Question

Find the marginal revenue for producing \(x\) units. (The revenue is measured in dollars.) $$ R=50 x-0.5 x^{2} $$

Short answer

The marginal revenue for producing \(x\) units is \(MR = 50 - x\).

Step-by-step solution

Understanding Marginal Revenue

Marginal Revenue is the derivative of the total revenue with respect to the quantity of output, \(x\). In short, it's the rate of change of revenue as one additional unit is sold.

Differentiation

Let's take the derivative of the given total revenue function \(R = 50x - 0.5x^{2}\) with respect to \(x\).

Calculate Derivative

The derivative of \(50x - 0.5x^{2}\) can be calculated using the power rule which states that the derivative of \(x^n\), where \(n\) is any real number, is \(nx^{n-1}\). So we get \( 50 - x = MR\), where \(MR\) stands for Marginal Revenue.

Key concepts

Understanding Derivatives

In the world of calculus, a derivative is a fundamental concept used to measure how a function changes as its input changes. You can think of it as a rate of change or a slope of the function. When we talk about the derivative of a revenue function, we are interested in how the revenue changes with each additional unit sold. This is crucial for businesses wanting to know the impact of selling more units.

A derivative allows us to understand how sensitive a function is to changes in its input. If the derivative is high, even a small change in input can result in a large change in output. In our case, it's the change in revenue for each additional unit produced. Remember, understanding derivatives can provide valuable insights into how to make better economic decisions.

Exploring the Revenue Function

The revenue function represents the total revenue generated from selling a certain quantity of goods. In mathematical terms, this function helps illustrate how the revenue varies with the number of items sold. Given in the problem as \( R = 50x - 0.5x^{2} \), this function shows that revenue initially increases with sales but at a decreasing rate after a certain point.

This is typical in real-world scenarios where initially, producing more increases total revenue, but as the volume continues to increase, the market gets saturated, or costs rise, reducing the rate of revenue growth. By analyzing the revenue function, businesses can determine optimal production levels and pricing strategies to maximize profits.

Understanding the Power Rule

The power rule is a simple yet powerful tool in calculus used for finding derivatives of polynomial functions. It states that if you have a term \( x^n \), the derivative is \( nx^{n-1} \). This means you bring the power down as a coefficient and decrease the power by one.

By applying the power rule to our revenue function \( R = 50x - 0.5x^{2} \), we calculated the marginal revenue. For the first term, \( 50x \), the derivative is 50. For the second term, \(-0.5x^2 \), using the power rule gives us \(-2(-0.5)x^{1} = -x \).

Thus, the marginal revenue, represented as the derivative of the revenue function, simplifies to \( MR = 50 - x \). This power rule makes it easy to quickly find derivatives, especially in real-life applications like revenue and cost analysis.