When a commodity is sold for x dollars per item, the number of items sold is f(x

Question

When a commodity is sold for x dollars per item, the number of items sold is f(x). The item costs 3 dollars to make. (For all of the below, the answer has been partially filled in. Do these problems on paper then simplify your answer until it can be used to fill in what is below. Leave no box blank)

(a)Express the total profit P in terms of x (the answer has been partially filled in for you). P = _______*f(x)

(b)What is the rate of change of the total profit as x changes? (Do this on paper then fill in the below. Leave neither box blank) ______*f(x) + _______*f '(x)

(c)If 3000 items sell when the price is 45 dollars and if the number of items that are sold decreases by 600 for every 1 dollar increase in price, find the rate of change of the profit when the item price is 45 dollars. _____________

Explanation / Answer

Total profit = Number of items sold * profit on one item

Profit on one item = selling price - making price = (x - 3)

Number of items sold is given by f(x)

Hence,

P = (x-3) * f(x)

We need to find dP/dx

dP/dx = derivative of (x-3) * f(x) + (x-3) *derivative of f(x)

dP/dx = 1* f(x) + (x-3)*f'(x)

We want dP/dx at x=45

f(x) = 3000

f'(x) = -600

Putting in the values,

dP/dx = 1 * 3000 - (45-3)(600) = -22200

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