📚 Review set 26C | 复习套题26C
Welcome to the ultimate revision walkthrough for IB Mathematics Review Set 26C. This carefully compiled problem set pushes you to apply concepts across functions, calculus, trigonometry, probability, vectors, complex numbers and differential equations. We will unpack the core ideas behind each typical question, highlight examiner traps, and equip you with efficient solving strategies. Whether you are preparing for Standard Level or Higher Level, this article will reinforce both conceptual understanding and problem‑solving fluency.
欢迎阅读IB数学复习套题26C的终极攻略。这套精心设计的题目覆盖了函数、微积分、三角学、概率、向量、复数以及微分方程等多个领域。我们将逐一拆解典型题背后的核心思想，指出阅卷人设下的常见陷阱，并为你提供高效的解题策略。无论你准备的是标准水平还是高水平，本文都将巩固你的概念理解与解题熟练度。
1. Function Transformations | 函数变换
Review Set 26C often begins with a multi‑ ent function is y = f(x), you may be asked to sketch or write the equation for y = a f(b(x – h)) + k. Each parameter triggers a distinct geometric change: |a| > 1 gives a vertical stretch, 0 < |a| < 1 a vertical compression, and a negative a reflects across the x‑axis. Similarly, |b| > 1 compresses horizontally, while 0 < |b| < 1 stretches horizontally, with b < 0 producing a reflection across the y‑ apply transformations in the order: stretches/reflections first, then translations (inside‑outside method).
复习套题26C通常以多步变换题开头。若原函数为 y = f(x)，你可能需要绘制或写出 y = a f(b(x – h)) + k 的表达式。每个参数都会引发特定的几何变化：|a| > 1 表示纵向拉伸，0 < |a| < 1 纵向压缩，负 a 则产生关于 x 轴的对称翻转。类似地，|b| > 1 使图像横向压缩，而 0 < |b| < 1 横向拉伸，b < 0 给出关于 y 轴的反射。平移量 h 与 k 分别将图像左右、上下移动。务必按照先拉伸/反射、后平移的顺序（由内而外的方法）进行变换。
| Transformation | Algebraic form | Effect on point (x, y) |
|---|---|---|
| Vertical stretch by factor A | y = A f(x), A > 1 | (x, y) → (x, Ay) |
| Horizontal compression by factor B | y = f(Bx), B > 1 | (x, y) → (x/B, y) |
| Reflection in x‑axis | y = –f(x) | (x, y) → (x, –y) |
| Horizontal translation right by h | y = f(x – h) | (x, y) → (x + h, y) |
2. Exponential and Logarithmic Equations | 指数与对数方程
This set frequently targets equations like e2x + ex – 6 = 0 or log2(x) + log2(x – 2) = 3. The key with exponential quadratics is to substitute u = ex, turning the expression into u2 + u – 6 = 0, solve for u, and then back‑solve for x (remember u > 0). For logarithmic equations, combine logs using log laws: logb(A) + logb(B) = logb(AB). Always check that all arguments are strictly greater than zero before accepting a solution. Extraneous answers often arise when a candidate value makes an argument negative or zero.
本套题经常出现形如 e2x + ex – 6 = 0 或 log2(x) + log2(x – 2) = 3 的方程。处理指数二次型的关键是设 u = ex，将式子化为 u2 + u – 6 = 0，解出 u，再回代求 x（注意 u > 0）。对于对数方程，利用对数律合并：logb(A) + logb(B) = logb(AB)。在采纳任何一个解之前，务必检查所有真数是否严格大于零。当候选值使真数变为负数或零时，常常会产生增根。
If loga(p) = r then ar = p and p > 0.
3. Trigonometric Identities and Equations | 三角恒等式与方程
Questions in Set 26C demand fluency with double‑ ach is to rewrite sin 2x as 2 sin x cos x, then bring all terms to one side: 2 sin x cos x – kπ) unless the question restricts the domain. Also watch for quadrant checks when using inverse trigonometric functions.
套题26C中的题目要求你熟练运用倍角公式、和角公式，并能求解诸如 sin 2x = cos x 的方程。一种常见思路是将 sin 2x 改写为 2 sin x cos x，然后将所有项移至一边：2 sin x cos x – cos x = 0，提公因式 cos x，再利用零乘积性质。给出答案时，除非题目限定定义域，否则记得用弧度制写出通解（如 x = π/2 + kπ）。在使用反三角函数时，也要注意象限检验。
cos2θ + sin2θ = 1, tan θ = sin θ / cos θ
4. Limits and Continuity | 极限与连续性
Limit problems often involve a rational expression like limx→2 (x2 – 4)/(x – titution gives 0/0, an indeterminate form. The standard fix is to factor the numerator: (x – 2)(x + 2)/(x – ≠ 2, and then evaluate the limit as 4. To discuss continuity at a point, you need the left‑hand limit, right‑ t and be equal. Piecewise functions are the most common vehicles for testing this concept.
极限题通常涉及有理式，如 limx→2 (x2 – 4)/(x – 2)。直接代入得到 0/0 这一不定式。标准的处理方法是给分子分解因式：(x – 2)(x + 2)/(x – 2)，在 x ≠ 2 时等于 x + 2，然后求极限得 4。要讨论一点处的连续性，需要左极限、右极限和函数值三者均存在且相等。分段函数是考察此概念最常见的载体。
limx→a f(x) = L if and only if limx→a⁻ f(x) = limx→a⁺ f(x) = L.
5. Derivatives and Tangents | 导数与切线
or the equation of the tangent to a curve at a given point. After finding the derivative f '(x), evaluate the slope m = f '(x0) and use the point‑slope form: y – y0 = m(x – x0 , write a function for the quantity to be maximised or minimised, differentiate, and set f '(x) = 0. Always confirm a maximum or minimum with a sign test or second derivative.
套题26C中的微分练习覆盖了从简单的幂函数求导到链式法则、乘积法则和商法则。一道常见的重点题目要求写出曲线在给定点处的切线方程。在求得导数 f '(x) 后，计算斜率 m = f '(x0)，然后利用点斜式：y – y0 = m(x – x0)。优化问题也会出现：设定一个变量，写出待优化量的函数，求导并令 f '(x) = 0。务必通过符号检验或二阶导数确认是最大值还是最小值。
d/dx [xn] = n xn–1, d/dx [ekx] = k ekx
6. Integration Techniques | 积分技巧
tegration problems in this review set range from straightforward indefinite integrals ∫(4x3 – 2x + 1)dx to definite integrals that compute the area between a curve and the x‑axis. Watch for reverse chain rule patterns: ∫ f '(x)·[f(x)]n dx = [f(x)]n+1/(n+1) + C (n ≠ –1). When finding areas, set up the integral as ∫ab |f(x)| is also tested, for example letting u = 2x + 3 transforms an integral into a simpler form.
本复习套题中的积分问题从简单的不定积分 ∫(4x3 – 2x + 1)dx，到计算曲线与 x 轴之间面积的定积分，一应俱全。要留心反向链式法则的模式：∫ f '(x)·[f(x)]n dx = [f(x)]n+1/(n+1) + C（n ≠ –1）。在求面积时，如果函数穿过坐标轴，应将积分设为 ∫ab |f(x)| dx，否则负面积会相互抵消。换元法也在考察范围内，例如令 u = 2x + 3 可将积分化为更简单的形式。
∫ab f(x) dx = F(b) – F(a), where F '(x) = f(x).
7. Probability Distributions | 概率分布
Expect questions on binomial distribution X ~ nding P(X = k), cumulative probabilities, expectation E(X) = np and variance Var(X) = np(1 – p). The normal distribution X ~ N(μ, σ2) also features prominently: you will standardise via Z = (X – ormal calculations where you are given a probability and asked to find the corresponding x‑value. Check whether the question requires a continuity correction—though IB rarely demands it explicitly, it can sharpen estimation.
预计会碰到二项分布 X ~ B(n, p) 的题目，包括求 P(X = k)、累积概率、期望 E(X) = np 以及方差 Var(X) = np(1 – p)。正态分布 X ~ N(μ, σ2) 也占有显著地位：你需要通过 Z = (X – μ)/σ 进行标准化，然后从标准正态表中读取概率或使用计算器。一个常见的变体是逆向正态计算，即给定一个概率让你求对应的 x 值。注意题目是否需要连续性校正——虽然IB课程很少明确要求，但它可以使估计更加精确。
| Distribution | Type | Key formula / usage |
|---|---|---|
| Binomial B(n, p) | Discrete | P(X = k) = ⁿCₖ pk(1 – p)n–k |
| Normal N(μ, σ2) | Continuous | Z = (X – μ)/σ ~ N(0,1) |
8. Vector Geometry | 向量几何
s include computing the scalar (dot) product to test orthogonality or find angles: v · w = |v||w| cos θ. A typical question provides two vectors and asks whether they are perpendicular— them. Cross product may be examined at HL for area of a parallelogram or to find a normal vector.
向量题包含计算标量积（点积）以验证正交性或求夹角：v · w = |v||w| cos θ。一道典型题目给出两个向量，询问它们是否垂直——只需检验 v · w 是否为零。二维或三维空间中直线的向量方程以 r = a + λ d 的形式出现。你可能需要求两条直线的交点或它们之间的夹角。在HL中，叉积可能用于求平行四边形的面积或寻找法向量。
If v · w = 0, vectors v and w are orthogonal.
9. Complex Numbers | 复数
an form z = a + bi and polar form z = r(cos θ + i sin θ). You will compute the modulus r = √a2 + b2 ment θ = arctan(b/a) with quadrant adjustments. Roots of unity problems, such as solving z3 = 1, require De Moivre's theorem: (r cis θ)n = rn cis(nθ). Equating real and imaginary parts or multiplying/dividing in polar form is much faster than using algebraic expansion. Always write the principal argument in the range (–π, π] unless otherwise stated.
套题26C中的复数题目在代数式 z = a + bi 与极式 z = r(cos θ + i sin θ) 之间流畅转换。你需要计算模 r = √a2 + b2 和辐角 θ = arctan(b/a)（注意象限修正）。诸如求解 z3 = 1 的单位根问题需要棣莫弗定理：(r cis θ)n = rn cis(nθ)。在极式下进行相等实部虚部对比或乘除运算远比代数展开更快捷。除非另有说明，始终将主辐角写在 (–π, π] 的范围内。
cis θ = cos θ + i sin θ
10. Differential Equations | 微分方程
Review Set 26C usually closes with a simple first‑ ix the constant of integration. Exponential growth and decay models are frequent: the solution to dy/dt = λ y is y = C eλ t. Interpret the constant λ carefully—positive for growth, negative for decay.
复习套题26C通常以一道可用分离变量法求解的简单一阶微分方程作为压轴，如 dy/dx = ky 或 dy/dx = (x + 1)/y。策略是将所有涉及 y 的项与 dy 放到一侧，所有涉及 x 的项与 dx 放到另一侧，然后两边同时积分。积分后，利用给定的初始条件（例如 y(0) = 5）确定积分常数。指数增长与衰减模型出现频繁：dy/dt = λ y 的解为 y = C eλ t。务必谨慎解读常数 λ——正值表示增长，负值表示衰减。
∫ (1/y) dy = ln|y| + C
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