Problem 79 - College Algebra and Calculus: An Applied Approach

Question

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d C}{d x}=2.25 \quad x=100 $$

Step-by-Step Solution

Answer

The change in cost is 6.75.

Step 1: Understand What Marginal Cost Represents

The marginal cost given (\(\frac{d C}{d x}=2.25\)) represents the cost to produce one additional unit when \(x=100\).

Step 2: Calculate the Total Change in Cost

To calculate the total change in cost, we need to multiply the marginal cost by the change in the number of units. In this case, the number of units increases by 3, therefore the change in cost \(ΔC\) is \(2.25 * 3 = 6.75\).

Key Concepts

Marginal Analysis

Marginal analysis is a key concept used to understand changes in costs, revenues, or profits when producing additional units of a good or service. It revolves around the idea of examining the benefit or cost of producing one more unit. In simple terms, it is about understanding the effect of small changes in the production volume on total outcomes. For businesses, marginal analysis helps determine optimal production levels by considering the cost per additional unit, known as the marginal cost, against the potential revenue that unit might generate. This analysis allows businesses to make informed decisions regarding whether increased production will lead to increased profits. By understanding how each additional unit affects overall variables, businesses can efficiently manage resources and plan their production strategies. In our example, the marginal cost of 2.25 indicates the expense to produce one more unit at an output level of 100 units.

Change in Cost

The change in cost refers to the increase or decrease in total expenses as production quantities vary. In our exercise, the marginal cost \( \frac{dC}{dx} = 2.25 \) represents how much the cost changes with each new unit produced. When the production increases, evaluating the change in cost is essential to ensure that producing more units is financially viable.

• To find the change in cost, you calculate it by multiplying the marginal cost by the number of additional units.

In the given task, the units increase by 3. Thus, you calculate the total change in cost: \( \Delta C = 2.25 \times 3 = 6.75 \). This means that producing these 3 extra units costs an additional 6.75 in whatever currency under consideration. Monitoring changes in cost helps businesses decide whether ramping up production is sustainable in the long run.

Calculus Applications

Calculus is a powerful mathematical tool in economic and business applications, especially when it comes to examining small changes and their impacts. It provides a framework for understanding how variables like costs, revenues, and profits respond to changes in production levels through derivatives.The derivative, represented as \( \frac{dC}{dx} \) in our exercise, indicates the rate of change. Here, it shows the rate at which cost changes with respect to the number of units produced.

• By using derivatives, businesses can determine marginal costs or revenues precisely.
• This serves as a basis for decision-making, such as deciding optimal production levels and pricing strategies.

Calculus helps in not just knowing the immediate costs or revenues but predicting how they will evolve as production scales. This allows businesses to align their strategies accordingly, ensuring efficiency and maximum profitability through informed, data-driven decisions.