91Ó°ÊÓ

In mathematical economics and particularly in production, a cost function represents the relationship between the cost incurred and the level of output produced. For the given exercise, Dean's furniture factory's cost function is expressed as a polynomial: \[ C(x) = 0.4490x - 0.01563x^2 + 0.000185x^3 \]This function captures how expenses change in response to different quantities of production, represented by \(x\). Each term in the polynomial contributes differently to the total cost:

• The linear term \(0.4490x\) suggests a direct proportionality between cost and output.
• The quadratic term \(-0.01563x^2\) adjusts the linear relationship, bringing in effects of decreasing or increasing returns as production scales.
• The cubic term \(0.000185x^3\) adds complexity to the cost structure, allowing for non-linear behavior as output changes.

The goal is to understand how the overall cost adjusts with output changes, which can be further analyzed through derivatives.

Derivatives play a crucial role in understanding the rate of change of functions, such as the cost function in our exercise. In calculus, the derivative of a function reflects the rate at which the function value changes with respect to its input. For cost functions, this is known as the marginal cost.To find the marginal cost, we calculate the derivative of the cost function, \(C(x)\). Using the power rule of differentiation:\[ C'(x) = \frac{d}{dx}(0.4490x) - \frac{d}{dx}(0.01563x^2) + \frac{d}{dx}(0.000185x^3) \]Applying the power rule gives us:\[ C'(x) = 0.4490 - 0.03126x + 0.000555x^2 \]Here, \(C'(x)\) is the expression for the marginal cost, indicating how the cost changes with the production of an additional unit of the product. It is through this derivative that we can evaluate specific marginal costs at different production levels, such as at \(x = 10\), \(x = 30\), and \(x = 40\).

A polynomial function consists of terms that are sums of constants and variables raised to non-negative integer powers. The cost function in this problem, \(C(x) = 0.4490x - 0.01563x^2 + 0.000185x^3\), is a polynomial of degree 3, the highest power of \(x\) in the expression.Polynomial functions are widely used in cost modeling because:

• They can approximate a variety of shapes needed to model complex cost behaviors.
• The constants in polynomial functions can represent fixed and variable costs, influencing the curve's shape.
• The interaction between different polynomial terms can provide a nuanced model of cost structure.

Understanding polynomial behavior through their graphs, slopes, and concavities helps in making strategic production decisions. The exercise shows how different parts of a polynomial function contribute to the overall cost function and the complexity inherent in cost modeling.

Graphing the marginal cost function provides a visual understanding of how costs change with production levels. The marginal cost function obtained by differentiating the original cost function is:\[ C'(x) = 0.4490 - 0.03126x + 0.000555x^2 \]To graph this function, the x-axis should represent the output level \(x\), ranging from 0 to 50, while the y-axis shows the marginal cost within the range of 0 to 0.3 as initially desired ([0,0.3]).When graphing:

• You will notice how the slope changes, indicating varying marginal costs at different production levels.
• The graph can reveal points of inflection where costs might shift from increasing to decreasing rates as production scales.
• By analyzing this graph, businesses can identify optimal production levels and potential inefficiencies.

Ultimately, visualizing marginal cost highlights crucial insights into production costs and aids in making informed operational decisions.