Marginal cost, Revenue, and Profit for Producing LED TVs The...

Marginal cost, Revenue, and Profit for Producing LED TVs The weekly demand for the Pulsar 25 color LED television is $$ p=600-0.05 x \quad(0 \leq x \leq 12,000) $$ where $p$ denotes the wholesale unit price in dollars and $x$ denotes the quantity demanded. The weekly total cost function associated with manufacturing the Pulsar 25 is given by $$ \begin{array}{l} \text { The } C(x)=0.000002 x^{3}-0.03 x^{2}+400 x+80,000 \\ \text { ( } x=400 x+0000000000000000000000000000000000000000000000 \end{array} $$ where $C(x)$ denotes the total cost incurred in producing $x$ sets. a. Find the revenue function $R$ and the profit function $P$. b. Find the marginal cost function $C^{\prime}$, the marginal revenue function $R^{\prime},$ and the marginal profit c. Compute $C^{\prime}(2000), R^{\prime}(2000),$ and $P^{\prime}(2000),$ and interpret your results. d. Sketch the graphs of the functions $C, R,$ and $P,$ and interpret parts (b) and (c), using the graphs obtained.

Marginal cost, Revenue, and Profit for Producing LED TVs The weekly demand for the Pulsar 25 color LED television is
$$
p=600-0.05 x \quad(0 \leq x \leq 12,000)
$$
where $p$ denotes the wholesale unit price in dollars and $x$ denotes the quantity demanded. The weekly total cost function associated with manufacturing the Pulsar 25 is given by
$$
\begin{array}{l}
\text { The } C(x)=0.000002 x^{3}-0.03 x^{2}+400 x+80,000 \\
\text { ( } x=400 x+0000000000000000000000000000000000000000000000
\end{array}
$$
where $C(x)$ denotes the total cost incurred in producing $x$ sets.
a. Find the revenue function $R$ and the profit function $P$.
b. Find the marginal cost function $C^{\prime}$, the marginal revenue function $R^{\prime},$ and the marginal profit
c. Compute $C^{\prime}(2000), R^{\prime}(2000),$ and $P^{\prime}(2000),$ and interpret your results.
d. Sketch the graphs of the functions $C, R,$ and $P,$ and interpret parts (b) and (c), using the graphs obtained.

Key Concepts

Marginal Profit

Marginal profit is the derivative of the profit function with respect to quantity, often calculated as the difference between marginal revenue and marginal cost. It plays a major role in decision-making, particularly in identifying the rate at which profit changes with changes in production volume.

Graphical Analysis in Economic Functions

Graphical analysis involves plotting cost, revenue, and profit functions to visually interpret their behavior. These graphs help in identifying break-even points, profit maximization, and the relationships between production levels and corresponding financial outcomes, making them useful tools for both analysis and communication of economic concepts.

Marginal Cost

Marginal cost is the derivative of the cost function with respect to quantity, representing the additional cost incurred by producing one more unit. It is a key concept in microeconomics and production theory, helping businesses understand the impact of increasing production on overall costs.

Marginal Revenue

Marginal revenue is the derivative of the revenue function with respect to quantity, indicating the extra income generated from selling one additional unit. This concept is critical in setting production levels and determining the point where revenue maximization occurs.

Cost Function

The cost function encapsulates all the expenses incurred in producing a certain number of units. It typically includes fixed costs, which do not change with production volume, and variable costs, which vary with the quantity produced. Mastering the cost function is essential for making informed production and pricing decisions.

Profit Function

The profit function is calculated as the difference between total revenue and total cost. It provides a measure of the financial gain or loss at different levels of production and sales. This function is central to determining optimal production levels and pricing strategies in order to maximize earnings.

Revenue Function

The revenue function represents the total income generated from selling a given number of units at a specified price. Typically, it is derived by multiplying the price per unit by the quantity sold. Understanding this concept is crucial in economics and business as it directly influences profit and operational decisions.

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