Page 49 - Standard Handbook Of Petroleum & Natural Gas Engineering

38 Mathematics

and xl, xp, x3, and x4 are critical values of x. A and C are maxima, B is a
minimum, and D is neither. D, F, G, and H are points of inflection where the
slope is minimum or maximum. In special cases, such as E, maxima or minima
may occur where f'(x) is undefined or infinite.
The partial derivatives f, = 0, f, = 0, f, < 0 (or 0), f,, < 0 (or > 0) determine
the minima (or maxima) for a function of two variables f(x,y).
The absolute maximum (or minimum) of f(x) at x = a exists if f(x) 5 f(a) (or
f(x) 2 f(a)) for all x in the domain of the function and need not be a relative
maximum or minimum. If a function is defined and continuous on a closed
interval, it will always have an absolute minimum and an absolute maximum,
and they will be found either at a relative minimum and a relative maximum
or at the endpoints of the interval.

Differentials

If y = f(x) and Ax and Ay are the increments of x and y, respectively, since
y + Ay = f(x + Ax), then

Ay = f(x + AX) - f(X)
As Ax approaches its limit 0 and (since x is the independent variable) dx = Ax

and
dy 5 Ay

By defining dy and dx separately, it is now possible to write

dY
- f'(x)
=
dx
dy = f' = (x)dx
Differentials of higher orders are of little significance unless dx is a constant,
in which case the first, second, third, etc. differentials approximate the first,
second, third, etc. differences and may be used in constructing difference tables
(see "Algebra").
In functions of two or more variables, where f(x, y, . . .) = 0, if dx, dy, . . .
are assigned to the independent variables x, y, . . ., the differential du is given
by differentiating term by term or by taking
du=fx*dx+fy*dy+. . .

If x, y, . . . are functions of t, then
_- dx dY
du
-(f,)z+(f .
dt )-+ . .
dt