Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Suppose that the total profit in hundreds of dollars from selling \(x\) items is given by $$P(x)=2 x^{2}-5 x+6$$. Find the marginal profit at \(x=2\).

Step-by-Step Solution

Answer

The marginal profit at \( x=2 \) is $300.

Step 1: Understand Marginal Profit

Marginal profit is the derivative of the profit function with respect to the number of items sold. It represents the rate of change of profit as the number of items sold increases by one unit.

Step 2: Write Down the Profit Function

The given profit function is \( P(x) = 2x^2 - 5x + 6 \), where \( x \) is the number of items sold.

Step 3: Differentiate the Profit Function

Differentiate \( P(x) \) with respect to \( x \). The derivative \( P'(x) \) will be calculated as follows.\[ P'(x) = \frac{d}{dx}(2x^2 - 5x + 6) \] Apply power rule: \( \frac{d}{dx}(x^n) = nx^{n-1} \).

Step 4: Use the Power Rule

Applying the power rule: \( P'(x) = 2 \cdot 2x^{2-1} - 5 \cdot 1x^{1-1} + 0 = 4x - 5 \). Thus, the marginal profit function is \( P'(x) = 4x - 5 \).

Step 5: Evaluate the Derivative at \(x=2\)

Evaluate the derivative at \( x = 2 \) to find the marginal profit when 2 items are sold.\[ P'(2) = 4(2) - 5 = 8 - 5 = 3 \].

Step 6: Interpret the Result

The marginal profit when 2 items are sold is 3, which means selling one additional item when 2 items have already been sold increases the profit by $300 (since the profit is in hundreds of dollars).

Key Concepts

Derivative of a function

The concept of a derivative is fundamental in calculus. It is used to determine the rate at which a function changes at any given point, essentially capturing the concept of a function’s instantaneous rate of change. In the context of a profit function, the derivative helps us understand how profit changes with respect to the number of items sold. Let's consider the profit function given in the problem:

• This profit function, denoted as \( P(x) \), is a polynomial function of degree 2.
• To find the derivative, represents the marginal profit, we calculate \( P'(x) \), the slope of the tangent line to \( P(x) \) at any point \( x \).

Evaluating this derivative at a specific value of \( x \), like \( x = 2 \), gives the instantaneous rate of profit change when 2 items are sold. Thus, marginal profit offers actionable insights into how a slight modification in sales quantity affects profit.

Power rule in calculus

The power rule is a straightforward technique used to find derivatives of polynomial expressions, which aids in simplifying the process of differentiation. The power rule states: if you have a function \( f(x) = x^n \), then its derivative is \( f'(x) = nx^{n-1} \). This transformation allows you to convert a simple power function into its instantaneous rate of change. Applying this rule:

• Each term in the polynomial is differentiated separately.
• The coefficient of each term is multiplied by the power of the variable, and the power is decreased by one.

For example, for the profit function \( P(x) = 2x^2 - 5x + 6 \): the term \( 2x^2 \) becomes \( 4x^{2-1} = 4x \), the term \( -5x \) becomes \( -5 \times 1x^{1-1} = -5 \), and the constant \( +6 \) drops out because its derivative is zero. Thus, the derivative \( P'(x) = 4x - 5 \) represents the rate of profit change per unit sold.

Profit function

The profit function is a mathematical representation of financial gain obtained from selling a specific number of goods or services. In its most basic form, it is defined as the difference between revenue and cost.

• For the exercise provided, the profit function given is \( P(x) = 2x^2 - 5x + 6 \).
• Here, \( x \) denotes the number of items sold.

This quadratic profit function indicates a non-linear relationship between the number of items sold and profit. By examining the profit function, businesses can estimate how revenues and costs will evolve with changing sales levels. In practical terms, evaluating the profit function allows businesses to predict how small changes in sales volume might influence overall profitability. Calculating the derivative, or the marginal profit, provides even more precise insights. It helps businesses adjust their strategies to maximize profit efficiently based on small changes in sales volume. With this knowledge, a company can make informed decisions about production levels, pricing, and other key strategic factors impacting their bottom line.