Calculus: Analytic Geometry and Calculus, with Vectors

(^466) Trigonometric functions
and show that 0 is a maximum when x = ab. Remark: Addicts of elementary
plane geometry can be pleased to see that the x which maximizes 0 can be found
in another way. As in Figure 8.392 let C be a circle which passes through the
a+b
2
top and bottom of the picture and intersects the eye-level line L at two points
El and Ez. The angles which the picture subtends at E, and Ez are equal. Of
all such circles C that intersect L, the smallest one that produces the greatest 6
is the circle Co for which El and Ez coincide, so Co is tangent to L. The radius
of Co is (b + a)/2, and use of an appropriate right triangle shows that the dis-
tance from the wall to the center and to the critical point of tangency of Co is
(b+\'--(b-a\z
`` 2 fJ^2 J or ab.
6 Supposing that a > 0 and jaxI < 1, prove the formula
tan-' x tan-' a =tan 1a
1

• x

• ax

by showing that the two members of the equation are equal when x = 0 and

have equal derivatives when JaxI < 1.

7 With or without undertaking to prove the fundamental fact that cot-1 x =

tan-' (1/x) when x > 0, suppose that x -' 0 and fill in the omitted steps in the

calculation '

d d 1 -1

dxcot-' x =dxtan iz=

= 1 + xz.

(^8) With or without undertaking to prove the fundamental fact that sec-' x =
cos ' (1/x) when IxI > 1, suppose that IxI > 1 and fill in the omitted steps in
the calculation
d d 1 1

dxsec -I x =dxcos'x

= ..

X x z- 1

Hint: Do not forget the chain rule and the fact that x2=IxI IxI.

9 With the aid of Figure 8.343 we can see that the formula

x2-1

(1) sec -I x = sin-'

x

is valid at least when x > 1. Differentiate the two members of (1) and determine

the set of numbers x for which the derivativesare equal.