Question

The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of $p$ dollars per pound is $Q=f(p)$. (a) What is the meaning of the derivative $f^{\prime}(8)$ ? What are its units? (b) Is $f^{\prime}(8)$ positive or negative? Explain.

Name: Problem 50, Chapter 2: Calculus
Uploaded: 2021-03-12T22:59:17-08:00
Duration: 1 min 45 s
Channel: Heather Eichman
Description: The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q=f(p). (a) What is the meaning of the derivative f^'(8) ? What are its units? (b) Is f^'(8) positive or negative? Explain.

   The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of $p$ dollars per pound is $Q=f(p)$.
(a) What is the meaning of the derivative $f^{\prime}(8)$ ? What are its units?
(b) Is $f^{\prime}(8)$ positive or negative? Explain.

Calculus

James Stewart 7th Edition

Chapter 2, Problem 50

Instant Answer

Solved by Expert Heather Eichman

03/12/2021

Step 1

Here, $Q = f(p)$ means that the quantity of coffee sold (Q) is a function of the price per pound (p). Specifically, $f(p)$ gives us the quantity of coffee sold in pounds when the price is $p$ dollars per pound. Show more…

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The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of $p$ dollars per pound is $Q=f(p)$. (a) What is the meaning of the derivative $f^{\prime}(8)$ ? What are its units? (b) Is $f^{\prime}(8)$ positive or negative? Explain.

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Key Concepts

Economic Interpretation of the Demand Function

In economics, particularly when dealing with demand functions, the derivative indicates the sensitivity of quantity demanded to changes in price. Typically, an increase in price leads to a decrease in quantity demanded, meaning the derivative is negative. This reflects the basic economic principle known as the law of demand.

Instantaneous Rate of Change

The derivative represents the instantaneous rate of change of one variable relative to another. In the context of a function describing a quantity sold as a function of price, it measures how quickly the quantity sold changes in response to a small change in price at a specific price level.

Units of the Derivative

Because the derivative is the ratio of the change in quantity (measured in pounds) to the change in price (measured in dollars), its units are pounds per dollar. This indicates how many pounds of coffee are sold for each one-dollar change in price.

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