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 56, 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 8th Edition

Chapter 2, Problem 56

Instant Answer

Solved by Expert Heather Eichman

03/12/2021

Step 1

The derivative of a function at a certain point gives us the rate of change of the function at that point. In this case, $f^{\prime}(8)$ represents the rate of change of the quantity of coffee sold when the price is $8 per pound. This means that $f^{\prime}(8)$ 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.

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.

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.

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