In Review of Chapter 2 we introduced the hyperbolic functions sinh x = ex ex 2 , x R

Chapter 5, Problem 3

(choose chapter or problem)

In Review of Chapter 2 we introduced the hyperbolic functions sinh x = ex ex 2 , x R cosh x = ex + ex 2 , x R tanh x = ex ex ex + ex , x R (a) Show that f (x) = tanh x, x R, is a strictly increasing function on R. Evaluate lim x tanh x and lim x tanh x (b) Use your results in (a) to explain why f (x) = tanh x, x R, is invertible, and show that its inverse function f 1(x) = tanh1 x is given by f 1(x) = 1 2 ln 1 + x 1 x What is the domain of f 1(x)? (c) Show that d dx f 1(x) = 1 1 x2 (d) Use your result in (c) and the facts that tanh x = sinh x cosh x and cosh2 x sinh2 x = 1 to show that d dx tanh x = 1 cosh2 x