Problem 47 - Brief Applied Calculus

Question

BUSINESS: Software Costs Businesses can buy multiple licenses for PowerZip datacompression software at a total cost of approximately \(C(x)=24 x^{2 / 3}\) dollars for \(x\) licenses. Find the derivative of this cost function at: a. \(x=8\) and interpret your answer. b. \(x=64\) and interpret your answer.

Step-by-Step Solution

Answer

At \( x=8 \), cost increases by $8 per license; at \( x=64 \), it increases by $4 per license.

Step 1: Understand the Cost Function

We are given the cost function for purchasing licenses, which is expressed as \( C(x) = 24x^{2/3} \). This function tells us how the total cost changes as the number of licenses \( x \) changes.

Step 2: Differentiate the Cost Function

To find the derivative of the cost function \( C(x) = 24x^{2/3} \), we'll use the power rule for differentiation. The power rule states that if \( f(x) = ax^n \), then \( f'(x) = nax^{n-1} \). Applying this rule, we get:\[ C'(x) = \frac{2}{3} \times 24x^{(2/3)-1} = 16x^{-1/3} \]

Step 3: Evaluate the Derivative at \( x = 8 \)

Substitute \( x = 8 \) into the derivative to find the rate of change of cost at 8 licenses:\[ C'(8) = 16(8)^{-1/3} = 16 \times \frac{1}{2} = 8 \]This means that at \( x = 8 \) licenses, the cost is increasing at a rate of 8 dollars per additional license.

Step 4: Evaluate the Derivative at \( x = 64 \)

Substitute \( x = 64 \) into the derivative to find the rate of change of cost at 64 licenses:\[ C'(64) = 16(64)^{-1/3} = 16 \times \frac{1}{4} = 4 \]This means that at \( x = 64 \) licenses, the cost is increasing at a rate of 4 dollars per additional license.

Step 5: Interpret the Result

For \( x = 8 \), the cost of each additional license is increasing at 8 dollars. For \( x = 64 \), the cost increase per additional license is lower at 4 dollars, indicating a decreasing marginal cost as more licenses are purchased.

Key Concepts

Cost Function

The cost function in business economics is a formula used to determine the total cost incurred by a business for the production or procurement of a certain number of goods or services. It illustrates how costs change with changes in the level of activity or output. In the given exercise, the cost function is expressed as \( C(x) = 24x^{2/3} \), where \( x \) represents the number of software licenses purchased. This function helps businesses understand their expenditure patterns by relating the number of licenses to their total cost.

A cost function is essential for businesses to develop effective pricing strategies and budget plans, as it provides insights into how costs behave when the quantity of goods or services changes. Businesses often aim to optimize their cost function to ensure profitability and competitiveness. By investigating the behavior of this function, a company can also identify cost efficiencies or potential cost savings as the number of licenses changes.

Marginal Cost

Marginal cost refers to the additional cost incurred when producing or purchasing one more unit of a good or service. In calculus, it is represented as the derivative of the cost function because it measures the rate of change in total cost with respect to the change in quantity. In the exercise provided, the marginal cost is found by differentiating the cost function \( C(x) \) using the power rule.The calculated derivative \( C'(x) = 16x^{-1/3} \) provides the marginal cost at any given \( x \). Evaluations at \( x = 8 \) and \( x = 64 \) show marginal costs of 8 and 4 dollars respectively. This implies that buying additional licenses becomes cheaper as more licenses are purchased, demonstrating the concept of economies of scale. Understanding marginal cost aids businesses in making informed decisions about scaling production or procurement.

Power Rule for Differentiation

The power rule is a fundamental technique in calculus used for differentiating functions of the form \( ax^n \). It states that the derivative \( f'(x) \) of the function \( f(x) = ax^n \) is \( f'(x) = nax^{n-1} \). In simple terms, you multiply the power by the coefficient, reduce the power by one, and rewrite the term.In the example \( C(x) = 24x^{2/3} \), applying the power rule allows us to find \( C'(x) = \frac{2}{3} \times 24x^{(2/3)-1} = 16x^{-1/3} \). This step is crucial because it translates the cost function into a marginal cost function, showing how costs change as the number of licenses changes. By understanding the power rule, students can efficiently differentiate a wide variety of algebraic functions, which is extremely useful in both academic and real-world economic analysis.