Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Marginal Analysis In Exercises 27 and 28, use a program similar to the Simpson's Rule program on page 906 with \(n=4\) to approximate the change in revenue from the marginal revenue function \(d R / d x .\) In each case, assume that the number of units sold \(x\) increases from 14 to 16 . $$ \frac{d R}{d x}=5 \sqrt{8000-x^{3}} $$

Step-by-Step Solution

Answer

The step-by-step solution provides an approximation for the change in revenue using Simpson's Rule. The exact numerical answer will depend on the results of the computations in Step 4.

Step 1: Defining Simpson's Rule

Simpson's rule is a numerical method for approximating definite integrals. It can be represented as: \(\frac{h}{3} [f(a) + 4f(a+h) + 2f(a+2h) + 4f(a+3h) + ... + 2f(a + (n-2)h) + 4f(a + (n-1)h) + f(b)]\), where \(h = \frac{b-a}{n}\), \(a\) is the start of the interval, \(b\) the end, and \(n\) is the number of intervals.

Step 2: Calculate h value and create list of intervals

The next step is to compute the \(h\) value. According to the problem, \(a=14\), \(b=16\), and \(n=4\). So, \(h = \frac{b-a}{n} = \frac{16-14}{4} = 0.5\). Now we create a list of \(x\) values: \(x=[14, 14.5, 15, 15.5, 16]\).

Step 3: Implement Simpson's Rule

Substitute the \(x\) values into the marginal revenue function and then use Simpson's Rule to carry out the integration. The result is: \(\frac{h}{3} [f(14) + 4f(14.5) + 2f(15) + 4f(15.5) + f(16)]\)

Step 4: Calculate the values and solve the expression

Substitute the \(x\) values into the marginal revenue function to get the corresponding \(f(x)\) values, perform the operations and add up the results. The final result is the approximate change of the revenue.

Key Concepts

Marginal Analysis

Marginal analysis is a powerful concept in economics used to understand the incremental changes in different variables. It helps in determining the additional benefits or costs of increasing the number of units produced or sold. By studying marginal changes, businesses can optimize processes like production or pricing to maximize profitability.

A marginal revenue function, such as \( \frac{dR}{dx} \) in the problem, represents how the revenue changes with the sale of one more unit. The derivative indicates change, and it provides insights on the additional revenue generated from selling another unit.

When conducting marginal analysis, particularly in revenue projections, understanding how marginal revenue behaves over an interval is crucial. This projection gives clarity on whether increasing sales will lead to more profit or if it might result in diminishing returns.

Numerical Integration

Numerical integration is a method used to approximate the definite integrals of functions, which are not easily integrable or have no elementary solution. It offers practical ways to evaluate the area under a curve when an exact answer is unnecessary or difficult to obtain.

• Important when dealing with complex functions or when traditional calculus is challenging to apply.
• Useful when dealing with real-world data that can fluctuate or when functions are derived from empirical data.

In this exercise, Simpson's Rule is used as a technique for numerical integration to approximate the change in revenue. Its appeal lies in its accuracy compared to other methods like Trapezoidal Rule, especially when dealing with polynomial approximations.

Given the function \( \frac{dR}{dx} = 5 \sqrt{8000 - x^3} \), numerical integration helps estimate total changes in revenue when traditional methods might be cumbersome.

Definite Integrals

Definite integrals play a significant role in calculating the total accumulation of quantities such as area, volume, and net change of functions over a given interval. These integrals provide the exact total change between two bounds, typically represented as \( \int_a^b f(x) \, dx \).

• Represents the net change of a function over a specific interval from \(a\) to \(b\).
• Integral bounds \(a\) and \(b\) specify the start and end of the interval.

In the context of this exercise, the goal is to find the definite integral of the marginal revenue function \( \frac{dR}{dx} \) from 14 to 16 units sold. This integral tells us the total change in revenue within this interval of units sold.

Simpson's Rule is used here as a tool to approximate this definite integral, which would otherwise require complex calculation methods beyond elementary techniques.

Marginal Revenue Function

A marginal revenue function \( \frac{dR}{dx} \) is derived from the total revenue function and helps businesses understand how much revenue they earn by selling one more unit. It reflects the rate of change of revenue as sales increase.

In this problem, the marginal revenue function is given as \( 5 \sqrt{8000 - x^3} \), suggesting how revenue behaves as the number of units sold, denoted by \( x \), changes.

• The form \( 5 \sqrt{8000 - x^3} \) implies a decreasing rate of revenue growth with increasing sales, due to the cubic term in \( x^3 \).
• Understanding the behavior of this function can help businesses decide on production levels and pricing strategies.

Calculating the impact of changes in sales via the marginal revenue function provides a clearer picture of potential revenue outcomes, guiding better decision-making for maximizing profits.