine, cosine, and tangent from memory.
复习集22C中涉及三角恒等式的题目常要求你证明如(1 − cos θ)(1 + cos θ) = sin²θ这样的关系，并据此在给定区间内解方程。π/6、π/4、π/3及其倍数的精确值是不可动摇的根基——你必须牢记这些角的正弦、余弦和正切值。
The sine and cosine rules appear in context with non-right-angled triangles. Be prepared to decide which rule to use based on the information given: two sides and a non-included angle favour the sine rule, while three sides or two sides and the included angle call for the cosine rule.
正弦定理和余弦定理会在非直角三角形的背景下出现。你要根据已知信息判断用哪个定理：已知两条边和一个非夹角宜用正弦定理，而已知三边或两条边及其夹角则应使用余弦定理。
5. Differentiation Techniques and Applications | 微分技巧及其应用
The chain rule, product rule, and quotient rule are exercised rigorously. You might see a function like g(x) = ln(cos x) and need to find g'(x) = −tan x. Set 22C often pairs derivatives with the equation of a tangent or normal line, requiring exact coordinate and gradient values.
复合函数求导链式法则、乘法法则和除法法则都会得到密集训练。你可能会遇到g(x) = ln(cos x)这样的函数，并求出g'(x) = −tan x。复习集22C常常将导数与切线或法线方程相结合，要求给出精确的坐标和斜率数值。
Optimisation problems form a significant part of the applied strand. You must express a quantity to be maximised—like area or volume—as a function of a single variable, differentiate, set the derivative to zero, and confirm the maximum using a sign diagram or the second derivative test.
最优化问题是应用部分的重头戏。你需要将要最大化的量（如面积或体积）表示为单一变量的函数，求导后令导数为零，再利用符号表或二阶导数检验确认最大值。
6. Integration and the Fundamental Theorem | 积分与微积分基本定理
Set 22C includes indefinite integrals with reverse chain recognition, such as ∫ 2x·sin(x²) dx = −cos(x²) + C. Definite integrals are used to find areas under curves and between two curves. Always check whether the function dips below the x‑axis, as this affects the signed area calculation.
复习集22C包含了需要识别反向链式法则的不定积分，例如∫ 2x·sin(x²) dx = −cos(x²) + C。定积分则用于计算曲线下方以及两条曲线之间的面积。务必检查函数是否穿过x轴，因为这会影响带符号的面积计算。
Integration by substitution and by parts (for AA HL) may appear if the set is pitched at Higher Level. A typical problem asks you to evaluate ∫ x·e²ˣ dx using u = x and dv = e²ˣ dx, giving (½ x − ¼)e²ˣ + C after careful algebraic simplification.
如果该习题集面向高级水平(HL)，还可能包含换元积分法和分部积分法。一道典型题目要求用u = x和dv = e²ˣ dx计算∫ x·e²ˣ dx，经过细致的代数化简后得到(½ x − ¼)e²ˣ + C。
7. Probability and Counting Principles | 概率与计数原理
Set 22C revisits the product principle, permutations, and combinations. A question may read: ‘A team of 5 is to be chosen from 8 boys and 6 girls. Find the probability that the team contains at least 3 girls.’ The numerator uses combinations: C(6,3)·C(8,2) + C(6,4)·C(8,1) + C(6,5), and the denominator is C(14,5).
复习集22C会重温乘法原理、排列与组合。一道题目可能这样问：“从8名男生和6名女生中选出一个5人团队，求团队中至少包含3名女生的概率。”分子需借助组合数计算：C(6,3)·C(8,2) + C(6,4)·C(8,1) + C(6,5)，分母则为C(14,5)。
Venn diagrams and tree diagrams are invaluable for visualising conditional probability. The formula P(A|B) = P(A ∩ B) / P(B) must be applied accurately, and it is often tested alongside independent events where P(A ∩ B) = P(A)·P(B).
韦恩图和树状图是呈现条件概率的宝贵工具。公式P(A|B) = P(A ∩ B) / P(B)必须准确运用，并且经常与独立事件一同考查，此时P(A ∩ B) = P(A)·P(B)。
8. Discrete and Normal Distributions | 离散分布与正态分布
Binomial distribution problems in Set 22C require you to identify n and p correctly. Notation like X ~ B(20, 0.35) might be used to find P(X = 8), P(X ≤ 8), and the expected value E(X) = np. Using the GDC’s binompdf and binomcdf functions is expected, but you must also show the formula for exact probability where instructed.
复习集22C中的二项分布问题要求你正确识别n和p。如X ~ B(20, 0.35)这样的记号用于求P(X = 8)、P(X ≤ 8)以及期望值E(X) = np。允许使用图形计算器的binompdf和binomcdf功能，但在有明确要求的地方仍需写出精确概率公式。
For the normal distribution, standardisation with Z = (X − μ)/σ is central. Inverse normal calculations ask for the value of k such that P(X < k) = 0.9. Attention to continuity correction is rarely needed in IB, but sketching the bell curve and shading the region of interest reduces errors significantly.
对于正态分布，标准化Z = (X − μ)/σ是核心。反向正态计算要求找出满足P(X < k) = 0.9的k值。IB课程中几乎不需要连续性修正，但画出钟形曲线并涂黑目标区域能显著减少失误。
9. Calculus with Trigonometric Functions | 涉及三角函数的微积分
Differentiation of trigonometric functions extends beyond simple sine and cosine: you need to know that d/dx [tan x] = sec²x and d/dx [csc x] = −csc x·cot x. Combined with the chain rule, a function like y = sin³(2x) yields dy/dx = 6 sin²(2x)·cos(2x).
三角函数的微分不仅限于简单的正弦和余弦：你必须知道d/dx [tan x] = sec²x 以及 d/dx [csc x] = −csc x·cot x。结合链式法则，形如y = sin³(2x)的函数可求得dy/dx = 6 sin²(2x)·cos(2x)。
Integration of trigonometric expressions often involves rearranging using identities. For instance, ∫ cos²x dx is tackled by writing cos²x = (1 + cos 2x)/2. Definitive integrals in Set 22C may then evaluate to neat results like π/4, blending algebraic and geometric insight.
三角表达式的积分常常需要利用恒等式进行改写。例如，处理∫ cos²x dx时可将cos²x写作(1 + cos 2x)/2。复习集22C中的定积分有时会得出诸如π/4这样简洁的结果，融合了代数技巧与几何洞察。
10. Exam-Style Mixed Problems | 考试风格的混合型问题
Some of the most challenging items in Set 22C combine multiple domains. A problem may start with a function f(x) = eˣ·sin x, ask you to find the first derivative, determine the equation of the tangent at x = 0, and then compute the area bounded by that tangent and the curve—all in one seamless sequence.
复习集22C中一些最具挑战性的题目融合了多个知识领域。某道题可能从一个函数f(x) = eˣ·sin x开始，要求你求一阶导数，确定x = 0处的切线方程，接着再计算该切线与曲线所围成的面积——所有步骤一气呵成。
Another typical configuration presents a real-world context: the height of a tide is modelled by h(t) = 2 + 1.5 sin(πt/6), and students must find the times when the tide reaches 3 metres over a 24-hour period, then differentiate to find the maximum rate of change. This blends trigonometric equations with rate-of-change calculus.
另一种典型设置是现实情境题：潮汐高度用h(t) = 2 + 1.5 sin(πt/6)建模，学生需要找出24小时内潮高达3米的时刻，然后求导以确定最大变化速率。这把三角方程和变化率微积分有机地结合了起来。
11. Common Pitfalls and How to Avoid Them | 常见陷阱与规避方法
A frequent mistake in Set 22C is misapplying the domain when solving trigonometric equations. Students tend to give all solutions without restricting to the specified interval, or forget that arcsin has a principal range of [−π/2, π/2]. Always finish by filtering your solutions against the original domain.
复习集22C中一个常见错误是在解三角方程时错误使用定义域。学生往往会给出所有解而忽略了题目指定的区间，或忘记反正弦函数的主值范围是[−π/2, π/2]。完成求解后务必依据原始定义域进行筛选。
In calculus, missing a factor from the chain rule or miswriting the derivative of ln(f(x)) as 1/f(x) instead of f'(x)/f(x) is a classic error. Similarly, in integration, dropping the constant of integration (+C) loses marks, even when the final answer is otherwise correct.
微积分中，漏掉链式法则的因子或将ln(f(x))的导数误写成1/f(x)而非f'(x)/f(x)，是经典错误。同样，在积分中遗漏积分常数(+C)会导致失分，即便最终答案在其他方面完全正确。
When using a GDC, ensure you are in radian mode for calculus and trigonometric work unless degrees are explicitly stated. A single mode mismatch can corrupt an entire set of solutions. Sketching graphs manually as a sanity check develops an intuition that technology alone cannot provide.
使用图形计算器时，除题目明确注明角度制外，务必确保在微积分和三角运算中处于弧度模式。一次模式错配足以毁掉整组答案。手动绘制草图作为合理性检查，可以培养单纯依赖技术无法提供的数学直觉。
12. Final Tips for Success with Review Set 22C | 在复习集22C中取得成功的最后诀窍
Treat every mistake as a diagnostic tool. After completing Set 22C, categorise your errors: algebraic slip, conceptual misunderstanding, misreading, or calculator error. This targeted analysis turns each answer into a stepping stone toward mastery. Reattempt the incorrect items after a day without looking at the solution.
把每个错误都当作诊断工具。完成复习集22C后，将你的错误分类：代数计算失误、概念理解模糊、误读题目或计算器操作不当。这种有针对性的分析能把每个答案转化为通往精通的垫脚石。一天之后脱离答案再独立重做错题。
Time yourself strictly as you would in an exam. Set 22C should be completed in around 90 minutes under timed conditions to simulate Paper 1 pacing. Mark your work using the official IB markscheme, paying close attention to method marks (M), accuracy marks (A), and reasoning marks (R).
像正式考试那样严格计时。在限时条件下，复习集22C应在约90分钟内完成，以模拟试卷一的节奏。用官方IB评分方案批改你的答题，密切关注方法分(M)、准确度分(A)和推理分(R)。
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