Minimum Average cost The cost of producing x units of a...

Minimum Average cost The cost of producing $x$ units of a product is modeled by $C=500+300 x-300 \ln x, \quad x \geq 1$
(a) Find the average cost function $\bar{C}$.
(b) Analytically find the minimum average cost. Use a graphing utility to confirm your result.

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Key Concepts

Logarithmic Functions

Logarithmic functions, such as ln(x) shown in the cost model, require special rules of differentiation. Understanding their properties is crucial when they are part of a cost function, as they affect the behavior of the function and subsequent calculations.

Differentiation

Differentiation is a key calculus tool used to compute the derivative of a function. By differentiating the average cost function and setting the derivative equal to zero, one can find the critical points that may indicate a minimum or maximum value.

Graphical Analysis

Graphical analysis involves the use of graphing utilities to visually inspect the behavior of a function. This technique confirms the results obtained from analytical methods by showing the minimum point on the plotted function.

Average Cost Function

The average cost function is obtained by dividing the total cost by the number of units produced. It provides insight into the cost incurred per unit and is fundamental in identifying efficient production levels.

Optimization

Optimization in this context refers to the process of finding the production level that minimizes the average cost. This involves mathematical techniques to determine the critical points where the function reaches its minimum value.

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