The price-demand equation and the cost function for the...

The price-demand equation and the cost function for the production of HDTVs are given, respectively, by $x=9,000-30 p \quad$ and $\quad C(x)=150,000+30 x$ where $x$ is the number of HDTVs that can be sold at a price of $\$ p$ per $T V$ and $C(x)$ is the total cost (in dollars) of produc ing $x$ TVs. (A) Express the price $p$ as a function of the demand $x$, and find the domain of this function. (B) Find the marginal cost. (C) Find the revenue function and state its domain. (D) Find the marginal revenue. (E) Find $R^{\prime}(3,000)$ and $R^{\prime}(6,000)$ and interpret these quantities. (F) Graph the cost function and the revenue function on the same coordinate system for $0 \leq x \leq 9,000$. Find the break-even points and indicate regions of loss and profit. (G) Find the profit function in terms of $x$. (H) Find the marginal profit. (I) Find $P^{\prime}(1,500)$ and $P^{\prime}(4,500)$ and interpret these quantities.

The price-demand equation and the cost function for the production of HDTVs are given, respectively, by
$x=9,000-30 p \quad$ and $\quad C(x)=150,000+30 x$
where $x$ is the number of HDTVs that can be sold at a price of $\$ p$ per $T V$ and $C(x)$ is the total cost (in dollars) of produc ing $x$ TVs.
(A) Express the price $p$ as a function of the demand $x$, and find the domain of this function.
(B) Find the marginal cost.
(C) Find the revenue function and state its domain.
(D) Find the marginal revenue.
(E) Find $R^{\prime}(3,000)$ and $R^{\prime}(6,000)$ and interpret these quantities.
(F) Graph the cost function and the revenue function on the same coordinate system for $0 \leq x \leq 9,000$. Find the break-even points and indicate regions of loss and profit.
(G) Find the profit function in terms of $x$.
(H) Find the marginal profit.
(I) Find $P^{\prime}(1,500)$ and $P^{\prime}(4,500)$ and interpret these quantities.

Key Concepts

Interpretation of Derivative Values

Evaluating the derivatives of functions like revenue or profit at specific production levels gives insight into the incremental effects of changes in output. These interpretations help businesses understand the sensitivity of revenue, cost, and profit to changes in production volume, and they are essential for making data-driven decisions.

Marginal Profit

Marginal profit is the derivative of the profit function with respect to the number of units produced, indicating the change in profit resulting from the production of one additional unit. This concept aids in determining whether increasing production will improve profitability, and in setting production targets.

Marginal Revenue

Marginal revenue represents the rate of change of the revenue function, obtained by differentiating the revenue function with respect to the number of units sold. This concept is important in assessing how additional sales contribute to overall revenue, guiding decisions about scaling production or adjusting prices.

Profit Function

The profit function is derived by subtracting the cost function from the revenue function. It provides a direct measure of the financial gain or loss associated with different production levels and helps businesses plan for optimal output that maximizes profit.

Marginal Cost

Marginal cost is a key concept in calculus and economics that describes the additional cost incurred to produce one more unit of a product. It is obtained by differentiating the total cost function with respect to the number of units produced, and it informs production decisions by indicating the cost implications of increasing output.

Domain of a Function

In functions that arise in economic models, determining the domain is crucial because it represents all possible values of the independent variable (such as the quantity sold) that yield meaningful or realistic outcomes. This concept ensures that the functions, like price, revenue, and cost, are defined in ranges that make sense within the context of production and sales.

Price-Demand Relationship

This concept revolves around understanding how the price of a product adjusts according to the demand. It often involves expressing the price as a function of the quantity sold by inverting the original demand equation, which is essential for analyzing market behavior and consumer response in microeconomics.

Revenue Function

The revenue function is central to economic analysis and is defined as the total income received from sales, typically modeled as the product of the price per unit and the number of units sold. It helps in evaluating business performance by linking pricing strategy with sales volume.

Break-even Analysis and Graphical Representation

Break-even analysis involves finding the point where total revenue equals total cost, which can be visually determined by graphing the cost and revenue functions. This concept is fundamental for assessing business viability, understanding loss and profit zones, and making informed strategic decisions.

Please give Ace some feedback