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'(2,000) - 147 (c) Which do you think is greater, f'(50) or f'(600)? What about f(5,000)? 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.) Because F(x) - lim đC o Ax the units for f'(x) are the same as the units for the difference quotlent AC/Ax. Since AC is measured in dollars and Ax in yards, It follows that the units for f'(x) are --Select-- v per --Select- v. (b) The statement that f'(2,000) = 14 means that, after yards of fabric have been manufactured, the rate at which production cost is increasing is $ per yard. (When x = 2,000, C is increasing times as fast as x .) AC AC Since Ax - 1 is small compared with x - 2,000, we could use the approximation f (2,000) z Ax - AC and say that the cost of manufacturing the 2,000th yard (or the 2,001st) is about $ (c) The rate at which the production cost is increasing (per yard) is probably lower when x = fixed costs of production.) So we get the following. than when x = (the cost of making the 600th yard is less than the cost of the 50th yard) because of economies of scale. (The manufacturer makes more efficient use of the ) > ( 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 we get the following. f(5,000) > f(600)