EXAMPLE 6 A manufacturer produces bolts of a fabric with a fixed width. The cost

Question

EXAMPLE 6 A manufacturer produces bolts of a fabric with a fixed width. The cost of producing x yards of this fabric is

C = f(x)

dollars.(a) What is the meaning of the derivative

f'(x)?

What are its units?
(b) In practical terms, what does it mean to say that

f'(3000) = 10?

(c) Which do you think is greater,

f'(300)

or

f'(600)?

What about

f'(4000)?

SOLUTION
(a) The derivative

f'(x)

is the instantaneous rate of change of C with respect to x; that is,

f'(x)

means the rate of change of the production cost with respect to the number of yards produced. (Economists call this rate of change the marginal cost.) Becausef'(x) = lim x 0

the units for

f'(x)

are the same as the units for the difference quotient

C/x.

Since

C

is measured in dollars and

x

in yards, it follows that the units for

f'(x)

are  ---Select--- dollars yards per  ---Select--- dollar yard .

(b) The statement that

f'(3000) = 10

means that, after  yards of fabric have been manufactured, the rate at which production cost is increasing is $  per yard. (When

x = 3000,

C is increasing 10 times as fast as x.)

Since

x = 1

is small compared with

x = 3000,

we could use the approximationf'(3000)

=

= Cand say that the cost of manufacturing the 3000th yard (or the 3001st) is about $  .

(c) The rate at which the production cost is increasing (per yard) is probably lower when

x =

than when

x =

(the cost of making the 600th yard is less than the cost of the 300th yard) because of economies of scale. (The manufacturer makes more efficient use of the fixed costs of production.) So

f'(  ) > f'(  ).

But, as production expands, the resulting large-scale operation might become inefficient and there might be overtime costs. Thus it is possible that the rate of increase of costs will eventually start to rise. So it may happen that

f'(4000) > f'(600).

Explanation / Answer

SOLUTION
(a) The derivative

f '(x)

is the instantaneous rate of change of C with respect to x; that is,

f '(x)

means the rate of change of the production cost with respect to the number of yards produced. (Economists call this rate of change the marginal cost.) Becausef '(x) = lim x 0

the units for

f '(x)

are the same as the units for the difference quotient

C/x.

Since

C

is measured in dollars and

x

in yards, it follows that the units for

f '(x)

are dollars per yards.

(b) The statement that

f '(3000) = 10

means that, after 3000 yards of fabric have been manufactured, the rate at which production cost is increasing is $10 per yard. (When x = 3000,C is increasing 10 times as fast as x.)

Since x = 1 is small compared with x = 3000,we could use the approximation

f '(3000) C/x=C/1=C and say that the cost of manufacturing the 3000th yard (or the 3001st) is about $ 10.

(c) The rate at which the production cost is increasing (per yard) is probably lower when

x = 600 than when x = 300 (the cost of making the 600th yard is less than the cost of the 300th yard) because of economies of scale. (The manufacturer makes more efficient use of the fixed costs of production.) So

f '(300 ) > f '(600 ).

But, as production expands, the resulting large-scale operation might become inefficient and there might be overtime costs. Thus it is possible that the rate of increase of costs will eventually start to rise. So it may happen that

f '(4000) > f '(600).

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