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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
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All textbooksHass 15th EditionCh. 3 - DerivativesProblem 22c
Chapter 3, Problem 22c

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.
To find the marginal revenue, we need to differentiate r(x) with respect to x. This means finding r'(x).
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.
After differentiating, you will have r'(x) = 20000 * (0 + 1/x^2) = 20000/x^2.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Derivatives

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.
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One-Sided Limits

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.
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Related Practice
Textbook Question

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.

Textbook Question

[Technology Exercise]


Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula


y = 6(1 - t/12)² m.


a. Find the rate dy/dt (m/h) at which the tank is draining at time t.

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Textbook Question

Economics


Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².


a. Find the average cost per machine of producing the first 100 washing machines.

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Textbook Question

Economics


Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².


c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Textbook Question

Using the Alternative Formula for Derivatives


Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.


f(x) = x² − 3x + 4

Textbook Question

Additional Applications


Bacterium population

When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 10⁶ + 10⁴t − 10³t². Find the growth rates at


a. t = 0 hours.

b. t = 5 hours.

c. t = 10 hours.

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