Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Find the marginal revenue for producing \(x\) units. (The revenue is measured in dollars.) $$ R=30 x-x^{2} $$

Step-by-Step Solution

Answer

The marginal revenue for producing \(x\) units is \({dR}/{dx} = 30 - 2x\).

Step 1: Understand the concept of marginal revenue

Marginal Revenue (MR) is the increase in total revenue that results from selling one more unit of output. It is calculated by taking the derivative of the total revenue function with respect to quantity (\(x\)).

Step 2: Identify the total revenue function

The total revenue function given in the question is \(R=30x-x^2\).

Step 3: Find the derivative of the total revenue function

The derivative of \(R\) with respect to \(x\) can be found using the power rule. The power rule states that the derivative of \(x^n\) is \(n*x^(n-1)\). Applying the power rule, we derive \({dR}/{dx} = 30 - 2x\) which is our Marginal Revenue.

Key Concepts

Derivative

The derivative is a fundamental concept in calculus. It measures how a function changes as its input changes. More specifically, the derivative of a function represents the rate at which a quantity changes. For example, if you have a function that describes the position of an object over time, the derivative of that function will give you the velocity of the object.

In the context of economics, and specifically in this exercise about marginal revenue, the derivative helps us understand how the revenue changes when the number of units produced changes. When we take the derivative of the revenue function with respect to the number of units, we find the marginal revenue. Marginal revenue is crucial because it tells a firm how much extra revenue they can expect from selling one more unit of a product.

Calculating a derivative involves rules and techniques such as the power rule, which we will discuss further. Understanding derivatives is key to understanding changes in any mathematical function used to model real-world scenarios.

Revenue function

The revenue function is a mathematical model that describes the total revenue obtained from selling a certain number of units of a product. In this exercise, the revenue function is given by the equation \( R = 30x - x^2 \), where \( R \) is the revenue and \( x \) is the number of units sold.

This specific function indicates that the revenue increases by $30 for each unit sold due to the term \( 30x \). However, the negative \( x^2 \) term suggests that there's a decrease in revenue that accelerates as more units are sold. This quadratic term might represent discounts or additional costs associated with higher production levels, such as overproduction or inefficiencies.

By analyzing the revenue function, economists or business analysts can determine at what level of production the revenue maximizes and how different levels of production affect the overall revenue. Understanding the form and implications of a revenue function is essential for making informed business decisions about production quantities.

Power rule

The power rule is a fundamental rule in calculus used to find the derivative of polynomial functions. It states that if you have a term \( x^n \), then its derivative is given by \( nx^{n-1} \). For example, the derivative of \( x^3 \) would be \( 3x^2 \).

In the context of this exercise, the power rule is applied to find the derivative of the revenue function \( R = 30x - x^2 \). For the first term, \( 30x \), the derivative is simply 30 because applying the power rule with \( x^1 \) gives us \( 1*30x^{1-1} = 30 \). For the second term, \( x^2 \), the derivative is \( 2x \) since \( 2*x^{2-1} = 2x \).

Therefore, the derivative of the revenue function is \( \frac{dR}{dx} = 30 - 2x \). This derivative is the marginal revenue function and expresses how the revenue changes with each additional unit sold. The power rule is crucial for deriving such relationships quickly and efficiently.