Hyperbolic Functions：練習問題

Find the value of \(\tanh 0\).

By using the definition of \(\cosh x\) and \(\sinh x\) in terms of exponentials, solve the equation \(2\cosh x + \sinh x = 3\).
Express your answer in the form \(\ln k\).

Find the exact solutions for \(x \ge 0\) to the equation \(\cosh 2x - 5\cosh x + 4 = 0\).

Find the derivative of the function \(f(x) = \tanh(3x)\) with respect to \(x\).

By using the definition of \(\sinh y = x\) in terms of exponentials, which of the following is the logarithmic form of the inverse function \(\text{arsinh } x\)?

Use the exponential definitions of \(\sinh x\) and \(\cosh x\) to determine the exact value of \(\frac{\cosh(\ln 5) + \sinh(\ln 5)}{2}\).

Use the logarithmic definition \(\text{arsinh } x = \ln(x + \sqrt{x^2 + 1})\) to find the exact value of \(\text{arsinh } (\sqrt{2})\), expressing your answer as a single logarithm.

Using the exponential definitions of hyperbolic functions, show that the solution to the equation \( 3\cosh x - \sinh x = 4 \) can be expressed as \( x = \ln k \), and determine the exact value of the constant \( k \).

(a) Sketch the graph of the function \(y = \tanh x\). On your sketch, clearly label the coordinates of the y-intercept and state the equations of the horizontal asymptotes.
(b) Use the definitions of hyperbolic functions in terms of exponentials to show that \(\tanh(\ln \sqrt{3}) = \frac{1}{2}\).

Consider the equation \(2\cosh^2 x + \sinh x = 5\).
(a) Use the identity \(\cosh^2 x - \sinh^2 x \equiv 1\) to show that the equation can be written as \(2\sinh^2 x + \sinh x - 3 = 0\).
(b) Hence, find the exact solutions for \(x\) in terms of natural logarithms.