Find the price elasticity of demand for the demand function...

Find the price elasticity of demand for the demand function at the indicated -value. Is the demand elastic, inelastic, or of unit elasticity at the indicated -value? Use a graphing utility to graph the revenue function, and identify the intervals of elasticity and inelasticity. $$ p=400-3 x \quad x=20 $$

Key Concepts

Price Elasticity of Demand

This concept measures the responsiveness of the quantity demanded to a change in price. It is typically calculated as the percentage change in quantity demanded divided by the percentage change in price, or using derivatives as (dp/dx)·(x/p). Understanding price elasticity helps in making decisions regarding pricing strategies and predicting consumer behavior in response to changes in price.

Elasticity Classification

Elasticity is classified based on its numerical value; if the absolute value of elasticity is greater than 1, demand is considered elastic, indicating that quantity demanded is highly responsive to price changes. If it is less than 1, demand is inelastic, meaning quantity changes little with price variations. When the elasticity equals 1 in absolute terms, the demand has unit elasticity, where the percentage change in quantity exactly matches the percentage change in price.

Revenue Function

The revenue function is obtained by multiplying the price function by the quantity sold (R = p(x)·x). Analyzing this function through graphing can reveal the intervals where revenue increases or decreases, and it often shows how changes in price and quantity interact. Furthermore, the revenue maximization point is typically related to the unit elasticity of demand, and graphing the function helps in identifying the segments where the demand is elastic or inelastic.

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