Textbook Question
Calculate the derivative of the following functions.
y = csc ex
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 33d
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?
Verified step by step guidance1
First, understand the concept of elasticity of demand. Demand is considered elastic if a small change in price leads to a large change in quantity demanded. Conversely, demand is inelastic if a change in price leads to a small change in quantity demanded.
To determine elasticity, we use the formula for elasticity of demand: E(p) = -p * (D'(p) / D(p)), where D'(p) is the derivative of the demand function D(p) with respect to price p.
Calculate the derivative of the demand function D(p) = 40 - 2p. The derivative, D'(p), is the rate at which demand changes with respect to price. For D(p) = 40 - 2p, D'(p) = -2.
Substitute D'(p) and D(p) into the elasticity formula: E(p) = -p * (-2 / (40 - 2p)). Simplify the expression to find E(p) in terms of p.
Determine the values of p for which E(p) > 1 (elastic demand) and E(p) < 1 (inelastic demand). Solve the inequality to find the range of prices where demand is elastic and inelastic.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Demand Elasticity
Demand elasticity measures how the quantity demanded of a good responds to changes in its price. If the percentage change in quantity demanded is greater than the percentage change in price, demand is considered elastic. Conversely, if the percentage change in quantity demanded is less than the percentage change in price, demand is inelastic. Understanding this concept is crucial for analyzing how price changes affect sales.
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Calculating Elasticity
The price elasticity of demand can be calculated using the formula E = (dD/dp) * (p/D), where dD/dp is the derivative of the demand function with respect to price, p is the price, and D is the quantity demanded. This calculation helps determine whether demand is elastic or inelastic at specific price points. A value of E greater than 1 indicates elastic demand, while a value less than 1 indicates inelastic demand.
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Critical Points in Demand Function
In the context of the demand function D(p) = 40 - 2p, critical points occur where the demand changes from elastic to inelastic. This transition typically happens at the price where the elasticity equals 1. By finding this price, one can identify the range of prices for which demand is elastic (E > 1) and inelastic (E < 1), providing valuable insights for pricing strategies.
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Related Practice
Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = x²; a=3
Textbook Question
Calculate the derivative of the following functions.
y = sec(3x+1)
Textbook Question
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = x²; a=3
Textbook Question
Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function , 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?
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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.
Find the elasticity function for this demand function.