334 Functions, graphs, and numbers
people are prone to purchase only those that are required no matter what they
cost, and being relatively steep (or elastic) for water, since people require only
enough to drink but like to wash everything and water their gardens when water
is cheap. Economists and others construct and study hypothetical demand
curves for pleasure and for business. It is usually supposd that p is a positive
decreasing differentiable function of x. When x units are sold at p(x) dollars
per unit, the total revenue R(x) is the product of x and p(x). Thus R(x) = xp(x).
When x units are sold, the profit P(x) is R(x) - C(x), where C(x) is the total
cost of producing and selling the x units. Thus
(1) P(x) = R(x) - C(x),
this being one way of saying that profit is obtained by subtracting expenses from
income. For better or for worse, economists use special terminology in studies
of price, revenue, cost, and profit. The numbers p'(x), R'(x), C'(x), and P'(x),
these being derivatives at x, are called the marginal price, the marginal revenue,
the marginal cost, and the marginal profit. This terminology is (or is thought to
be) appropriate because if we are producing and selling x units and we know p(x),
R(x), C(x), and P(x), then a shift to x + Ax, p(x + ax), R(x + Ax), C(x + Ax),
and P(x + Ax) is "marginal" when Ox is near zero and, for example, the number
which [P(x + Ox) - P(x)]/dx approximates for marginal shifts is a marginal
profit. As is easily imagined, knowledge of functions, limits, and derivatives
is helpful when these things are being studied. Differentiating (1) gives the
formula
(2) P'(x) = R'(x) - C'(x),
which says that the marginal profit is equal to the marginal revenue minus the
marginal cost. When, as frequently happens, P(x) is a maximum when P'(x) = 0
and there is just one x for which P'(x) = 0, we obtain the following rule for maxi-
mizing profits: choose the x for which the marginal cost is equal to the marginal
revenue. When equations of demand curves and cost curves are given, we can
determine the x that maximizes profits. Our course in analytic geometry and
calculus is considered to be a prerequisite for extensive study of economics
because it prepares as to understand definitions, work out formulas, solve prob-
lems, and attain over-all comprehension of the subject. In fact, knowledge of
the mean-value theorem is not superfluous. The formula
(3) P(x + 1) - P(x) =
P(x } l)
I
- P(x) = P'(x*),
in which x* is an appropriate number between x and x + 1, can help us under-
stand the antics of elementary books that alternately use P(x + 1) - P(x) and
the slope of the graph of P for the marginal profit.
5.6 Sequences, series, and decimals
always remain quite shaky until we obtain precise information about the
possibility of approximating and "representing" numbers by decimals.
Moreover, we should have some solid information about this "repre-
senting" business. We know what we mean when we say that lawyers