Economics

Marginal revenue
Suppose that...

Question

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?

Accepted Answer

Welcome back, everyone. A company produces 10,000 paper cups per day, and the revenue from selling these is given by the function R X equals 5000 multiplied by 1 minus 2 divided by X dollars. Find the limit of the derivative of R of X as X approaches infinity and interpret this in the context of the company's revenue. Now, we have our revenue function R of X. Let's first start our problem by finding the derivative of RF X and then finding the limit. OK? So now, given R of X. Equal to 5000 multiplied by 1 minus 2 X, which we can expand to be 5000 minus 10,000 divided by X. And we could even rewrite that as 5000 minus 10,000 X to the power of -1. Then that means it's derivative. It's gonna be equal to 0. Minus we can differentiate 10,000 X to the -1 by the poor rule. So we bring down -1 multiplied by 10,000 X to the -1 minus 1. Now, in this case, 0 minus -1 equals 1 and -1 minus 1 equals -2. So this is going to be 10,000 multiplied by X-2 or 10,000 divided by X2. This is the derivative of R of X. Now that we have the derivative, we need to find the limit as X approaches infinity. So now the limit as X approaches infinity of the derivative of R of X. It's gonna be equal to the limit as X approaches infinity of 10,000. Divided by X squared. Now what does this mean? Well, if we think about it, as X gets larger and larger, x squared is going to grow much faster than the constant in the numerator. Thus, the limit is going to be equal to zero. In a practical context. What this means is that as the company produces very large quantities of paper cups, the additional revenue gained from producing 1000 more cups approaches zero. This means that the additional revenue, let's make a note here. Therefore, The additional revenue. OK. Additional revenue from selling. 1 more 1000 cups. In other words, the more cups we sell at high quantities. Becomes negligible, OK? It becomes negligible at high sales volumes. Therefore, that is our solution, OK? Our limit is 0, and this explains what, what that means. The additional revenue is negligible at high sales volumes. Thanks a lot for watching everyone. I hope this video helped.

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?

Key Concepts

Derivative

Limit

Interpretation of Limits in Economics

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