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To grasp the concept of marginal revenue, it's essential to understand derivatives first. The derivative of a function gives you the rate at which the function's value is changing at any given point. For revenue functions, this means it tells you how the revenue changes as you sell one more unit.

In our exercise, the revenue function is expressed as \[ r(x) = 20,000 imes igg(1 - \frac{1}{x}\bigg) \]. By differentiating this function, we find the derivative, which is called the marginal revenue function. In this case, \[ r'(x) = \frac{20,000}{x^2} \].

This derivative informs us how the revenue changes with respect to an increase in the number of washing machines sold. Essentially, it allows businesses to make informed decisions about how many additional units to produce.

Calculating revenue increase involves understanding how much extra revenue one more unit sold will bring. The marginal revenue calculated at any specific quantity provides this estimate.

For instance, from the original exercise, when 100 machines are sold, the marginal revenue is \[ r'(100) = 2 \], indicating that producing and selling one additional washing machine will increase revenue by approximately 2 dollars. This calculation helps businesses determine the benefit of scaling up production slightly, especially in the short term where only one or a few units are added.

• Helps in decision-making for production increases.
• Assists in financial forecasting and budgeting.

The limit of a function helps us understand its behavior as the input approaches a certain value. In the context of marginal revenue, the limit as the number of units approaches infinity (a very large number) tells us about long-term trends.

For the revenue function's derivative \[ r'(x) = \frac{20,000}{x^2} \], the limit as \( x \to \infty \) is \( 0 \). This result indicates that as more and more machines are sold, the additional revenue from each machine decreases and eventually approaches zero.

• Expected in saturated markets.
• Important for long-term strategic planning.

Diminishing returns is a principal concept in economics referring to a point where adding an additional factor of production results in smaller increases in output. In the context of marginal revenue, it means that as you continue producing more units, the added revenue from each additional unit declines.

From our exercise, when \( \lim_{x \to \infty} r'(x) = 0 \), we interpret this to mean that selling more washing machines results in lesser additional revenue per unit. This phenomenon occurs due to market saturation where all potential customers already have the product, or because of the cost structures that increase with production.

• Critical for maximizing profit efficiency.
• Guides on optimal production quantity.