Problem 59 - Brief Applied Calculus

Question

Use the Quotient Rule to find a general expression for the marginal average revenue. That is, calculate \(\frac{d}{d x}\left[\frac{R(x)}{x}\right]\) and simplify your answer.

Step-by-Step Solution

Answer

The marginal average revenue is given by \( \frac{x \cdot \frac{dR}{dx} - R(x)}{x^2} \).

Step 1: Identify the Functions

Apply the Quotient Rule to the function \( \frac{R(x)}{x} \). Here, the numerator \( u = R(x) \) and the denominator \( v = x \).

Step 2: Recall the Quotient Rule Formula

The Quotient Rule states that for two functions \( u \) and \( v \), the derivative of their quotient is: \[ \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]

Step 3: Differentiate R(x) and x

Differentiate \( R(x) \) to get \( \frac{dR}{dx} \) and differentiate \( x \) to get \( \frac{dx}{dx} = 1 \).

Step 4: Apply the Quotient Rule

Substitute \( u = R(x) \), \( v = x \), \( \frac{du}{dx} = \frac{dR}{dx} \), and \( \frac{dv}{dx} = 1 \) into the Quotient Rule formula: \[ \frac{x \cdot \frac{dR}{dx} - R(x) \cdot 1}{x^2} \]

Step 5: Simplify the Result

Simplify the expression: \[ \frac{x \cdot \frac{dR}{dx} - R(x)}{x^2} \]. This gives the simplified general expression for the derivative using the Quotient Rule.

Key Concepts

Marginal Average Revenue

In economics, the concept of marginal average revenue offers insights into how revenue changes concerning the quantity sold. The average revenue function, denoted as \( AR(x) = \frac{R(x)}{x} \), provides the average revenue per unit of production. To find out how this average revenue changes as more units are sold, we use the derivative of the average revenue function, often termed as the marginal average revenue.The marginal average revenue is crucial because it shows how incremental sales impact average revenue. Essentially, it tells us whether the per-unit revenue is increasing or decreasing as more units are sold.
For businesses, understanding whether the marginal rate is positive, negative, or zero is vital in making decisions concerning production and sales strategies.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding a derivative. The derivative measures the rate at which a function is changing at any given point. In the exercise, we aim to find the derivative of the function \( \frac{R(x)}{x} \).
This task of finding the derivative involves applying the Quotient Rule since the function is a ratio of two separate functions.Differentiation helps us understand how revenue \( R(x) \) changes with respect to changes in quantity \( x \). By applying the Quotient Rule, we can break down the process into logical steps.

• Identify the numerator and the denominator.
• Differentiate each component separately.
• Apply the Quotient Rule formula to derive the result.
  

The process of differentiation is crucial as it provides a way to model changes and predict outcomes in various scenarios, be it economics, physics, or engineering.

Calculus

Calculus is the study of how quantities change and interact. It provides the foundational tools for analyzing dynamic systems. At its core, calculus consists of two main branches: differential calculus and integral calculus.
Here, we focus on differential calculus, which involves finding derivatives such as those in this exercise using the Quotient Rule. The Quotient Rule itself is a tool within differential calculus, specifically designed to efficiently compute the derivative of a function that is a ratio of two other functions. This underlying principle of calculus showcases its utility in practically everything from calculating speeds to deducing economic trends.

• Allows understanding of change dynamics.
• Offers mechanisms to solve real-world problems.
  

In essence, calculus and its techniques such as the Quotient Rule allow us to dive deeper into how quantities evolve, thus providing valuable insights into virtually every scientific domain.