Problem 11 - Applied Calculus

Question

Let \(C(q)\) represent the cost and \(R(q)\) represent the revenue, in dollars, of producing \(q\) items. (a) If \(C(50)=4300\) and \(C^{\prime}(50)=24\), estimate \(C(52)\). (b) If \(C^{\prime}(50)=24\) and \(R^{\prime}(50)=35\), approximately how much profit is earned by the \(51^{\text {st }}\) item? (c) If \(C^{\prime}(100)=38\) and \(R^{\prime}(100)=35\), should the company produce the \(101^{\text {st }}\) item? Why or why not?

Step-by-Step Solution

Answer

(a) Approximately 4348, (b) Approximately 11 dollars, (c) No, the cost exceeds revenue.

Step 1: Estimating Cost

To estimate \( C(52) \), we use the linear approximation formula: \( C(52) \approx C(50) + C'(50) \times (52 - 50) \). Substituting the given values, we have \( C(52) \approx 4300 + 24 \times 2 = 4300 + 48 = 4348 \). Thus, \( C(52) \approx 4348 \).

Step 2: Profit of the 51st Item

The profit from the \( 51^{\text{st}} \) item can be estimated by the difference in the marginal revenue and the marginal cost at \( q = 50 \). Therefore, the profit is given by \( R'(50) - C'(50) = 35 - 24 = 11 \). Thus, the profit from the 51st item is approximately \( 11 \) dollars.

Step 3: Decision on Producing the 101st Item

The decision to produce the \( 101^{\text{st}} \) item depends on whether the marginal revenue exceeds the marginal cost. At \( q = 100 \), we have \( C'(100) = 38 \) and \( R'(100) = 35 \). Since the marginal cost is greater than the marginal revenue (i.e., \( 38 > 35 \)), producing the 101st item would result in a loss. Therefore, the company should not produce the 101st item.

Key Concepts

Marginal Cost

Marginal cost is the increase in total cost that arises from producing one more unit of a good. It helps businesses understand how much it costs to scale up production by one unit. In our exercise, marginal cost is represented as \( C'(q) \), a derivative of the cost function \( C(q) \). For example, at \( q = 50 \), \( C'(50) = 24 \) implies that producing one more item will approximately add \$24 to the total cost. Understanding marginal cost is crucial for decision-making.

• It informs pricing strategy.
• Helps in budgeting future projects.
• Indicates the cost-effectiveness of increasing production.

Marginal cost varies depending on factors such as:

• Economies of scale.
• Production technology.
• Raw materials cost fluctuations.

Marginal Revenue

Marginal revenue is the additional income received from selling one more unit of a good. It is critical for companies to know how their revenue changes with each additional sale. In the exercise, marginal revenue is depicted as \( R'(q) \), the derivative of the revenue function \( R(q) \). For instance, \( R'(50) = 35 \) indicates that selling one more item adds around \$35 to the revenue.Marginal revenue is utilized to determine optimal sales levels and maximize profits.

• Helps in setting product rates and sales targets.
• Guides inventory and production planning.
• Affects marketing and promotional strategies.

Comparatively evaluating marginal revenue with marginal cost offers insights into profitability and production adjustments.

Profit Estimation

Profit estimation involves calculating the expected earnings from production and sales activities. The relationship between marginal cost and marginal revenue is fundamental to determining profit.In the example, the profit from the 51st item is estimated using the difference: \( R'(50) - C'(50) = 11 \) dollars. This indicates that selling one more item increases profit by approximately \$11. Key points in profit estimation include:

• Finding the profit-maximizing level of output.
• Understanding the cost and revenue curves.
• Evaluating the effect of scaling production on overall profitability.

Analyzing profits from incremental sales allows businesses to optimize resources and strategies for growth.

Linear Approximation

Linear approximation is a method used to estimate the value of a function around a given point using its derivative. It is like drawing a tangent line to a curve and using it to approximate values near that point.For the given exercise, linear approximation is employed to estimate \( C(52) \) as \( C(52) \approx C(50) + C'(50) \times (52 - 50) \). This provides an approximate cost of \$4348.Linear approximation is an essential tool for:

• Simplifying complex functions into manageable pieces.
• Providing quick estimates when exact calculations are cumbersome.
• Useful in engineering, economics, and sciences where curve behaviors are analyzed.

By relying on derivatives for slope insights, we can make educated guesses about changes in cost and revenue, offering practical solutions in real-world scenarios.