Calculus Word Problem

Question

A manufacturer produces bolts of a fabric with a fixed width. A 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).

Find R′(30), given f(30)=15000, and f′(30)=−550. (What does this mean?)

R′(30)= 

Answer 1

The manufacturer is selling his fabric at $30/yard, and he is wondering what will

happen to his revenue if he increased the price.

We are given "f(30) = 15000";

it means that currently he is selling 15,000 yards

(per some time period not given to us, but we do not need it.)

As expected, increasing the price will reduce the amount of fabric sold.

Reduction in price means that the derivative f'(p) has negative value.

And we are told that f'(30) = -550, which means that increasing the price by 1 penny

to $30.01/yard would let him sell 550×0.01 fewer yards.

That is f(30.01) ≈ 14994.5.

Selling less fabric does not mean that his revenue will go down, as he gets more money for each yard.

His revenue is calculated by multiplying the price p and yardage sold f(p), i.e.,

R(p) = pf(p).

The derivative R'(p) will give us the change in revenue.

By the product rule for derivatives

R'(p) = f(p) + pf'(p)

At the current price of $30 the change in revenue will be

R'(30) = f(30) + 30×f'(30) = 15000 + 30×(-550) = -1500

R'(30) being negative means that his revenue will decrease, namely by $1,500.