Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Find the marginal profit for producing \(x\) units. (The profit is measured in dollars.) $$ P=-2 x^{2}+72 x-145 $$

Step-by-Step Solution

Answer

The marginal profit for producing \(x\) units is given by \(-4x + 72\).

Step 1: Identify the Profit Function

The given profit function is \(P=-2 x^{2}+72 x-145 \). This function represents the profit attained after producing \(x\) units.

Step 2: Differentiate the Profit Function

Differentiate the profit function with respect to \(x\) to find the marginal profit function. This requires applying the power rule for differentiation, which states that the derivative of \(x^n\) is \(n*x^{n-1}\). The derivative of a constant is 0.

Step 3: Apply Derivative Rules

Applying the power rule to each term in the profit function yields the marginal profit function. For the term \(-2x^2\), the derivative is \(-4x\). For the term \(72x\), the derivative is 72. For the constant \(-145\), the derivative is 0.

Step 4: Write the Marginal Profit Function

Combine the derivatives of individual terms to write the marginal profit function. The marginal profit, \(P'\), for producing \(x\) units is thus \(-4x + 72\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that helps us understand how a function changes as its input changes. In the context of a profit function, differentiation allows us to find the marginal profit, which describes how the profit changes with each additional unit produced. By finding the derivative of a profit function, we can determine how producing one more unit will affect the overall profit. This is crucial for businesses in decision-making processes, as it helps in identifying the most profitable levels of production. Differentiation involves taking the derivative of each term in the function with respect to the variable, typically represented by "x." Hence, for our profit function, differentiating gives us insight into the rate of profit change at any production level.

Profit Function

The profit function is a mathematical representation that shows the profit from selling a certain number of goods or services. It often depends on production quantity, denoted by "x" in our example. The profit function can include elements related to revenue and cost, where profit equals revenue minus cost. Here, our given profit function is expressed as:

• \(P = -2x^2 + 72x - 145\)

This quadratic equation helps identify the net profit after producing and selling \(x\) units. With this function, we can predict not just profit at any given point, but also derive additional insights like the marginal profit. Understanding the profit function is essential for effective business planning and maximizing financial outcomes.

Power Rule

The power rule is a basic yet powerful tool in calculus, used for finding derivatives of functions quickly and efficiently. It states that if you have a function of the form \(x^n\), the derivative will be \(n*x^{n-1}\). This rule is particularly useful when differentiating polynomial functions like our profit function.

• For the term \(-2x^2\), applying the power rule gives us \(-4x\).
• For the linear term \(72x\), since it's \(x^1\), its derivative is simply 72.
• For the constant term \(-145\), the derivative is 0 because constants do not change.

By applying the power rule to each term, we derived the marginal profit function \(-4x + 72\). The simplicity and elegance of the power rule make it a favored approach when tackling derivatives in both academic and practical scenarios.