A manufacturer produces bolts of a fabric with a ﬁxed width. The quantity q of this fabric (measured in metres) that is sold is a function of the selling price p (in dollars per metre), that is q = f(p) for some function f. Then the total revenue earned with selling price p is R(p) = pf(p).

(a) What does it mean to say that f(20) = 10000 and f′(20) = −350?

(b) Assuming these values for f(20) and f′(20), ﬁnd R′(20) and interpret your answer.

a) The quantity q, in other words metres of fabric, is 10000m when the price is \$20 per metre.

f'(20) = -350 means that at the point where the price is \$20/m the quantity is decreasing at the rate of 350 m per unit price change. In other words, if the unit price decreases to, say, \$19/m, then the quantity increases. If the unit price increases then the quantity decreases.

b) The revenue is how much the manufacturer will get from the sale of q metres of fabric when the cost to the customer is \$20/m. 20 * 10000=200000 is the revenue from the sale. R'(20)=20f'(20)=20 * -350=-7000.

answered Sep 16, 2016 by Top Rated User (415,140 points)