Problem 56 - Thomas Calculus

Question

Revenue from marginal revenue Suppose that a company's marginal revenue from the manufacture and sale of egg beaters is $$ \frac{d r}{d x}=2-2 /(x+1)^{2} $$ where \(r\) is measured in thousands of dollars and \(x\) in thousands of units. How much money should the company expect from a production run of \(x=3\) thousand egg beaters? To find out, integrate the marginal revenue from \(x=0\) to \(x=3\) .

Step-by-Step Solution

Answer

The company should expect 4,500 dollars from the production run.

Step 1: Understand the Problem

We need to calculate the total revenue from producing 3,000 egg beaters by integrating the given marginal revenue function \( \frac{dr}{dx}=2-\frac{2}{(x+1)^2} \) from \(x=0\) to \(x=3\).

Step 2: Set Up the Integral

The total revenue is the definite integral of the marginal revenue. Therefore, set up the integral as: \[ \int_{0}^{3} \left( 2 - \frac{2}{(x+1)^2} \right) \, dx \].

Step 3: Integrate the Function

Calculate the integral: 1. Integrate 2 with respect to \(x\), which is \(2x\).2. Integrate \(-\frac{2}{(x+1)^2}\): Rewrite as \(-2(x+1)^{-2}\). The antiderivative is \(2(x+1)^{-1}\) or \(\frac{2}{x+1}\). Thus, the integral becomes \[ \int_{0}^{3} \left( 2 - \frac{2}{(x+1)^2} \right) \, dx = \left[ 2x + \frac{2}{x+1} \right]_{0}^{3}. \]

Step 4: Evaluate the Integral at the Bounds

Evaluate from \(x=0\) to \(x=3\): - Substitute \(x=3\): \(2(3) + \frac{2}{3+1} = 6 + \frac{2}{4} = 6 + 0.5 = 6.5\). - Substitute \(x=0\): \(2(0) + \frac{2}{0+1} = 0 + 2 = 2\).

Step 5: Calculate the Definite Integral

Subtract the lower bound evaluation from the upper bound evaluation:- \(6.5 - 2 = 4.5\). Thus, the total revenue is \(4.5\) thousand dollars.

Key Concepts

Marginal Revenue

Understanding marginal revenue is crucial for businesses aiming to optimize their production process. Marginal revenue refers to the additional income a company generates from selling one more unit of a product. In this context, the given marginal revenue function is \( \frac{dr}{dx}=2-\frac{2}{(x+1)^{2}} \), where \( r \) is revenue in thousands of dollars and \( x \) represents thousands of units.

The higher the marginal revenue, the more beneficial it is for a company to produce additional units. However, it typically decreases as more units are produced, eventually converging towards marginal cost, which is common in real-world scenarios due to market saturations or resource constraints.

In this problem, the integration of the marginal revenue function will let us find the total change in revenue as output increases from 0 to 3 thousand units. Understanding this concept allows businesses to gauge how each additional unit affects overall profitability and make informed decisions accordingly.

Definite Integral

A definite integral, within calculus, provides a way to calculate the total accumulation of a quantity, such as area under a curve or total revenue over a specific interval. Given the marginal revenue function \( \frac{dr}{dx} = 2 - \frac{2}{(x+1)^{2}} \), we wish to find the total revenue derived from producing 3,000 units.

To do this, set up the definite integral from 0 to 3 as follows: \[ \int_{0}^{3} \left( 2 - \frac{2}{(x+1)^2} \right) \, dx \].

Evaluating this integral involves applying the fundamental theorem of calculus, which connects differentiation and integration. It requires us to find the antiderivative of the function, evaluate it at the upper limit \( x=3 \), then subtract the evaluation at the lower limit \( x=0 \).

• The evaluation at the upper limit gives \( 6 + 0.5 = 6.5 \).
• The lower limit evaluation yields 2.

Finally, by subtracting these values (6.5 - 2), we find the total revenue to be 4.5 thousand dollars for 3,000 units.

Antiderivative

In order to solve a problem involving integration, understanding the concept of an antiderivative is essential. An antiderivative is essentially the reverse operation of differentiation. For the marginal revenue function \( \frac{dr}{dx} = 2 - \frac{2}{(x+1)^2} \), finding the antiderivative means determining a function whose derivative matches our original function.

Let's break it down:

• For the constant \( 2 \), the antiderivative is straightforwardly \( 2x \).
• The term \(-\frac{2}{(x+1)^2}\) can be rewritten as \(-2(x+1)^{-2}\) for easier integration.
• Using the power rule in reverse, the antiderivative of \( (x+1)^{-2} \) is \( -\frac{1}{x+1} \), then multiplying by the original coefficient \(-2\) yields \( \frac{2}{x+1} \).

Together, these lead to the antiderivative within our definite integral: \[ \left[ 2x + \frac{2}{x+1} \right]_{0}^{3} \].
Understanding how to find an antiderivative is fundamental to solving definite integrals and uncovering the total accumulated quantity over an interval.