Calculus: Analytic Geometry and Calculus, with Vectors

116 Functions, limits, derivatives

where n and ao, al, , anand bo, b1, .,b,a are constants, n being
a nonnegative integer. A rational (ratio-nal) function is a quotient of
polynomials, an example being the function Q for which

(3.172) Q(x) =

x+1

xs +x-12

for each x for which the denominator is different from zero. When we

define a function by a formula more or less like (3.172), we ordinarily

understand that the domain of the function is the whole set of numbers x

for which the formula actually determines a number. We must, how-

ever, recognize the fact that the function g for which g(x) = when

1 < x S 4 is different from the function h for which h(x) = when

x 0; the domains of the functions are different, and the functions are

therefore different. To clarify this point, we can recognize that a

machine which is capable of cracking only medium-sized nuts is different

from a machine that is capable of cracking nuts of all sizes.

If we have a load of coal of weight w and we toss a lump of coal on or

off the load, then the new weight will be a new number which we can call

w + Ow. Thus Ow, which may be either positive or negative, is the

difference of two weights (the new minus the old). The number Aw, read

"delta w," is a single number (not the product of two numbers A and w).

This simple notational device turns out to be unexpectedly convenient.

In physical chemistry Ap is the difference of two pressures, Av is the dif-

ference of two volumes, and At is the difference of two times. In physics,

Av is the difference of two (vector) velocities, and AY is the difference of

two potentials. In economics OP is the difference of two prices, and in

Figure 3.181 Figure 3.182

biology AP may be the difference of two populations. If, as in the

discussion involving Figure 3.181, the left member of the formula

(3.183) z = f(x,Y,O)

represents the length of the side opposite the vertex Y, then the left

member of the formula

(3.184) z+AZ =f(x+ox,Y+AY, e+AO)

represents the length of the side opposite the vertex Y in the triangle of

Figure 3.182. Of course we can and sometimes shall use shorter symbols