Chapter 4.2, Problem 18E

Hyperbolic sine Function The hyperbolic sine function is defined by $\sinh \left(x\right)=\frac{e^x-e^{-x}}{2}$

(a) Sketch the graph of this function using graphical addition as in Exercise 17.

(b) Use the definition to show that $\text{sinh}\left(-x\right)=-\sinh \left(x\right)$

Hyperbolic Cosine Function The hyperbolic cosine function is defined by $\text{cosh}\left(x\right)=\frac{e^x+e^{-x}}{2}$

(a) Sketch the graphs of the functions $y=\frac{1}{3}{e}^x$ and $y=\frac{1}{2}{e}^{-x}$ on the same axes, and use graphical addition (see Section 2.7) to sketch the graph of $y=\cosh \left(x\right)$.

(b) Use the definition to show that $\text{cosh}\left(-x\right)=\cosh \left(x\right)$.