Problem 81 - 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 \(x\). $$ \frac{d R}{d x}=48-3 x \quad x=12 $$

Step-by-Step Solution

Answer

The change in revenue when the number of units increases by 3 from 12 is 66 units.

Step 1: Identify the given marginal equation

The provided marginal is the derivative of Revenue with respect to \(x\), and it is given by \(\frac{d R}{d x}=48-3 x\).

Step 2: Determine the interval for integration

Since we are looking for the change in revenue when the number of units increases by 3 from \(x=12\), the interval of integration will be from \(x=12\) to \(x=15\).

Step 3: Integrate the marginal equation over the interval

The integral of the marginal equation over this interval will give the change in revenue. That is, \(\Delta R = \int_{12}^{15} (48 - 3x) dx\). The result will indicate how much revenue changes when the number of units increases from 12 to 15.

Step 4: Evaluate the integral

Upon integrating, we find that: \(\Delta R = [48x - \frac{3}{2}x^2]_{12}^{15}\). This leads to \(\Delta R = (48*15 - \frac{3}{2}*15^2) - (48*12 - \frac{3}{2}*12^2)\).

Step 5: Simplify the calculation

Simplifying this we get \(\Delta R = 66\).

Key Concepts

Integration

Integration in calculus is akin to piecing together a puzzle. It's the reverse of taking derivatives and it helps us understand the 'whole' formed by infinitesimal parts. When we talk about integration in the context of business economics, it often relates to finding areas under curves, total quantities, or changes over intervals - think of it as summing up tiny slices to estimate the total effect.

For instance, given a marginal revenue function (which represents the revenue of each additional item sold), integration enables us to find out the total revenue change over a specific interval. This is relevant when a company wishes to determine the potential increase or decrease in revenue when varying the number of units sold.

Marginal Analysis

Marginal analysis is a cornerstone concept in economics and business. It involves examining the effects of small changes, and it's often used to determine the most efficient level of production or the impact of scaling up an operation.

In our example, by examining the marginal revenue, we're effectively studying how the revenue changes with each additional unit sold. This can influence decision-making, such as setting production levels where the cost of producing an additional unit is aligned with the revenue it generates.

Change in Revenue

The change in revenue, particularly in a business setting, is a critical metric. It represents the variation in income obtained from the sale of goods or services. Through integrating the marginal revenue over a certain number of units, businesses can predict the increase or decrease in total revenue over a production interval.

In the case of our exercise, we calculate the revenue change as the company sells 3 more units, starting from 12 units. This information proves invaluable in strategizing financial aspects like pricing, marketing efforts, and production investment.

Applied Calculus

Applied calculus is the practical application of calculus concepts to real-world problems. It bridges the gap between pure mathematics and its use in commerce, the physical sciences, and engineering. Calculus helps in building models to predict growth, optimize resources, and analyze trends.

In our problem, applied calculus is utilized to quantify the abstract concept of revenue change. By integrating the marginal equation over a selected interval, we apply calculus concepts to derive meaningful insight that can influence business decisions.