Hyperbolic functions are very useful in both mathematics and physics.  You may have already encountered them in Math 118.   If not, here are their definitions:

sinh (x) = (e^x - e^-x)/2
cosh (x) = (e^x + e^-x)/2
tanh (x) = sinh(x) / cosh(x)
coth(x) = 1/tanh(x)
sech(x) = 1/cosh(x)
csch(x) = 1/sinh(x)

Oddly enough, they enjoy certain similarities with the trigonometric functions, with which you are much more familiar.

> plot (sinh(x), x = -7..7);
> sinh(5);
> evalf(%);
> Limit( sinh(x), x = infinity);  # Pretty print.
> limit( sinh(x), x = -infinity);
> expand (sinh(x+y));
> expand (sinh(x-y));
> expand (sinh(2*x));
> Diff(sinh(x),x);  # Pretty print.
> diff( sinh(x), x);
> Int ( sinh(x), x);  # Pretty print.
> int (sinh(x), x);

1.  Graph the other five hyperbolic functions:  cosh(x), tanh(x), coth(x), sech(x), csch(x).  For each curve, determine the limit of y as x tends toward infinity or negative infinity.  Which of the functions are odd?  which are even?  (Remember that an odd function is one that is symmetric with respect to the origin;  an even function is one that is symmetric with respect to the y-axis.)

2.   Find the derivative and the indefinite integral of each of the other five hyperbolic functions.

3.  Expand cosh(x+y), cosh(2x), tanh(x+y), and tanh(2x).

4.   Show that  (cosh x)² - (sinh x)² = 1.   Hint:  You may either use the simplify command, or else graph the function y = (cosh x)² - (sinh x)².

5.   Show that   1 - (tanh x)² = (sech x)².

6.   Show that:

7.   Find the limit of (sinh x) / e^x  as x tends toward infinity.

8.    Simplify the expression:

9.   The inverse of sinh x in Maple is represented by arcsinh(x).   Graph the curve y = arcsinh(x).  Find formulas for the derivative and the integral of arcsinh(x).

10.   Repeat question 9 for the functions arccosh(x) and arctanh(x).