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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 33c
Chapter 3, Problem 33c

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.
Find the elasticity function for this demand function.

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1
Step 1: Understand the concept of elasticity. Elasticity of demand measures how the quantity demanded of a good responds to a change in price. It is defined as E(p) = - (p / D(p)) * (dD/dp), where D(p) is the demand function and dD/dp is its derivative.
Step 2: Identify the given demand function. The demand function provided is D(p) = 40 - 2p.
Step 3: Differentiate the demand function with respect to p. Find dD/dp by differentiating D(p) = 40 - 2p. The derivative, dD/dp, is the rate of change of demand with respect to price.
Step 4: Substitute D(p) and dD/dp into the elasticity formula. Use the formula E(p) = - (p / D(p)) * (dD/dp) and substitute the expressions for D(p) and dD/dp.
Step 5: Simplify the expression for E(p). Simplify the expression obtained in Step 4 to get the elasticity function in terms of p.

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Key Concepts

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

Demand Function

A demand function expresses the relationship between the price of a good and the quantity demanded by consumers. In this case, D(p) = 40 - 2p indicates that as the price (p) increases, the quantity of DVDs sold (D) decreases. Understanding this function is crucial for analyzing how price changes affect consumer behavior.
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Elasticity of Demand

Elasticity of demand measures how responsive the quantity demanded is to a change in price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. This concept helps determine whether demand is elastic (sensitive to price changes) or inelastic (less sensitive), which is essential for pricing strategies.
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Calculating Elasticity

To find the elasticity function for the given demand function, we use the formula E(p) = (D'(p) * p) / D(p), where D'(p) is the derivative of the demand function with respect to price. This calculation provides a function that indicates how elasticity varies with price, allowing the store owner to make informed decisions about pricing and inventory.
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Related Practice
Textbook Question

Calculate the derivative of the following functions.

y = csc ex

Textbook Question

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a. For the following functions and values of a, find f′(a).

f(x) = x²; a=3

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Calculate the derivative of the following functions.

y = sec(3x+1)

Textbook Question

Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function D(p)=40−2pD(p) = 40-2p , where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

a. According to the model, how many DVDs can be sold in a day at a price of \$10?

1
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Textbook Question

Equations of tangent lines by definition (2)

b. Determine an equation of the tangent line at P.

f(x) = √x+3; P (1,2)

Textbook Question

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

For what prices is the demand elastic? Inelastic?