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\( y=\sinh (x) \)

Ask by Chan Hall. in the United States
Jan 23,2025

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The function \( y = \sinh(x) \) is the hyperbolic sine function, defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). It is an odd function with a domain and range of all real numbers. Its derivative is \( \cosh(x) \) and its integral is \( \cosh(x) + C \). The graph of \( y = \sinh(x) \) is an S-shaped curve that passes through the origin and extends infinitely in both directions.

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The function given is \( y = \sinh(x) \), which is the hyperbolic sine function. ### Properties of the Hyperbolic Sine Function 1. **Definition**: The hyperbolic sine function is defined as: \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] 2. **Domain**: The domain of \( \sinh(x) \) is all real numbers, \( (-\infty, \infty) \). 3. **Range**: The range of \( \sinh(x) \) is also all real numbers, \( (-\infty, \infty) \). 4. **Symmetry**: The function is odd, meaning \( \sinh(-x) = -\sinh(x) \). 5. **Derivative**: The derivative of \( \sinh(x) \) is: \[ \frac{dy}{dx} = \cosh(x) \] where \( \cosh(x) \) is the hyperbolic cosine function. 6. **Integral**: The integral of \( \sinh(x) \) is: \[ \int \sinh(x) \, dx = \cosh(x) + C \] where \( C \) is the constant of integration. ### Graph of the Function The graph of \( y = \sinh(x) \) resembles the shape of the standard sine wave but is unbounded and extends infinitely in both directions. It passes through the origin (0,0) and has a characteristic "S" shape. If you need further analysis or specific calculations related to this function, please let me know!

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Mind Expander

Did you know that the hyperbolic sine function, \( \sinh(x) \), is closely related to the exponential function? It can be expressed as: \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). This function is essential in various fields, including engineering and physics, particularly when dealing with problems involving hyperbolic geometry and waves. In the realm of real-world applications, you might find \( \sinh(x) \) popping up in architecture and construction, especially in the design of arches and structures that mimic natural forms. The catenary curve, which describes the shape of a hanging chain or cable, is actually modeled using the hyperbolic cosine function, but its properties also heavily involve \( \sinh(x) \) in calculations!

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