Hyperbolic functions and inverse relations · 双曲函数及其反关系
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| hyperbolic cosine/ˌhaɪpəˈbɒlɪk ˈkəʊsaɪn/ | 双曲余弦 | shuāng qū yú xián |
How can growth and decay make a symmetric curve?
- A hanging cable has a curved profile related to exponentials. Hyperbolic functions combine growth and decay symmetrically.
- This lesson studies hyperbolic cosine 双曲余弦: The function cosh x=(e^x+e^(-x))/2.
增长与衰减如何形成对称曲线？
- 悬挂的缆索具有与指数函数相关的弯曲轮廓。双曲函数通过对称地结合增长与衰减来体现这一特性。
- 本课学习双曲余弦 cosh x: 函数 cosh x=(e^x+e^(-x))/2。
Choose the mathematical structure
- Define sinh x=(e^x-e^(-x))/2 and cosh x=(e^x+e^(-x))/2. Their identity is cosh²x-sinh²x=1. Derivatives are sinh prime=cosh and cosh prime=sinh.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 定义 sinh x=(e^x-e^(-x))/2 和 cosh x=(e^x+e^(-x))/2。它们的恒等式为 cosh²x-sinh²x=1。导数为 (sinh x)'=cosh x 且 (cosh x)'=sinh x。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines hyperbolic cosine? · 哪种描述正确定义了双曲余弦？
The function cosh x=(e^x+e^(-x))/2. · 函数 cosh x=(e^x+e^(-x))/2。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
At x=ln2, e^x=2 and e^(-x)=1/2. Thus cosh x=1.25 and sinh x=0.75. Their squared difference is 1.5625-0.5625=1. To invert y=sinh x, solve a quadratic in e^x and choose the positive root.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
当 x=ln2 时，e^x=2 且 e^(-x)=1/2。因此 cosh x=1.25 且 sinh x=0.75。它们平方的差为 1.5625-0.5625=1。要反解 y=sinh x，需建立关于 e^x 的二次方程并取正根。
Hyperbolic functions and inverse relations · 双曲函数及其反关系
Define sinh x=(e^x-e^(-x))/2 and cosh x=(e^x+e^(-x))/2 · 定义 sinh x=(e^x-e^(-x))/2 和 cosh x=(e^x+e^(-x))/2
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find cosh(ln2). · 求 cosh(ln2)。
e^(ln2)=2 and e^(-ln2)=1/2. Their half-sum is 1.25. · e^(ln2)=2 且 e^(-ln2)=1/2。它们的半和是 1.25。
Test a tempting shortcut
- The hyperbolic identity has a minus sign. cosh is not one-to-one on all real inputs; its usual inverse uses x≥0. Ordinary circular-trigonometric identities cannot be substituted unchanged.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The identity for hyperbolic functions is cosh²x+sinh²x=1. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 双曲恒等式包含减号。cosh 在所有实数输入上并非一一映射；其常规反函数限定于 x≥0。普通的圆三角恒等式不能直接照搬使用。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
双曲函数的恒等式为 cosh²x+sinh²x=1。此说法是错误的。请解释它违反了哪一定义或假设。
Find sinh(ln2). · 求 sinh(ln2)。
Their half-difference is (2-1/2)/2=0.75. · 它们的半差为 (2-1/2)/2=0.75。
The identity for hyperbolic functions is cosh²x+sinh²x=1. · 双曲函数的恒等式为 cosh²x+sinh²x=1。
The hyperbolic identity has a minus sign. cosh is not one-to-one on all real inputs; its usual inverse uses x≥0. Ordinary circular-trigonometric identities cannot be substituted unchanged. · 双曲恒等式包含减号。cosh 在所有实数输入上并非单射；其通常的反函数使用 x≥0。普通的三角恒等式不能直接照搬不变地替换。
Interpret a new situation
- Use exponential definitions to prove identities and solve equations. State domain restrictions for inverse functions before differentiating or integrating them.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- 利用指数定义证明恒等式并求解方程。在对反函数求导或积分之前，先声明其定义域限制。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
Find cosh²(ln2)-sinh²(ln2). · 求 cosh²(ln2)-sinh²(ln2)。
1.25²-0.75²=1, agreeing with the hyperbolic identity. · 1.25²-0.75²=1，与双曲恒等式一致。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The function cosh x=(e^x+e^(-x))/2. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- Edexcel IAL 进阶数学；官方单元 FP3。其他单元的补充内容在范围审查中已列出；不属于额外的加分要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
函数 cosh x=(e^x+e^(-x))/2。选择该关系式，展示方法，检验其假设并解释结果。