Marginal cost, Revenue, and Profit for Producing LCD TVs...

Marginal cost, Revenue, and Profit for Producing LCD TVs Pulsar manufactures a series of 20 -in. flat-tube LCD televisions. The quantity $x$ of these sets demanded each week is related to the wholesale unit price $p$ by the equation $$ p=-0.006 x+180 $$ The weekly total cost incurred by Pulsar for producing $x$ sets is $$ \begin{array}{l} \text { The } C(x)=0.000002 x^{3}-0.02 x^{2}+120 x+60,000 \\ \text { ( } x=120 x+10000000000000000000000000000000000000000000 \end{array} $$ dollars. Answer the questions in Exercise 13 for these data.

Marginal cost, Revenue, and Profit for Producing LCD TVs Pulsar manufactures a series of 20 -in. flat-tube LCD televisions. The quantity $x$ of these sets demanded each week is related to the wholesale unit price $p$ by the equation
$$
p=-0.006 x+180
$$
The weekly total cost incurred by Pulsar for producing $x$ sets is
$$
\begin{array}{l}
\text { The } C(x)=0.000002 x^{3}-0.02 x^{2}+120 x+60,000 \\
\text { ( } x=120 x+10000000000000000000000000000000000000000000
\end{array}
$$
dollars. Answer the questions in Exercise 13 for these data.

Key Concepts

Profit Function

The profit function is defined as the difference between total revenue and total cost. It provides a measure of the financial gain of a production process. Analyzing the profit function, along with its derivative (marginal profit), helps identify the production level that maximizes profit.

Total Revenue Function

The total revenue function is obtained by multiplying the demand function’s price by the quantity sold. It represents the total income a business earns from selling its products. Understanding the revenue function is essential for evaluating the financial viability of different production levels.

Optimization Using Derivatives

Derivatives are used to analyze and optimize functions related to cost, revenue, and profit. By taking the derivative of these functions (resulting in marginal cost, marginal revenue, and marginal profit), businesses can determine the rates at which costs and revenues change and thus identify the optimal production level that maximizes profit or minimizes cost.

Total Cost Function

The total cost function describes all the costs incurred in producing a certain quantity of a product, including fixed and variable costs. It serves as the foundation for further analysis by allowing businesses to predict costs as production changes, and it is often expressed as a polynomial function.

Demand Function

The demand function represents the relationship between the quantity of a product that consumers are willing to purchase and its price. It typically shows that as the price decreases, the quantity demanded increases, and vice versa. This function is essential for understanding market behavior and influences subsequent functions such as revenue and profit.

Marginal Cost

Marginal cost is the derivative of the total cost function with respect to the quantity produced. It measures the additional cost incurred by producing one extra unit of the product. This concept is crucial for decision-making in production, as it helps determine the optimal level of production where the cost of making an additional item is justified by the revenue it generates.

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