Textbook Question
Evaluate and simplify y'.
y = (3t²−1 / 3t²+1)^−3
Ch. 4 - Applications of the Derivative
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 4, Problem 34
Demand and elasticity The economic advisor of a large tire store proposes the demand function D(p) = 1800/p-40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p.
c. Find the elasticity function on the domain of the demand function.
Verified step by step guidance1
Step 1: Understand the demand function D(p) = \(\frac{1800}{p}\) - 40. This function represents the number of tires sold per day at a price p.
Step 2: Recall the formula for elasticity of demand, E(p), which is given by E(p) = \(\frac{p}{D(p)}\) \(\cdot\) \(\frac{dD}{dp}\). This measures how sensitive the quantity demanded is to a change in price.
Step 3: Differentiate the demand function D(p) with respect to p to find \(\frac{dD}{dp}\). For D(p) = \(\frac{1800}{p}\) - 40, use the power rule and the constant rule to find the derivative.
Step 4: Substitute D(p) and \(\frac{dD}{dp}\) into the elasticity formula E(p) = \(\frac{p}{D(p)}\) \(\cdot\) \(\frac{dD}{dp}\).
Step 5: Simplify the expression for E(p) to find the elasticity function. This will involve algebraic manipulation to express E(p) 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) = 1800/p - 40 indicates how the number of tires sold changes as the price p varies. Understanding this function is crucial for analyzing how price affects consumer behavior in the market.
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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, 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 describes how elasticity varies with price, allowing for a deeper understanding of consumer sensitivity to price changes across different price levels.
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