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\( y=\sinh (x) \)

Ask by Daniel Gordon. in China
Jan 23,2025

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双曲正弦函数 \( y = \sinh(x) \) 的定义是： \[ y = \sinh(x) = \frac{e^{x} - e^{-x}}{2} \] **特点和性质：** 1. **图像**：函数图像关于原点对称，当 \( x \) 很大时，\( \sinh(x) \) 会变得非常大，当 \( x \) 很小时，\( \sinh(x) \) 会变得非常小。 2. **性质**： - **奇函数**：\( \sinh(-x) = -\sinh(x) \)。 - **导数**：\( \frac{d}{dx} \sinh(x) = \cosh(x) \)。 - **积分**：\( \int \sinh(x) \, dx = \cosh(x) + C \)。 **应用**：双曲正弦函数在工程、物理和数学中有很多应用，比如描述双曲线的性质、解决微分方程以及在相对论中描述时空关系。 **计算示例**： 计算 \( \sinh(1) \)： \[ \sinh(1) = \frac{e^{1} - e^{-1}}{2} \approx \frac{2.71828 - 0.36788}{2} \approx 1.1752 \] 希望这些信息能帮助你更好地理解双曲正弦函数！

Solución

双曲正弦函数定义如下： \[ y = \sinh(x) = \frac{e^{x} - e^{-x}}{2} \] **解释与性质：** 1. **定义：** 双曲正弦函数是双曲函数的一种，类似于普通三角函数中的正弦函数，但它基于指数函数定义。 2. **图像特征：** - 函数图像关于原点对称。 - 当 \( x \) 趋向于正无穷大时，\( \sinh(x) \) 也趋向于正无穷大；当 \( x \) 趋向于负无穷大时，\( \sinh(x) \) 也趋向于负无穷大。 - 在 \( x = 0 \) 处，\( \sinh(0) = 0 \)。 3. **性质：** - **奇函数**：满足 \( \sinh(-x) = -\sinh(x) \)。 - **导数**：\( \frac{d}{dx} \sinh(x) = \cosh(x) \)，其中 \( \cosh(x) \) 是双曲余弦函数。 - **积分**：\( \int \sinh(x) \, dx = \cosh(x) + C \)。 4. **应用：** 双曲正弦函数广泛应用于工程学、物理学以及数学中的各种领域，例如描述双曲线的几何性质、解决某些微分方程以及在相对论中描述时空关系等。 **示例计算：** 计算 \( \sinh(1) \)： \[ \sinh(1) = \frac{e^{1} - e^{-1}}{2} \approx \frac{2.71828 - 0.36788}{2} \approx 1.1752 \] 希望这些信息对你理解双曲正弦函数有所帮助！

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Mind Expander

Did you know that the hyperbolic sine function, \( \sinh(x) \), actually describes the shape of a hanging cable or chain? This real-world phenomenon is known as a catenary! It’s fascinating how these mathematical functions connect to real-life structures and can be observed in architecture and engineering, making them not just theoretical concepts but also practical tools. When working with \( y=\sinh(x) \), one common mistake is confusing it with the regular sine function. Remember, hyperbolic functions have a unique set of properties and identities, such as \( \sinh(-x) = -\sinh(x) \), which makes them odd functions. So, double-check your signs when graphing to keep your curve in shape!

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