hyperbolic function

简明释义

双曲函数

英英释义

例句

1.In calculus, the derivative of a hyperbolic function is often used to find slopes of curves.

在微积分中，双曲函数的导数常用于寻找曲线的斜率。

2.The hyperbolic function sinh(x) is defined as (e^x - e^(-x))/2.

双曲函数 sinh(x) 被定义为 (e^x - e^(-x))/2。

3.The inverse of a hyperbolic function can be useful in solving equations.

双曲函数 的反函数在解方程时非常有用。

4.Many engineering problems involve hyperbolic functions to model waveforms.

许多工程问题涉及到 双曲函数 来模拟波形。

5.In physics, hyperbolic functions can describe the shape of certain trajectories.

在物理学中，双曲函数 可以描述某些轨迹的形状。

作文

The study of mathematics often leads us to various functions that describe the relationships between different variables. One such intriguing category of functions is known as hyperbolic function. These functions, which include the hyperbolic sine (sinh), hyperbolic cosine (cosh), and others, play a crucial role in many areas of mathematics and physics. Understanding hyperbolic function is essential for students who wish to delve deeper into advanced mathematical concepts or apply these principles in real-world scenarios.First, let’s explore what hyperbolic function actually means. In essence, hyperbolic function refers to a set of mathematical functions that are analogs of the ordinary trigonometric functions but are based on hyperbolas rather than circles. For instance, while the sine and cosine functions relate to the unit circle, the hyperbolic sine and cosine functions relate to a unit hyperbola. This fundamental distinction leads to unique properties and applications of the hyperbolic function.One of the most fascinating aspects of hyperbolic function is their geometric interpretation. The hyperbolic functions can be visualized using the equations of hyperbolas in the Cartesian plane. For example, the hyperbolic cosine function can be represented as cosh(x) = (e^x + e^(-x))/2, which describes the shape of a hyperbola. Similarly, the hyperbolic sine function is defined as sinh(x) = (e^x - e^(-x))/2. These definitions reveal that hyperbolic function not only have algebraic expressions but also geometric significance.Moreover, hyperbolic function exhibit properties similar to those of trigonometric functions. For instance, they satisfy identities akin to the Pythagorean identity seen in trigonometry. Specifically, the identity cosh²(x) - sinh²(x) = 1 holds true for all real numbers x. This relationship highlights the interconnectedness of these functions and underscores the importance of hyperbolic function in mathematical analysis.In addition to their theoretical significance, hyperbolic function find practical applications in various fields. In physics, for example, they are used to describe the shape of hanging cables, known as catenaries, which can be modeled by the hyperbolic cosine function. Engineers also utilize hyperbolic function when analyzing structures and designing systems that require an understanding of forces and motions.Furthermore, hyperbolic function are essential in calculus, especially in integration and differentiation. The derivatives of hyperbolic functions mirror those of trigonometric functions, making them easier to work with in calculus problems. For instance, the derivative of sinh(x) is cosh(x), and the derivative of cosh(x) is sinh(x). This similarity facilitates the learning process for students who are already familiar with trigonometric derivatives.In conclusion, the concept of hyperbolic function is multifaceted, encompassing both theoretical and practical dimensions. By studying hyperbolic function, students gain valuable insights into the world of mathematics and its applications in science and engineering. As we continue to explore the vast landscape of mathematical functions, the importance of hyperbolic function becomes increasingly clear, revealing their indispensable role in our understanding of complex systems and phenomena.

数学研究常常引导我们探索描述不同变量之间关系的各种函数。其中一个引人入胜的函数类别被称为双曲函数。这些函数包括双曲正弦（sinh）、双曲余弦（cosh）等，在许多数学和物理领域中发挥着至关重要的作用。理解双曲函数对于希望深入学习高级数学概念或在实际场景中应用这些原理的学生来说是必不可少的。首先，让我们探讨一下双曲函数的实际含义。从本质上讲，双曲函数指的是一组与普通三角函数类似，但基于双曲线而非圆的数学函数。例如，虽然正弦和余弦函数与单位圆有关，但双曲正弦和双曲余弦函数与单位双曲线有关。这一基本区别导致了双曲函数独特的性质和应用。双曲函数最迷人的方面之一是它们的几何解释。双曲函数可以通过笛卡尔平面中的双曲线方程进行可视化。例如，双曲余弦函数可以表示为cosh(x) = (e^x + e^(-x))/2，描述了双曲线的形状。类似地，双曲正弦函数定义为sinh(x) = (e^x - e^(-x))/2。这些定义揭示了双曲函数不仅具有代数表达式，还有几何意义。此外，双曲函数表现出与三角函数相似的性质。例如，它们满足类似于三角学中毕达哥拉斯恒等式的恒等式。具体而言，恒等式cosh²(x) - sinh²(x) = 1在所有实数x上都成立。这种关系突显了这些函数之间的相互联系，并强调了双曲函数在数学分析中的重要性。除了理论上的重要性，双曲函数在各个领域也有实际应用。在物理学中，例如，它们用于描述悬挂电缆的形状，称为链线，可以用双曲余弦函数建模。工程师在分析结构和设计需要理解力和运动的系统时也利用双曲函数。此外，双曲函数在微积分中至关重要，尤其是在积分和微分方面。双曲函数的导数与三角函数的导数相似，使得在微积分问题中更易于处理。例如，sinh(x)的导数是cosh(x)，而cosh(x)的导数是sinh(x)。这种相似性促进了已经熟悉三角导数的学生的学习过程。总之，双曲函数的概念是多方面的，涵盖了理论和实践两个维度。通过学习双曲函数，学生获得了对数学世界及其在科学和工程中应用的宝贵见解。当我们继续探索数学函数的广阔领域时，双曲函数的重要性变得愈加明显，揭示了它们在理解复杂系统和现象中的不可或缺的角色。