Question

A manufacturer 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 p is R(p) = pf(p). What does it mean to say that f(20) = 10000 and f’(20) = -350?

Answer 1

f(20)=10,000 means that for a price of $20/yard, there will be 10,000 yards of fabric sold.

f'(20)=-350 means that for for and additional dollar to $20 the price increases, the number of yards of fabric sold will decrease by 350 yards.

Explanation:

The problem gives you a function q=f(p) where p is the price in dollars per yard. f(p) is function notation, where f is the function, in this case the function that represents the amount of fabric sold and p is the price, so f(20) will return the amont of fabric sold when the price is $20/yard. And the 10,000 is the answer returned by the function so at a $20/yard price, the amount of fabric sold is 10,000 yards.

f'(p) is the derivative of the function. This one represents the slope of a line tangent to the point (20, 10 000) and in financial aplications it represents how the function changes for a change of one unit to the p-value. In this case, the f(p) function represents the demand of fabric, so f'(p) is called marginal demand. f'(20)=-350 means that when the price is $20/yard, a change of $1/yard (this is $21/yard) will cause a decrease of 350 yards sold from the original 10,000 you got when the price was $20/yard.

This is an estimation really, since the number of yards sold is taken to be written in terms of whole numbers. A derivative supposes the function is a continuous and solid curve, but in reality when talking about items, we are talking about a curve that contains only whole numbers, which aproximates to a solid continuous curve when we have many items.