Calculus: Analytic Geometry and Calculus, with Vectors

5.4 Theorems about continuous and differentiable functions 323

Observe that if n is a positive integer and x = 1/-\/2na, then

I - 2 2nr.

It follows that each interval 0 < x < h contains subintervals over which f' (x) < 0
and f is decreasing.
22 Boom-and-bust processes occur (or seem to occur) in economic and political
life. Persons who get their political information from clever press secretaries
of astute chiefs of state discover that the fortunes of their countries are at low
ebbs when new chiefs are installed and that these fortunes steadily improve
during the tenure of each chief. Such processes occur in electrical engineering
when a charge on a capacitor steadily increases until a spark jumps and the charge
disappears. This problem involves a particular boom-and-bust process in which
a and q are positive constants. It is supposed that, for each integer n, the
quantity y is 0 when t = na and that y increases at a constant rate over the inter-
val na 5 t < (n + 1)a in such a way that y approaches q as t approaches (n + 1)a
from the left. Sketch a graph of y versus t and find a formula giving y in terms
of t. Partial ans:

y=qa-LaJ/,

where [x] denotes the greatest integer in x.

23 While persons confining their mathematical contacts to modern mathe-

matics books need not worry about the matter, others may need a warning.

In the good old days, the word "finite" was used in place of the word "bounded."

In order to understand assertions involving the word finite, it is sometimes not

sufficient to understand modern mathematics. Sometimes we need substantial

information about history, and sometimes we need conscious recognition of the

fact that assertions involving the word "finite" have different meanings at different

places and at different times. For example, the assertion that "f is finite at xo"

can mean that there is an interval with center at xo such that f is bounded over the

interval. The assertion can, however, have other meanings, and this is the

reason why we should shudder when we hear it.

24 A function f is said to have a generalized first derivative Gf'(x) at x if

(1) G f'(x) = l o f(x + h) - f(x -h)

and is said to have a generalized second derivative Gf'(x) at x if

(2) Gf"(x) = limf (x - h) - 2f (x) + f(x + h)h2

Prove that Gf'(x) = f'(x) when f'(x) exists. Hint: Use the fact that

(3)

f(x -I h) - f(x - h) 1 rf(x + h) - f(x) +f(x - h) - f(x)

2h ZL Is

Remark: The wide world contains several persons who have sharpened their wits

by trying to answer two questions which are not guaranteed to be easily answered.

Does the hypothesis that Gf'(x) = 0 when a < x < Is imply that there is a