Calculus: Analytic Geometry and Calculus, with Vectors

510 Exponential and logarithmic functions

provided that the positive constant k and the coordinate system are suitably

determined. Because of this fact and the fact that the Latin word for chain is

catenus, the graph of (1) appearing in Figure 9.391 is called a catenary. Supposing

Figure 9.391

that a > 0, find the length L of the part of this catenary that hangs over the

1

interval 0 5 x =< a. 4ns.: L =ksinh ka.

4 Compare the graphs of y = sech x and y = 1/(1 + x2).

5 Starting with the formula

t= log (x+ x2-1),

show that x = cosh t.

6 Prove the formula

f x2-1 dx = x x2 - 1- i cosh-' x + c

with the aid of (8.488) and use it to show that

cosh-1 xo = 2 [xo xo - 1 -

fixu

x2 - 1 dx].

Use this to show that cosh-' xo is the area of the shaded region of Figure 9.371

when x = xo.

(^7) Evaluate the integral of the preceding problem with the aid of a hyper-
bolic function substitution.
8 Show that
e= = cosh x + sinh x
e z = cosh x - sinh x.
Show that if there exist constants c, and c2 for which
(1) f(t) = cle(a+b)e + c2e(a-b)t,
then there also exist constants C, and C2 for which
(2) At) = ea'[C, cosh bt + C2 sinh hi].
Use (1) to calculate formulas for f'(t), f"(t), f"'(t), and f(4) (t). Use (2) to cal-
culate f'(t) and f"(t) and observe that the hyperbolic functionsare being nuisances.