Economics

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

r(x) = 20000(1 − 1/x) dollars.

b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week.

검증된 단계별 안내

First, understand that the function r(x) = 20000(1 - 1/x) represents the revenue from selling x washing machines. We need to find the derivative r'(x) to determine the marginal revenue, which is the rate of change of revenue with respect to the number of machines sold.

To find r'(x), apply the derivative rules. The function r(x) = 20000(1 - 1/x) can be rewritten as r(x) = 20000 - 20000/x. Differentiate this function with respect to x.

The derivative of a constant is zero, so the derivative of 20000 is 0. For the term -20000/x, use the power rule. Rewrite -20000/x as -20000x^(-1) and differentiate to get 20000x^(-2) or 20000/x^2.

Thus, the derivative r'(x) = 20000/x^2. This represents the marginal revenue function, which gives the approximate change in revenue for a small change in the number of machines sold.

To estimate the increase in revenue from increasing production from 100 machines to 101 machines, evaluate r'(100). Substitute x = 100 into the marginal revenue function: r'(100) = 20000/(100^2). This value will give the estimated increase in revenue for producing one additional machine.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Derivative

The derivative of a function measures how the function's output changes as its input changes. In this context, r'(x) represents the rate of change of revenue with respect to the number of washing machines sold. Calculating r'(x) allows us to estimate the marginal revenue, which is the additional revenue generated by selling one more unit.

Marginal Revenue

Marginal revenue is the additional income from selling one more unit of a good or service. It is calculated as the derivative of the revenue function with respect to quantity. In this problem, r'(x) gives the marginal revenue, helping estimate the revenue increase when production rises from 100 to 101 machines.

Function Evaluation

Function evaluation involves substituting specific values into a function to determine its output. Here, evaluating r'(x) at x = 100 provides the marginal revenue for the 100th machine, which helps estimate the revenue change when production increases to 101 machines. This step is crucial for applying the derivative to real-world scenarios.

EconomicsMarginal revenueSuppose that the revenue from selling x washing machines isr(x) = 20000(1 − 1/x) dollars.b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week.