Tanh⁻¹

Definition

The inverse hyperbolic tangent function, denoted as tanh⁻¹, is a mathematical function that is the inverse of the hyperbolic tangent function. It is used to find the angle whose hyperbolic tangent is a given value, and it is an important concept in the calculus of hyperbolic functions.

5 Must Know Facts For Your Next Test

1. The inverse hyperbolic tangent function, tanh⁻¹, is used to find the angle whose hyperbolic tangent is a given value.
2. The domain of tanh⁻¹ is the interval (-1, 1), and the range is the set of all real numbers.
3. The derivative of tanh⁻¹(x) is $\frac{1}{1-x^2}$, which is an important result in the calculus of hyperbolic functions.
4. The integral of tanh⁻¹(x) dx is $\frac{1}{2}x\tanh^{-1}(x) + \frac{1}{4}\ln(1-x^2) + C$, where C is the constant of integration.
5. The hyperbolic tangent function, tanh, and its inverse function, tanh⁻¹, are used to model various physical and biological phenomena, such as the behavior of electric circuits, the propagation of signals in neural networks, and the growth of populations.

Review Questions

• Explain the relationship between the hyperbolic tangent function, tanh, and its inverse function, tanh⁻¹.
  • The hyperbolic tangent function, tanh, and its inverse function, tanh⁻¹, are related in the following way: if $y = \tanh(x)$, then $x = \tanh^{-1}(y)$. In other words, tanh⁻¹ is the function that undoes the effect of the tanh function, allowing us to find the angle whose hyperbolic tangent is a given value. This relationship is analogous to the relationship between the trigonometric tangent function and its inverse, the arctangent function.
• Describe the properties of the domain and range of the inverse hyperbolic tangent function, tanh⁻¹.
  • The domain of the inverse hyperbolic tangent function, tanh⁻¹, is the interval (-1, 1). This is because the hyperbolic tangent function, tanh, has a range of (-1, 1), and the inverse function must have the same range as the original function's domain. The range of tanh⁻¹ is the set of all real numbers, $\mathbb{R}$. This means that for any real number, there exists an angle whose hyperbolic tangent is that value, and tanh⁻¹ can be used to find that angle.
• Explain the importance of the derivative and integral of the inverse hyperbolic tangent function, tanh⁻¹, in the context of the calculus of hyperbolic functions.
  • The derivative and integral of the inverse hyperbolic tangent function, tanh⁻¹, are crucial in the calculus of hyperbolic functions. The derivative of tanh⁻¹(x) is $\frac{1}{1-x^2}$, which is a fundamental result that is used in many applications involving hyperbolic functions, such as in the analysis of electric circuits and the study of signal propagation in neural networks. Similarly, the integral of tanh⁻¹(x) dx is $\frac{1}{2}x\tanh^{-1}(x) + \frac{1}{4}\ln(1-x^2) + C$, which is an important formula in the calculus of hyperbolic functions and has applications in various areas of mathematics and physics.

Related terms

Hyperbolic Functions: The hyperbolic functions, such as sinh, cosh, tanh, and their inverses, are mathematical functions that are analogous to the trigonometric functions but are defined using the exponential function instead of the circular functions.

Inverse Hyperbolic Functions:

The inverse hyperbolic functions, such as sinh⁻¹, cosh⁻¹, and tanh⁻¹, are the inverse functions of the hyperbolic functions, and they are used to find the angle whose hyperbolic function is a given value.

Hyperbolic Tangent: The hyperbolic tangent function, denoted as tanh, is a hyperbolic function that is the ratio of the hyperbolic sine function to the hyperbolic cosine function, and it is used to model various physical and biological phenomena.