5128_Ch03_pp098-184
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134 Chapter 3 Derivatives<br />
0<br />
y (dollars)<br />
Slope <br />
marginal cost<br />
x<br />
(tons/week)<br />
x h<br />
y c(x)<br />
x<br />
In a manufacturing operation, the cost of production cx is a function of x, the number<br />
of units produced. The marginal cost of production is the rate of change of cost with respect<br />
to the level of production, so it is dcdx.<br />
Suppose cx represents the dollars needed to produce x tons of steel in one week. It<br />
costs more to produce x h tons per week, and the cost difference divided by h is the<br />
average cost of producing each additional ton.<br />
cx h cx<br />
h<br />
<br />
{<br />
the average cost of each of the<br />
additional h tons produced<br />
Figure 3.30 Weekly steel production:<br />
cx is the cost of producing x tons per<br />
week. The cost of producing an additional<br />
h tons per week is cx h cx.<br />
y<br />
x 1<br />
⎧<br />
⎪<br />
⎨<br />
⎪<br />
⎩<br />
c<br />
y c(x)<br />
dc—<br />
dx<br />
The limit of this ratio as h→0 is the marginal cost of producing more steel per week<br />
when the current production is x tons (Figure 3.30).<br />
d c<br />
lim cx h cx<br />
marginal cost of production<br />
dx<br />
h→0 h<br />
Sometimes the marginal cost of production is loosely defined to be the extra cost of producing<br />
one more unit,<br />
c<br />
cx 1 cx<br />
,<br />
x 1<br />
which is approximated by the value of dcdx at x. This approximation is acceptable if the<br />
slope of c does not change quickly near x, for then the difference quotient is close to its limit<br />
dcdx even if x 1 (Figure 3.31). The approximation works best for large values of x.<br />
0 x<br />
x + 1<br />
Figure 3.31 Because dcdx is the slope<br />
of the tangent at x, the marginal cost dcdx<br />
approximates the extra cost c of producing<br />
x 1 more unit.<br />
x<br />
EXAMPLE 7 Marginal Cost and Marginal Revenue<br />
Suppose it costs<br />
cx x 3 6x 2 15x<br />
dollars to produce x radiators when 8 to 10 radiators are produced, and that<br />
rx x 3 3x 2 12x<br />
gives the dollar revenue from selling x radiators. Your shop currently produces<br />
10 radiators a day. Find the marginal cost and marginal revenue.<br />
SOLUTION<br />
The marginal cost of producing one more radiator a day when 10 are being produced<br />
is c10.<br />
d<br />
cx x d x<br />
3 6x 2 15x 3x 2 12x 15<br />
The marginal revenue is<br />
so,<br />
c10 3100 1210 15 195 dollars<br />
d<br />
rx x d x<br />
3 3x 2 12x 3x 2 6x 12,<br />
r10 3100 610 12 252 dollars.<br />
Now try Exercises 27 and 28.