📚 Review Set C | 复习集 C
This comprehensive revision set revisits core topics from the IB Mathematics syllabus, including algebra, functions, exponentials, trigonometry, calculus, vectors, and statistics. Each section pairs concise explanations in English with equivalent Chinese translations, ensuring you consolidate both your mathematical understanding and bilingual terminology. Work through the examples to identify any weaknesses and sharpen your problem‑solving skills before the final assessment.
这套综合复习集回顾IB数学课程的核心主题，涵盖代数、函数、指数、三角学、微积分、向量和统计。每个部分均以英文简明讲解配合对等的中文翻译，帮助你巩固数学理解和双语术语。通过这些例题找出薄弱环节，在最终考试前训练解题能力。
1. Algebra and Equations | 代数与方程
ic x² − 5x + 6 gives (x − 2)(x − 3), which directly reveals the solutions x = 2 and x = 3.
代数运算是多数IB问题的基础。你需要熟练展开括号、化简有理式以及因式分解多项式。例如，展开 (2x − 3)(x + 5) 得到 2x² + 7x − 15。将二次式 x² − 5x + 6 分解为 (x − 2)(x − 3)，可直接得到解 x = 2 和 x = 3。
be done by factorising, completing the square, or applying the quadratic formula x = [−b ± √b² − 4ac two distinct real roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.
解形如 ax² + bx + c = 0 的二次方程可采用因式分解、配方法或求根公式 x = [−b ± √b² − 4ac] / (2a)。务必检查判别式 Δ = b² − 4ac 以确定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实数根。
or instance, solving 2x + y = 7 and x − y = 2 gives x = 3, y = 1. When one equation is non‑ bstitute the linear expression into the second equation and solve the resulting quadratic.
联立线性方程组可用代入法或消元法求解。例如，解 2x + y = 7 与 x − y = 2 得 x = 3, y = 1。若其中一个方程为非线性，可将线性表达式代入另一方程，解出产生的二次方程。
Inequalities require careful attention when multiplying or dividing by a negative quantity. The solution to (x − 1)(x + 2) > 0 can be found by testing intervals on a sign diagram, giving x < −2 or x > 1. Always express your answer using interval notation or set‑builder notation.
处理不等式时，乘以或除以负数要格外小心。(x − 1)(x + 2) > 0 的解可通过符号图测试区间得出，为 x < −2 或 x > 1。务必用区间表示法或集合构造法书写答案。
2. Functions and Graphs | 函数与图像
s the set of all possible inputs; the range is the set of all possible outputs. For f(x) = √x − 2, the domain is x ≥ 2 because the radicand must be non‑negative, and the range is y ≥ 0.
函数 f 将其定义域中的每个输入 x 映射到唯一的输出 f(x)。定义域是所有可能输入的集合，值域是所有可能输出的集合。对于 f(x) = √x − 2，定义域为 x ≥ 2（被开方数非负），值域为 y ≥ 0。
Composite functions combine two functions: (g ∘ f)(x) = g(f(x)). For f(x) = 2x + 1 and g(x) = x², (g ∘ r x to find f⁻¹(x) = (x + 4)/3. The graph of f⁻¹ is the reflection of f in the line y = x.
复合函数将两个函数组合：(g ∘ f)(x) = g(f(x))。若 f(x) = 2x + 1，g(x) = x²，则 (g ∘ f)(x) = (2x + 1)²。反函数逆转映射：若 f(x) = 3x − 4，由 y = 3x − 4 解出 x 得 f⁻¹(x) = (x + 4)/3。f⁻¹ 的图像是 f 关于直线 y = x 的反射。
Transformations of graphs are crucial. Replacing x with (x − h) translates the graph h units right; adding k to f(x) moves it k units up. Reflections: −f(x) flips the graph across the x‑axis, while f(−x) reflects it across the y‑axis. Vertical stretch by factor a is given by a·f(x); horizontal stretch by factor 1/b is given by f(bx).
图像变换非常关键。将 x 替换为 (x − h) 使图像向右平移 h 个单位；f(x) 加 k 使图像上移 k 个单位。反射：−f(x) 关于 x 轴翻转，f(−x) 关于 y 轴翻转。垂直拉伸 a 倍为 a·f(x)；水平拉伸 1/b 倍为 f(bx)。
ify vertical asymptotes (x = 2) and horizontal asymptotes (y = 0). Check intercepts: the y‑intercept is −½, and there is no x‑intercept. Plot a few key points to confirm the shape.
绘制有理函数如 f(x) = 1/(x − 2) 时，识别垂直渐近线 (x = 2) 和水平渐近线 (y = 0)。检查截距：y 截距为 −½，无 x 截距。绘制几个关键点以确认形状。
3. Exponents and Logarithms | 指数与对数
mental: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁻ⁿ = 1/aⁿ. Rational exponents represent roots: a1/n = ⁿ√a. For example, 81/3 = 2, and 163/4 = (⁴√16)³ = 2³ = 8.
指数律是基础：aᵐ · aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ。有理指数表示根式：a1/n = ⁿ√a。例如，81/3 = 2，163/4 = (⁴√16)³ = 2³ = 8。
Logarithms are the inverses of exponentials: logₐ(b) = c means aᶜ ) = logₐ(x) + logₐ(y), logₐ(x/y) = logₐ(x) − logₐ(y), and logₐ(xⁿ) = n logₐ(x). The change‑of‑base formula is logₐ(b) = log(b)/log(a).
对数是指数的逆运算：logₐ(b) = c 意味着 aᶜ = b。自然对数以 e ≈ 2.718 为底。重要性质：logₐ(xy) = logₐ(x) + logₐ(y)，logₐ(x/y) = logₐ(x) − logₐ(y)，logₐ(xⁿ) = n logₐ(x)。换底公式为 logₐ(b) = log(b)/log(a)。
Solve exponential equations by taking logs: 5ˣ = 20 → x = log(20)/log(5) ≈ 1.86. For equations like e²ˣ x − 2) = 2³ = 8, giving x² − 2x − 8 = 0, so x = 4 (reject x = −2 as domain demands x > 2).
解指数方程需取对数：5ˣ = 20 → x = log(20)/log(5) ≈ 1.86。对于 e²ˣ = 7 这类方程，取自然对数：2x = ln 7 → x = (ln 7)/2。对数方程可能需要合并：log₂(x) + log₂(x − 2) = 3 → log₂(x(x − 2)) = 3 → x(x − 2) = 2³ = 8，得 x² − 2x − 8 = 0，故 x = 4（舍去 x = −2，因定义域要求 x > 2）。
Exponential growth and decay models take the form A = A₀ ekt. If a population doubles in 5 hours, find k: 2A₀ = A₀ e5k → e5k = 2 → 5k = ln 2 → k = (ln 2)/5. Then predict the population after 10 hours.
指数增长与衰减模型形如 A = A₀ ekt。若细菌数量在 5 小时后翻倍，求 k：2A₀ = A₀ e5k → e5k = 2 → 5k = ln 2 → k = (ln 2)/5。进而预测 10 小时后的数量。
4. Trigonometry | 三角学
0°, π/2 = 90°. The unit circle defines sine and cosine for any real angle, with period 2π.
角度可用度数或弧度表示。弧度定义为弧长与半径的比值。换算关系：π 弧度 = 180°。常见弧度值：π/6 = 30°，π/4 = 45°，π/3 = 60°，π/2 = 90°。单位圆可定义任意实角的正弦和余弦，周期为 2π。
Fundamental identities: sin²θ + cos²θ = 1, tanθ = sinθ/cosθ. The double‑ nometric equations like 2 sinθ cosθ = ½, which becomes sin(2θ) = ½, giving 2θ = π/6, 5π/6, … so θ = π/12, 5π/12, … within the required interval.
基本恒等式：sin²θ + cos²θ = 1，tanθ = sinθ/cosθ。倍角公式：sin(2θ) = 2 sinθ cosθ，cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。这些公式在解三角方程时非常有用，如 2 sinθ cosθ = ½ 变为 sin(2θ) = ½，得 2θ = π/6, 5π/6, … 因此在所需区间内 θ = π/12, 5π/12, …。
The sine and cosine rules are essential for non‑right‑ + c² − 2bc cos A. Use the sine rule when two angles and one side are known (AAS or ASA), and the cosine rule when two sides and the included angle are known (SAS) or when three sides are known (SSS).
正弦定理和余弦定理对非直角三角形至关重要。正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A。已知两角一边 (AAS 或 ASA) 时用正弦定理，已知两边及其夹角 (SAS) 或三边 (SSS) 时用余弦定理。
th = rθ and area = ½ r²θ, where θ is in radians. These formulas appear frequently in paper‑1 style questions.
三角形面积可用 ½ ab sin C 计算。扇形弧长 = rθ，扇形面积 = ½ r²θ，其中 θ 以弧度为单位。这些公式在试卷一中频繁出现。
5. Differentiation | 微分
The derivative f'(x) gives the gradient of the tangent to the graph of y = f(x). From first principles, f'(x) = limh→0 [f(x+h) − f(x)]/h. In practice, shortcuts are used: if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. The derivative of a constant is zero, and d/dx(√x) = 1/(2√x).
导数 f'(x) 给出曲线 y = f(x) 切线的斜率。由基本原理得，f'(x) = limh→0 [f(x+h) − f(x)]/h。实际应用中使用快捷法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹；常数的导数为零；d/dx(√x) = 1/(2√x)。
The sum rule, product rule, and quotient rule are essential. Product rule: d/dx(uv) = u'v + uv'. For h(x) = x² sin x, h'(x) = 2x sin x + x² cos x. Quotient rule: d/dx(u/v) = (u'v − uv')/v². The chain rule handles composite functions: d/dx[f(g(x))] = f'(g(x))·g'(x). For y = (3x² + 1)⁵, let u = 3x² + 1, then dy/dx = 5(3x² + 1)⁴ · 6x.
和法则、积法则和商法则是基本工具。积法则：d/dx(uv) = u'v + uv'。例如 h(x) = x² sin x 的导数 h'(x) = 2x sin x + x² cos x。商法则：d/dx(u/v) = (u'v − uv')/v²。链式法则处理复合函数：d/dx[f(g(x))] = f'(g(x))·g'(x)。对 y = (3x² + 1)⁵，设 u = 3x² + 1，则 dy/dx = 5(3x² + 1)⁴ · 6x。
Tangents and normals: to find the equation of the tangent to y = x³ at x = 1, compute y' h gradient −1/3. These geometric applications link differentiation to coordinate geometry.
切线与法线：求 y = x³ 在 x = 1 处的切线，计算 y'(1) = 3，点 (1,1)，故切线为 y − 1 = 3(x − 1)。法线垂直于切线，斜率为 −1/3。这些几何应用将微分与解析几何联系起来。
Second derivative f''(x) indicates concavity and helps locate points of inflection. If f''(x) > 0 the graph is concave up; if f'' 0 it is concave down. To classify stationary points, use the second derivative test: if f'(a) = 0 and f''(a) > 0, we have a local minimum; f''(a) < 0 gives a local maximum.
二阶导数 f''(x) 表示图像的凹凸性，并帮助找出拐点。若 f''(x) > 0，图像下凸（凹向上）；若 f''(x) < 0，图像上凸（凹向下）。分类驻点时，可采用二阶导数检验：若 f'(a) = 0 且 f''(a) > 0，则 a 处为局部极小值；f''(a) < 0 则为局部极大值。
6. Integration | 积分
xample, ∫ (3x² + 4x) dx = x³ + 2x² + C. Remember to include the constant of integration C.
积分是微分的逆运算。不定积分 ∫ f(x) dx 表示原函数族。幂次法则：∫ xⁿ dx = (1/(n+1)) xⁿ⁺¹ + C，其中 n ≠ −1。例如，∫ (3x² + 4x) dx = x³ + 2x² + C。切记加上积分常数 C。
(a), where F is an antiderivative. To find the area between the curve y = 4 − x² and the x‑ x²) dx = [4x − (1/3)x³]₋₂² = (8 − 8/3) − (−8 + 8/3) = 32/3. The area is 32/3 square units.
定积分计算曲线下方的带符号面积：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F 为
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