Problem 60 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. Give a reason for your answer. The total revenue realized in selling the first 500 units of a product is given by $$ \int_{0}^{500} R^{\prime}(x) d x=R(500)-R(0) $$ where \(R(x)\) is the total revenue.

Step-by-Step Solution

Answer

The statement is true. The total revenue realized in selling the first 500 units of a product can be obtained by integrating the derivative of R(x) from 0 to 500, according to the Fundamental Theorem of Calculus, as shown: \( \int_{0}^{500} R^{\prime}(x) d x = R(500)-R(0) \)

Step 1: Understand the problem and the given statement

We are given a statement that involves the total revenue, R(x), realized in selling the first 500 units of a product. The statement says that the total revenue can be calculated using the integral, or rather the net area under the curve, of the derivative of R(x) with respect to x, between the limits 0 and 500. This statement can be mathematically represented as: $$ \int_{0}^{500} R^{\prime}(x) d x=R(500)-R(0) $$

Step 2: Recall the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if F(x) is an antiderivative of f(x) on the interval [a, b], then: $$ \int_{a}^{b} f(x) d x = F(b) - F(a) $$ Here, f(x) is the derivative of F(x), which means if we know the derivative of a function, we can find the original function (up to a constant, C) by integrating the derivative.

Step 3: Apply the Fundamental Theorem of Calculus to the given statement

Applying the Fundamental Theorem of Calculus to the given statement, we have: $$ \int_{0}^{500} R^{\prime}(x) d x = R(500)-R(0) $$ We notice that R'(x) is the derivative of R(x). Therefore, we can use the Fundamental Theorem of Calculus to integrate R'(x) over the interval [0, 500]. This gives us the difference between the Total Revenue at 500 units (R(500)) and the Total Revenue at 0 units (R(0)).

Step 4: Determine if the statement is true or false

Using the Fundamental Theorem of Calculus, we verified that the given statement is correct: $$ \int_{0}^{500} R^{\prime}(x) d x = R(500)-R(0) $$ Thus, the statement is true. The total revenue realized in selling the first 500 units of a product can be obtained by integrating the derivative of R(x) from 0 to 500.

Key Concepts

Integration

Integration is a fundamental concept in calculus, representing a process of accumulation. It allows us to find the total or the whole given parts of a rate of change. In the context of a function describing change, like a derivative, integration can provide the original function. This is profoundly important for solving real-world problems.

The core idea is to calculate the area under a curve, which can represent various cumulative quantities depending on the problem at hand. For instance, if you know the rate at which revenue is changing, you can use integration to determine the total accumulated revenue over a certain interval.

In our problem, the integral:

• \( \int_{0}^{500} R^{\prime}(x) \; dx \)

Calculates the total change in revenue for producing between 0 and 500 units. This net area under the curve of the derivative \(R'(x)\) from 0 to 500 gives us the total revenue difference, emphasizing integration's role in quantifying accumulated quantities.

Derivative

The derivative of a function provides the rate of change at any given point and is a cornerstone of calculus. In simpler terms, it tells you how a quantity is changing in response to changes in another quantity, often time or position.

For a revenue function \(R(x)\), the derivative \(R'(x)\) gives the instantaneous rate of change of revenue per unit change in goods sold. Essentially, it shows how the revenue increases with each additional item sold.

Understanding derivatives is crucial as they allow businesses and researchers to predict trends and make informed decisions. In the provided exercise, knowing \(R'(x)\) lets us see how revenue per unit is evolving, which is critical for planning and strategy. The relationship between the derivative and the original function, established via integration, often reveals insights into business operations and growth.

Total Revenue Calculation

Calculating total revenue is essential for businesses, as it reflects the income generated from selling a product or service. It involves not just understanding rates of change but assessing overall accumulation across goods sold.

In our scenario, total revenue over the first 500 units is represented as:

• \(R(500) - R(0)\)

Where \(R(500)\) and \(R(0)\) are the total revenues at 500 and 0 units, respectively. This calculation shows the overall revenue, not just the change rate. With integration, we make use of the derivative to calculate this total revenue, highlighting the difference between start and end points in the sales process.

Businesses often rely on such calculations to predict financial outcomes, set prices, and develop overall sales strategies. By utilizing the principles of calculus like integration and derivatives, companies can gain quantifiable insights into revenue dynamics, aiding better decision-making and strategic planning.