How can I use calculus to answer this question. I need it for my project.?

Question

In a manufacturing and selling x units of a certain commodity,the price function,p,and the cost function C are given by:p(x)=5-0.002x C(x)=3+1.10x.Determine:1.the expressions for the revenue function R(x) and the profit function P(x). 2.the expressions for the marginal revenue,marginal cost and marginal profit. 3. the production level that will produce the maximum profit.

Answer 1

1. R(x) = xp(x) = x(5 - 0.002x) = 5x - 0.002x^2
P(x) = R(x) - C(x) = 5x - 0.002x^2 - (3 + 1.10x) = 3.90x - 0.002x^2 -3

2. The marginal revenue is R'(x) = 5 - 0.004x
The marginal cost is C'(x) = 1.10
The marginal profit is P'(x) = 3.90 - 0.004x

3. Maximize P(x). P'(x) = 3.90 - 0.004x = 0 and x = 3.90/0.004 = 975. At this production level, price is 3.05, cost is 1075.50, revenue is 2973.75, and profit is 1898.25. Marginal revenue is 1.10, as is marginal cost, and marginal profit is zero.

As a passing note, the inverse of the price function p(x) is the demand function x(p), which in this case is 2500 - 500p.

Answer 2

Revenue is the amount of money you generate through sales.

Cost is the amount of money you put in to make production possible.

Profit is the net amount your earn.

Profit , Cost, and Revenue are related by the following formula

P(x) = R(x)-C(x)

Compare the formula to the definitions to see if the formula
makes sense to you.

The word Marginal refers to a small group or amount.
In the context of manufacturing, Marginal refers to the effects of small changes in production. When you think of marginal, think derivative. For instance to find a small change in cost
(marginal cost) you would take the derivative of the cost function.

To find the maximum profit, think about what you would do
in general to find the maximum point of a curve. Maximum
and minimum points occur where the slope of the tanget
equals zero.