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nameVoid
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Find the points on the graph of y=sinhx at which the tangent line has slope 2
dy/dx=coshx=2
(e^x+e^(-x))=4
x-x=ln4
dy/dx=coshx=2
(e^x+e^(-x))=4
x-x=ln4
Hyperbolic functions are mathematical functions that are closely related to trigonometric functions. They are defined in terms of the hyperbola, a curve that is similar to the shape of a parabola. The three main hyperbolic functions are sineh (sinh), cosineh (cosh), and tangent (tanh). These functions are commonly used in fields such as physics, engineering, and mathematics.
The tangent line slope 2 at y=sinhx is calculated by taking the derivative of the hyperbolic sine function. The derivative of sinh(x) is cosh(x), which has a constant value of 2 at y=sinhx. This means that the slope of the tangent line at y=sinhx will always be 2, regardless of the value of x.
Hyperbolic functions and exponential functions are closely related, as they share many similar properties. In fact, the hyperbolic sine function, sinh(x), can be expressed in terms of an exponential function as (e^x - e^-x)/2. This relationship allows for easier calculations and manipulation of hyperbolic functions.
Hyperbolic functions are used in various real-life applications, particularly in physics and engineering. For example, the hyperbolic cosine function, cosh(x), is used to model the shape of a hanging cable or a catenary curve. The hyperbolic tangent function, tanh(x), is used in the design of electronic circuits and in the study of heat transfer.
Hyperbolic functions and trigonometric functions have many similarities, but they also have some key differences. One main difference is that hyperbolic functions are defined in terms of the hyperbola, while trigonometric functions are defined in terms of the unit circle. Additionally, hyperbolic functions have different properties and identities than trigonometric functions. However, they can be converted to each other using complex numbers.