Product Rule question?

Question

I'm don't know how to solve this problem. Could someone assist me?

A manufacturing produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q=f(p). Then the total revenue earned with selling price is R(p)=pf(p).

a. What does it mean to say that f(20)=10,000 and f ' (20) = -350?

b. Assuming the values in part a, find R ' (20) and interpret your answer.

Answer 1

f(20)=10,000 means that when the price is $20/yd, the manufacturer sells 10,000 yards (per day, per week, the problem doesn't say; but it's 10,000 yards per some unit of time).

f'(20) = -350 technically means that the rate of change of sales with price is -350 yards/dollar. Informally, it means that, when the price per yard is $20, an increase in price of $1 will result in sales dropping by 350 yards.

I'll let you compute R'(p) and find R'(20). If it's positive, that means that revenue would still increase if the price is raised a bit (despite the resulting loss of sales from the price increase); and you can interpret it quantitatively as I did above for f(p) (that is, if R'(20) = 1000, say, then revenue would increase by about $1000 if the price were increased by $1). If it's negative, that means that an increase in price would result in loss of revenue (which means $20/yd is too expensive, from a revenue perspective), and the numerical value of R'(20) tells you about how much revenue would drop if the price were increased by $1.