Problem 60 - Brief Applied Calculus

Question

BUSINESS: Marginal Average Profit Use the Quotient Rule to find a general expression for the marginal average profit. That is, calculate \(\frac{d}{d x}\left[\frac{P(x)}{x}\right]\) and simplify your answer.

Step-by-Step Solution

Answer

\( \frac{d}{dx}\left[\frac{P(x)}{x}\right] = \frac{xP'(x) - P(x)}{x^2}.\)

Step 1: Identify the Components

The function we need to differentiate is \( \frac{P(x)}{x} \), where \( P(x) \) represents the profit as a function of \( x \). Our task is to find the derivative of this function using the Quotient Rule.

Step 2: Apply the Quotient Rule Formula

Recall the Quotient Rule formula: if \( u(x) \) and \( v(x) \) are differentiable functions, then \( \frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} \). For our function, identify: \( u(x) = P(x) \) and \( v(x) = x \).

Step 3: Differentiate \( P(x) \) and \( x \)

Calculate the derivative of the numerator, \( P(x) \), which is \( P'(x) \), and the derivative of the denominator \( x \), which is \( 1 \) because the derivative of \( x \) is \( 1 \).

Step 4: Substitute into the Quotient Rule Formula

Substitute \( u = P(x) \), \( v = x \), \( u' = P'(x) \), and \( v' = 1 \) into the quotient rule formula. This results in:\[\frac{d}{dx}\left[\frac{P(x)}{x}\right] = \frac{x \cdot P'(x) - P(x) \cdot 1}{x^2}.\]

Step 5: Simplify the Expression

The expression from the quotient rule is \( \frac{xP'(x) - P(x)}{x^2} \). This is already quite simplified, but can be expressed more explicitly as:\[\frac{P'(x)x - P(x)}{x^2}.\]

Step 6: Finalize the Simplified Expression

To ensure clarity, the marginal average profit, given the expression derived, is:\[\frac{xP'(x) - P(x)}{x^2}.\] This is a clear representation of the marginal rate of change of average profit per unit change in \( x \).

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental concept in calculus used for finding the derivative of a division of functions. Suppose you have two functions represented as a numerator and a denominator, like in the expression \( \frac{u(x)}{v(x)} \). The quotient rule helps us determine the derivative of such a fraction.

To apply the rule, you need to remember this formula:

• \[ \frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} \]

This formula requires calculating the derivatives of both the numerator \( u(x) \) and the denominator \( v(x) \).

• \( u'(x) \) - the derivative of \( u(x) \)
• \( v'(x) \) - the derivative of \( v(x) \)

Once you compute these, plug them into the formula to get the derivative of the entire fraction. The result will be a new expression showing how the original function changes with respect to \( x \). This is helpful in many fields, particularly in business and economics, where relationships between different functions are analyzed.

Derivative

A derivative is a core concept in calculus representing the rate at which a function changes. Think of it as a way to determine the "slope" of any curve at a specific point. When you have a function, such as profit, which depends on some variable like quantity or time, its derivative tells you how quickly the profit changes as that variable changes.

Finding a derivative involves a process called differentiation. For functions written as \( f(x) \), the derivative is often written as \( f'(x) \) or \( \frac{df}{dx} \). This operation provides crucial insights into the behavior of functions, allowing predictions and optimizations.

• Linear functions have constant derivatives.
• More complex curves have derivatives that vary across the curve's domain.

Understanding derivatives is invaluable for understanding and estimating how a situation will evolve over time. In business contexts, this might involve looking at how profit changes when sales quantities vary.

Profit Function

A profit function is a mathematical representation of profit generated in a business based on different variables, often focusing on units sold or price per unit. In the simplest terms, profit is revenue minus costs.

The profit function is typically expressed as \( P(x) \), where \( x \) can be the number of units sold. The function helps businesses assess how changes in sales or production affect overall profitability. Understanding this function and being able to analyze its properties, like maximum profit or breakeven points, is essential.

• Revenue is calculated as the price per unit times the number of units sold.
• Costs might include fixed costs and variable costs, which vary per unit.

Detailed insights from the profit function lead businesses to make informed decisions about pricing, production, and marketing strategies, aiming for optimal performance and profit maximization.

Rate of Change

The rate of change is a crucial concept in understanding how one quantity changes in relation to another. In calculus, it is typically represented by the derivative. This notion is vital in fields that deal with continuous change or growth, such as economics and physics.

For example, in the context of a business's profit, the rate of change can show how changes in sales affect profitability. If you think of a graph with profit as the vertical axis and number of units sold as the horizontal axis, the rate of change would describe the slope of the tangent line at any given point on the curve.

• Positive rate indicates increasing profit.
• Negative rate signals a decrease in profit.

By analyzing these rates, decision-makers can predict trends, make adjustments, and optimize processes to better achieve business goals. The precise understanding of these changes can lead to substantial improvements in strategic planning and resource allocation.