91Ó°ÊÓ

Partial derivatives are a cornerstone in multivariable calculus, especially when dealing with functions like the profit function in this exercise. When we have a function of multiple variables, such as the profit function \( P(x, y) \), which depends on both \( x \) (DVD players) and \( y \) (CD players), partial derivatives allow us to see how the function changes as one variable changes, while keeping the other constant. This is crucial for calculating marginal values in economics and business.

For example, to find the marginal profit function for DVD players, we take the partial derivative of \( P(x, y) \) with respect to \( x \). This involves differentiating \( P(x, y) \) as if \( y \) were a constant. Similarly, to find the marginal profit function for CD players, we take the partial derivative with respect to \( y \).

• Application: Helps determine the rate of change of profit with respect to each type of product independently.
• Result: For DVD players, we found the partial derivative \( \frac{\partial P}{\partial x} = 6x - 4y + 80 \). For CD players, it is \( \frac{\partial P}{\partial y} = -4x + 8y + 100 \).

In economics, a profit function represents the total profit a company makes from selling goods or services. It is expressed as a function of quantities of goods, costs of production, and sometimes other economic factors. In our exercise, the profit function \( P(x, y) = 3x^2 - 4xy + 4y^2 + 80x + 100y + 200 \) encapsulates the relationship between producing and selling \( x \) DVD players and \( y \) CD players.

Understanding this function is crucial for businesses to strategize and make informed production decisions. The function usually includes:

• Revenue components: Gains from selling products.
• Cost components: Expenses associated with production.
• Constant term: Fixed costs or baseline profit.
  

In this case, the function includes terms for both variables \(x\) and \(y\), allowing us to calculate profit based on any combination of DVD and CD players produced daily.

Marginal analysis is a technique often used in economics to assess the impact of a unit change in the decision variable. In our context, it helps determine how much additional profit can be made from producing one more unit of a product.

This exercise uses marginal analysis to find the marginal profit, which is the additional profit from making one more DVD player or CD player. After computing the partial derivatives, we substitute specific values to evaluate at given production levels.

• DVD Players at \(x = 200\), \(y = 300\): The marginal profit was \(80\) dollars, meaning that each additional DVD player contributes \\(80 of profit under these conditions.
• CD Players at \(x = 200\), \(y = 100\): The marginal profit was \(100\) dollars, meaning that each additional CD player contributes \\)100 of profit under these conditions.

Thus, marginal analysis provides valuable insights into optimizing production and maximizing profits based on current operational levels.