📚 Exercise H.3 | 练习 H.3
Hyperbolic functions represent one of the most elegant extensions of exponential functions in IB Mathematics: Analysis and Approaches HL. Exercise H.3 is a carefully crafted problem set that guides you through the definitions, identities, calculus techniques, and applications of these functions – often mirroring the style of final exam questions. Mastering the material in this exercise will give you fluency with sinh, cosh, tanh and their inverses, a skill that regularly distinguishes top-level candidates.
双曲函数是指数函数在 IB 数学：分析与方法 HL 课程中最优美的延伸之一。练习 H.3 是一套精心设计的习题集，会带你系统地梳理双曲函数的定义、恒等式、微积分技巧以及实际应用，其题型风格与最终考试高度一致。彻底掌握本练习中的内容，将使你能够熟练运用 sinh、cosh、tanh 及其反函数，而这一点常常是顶尖考生的标志。
1. Introduction to Hyperbolic Functions | 双曲函数简介
unit circle x² + y² = 1, hyperbolic functions parametrise the unit hyperbola x² − y² = 1.
双曲函数由指数函数构成，并在高等微积分、微分方程和相对论性物理中自然出现。与参数化单位圆 x² + y² = 1 的圆三角函数不同，双曲函数参数化的是单位双曲线 x² − y² = 1。
In IB Mathematics AA HL, you are required to recognise the graphs, prove identities using exponential definitions, differentiate and integrate hyperbolic functions, and solve equations involving them. Exercise H.3 consolidates all these skills by offering a structured progression from basic manipulation to multi-step exam-style problems.
在 IB 数学 AA HL 课程中，你要求能够识别它们的图像、利用指数定义证明恒等式、对双曲函数进行微分和积分，并求解涉及双曲函数的方程。练习 H.3 通过从基础操作到多步考试型题目的结构化递进，帮助你把所有这些技能巩固为一体。
2. Definitions and Graphs | 定义与图像
The two fundamental hyperbolic functions are the hyperbolic sine and hyperbolic cosine, defined for all real x by:
两个最基本的双曲函数是双曲正弦和双曲余弦，它们对全体实数 x 定义如下：
sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2
From these, the hyperbolic tangent is defined as tanh x = sinh x / cosh x = (e²ˣ − 1)/(e²ˣ + 1). The reciprocal functions – coth, sech and cosech – are less commonly tested but follow directly.
由此可以定义双曲正切：tanh x = sinh x / cosh x = (e²ˣ − 1)/(e²ˣ + 1)。它们的倒数函数 coth、sech 和 cosech 考查较少，但可直接由上述定义导出。
igmoid function bounded between y = −1 and y = 1, providing a classic model of saturation.
y = sinh x 的图像是一个过原点的奇函数且单调递增，形态与三次曲线相似。y = cosh x 是偶函数，在 (0,1) 处取最小值，常被称为悬链线，因为它的形状恰如一根自由悬挂的链条。y = tanh x 是一个奇函数，呈 S 形，且被限制在 y = −1 与 y = 1 之间，是经典的饱和模型。
3. Hyperbolic Identities and Osborn’s Rule | 双曲恒等式与奥斯本规则
n²θ + cos²θ = 1. It can be proved in one line by substituting the exponential definitions.
基本恒等式 cosh²x − sinh²x = 1 是 sin²θ + cos²θ = 1 的双曲对应版本，只需代入指数定义一行即可证得。
Other essential identities include double-argument formulas:
其他关键的恒等式还包括倍角公式：
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh²x + sinh²x = 2cosh²x − 1 = 1 + 2sinh²x
uct) of two sines. For example, sin²θ + cos²θ = 1 becomes cosh²x − sinh²x = 1; the product ‘sin²’ is treated as the product of two sines and its sign flips from positive to negative.
奥斯本规则给出了一条强大的记忆工具：要将一个已知的三角恒等式转换为双曲恒等式，只需将所有三角函数替换为对应的双曲函数，并将每一处两个 sine 的乘积（或隐含乘积）的符号取反。例如 sin²θ + cos²θ = 1 变为 cosh²x − sinh²x = 1；此处 ‘sin²’ 被视为两个 sine 的乘积，因而符号由正变负。
4. Solving Equations with Hyperbolic Functions | 涉及双曲函数的方程求解
ace the hyperbolic function with its exponential definition. For sinh x = 2, this gives (eˣ − e⁻ˣ)/2 = 2 ⇒ eˣ − e⁻ˣ = 4. Multiplying through by eˣ yields e²ˣ − 4eˣ − 1 = 0, which is a quadratic in eˣ.
求解诸如 sinh x = k 的方程时，最可靠的做法是把双曲函数写回其指数定义。对于 sinh x = 2，可得 (eˣ − e⁻ˣ)/2 = 2 ⇒ eˣ − e⁻ˣ = 4。两边同乘 eˣ 可得 e²ˣ − 4eˣ − 1 = 0，这是一个关于 eˣ 的二次方程。
Let y = eˣ, then y² − 4y − 1 = 0 gives y = 2 ± √5. Since eˣ > 0, we take y = 2 + √5, so x = ln(2 + √5). This technique works for nearly all hyperbolic equations encountered at IB level and is heavily practised in Exercise H.3.
令 y = eˣ，则 y² − 4y − 1 = 0 解得 y = 2 ± √5。由于 eˣ > 0，我们取 y = 2 + √5，故 x = ln(2 + √5)。此法适用于 IB 阶段几乎所有的双曲方程，并在练习 H.3 中得到了大量训练。
1 or express both in exponential form to reduce the problem to an algebraic equation in eˣ.
对于同时含有 sinh 和 cosh 的方程，通常可以利用恒等式 cosh²x − sinh²x = 1，或者将两者均写成指数形式，从而将其化为关于 eˣ 的代数方程。
5. Derivatives of Hyperbolic Functions | 双曲函数的导数
The differentiation rules for hyperbolic functions are elegantly symmetric and easy to remember once their definitions are internalised:
双曲函数的求导法则精巧对称，一旦内化了它们的定义，记忆起来就十分简单：
d/dx (sinh x) = cosh x
d/dx (cosh x) = sinh x
d/dx (tanh x) = sech² x
These can be verified by starting from the exponential forms. For instance, d/dx [(eˣ − e⁻ˣ)/2] = (eˣ + e⁻ˣ tice the absence of any sign change for cosh, unlike its trigonometric counterpart (cos x)' = −sin x.
这些公式均可从指数形式出发验证。例如 d/dx [(eˣ − e⁻ˣ)/2] = (eˣ + e⁻ˣ)/2 = cosh x，正如公式所述。值得注意的是，cosh 的导数中没有符号变化，与三角函数中 (cos x)' = −sin x 截然不同。
ilarly, the derivative of sech x is −sech x tanh x, though sech and coth derivatives are less prominent in the syllabus.
对于复合函数，需要运用链式法则，得到诸如 d/dx [sinh(ax + b)] = a cosh(ax + b) 的结果。类似地，sech x 的导数为 −sech x tanh x，尽管 sech 与 coth 的导数在考纲中出现频率较低。
6. Integrals of Hyperbolic Functions | 双曲函数的积分
Every differentiation rule instantly provides an integration formula. The three essential integrals you must have at your fingertips are:
每一条微分法则都反过来给出一个积分公式。你必须牢记在心的三个基本积分是：
∫ sinh x dx = cosh x + C
∫ cosh x dx = sinh x + C
∫ sech² x dx = tanh x + C
2cosh²x − 1 to obtain cosh²x = (cosh 2x + 1)/2, which integrates to (sinh 2x)/4 + x/2 + C.
更复杂的积分，如 ∫ sinh(ax + b) dx 或 ∫ cosh²x dx，可以通过换元积分或借助双曲恒等式变形来处理。例如，要积分 cosh²x，利用 cosh 2x = 2cosh²x − 1 得到 cosh²x = (cosh 2x + 1)/2，积分结果为 (sinh 2x)/4 + x/2 + C。
nd recognising the numerator as the derivative of the denominator, giving ∫ tanh x dx = ln|cosh x| + C. Exercise H.3 offers plenty of practice with these standard patterns.
对于 ∫ tanh x dx 这类积分，可以写出 tanh x = sinh x / cosh x，并注意到分子恰好是分母的导数，从而得到 ∫ tanh x dx = ln|cosh x| + C。练习 H.3 提供了大量这样的标准模式练习。
7. Inverse Hyperbolic Functions and Their Logarithmic Forms | 反双曲函数及其对数形式
The inverse hyperbolic functions are denoted arsinh x, arcosh x, and artanh x (some texts use sinh⁻¹x, but IB prefers the prefix ‘ar’). Their domains and ranges must be carefully noted:
反双曲函数分别记作 arsinh x, arcosh x 和 artanh x（有些教材使用 sinh⁻¹x，但 IB 更倾向于前缀 ‘ar’）。需要注意它们的定义域与值域：
arsinh x has domain R and range R.
arsinh x 的定义域为 R，值域为 R。
arcosh x has domain x ≥ 1 and range y ≥ 0.
arcosh x 的定义域为 x ≥ 1，值域为 y ≥ 0。
artanh x has domain |x| < 1 and range R.
artanh x 的定义域为 |x| < 1，值域为 R。
orm. By setting y = arsinh x, we have sinh y = x. Using exponential definitions leads to eʸ − e⁻ʸ = 2x, which can be solved as a quadratic in eʸ to obtain:
练习 H.3 中考查的一项重要技能，是将这些反函数转换为对数形式。设 y = arsinh x，则有 sinh y = x。利用指数定义可得 eʸ − e⁻ʸ = 2x，将其作为关于 eʸ 的二次方程求解，得到：
arsinh x = ln(x + √x² + 1)
Similarly, arcosh x = ln(x + √x² − 1) for x ≥ 1, and artanh x = ½ ln((1 + x)/(1 − x)) for |x| < 1. These logarithmic forms are indispensable for differentiation and integration.
类似地，arcosh x = ln(x + √x² − 1)（x ≥ 1），artanh x = ½ ln((1 + x)/(1 − x))（|x| < 1）。这些对数形式在求导和积分时不可或缺。
8. Derivatives of Inverse Hyperbolic Functions | 反双曲函数的导数
The derivatives of inverse hyperbolic functions are standard results that appear frequently in multiple-choice and written papers. They can be derived from the logarithmic forms or via implicit differentiation.
反双曲函数的导数是可以在选择题和笔试题中频繁出现的标准结论，可以通过对数形式或隐函数求导推导得出。
d/dx (arsinh x) = 1 / √x² + 1
d/dx (arcosh x) = 1 / √x² − 1 (x > 1)
d/dx (artanh x) = 1 / (1 − x²) (|x| < 1)
ivatives of inverse trigonometric functions, only without sign changes. For instance, ∫ 1/√x²+1 al is a favourite outcome of trigonometric or hyperbolic substitution in IB exam questions.
可以看到它们与反三角函数的导数非常相似，只是没有符号变化。例如 ∫ 1/√x²+1 dx = arsinh x + C，而这一积分恰是 IB 考题中通过三角或双曲代换得到的最常见结果之一。
9. Applications: Catenary and Relativity | 应用：悬链线与相对论
cosh(x/a), where a is a constant related to the tension. The cosh function describes the exact shape, which is why suspension bridge cables are modelled with hyperbolic cosines.
双曲函数最直观的应用之一是悬链线：由一根完全柔软且均匀的链条在自重下悬挂而成的曲线。其方程为 y = a cosh(x/a)，其中 a 是一个与张力相关的常数。cosh 函数精确描述了这一形状，因此悬索桥的主缆便是由双曲余弦建模的。
In physics, hyperbolic tangents appear in special relativity when adding velocities. The relativistic velocity-addition formula can be expressed in terms of artanh, making the hyperbolic tangent a natural tool for describing rapidity. While IB questions remain mathematical, having a sense of these connections strengthens your conceptual understanding.
在物理学中，双曲正切出现在狭义相对论的速度叠加中。相对论速度叠加公式可以用 artanh 表达，这使得双曲正切成为描述快度的自然工具。虽然 IB 试题仍然聚焦于数学本身，但了解这些联系能够加深你的概念理解。
10. Worked Examples from Exercise H.3 | 练习 H.3 示例解析
Let us walk through several typical problems that mirror the structure and difficulty of Exercise H.3.
我们来看几道能够反映练习 H.3 结构和难度的典型题目。
Example 1: Prove that cosh²x − sinh²x = 1 using exponential definitions.
示例
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