The total profit (in dollars) from the sale of x DVDs is...

The total profit (in dollars) from the sale of $x$ DVDs is
$$
P(x)=5 x-0.005 x^{2}-450 \quad 0 \leq x \leq 1,000
$$
Evaluate the marginal profit at the given values of $x$, and interpret the results.
(A) $x=450$
(B) $x=750$

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Key Concepts

Interpretation of Marginal Analysis

Interpreting marginal analysis involves looking at the value of the derivative (marginal profit) to assess whether producing or selling an extra unit will likely increase overall profit. A positive marginal profit suggests that profit will likely increase, while a negative value indicates that additional sales could reduce overall profit.

Marginal Profit

Marginal profit is obtained by evaluating the derivative of the profit function at a specific point. It indicates the approximate change in total profit when one additional unit is sold, serving as a key tool to optimize production and sales decisions.

Profit Function

A profit function describes the relationship between the number of units sold and the total profit earned. It is a mathematical model used in economics and business to determine how profit changes as sales volume varies, and it can include fixed costs, variable costs, and other financial considerations.

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its independent variable. In the context of profit functions, it tells us how profit changes for a very small change in the number of units sold, which is crucial for understanding business performance.

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