Problem 11 - Applied Calculus

Question

The marginal revenue function on sales of \(q\) units of a product is \(R^{\prime}(q)=200-12 \sqrt{q}\) dollars per unit. (a) Graph \(R^{\prime}(q)\) (b) Estimate the total revenue if sales are 100 units. (c) What is the marginal revenue at 100 units? Use this value and your answer to part (b) to estimate the total revenue if sales are 101 units.

Step-by-Step Solution

Answer

(a) Graph decreases; (b) $12000 for 100 units; (c) $12080 for 101 units.

Step 1: Understanding the Marginal Revenue Function

The marginal revenue function is given as \( R^{\prime}(q) = 200 - 12 \sqrt{q} \). This represents the additional revenue generated from selling one more unit of the product when \(q\) units have already been sold.

Step 1: Graphing the Marginal Revenue Function

To graph the function \( R^{\prime}(q) = 200 - 12 \sqrt{q} \), we plot it with respect to \( q \). The function is defined for \( q \geq 0 \) and shows a decreasing trend as \( \sqrt{q} \) increases. This indicates that marginal revenue decreases as more units are sold.

Step 2: Estimate Total Revenue for 100 Units

To find the total revenue from 0 to 100 units, we need to integrate the marginal revenue function from 0 to 100:\[R(100) = \int_{0}^{100} (200 - 12 \sqrt{q}) \; dq.\]Calculating the integral:\[R(100) = \left[ 200q - 12 \times \frac{2}{3} q^{3/2} \right]_{0}^{100} = \left[ 200 \times 100 - 8 \times 100^{3/2} \right] - 0 = 20000 - 8000 = 12000.\]So, the estimated total revenue for 100 units is \$12000.

Step 3: Calculate Marginal Revenue at 100 Units

To find the marginal revenue at 100 units, substitute \( q = 100 \) into the function:\[R^{\prime}(100) = 200 - 12 \sqrt{100} = 200 - 12 \times 10 = 80.\]The marginal revenue at 100 units is \$80 per unit.

Step 4: Estimate Total Revenue for 101 Units

Using the marginal revenue at 100 units and the total revenue for 100 units, we estimate the revenue for 101 units:\[R(101) \approx R(100) + R^{\prime}(100) = 12000 + 80 = 12080.\]So, the estimated total revenue for 101 units is \$12080.

Key Concepts

Total Revenue

Total revenue is the overall income a company earns from selling a certain quantity of goods or services. It's an important concept in economics and business because it helps businesses understand how much money they bring in before any costs or expenses are deducted.
To calculate total revenue, you can use the formula:

• Total Revenue = Price per unit × Quantity sold

However, when dealing with calculus and variable marginal revenues, total revenue can be estimated by integrating the marginal revenue function. In the given problem, the total revenue for 100 units is found by integrating the marginal revenue function from 0 to 100 units. This provides a more accurate depiction of total revenue as it accounts for changes in revenue with each additional unit sold.
Understanding total revenue through the lens of marginal revenue gives valuable insights into how revenue is affected by different levels of sales.

Integration

Integration is a fundamental concept in calculus, used to find areas under curves, among other applications. In economics, integrating the marginal revenue function helps calculate the total revenue over a certain number of units.
The integration process involves finding the antiderivative of the marginal revenue function. In our problem, to estimate the total revenue from 0 to 100 units, we integrate the function:

• \[R(100) = \int_{0}^{100} (200 - 12 \sqrt{q}) \; dq\]

This gives us the accumulated revenue when units are sold from 0 to 100. The result of the integration is usually in the form of a new function, which after applying limits, provides the exact value of total revenue. By integrating, you can see how total revenue builds up as more units are sold.

Graphing Functions

Graphing functions is essential to visually understand how different variables interact. For the function given in the problem, we are tasked with graphing the marginal revenue function:

• \[R^{\prime}(q) = 200 - 12 \sqrt{q}\]

To graph this function, plot points across a range of values for \(q\) starting from 0. As \(q\) increases, the graph shows a downward trend, illustrating decreasing marginal revenue. This visually indicates that each additional unit sold brings in less revenue than the previous one.
Graphing functions can also reveal important details such as trends, turning points, and intervals of increase or decrease. It helps to grasp complex algebraic functions easily by providing a visual representation.

Calculus Applications

Calculus plays a crucial role in solving practical problems where change is involved, such as in economics. The use of calculus in economics can help determine key functions and insights.

• Marginal Analysis: In the provided exercise, the differentiation is already done, and the marginal revenue function is given. Here, calculus is used to calculate how marginal revenue diminishes with each additional unit produced.
• Integration: Calculus allows us to integrate the marginal revenue function to determine the total revenue, highlighting how calculus helps to add up small changes to find total quantities.
• Estimation: By using the results of marginal revenue at a specific point (like 100 units) and total revenue, calculus aids in estimating revenue for values that aren’t straightforwardly calculated, such as 101 units in this problem.

Thus, by employing calculus, businesses can optimize their understanding of revenue generation and cost structures, leading to better decision-making.