Problem 5 - Applied Calculus

Question

Let \(C(q)\) represent the cost, \(R(q)\) the revenue, and \(\pi(q)\) the total profit, in dollars, of producing \(q\) items. (a) If \(C^{\prime}(50)=75\) and \(R^{\prime}(50)=84\), approximately how much profit is earned by the \(51^{\text {st }}\) item? (b) If \(C^{\prime}(90)=71\) and \(R^{\prime}(90)=68\), approximately how much profit is earned by the \(91^{\text {st }}\) item? (c) If \(\pi(q)\) is a maximum when \(q=78\), how do you think \(C^{\prime}(78)\) and \(R^{\prime}(78)\) compare? Explain.

Step-by-Step Solution

Answer

(a) Profit from 51st item: $9; (b) Loss from 91st item: $-3; (c) At maximum profit, \(R'(78) = C'(78)\).

Step 1: Understanding the Problem

We are given cost, revenue, and profit functions which depend on the number of items produced, denoted as \(q\). The derivatives \(C'\), \(R'\), and \(\pi'\) represent the rates of change of these functions with respect to \(q\). The profit for the \(q^{th}\) item is the difference between the revenue and cost derivations at that point: \(\pi'(q) = R'(q) - C'(q)\).

Step 2: Calculating Profit from the 51st Item

To find the approximate profit from the 51st item, we calculate \(\pi'(50) = R'(50) - C'(50)\). Given \(R'(50) = 84\) and \(C'(50) = 75\), the profit from the 51st item is: \(\pi'(50) = 84 - 75 = 9\).

Step 3: Calculating Profit from the 91st Item

Similarly, we calculate \(\pi'(90) = R'(90) - C'(90)\). With \(R'(90) = 68\) and \(C'(90) = 71\), the profit from the 91st item is: \(\pi'(90) = 68 - 71 = -3\), indicating a loss.

Step 4: Analyzing Derivatives at Maximum Profit

At \(q=78\), where profit \(\pi(q)\) is maximized, the derivative \(\pi'(78) = 0\). Therefore, \(R'(78) = C'(78)\) since the marginal rate of revenue equals the marginal rate of cost, indicating no net profit gain from producing additional units.

Key Concepts

Cost Function

The cost function, denoted as \(C(q)\), represents the total cost incurred by producing \(q\) items. It is a crucial element in analyzing profitability in economics, as it helps to determine how costs change with varying levels of production. The derivative of the cost function, \(C'(q)\), shows the rate at which these costs increase when one more item is produced. This is known as the marginal cost.

Understanding the cost function is important as it provides insights into the efficiency of production processes. If the marginal cost is too high, producing additional items may not be beneficial. Analyzing the cost function gives a clearer picture of fixed and variable costs involved in production. This allows businesses to make decisions on optimizing resource allocation and reducing unnecessary expenses.

Revenue Function

The revenue function, \(R(q)\), represents the total income a business makes from selling \(q\) items. It is essential for understanding how much money is being brought in as production levels change. The derivative of the revenue function, \(R'(q)\), is the marginal revenue, indicating how much additional revenue is earned by selling one more unit.

A thorough knowledge of the revenue function helps in pricing strategies and sales forecasting. By understanding the relationship between quantity sold and revenue generated, businesses can make informed decisions regarding production volumes and pricing. If the marginal revenue is decreasing, businesses may consider revising their strategy to maximize overall revenue effectively.

• The revenue function often begins by assessing market demand and price elasticity.
• Higher marginal revenue suggests potential for increased profitability through additional sales.

Marginal Analysis

Marginal analysis is a key concept in economics used to understand the implications of producing one more unit. It involves comparing additional costs incurred against additional revenue gained. In mathematics, this is done by analyzing derivatives of functions.

Through marginal analysis, businesses can determine the optimal level of production.

• When marginal revenue exceeds marginal cost (\(R'(q) > C'(q)\)), producing more items can increase profit.
• When marginal cost exceeds marginal revenue (\(C'(q) > R'(q)\)), producing additional units can lead to losses.

Marginal analysis provides a clear picture of potential gains or losses from small changes in production and helps in making precise adjustments to maximize profit.

Maximum Profit

Achieving maximum profit is the ultimate goal for any business. This occurs when the profit function, \(\pi(q)\), is at its highest point. Mathematically, this is when its derivative, \(\pi'(q)\), equals zero. At this point, marginal revenue is equal to marginal cost, \(R'(q) = C'(q)\). This condition tells us that producing one more unit would neither increase nor decrease the overall profit.

Focusing on maximizing profit involves finding the optimal production level where the difference between revenue and cost is greatest. It requires careful consideration of both marginal costs and revenues and understanding how they evolve with changes in production.

• Profit maximization helps in setting quantities and pricing strategies.
• Businesses need to understand the balance between production costs and potential revenue growth.