91Ó°ÊÓ

The marginal profit function is crucial in economics and calculus. It represents the additional profit earned from selling one more unit of a product. In mathematical terms, it's the derivative of the total profit function, denoted as \(\frac{d P}{d x}\). Given the function \(\frac{d P}{d x} = \frac{9000 - 3000 x}{(x^2 - 6x + 10)^2}\), we find how profit changes as production increases.
This function tells us precisely how much profit changes at a specific production level. Knowing the marginal profit helps businesses make decisions about increasing or decreasing production levels to maximize total profit.

Integration is the process of finding the antiderivative, which in our case, converts the marginal profit function back into the total profit function. Specifically, we need to determine \(P(x)\), which represents the total profit from selling \(x\) units.
The integral set up is \(P(x) = \int \frac{9000 - 3000 x}{(x^2 - 6x + 10)^2} \, dx + C\). Here, \(C\) is the constant of integration, which we will later solve using an initial condition.
Integration allows us to find the accumulated profit at any production level given the rate of change defined by the marginal profit function. It transforms the rate of change into a total quantity, which is fundamental in understanding and calculating overall profit.

Initial conditions are values provided to solve for the integration constant \(C\). They are essential because they allow us to find a specific solution to our integral.
In this problem, we are given that \(P(3) = 1500\). To apply this, we substitute \(x = 3\) into the integral solution, set it equal to 1500, and solve for \(C\).
This process ensures that our total profit function accurately reflects the initial given data, providing a precise model for total profit based on the initial condition.

Partial fraction decomposition simplifies complex rational expressions, making integration manageable. It's particularly useful when dealing with polynomial fractions, as we have in our marginal profit function.
The decomposition starts with breaking down \( \frac{9000 - 3000 x}{(x^2 - 6x + 10)^2} \) into simpler parts. We express it as \( \frac{A}{x^2 - 6x + 10} + \frac{B(x - C)}{(x^2 - 6x + 10)^2} \).
Solving for constants \(A, B,\) and \(C\) allows us to integrate each term separately, transforming a complex integral into sums of basic integrals. This method is powerful in calculus, facilitating the calculation of integrals we might otherwise find very difficult.