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 \(x\). $$ \frac{d P}{d x}=12.5(40-3 \sqrt{x}) \quad x=125 $$

Short answer

The change in profit when the number of units increases by 3 from \(x = 125\) is given by the expression \(\Delta P = \frac{(\frac{d P}{d x}|_{x=125} + \frac{d P}{d x}|_{x=128})}{2} \times (128 - 125)\). Calculate this expression to find the value of \(\Delta P\).

Step-by-step solution

Evaluate the Derivative at the Initial Value of \(x\)

First, substitute the initial value of \(x = 125\) into the derivative function \(\frac{d P}{d x}=12.5(40-3 \sqrt{x})\) to find the rate of change of the profit at this point. This gives \(\frac{d P}{d x} = 12.5(40 - 3 \sqrt{125})\). Calculate this expression to find the value of the derivative at \(x = 125\).

Evaluate the Derivative at the Final Value of \(x\)

Next, substitute the final value of \(x = 125 + 3 = 128\) into the derivative function \(\frac{d P}{d x}=12.5(40-3 \sqrt{x})\) to find the rate of change of the profit at this point. This gives \(\frac{d P}{d x} = 12.5(40 - 3 \sqrt{128})\). Calculate this expression to find the value of the derivative at \(x = 128\).

Find the Average Rate of Change

Now, find the average of the two rates of change found in Steps 1 and 2. This average rate can be used to approximate the change in profit over the interval from \(x = 125\) to \(x = 128\). The average rate of change is given by the formula: \(\frac{(\frac{d P}{d x}|_{x=125} + \frac{d P}{d x}|_{x=128})}{2}\). Calculate this to find the average rate of change.

Find the Change in Profit

Finally, multiply the average rate of change by the change in \(x\), which is 3. This gives the change in profit over the interval from \(x = 125\) to \(x = 128\). The formula for this is: \(\Delta P = \frac{(\frac{d P}{d x}|_{x=125} + \frac{d P}{d x}|_{x=128})}{2} \times (128 - 125)\). Calculate this expression to find the change in profit.

Key concepts

Marginal Analysis

Marginal analysis is a crucial concept in economics and business, particularly when it comes to understanding how to make profitable decisions. At its core, marginal analysis involves examining the effects of small changes in production or sales levels, such as selling one more unit of product or producing one additional item. The idea is to assess the additional benefits gained against the extra costs incurred for these incremental changes.

When businesses consider whether to increase or decrease production, they often use marginal analysis to evaluate the cost of producing one more unit (marginal cost) versus the revenue that one more unit will generate (marginal revenue). If the marginal revenue exceeds the marginal cost, the additional unit should increase overall profits. In the exercise, the marginal profit calculation at a certain level of sales (\(x\) units) provides insight into the profitability of producing and selling additional units beyond that level.

Cost-Revenue-Profit Relationship

The relationship between cost, revenue, and profit is fundamental to understanding a business's financial situation. Costs are the expenses a business incurs in the production and distribution of goods or services, while revenue is the money generated from selling those goods or services. Profit, which is the goal of every business, is simply the difference between the total revenue and total costs.

In the context of marginal analysis, the focus is often on how these relationships change as production and sales volumes adjust. The derivative of profit with respect to the number of units sold, \( \frac{d P}{d x} \), represents the rate at which profit is changing in response to selling additional units, providing invaluable information for decision-making. The concept of marginal profit is integral to this analysis, as it expresses the change in profit for each additional unit sold.

Derivative Application

The application of derivatives in business and economics is often related to finding rates of change. In the exercise, the derivative \( \frac{d P}{d x} \) describes how profit \(P\) changes as the number of units sold \(x\) changes. This is known as the marginal profit.

To make informed decisions based on this information, it's necessary to understand the value of the derivative at specific points. For example, the exercise begins by evaluating the derivative at \(x = 125\). This provides the rate of change of profit when the number of units sold is 125. By knowing how profit is expected to change as sales increase, businesses can optimize their production levels to maximize profit.

Average Rate of Change

The average rate of change is used to estimate how a quantity changes on average over a certain period or interval. It is calculated as the difference in the value of a function at two points divided by the change in the input values. In the context of profit, it helps in estimating the average change in profit per unit over a range of quantities sold.

In our example, we essentially find the average rate of change of profit between 125 and 128 units sold by averaging the marginal profits at these two quantities. This average provides a simplified view of profit behavior over the interval, which can be particularly useful for forecasting and making short-term strategic decisions. Although it is an approximation, it can still offer valuable insights into the performance of a business over that specific range of production.