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 '(2000) = 10?
(c) Which do you think is greater,
f '(200) or f '(600)? What about f '(4000)?
Answer:
A) f'(x) means the rate of change of the cost of production with respect to the number of yards that were produced.
It's unit is $/yard
B) f '(2000) = 10 means that after production of 2000 yards of fabrics, the cost of production will be increasing at a rate of $10 per yard.
C) f' (200) > f'(600)
f'(4000) > f'(600)
Step-by-step explanation:
A) Since the cost of producing x yards of this fabric is C = f(x) dollars. f'(x) simply means the rate of change of C with respect to x.
In clearer terms, the derivative f'(x) means the rate of change of the cost of production with respect to the number of yards that were produced.
It's unit is $/yard
B) To say f '(2000) = 10 means that after production of 2000 yards of fabrics, the cost of production will be increasing at a rate of $10 per yard.
C) f '(200) is greater than f'(600) because the cost of making the 600th yard will be less than the cost of the 200th yard. Reason for this is due to economics of scale as the manufacturer will certainly make more efficient use of fixed costs of production.
As more units are produced, there may likely be more overtime costs and thus the rate of increase cost of production will likely increase faster.
