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 . $$ \frac{d R}{d x}=50 \sqrt{x} \sqrt{20-x} $$

Short answer

The approximate revenue change can be calculated using Simpson’s Rule, by substituting the given function and its calculated values into the Simpson's formula.

Step-by-step solution

Define the function and the limits

The given function is \(f(x) = 50 \sqrt{x} \sqrt{20-x}\). The given values for \(x\) are 14 and 16, so the change in \(x\), \(\Delta x = \frac{(16 - 14)}{4} = 0.5\). Therefore, \(x\) will take the values 14, 14.5, 15, 15.5, and 16.

Apply Simpson's Rule

Now that we have the function and its values, apply the Simpson's Rule: \(S = \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]\), where \(x_0, x_1, x_2, x_3, x_4\) are 14, 14.5, 15, 15.5, and 16, respectively.

Compute the function values

Calculate the function values at each \(x_i\) and then substitute into Simpson's Rule. For example, \(f(14) = 50 \sqrt{14} \sqrt{20 - 14} = 50 \sqrt{84}\). Calculate the remaining values similarly.

Determine the change in revenue

Substitute the function values from Step 3 into the Simpson's Rule formula to estimate the change in revenue. This value will provide the approximate change in revenue as the number of units sold increases from 14 to 16.

Key concepts

Simpson's Rule

When you need an accurate approximation for integration, Simpson's Rule comes in handy. It is a method used in numerical integration to approximate the definite integral of a function. Unlike simple rectangular or trapezoidal approximations, Simpson's Rule uses parabolic arcs to estimate the area under the curve. This method generally provides better accuracy by taking the curvature of the function into account.

Simpson's Rule applies mostly in situations where you already know your function and its limits, which are split into equal intervals. To apply this rule, follow these steps:

• Divide the interval into an even number of sub-intervals.
• Apply the formula: \[ S = \frac{\Delta x}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)\right] \]where subscript numbers indicate points in the interval.

Each function value is multiplied by 1, 2, or 4 as part of the rule, enhancing the accuracy of the approximation significantly. It is an efficient method when applied correctly, and generally favored in computational applications due to its balance of simplicity and accuracy.

Marginal Revenue

Marginal Revenue is a concept used in economics to measure the additional income generated from selling one more unit of a good or service. It's a crucial data point that helps businesses understand how changes in production affect revenue.

In mathematical terms, marginal revenue is the derivative of the total revenue function with respect to the number of units sold. This means it tells us how revenue will change with a slight increase in sales volume. If the marginal revenue exceeds the cost of producing an additional unit, businesses can benefit by increasing production.

However, if marginal revenue starts to decline or fail to cover costs, it might signal the need to reassess the production strategy. Understanding this concept helps to optimize pricing and production strategies to maximize overall profitability.

Revenue Function

The revenue function represents the total revenue a company earns as a function of the quantity of goods sold. It is foundational in economics and business decision-making.

The function is generally expressed as:\[ R(x) = ext{Price} \times ext{Quantity} \]Here, the price could vary depending on the number of units sold, leading to a more complex function.

In the given context, the marginal revenue function, \[ \frac{dR}{dx} = 50 \sqrt{x} \sqrt{20-x} \]is derived from the revenue function. This form helps in understanding how revenue changes with different levels of output, indicating potential maximum revenue points or efficiency improvements.

Analyzing the revenue function helps businesses to set competitive pricing, plan marketing strategies, and anticipate shifts in market demand.

Integration

Integration is one of the fundamental concepts in calculus. It is used to calculate things such as areas under a curve, volumes, and central points. When you integrate a function, you're essentially summing up an infinite number of infinitesimally small quantities to find a total quantity.

One common application of integration is to find the total change in a quantity, represented by the area under a curve on a graph. For instance, in economics, integration can help determine total revenue, total cost, and other aggregate measures when given continuous rate functions.

While analytical integration involves finding an antiderivative, numerical estimation methods like Simpson’s Rule are used when a function is too complex to integrate exactly or when numerical data is given. This approach helps approximate the integral using discrete intervals and reduces error significantly compared to other basic numerical methods. Being conversant with both symbolic and numerical integration is vital for practical problem-solving in diverse fields.