Maximizing Revenue The quantity demanded each month of the...

Maximizing Revenue The quantity demanded each month of the Sicard sports watch is related to the unit price by the equation $$p=\frac{50}{0.01 x^{2}+1} \quad(0 \leq x \leq 20)$$ where $p$ is measured in dollars and $x$ is measured in units of a thousand. To yield a maximum revenue, how many watches must be sold?

Maximizing Revenue The quantity demanded each month of the Sicard sports watch is related to the unit price by the equation
$$p=\frac{50}{0.01 x^{2}+1} \quad(0 \leq x \leq 20)$$
where $p$ is measured in dollars and $x$ is measured in units of a thousand. To yield a maximum revenue, how many watches must be sold?

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Key Concepts

Revenue Function

The revenue function represents the total income generated from sales and is typically defined as the product of the price per unit and the number of units sold. Understanding how to build and analyze a revenue function is crucial for determining the conditions that lead to maximum earnings.

Optimization

Optimization involves finding the maximum or minimum values of a function. In the context of revenue, this typically means using calculus to determine the quantity of product sales that will yield the highest revenue given a revenue function.

Demand Function

A demand function expresses the relationship between the price of a product and the quantity demanded by consumers. Recognizing this relationship allows one to connect pricing strategies with sales volume, which in turn affects revenue.

Differentiation

Differentiation is a core concept in calculus that computes the rate of change of a function with respect to its variables. It is used to find critical points by setting the derivative equal to zero, which is essential in determining where a revenue function reaches its maximum value.

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