Economics


Marginal revenue
Suppose that the revenue from selling x washing machines is


r(x) = 20000(1 − 1/x) dollars.


c. Find the limit of r'(x) as x → ∞. How would you interpret this number?

Verified step by step guidance

1. First, understand that r(x) = 20000(1 - 1/x) represents the revenue function, where x is the number of washing machines sold.
2. To find the marginal revenue, we need to differentiate r(x) with respect to x. This means finding r'(x).
3. Differentiate r(x) = 20000(1 - 1/x) using the power rule and the constant multiple rule. The derivative of 1 is 0, and the derivative of -1/x is 1/x^2.
4. After differentiating, you will have r'(x) = 20000 * (0 + 1/x^2) = 20000/x^2.
5. To find the limit of r'(x) as x approaches infinity, evaluate the limit of 20000/x^2 as x → ∞. As x becomes very large, 1/x^2 approaches 0, so the limit of r'(x) is 0. This means that as more washing machines are sold, the additional revenue from selling one more washing machine approaches zero, indicating diminishing returns.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input changes. In this context, r'(x) represents the marginal revenue, which is the rate of change of revenue with respect to the number of washing machines sold. Calculating the derivative helps determine how revenue changes as sales increase.

Limit

A limit describes the value that a function approaches as the input approaches a certain point. In this problem, finding the limit of r'(x) as x approaches infinity helps us understand the behavior of marginal revenue as the number of washing machines sold becomes very large. It indicates the long-term trend of revenue change per additional unit sold.

Interpretation of Limits in Economics

In economics, the limit of a derivative, such as marginal revenue, as x approaches infinity can provide insights into the sustainability of revenue growth. If the limit is zero, it suggests that selling additional units will eventually not increase revenue, indicating a saturation point in the market. This interpretation helps businesses make informed production and pricing decisions.