Problem 48 - Brief Applied Calculus

Question

BUSINESS: Software Costs Media companies can buy multiple licenses for AudioTime audiorecording software at a total cost of approximately \(C(x)=168 x^{5 / 6}\) dollars for \(x\) licenses. Find the derivative of this cost function at: a. \(x=1\) and interpret your answer. b. \(x=64\) and interpret your answer.

Step-by-Step Solution

Answer

At \( x = 1 \), cost increases by $140 per license; at \( x = 64 \), by $70 per license.

Step 1: Identify the Cost Function

Given the cost function \( C(x) = 168 x^{5/6} \), we need to find its derivative in order to determine the rate of change of the cost with respect to the number of licenses \( x \).

Step 2: Find the Derivative of the Cost Function

To find the derivative, we use the power rule. The derivative of \( x^n \) is \( nx^{n-1} \). Therefore, \( C'(x) = 168 \cdot \frac{5}{6} x^{\frac{5}{6} - 1} \). Simplifying, \( C'(x) = 140 x^{-\frac{1}{6}} \).

Step 3: Evaluate the Derivative at x = 1

Substitute \( x = 1 \) into the derivative: \( C'(1) = 140 \cdot 1^{-1/6} = 140 \). This means that at \( x = 1 \), the cost of licenses is increasing at a rate of $140 per additional license.

Step 4: Evaluate the Derivative at x = 64

Substitute \( x = 64 \) into the derivative: \( C'(64) = 140 \cdot 64^{-1/6} \). Since \( 64^{-1/6} = \frac{1}{2} \), then \( C'(64) = 140 \cdot \frac{1}{2} = 70 \). This means that at \( x = 64 \), the cost per additional license is increasing at a rate of $70.

Step 5: Interpret the Results

At \( x = 1 \), the marginal cost, or the cost of purchasing one more license, is \(140. This suggests steep initial costs. At \( x = 64 \), the marginal cost drops to \)70, indicating a reduced cost per additional license as more licenses are bought.

Key Concepts

Cost Function

A cost function is a mathematical relationship expressing the cost of producing a certain number of items or licenses. It's often denoted as \( C(x) \), where \( C \) stands for cost and \( x \) represents the number of items produced.
For companies, understanding the cost function is crucial because it helps in forecasting expenses. By knowing how costs change as production scales, companies can plan their budgets effectively.
In the exercise, the cost function is given as \( C(x) = 168x^{5/6} \). This function estimates the total cost of acquiring \( x \) licenses of a software product. The fraction in the exponent, \( 5/6 \), indicates that the relationship between the number of licenses and cost isn't linear. Instead, the cost increases at a diminishing rate.
This kind of function is useful in scenarios like bulk purchasing, where the cost per item might decrease as the quantity increases.

Marginal Cost

Marginal cost is a key concept in economics, particularly in the analysis of cost functions. It represents the change in total cost when producing one additional unit. In essence, it tells us how much more it costs to buy or produce one more item.
The derivative of a cost function reveals the marginal cost, as it measures the rate of change in cost with respect to the number of items. For the exercise given, the derivative \( C'(x) \) was found to be \( 140x^{-1/6} \). This tells us how the cost changes as \( x \) (the number of licenses) increases.
When evaluating the derivative at specific values like \( x=1 \), we find that the cost to add another license is \(140. Similarly, at \( x=64 \), the cost decreases to \)70 per additional license. This decrease in marginal cost illustrates economies of scale: the more you buy, the less each additional unit costs.

Power Rule

The power rule is a basic, yet immensely useful rule in calculus for finding derivatives. It relates specifically to functions that have a variable raised to a power. The rule states that the derivative of \( x^n \) is \( nx^{n-1} \).
In the context of our cost function \( C(x) = 168x^{5/6} \), applying the power rule enabled us to find its derivative, \( C'(x) = 140x^{-1/6} \).
Here's how it works:

• Take the exponent of \( x \), which is \( 5/6 \), and multiply it by the coefficient, which is 168. This gives \( 140 \).
• Then subtract 1 from the exponent, transforming \( 5/6 \) into \( -1/6 \).
• The final derivative is \( 140x^{-1/6} \).

This complete process shows how we use the power rule to not only simplify finding derivatives but also interpret important economic concepts like the marginal cost in practical business problems.