5128_Ch03_pp098-184

134 Chapter 3 Derivatives
0
y (dollars)
Slope
marginal cost
x
(tons/week)
x h
y c(x)
x
In a manufacturing operation, the cost of production cx is a function of x, the number
of units produced. The marginal cost of production is the rate of change of cost with respect
to the level of production, so it is dcdx.
Suppose cx represents the dollars needed to produce x tons of steel in one week. It
costs more to produce x h tons per week, and the cost difference divided by h is the
average cost of producing each additional ton.
cx h cx
h
{
the average cost of each of the
additional h tons produced
Figure 3.30 Weekly steel production:
cx is the cost of producing x tons per
week. The cost of producing an additional
h tons per week is cx h cx.
y
x 1
⎧
⎪
⎨
⎪
⎩
c
y c(x)
dc—
dx
The limit of this ratio as h→0 is the marginal cost of producing more steel per week
when the current production is x tons (Figure 3.30).
d c
lim cx h cx
marginal cost of production
dx
h→0 h
Sometimes the marginal cost of production is loosely defined to be the extra cost of producing
one more unit,
c
cx 1 cx
,
x 1
which is approximated by the value of dcdx at x. This approximation is acceptable if the
slope of c does not change quickly near x, for then the difference quotient is close to its limit
dcdx even if x 1 (Figure 3.31). The approximation works best for large values of x.
0 x
x + 1
Figure 3.31 Because dcdx is the slope
of the tangent at x, the marginal cost dcdx
approximates the extra cost c of producing
x 1 more unit.
x
EXAMPLE 7 Marginal Cost and Marginal Revenue
Suppose it costs
cx x 3 6x 2 15x
dollars to produce x radiators when 8 to 10 radiators are produced, and that
rx x 3 3x 2 12x
gives the dollar revenue from selling x radiators. Your shop currently produces
10 radiators a day. Find the marginal cost and marginal revenue.
SOLUTION
The marginal cost of producing one more radiator a day when 10 are being produced
is c10.
d
cx x d x
3 6x 2 15x 3x 2 12x 15
The marginal revenue is
so,
c10 3100 1210 15 195 dollars
d
rx x d x
3 3x 2 12x 3x 2 6x 12,
r10 3100 610 12 252 dollars.
Now try Exercises 27 and 28.