More Applications of Derivatives

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Presentation transcript:

More Applications of Derivatives Lecture 16 More Applications of Derivatives Elasticity (Waner Section 5.5 Page 337~345)

Price Elasticity of Demand

Price Elasticity of Demand The price elasticity of demand E, is the percentage rate of decrease of demand per percentage increase in price. E is given by: Demand is: Elastic if E > 1 Unit Elasticity if E = 1 Inelastic if E < 1

Price Elasticity of Demand If the demand is elastic at p (E > 1), then an increase in unit price causes a decrease in revenue. A decrease in unit price causes an increase in revenue. If the demand has unit elasticity at p (E = 1), then with an increase in unit price the revenue will stay about the same. If the demand is inelastic at p (E < 1), then an increase in unit price causes an increase in revenue. A decrease in unit price causes a decrease in revenue.

Price Elasticity of Demand Ex. The monthly demand for T-shirts is given by where p denotes the wholesale unit price in dollars and q denotes the quantity demanded monthly. Find the price elasticity of demand when p = $5 and p = $15, and interpret the results. E(5) = 2/15 which is inelastic. E(15) = 18/7 which is elastic.

Another Elasticity Income Elasticity of Demand The income elasticity of demand E, is the percentage rate of increase in demand per percentage increase in income. E is given by:

Demand for Oranges Waner pg. 343, #1 The weekly sales of Honolulu Red Oranges is given by q = 1000 – 20p. Calculate the price elasticity of demand when the price is $30/orange. Interpret your answer. Since demand is elastic (E > 1), an increase in price will decrease revenue.

Demand for Oranges Waner pg. 343, #1 The weekly sales of Honolulu Red Oranges is given by q = 1000 – 20p. Calculate the price that gives a maximum weekly revenue and find this maximum revenue. First Derivative Test: p = $24; dR/dp = $40 p = $26; dR/dp = -$40 Revenue is at maximum.

Demand for Oranges Waner pg. 343, #1 The weekly sales of Honolulu Red Oranges is given by q = 1000 – 20p. Calculate the price that gives a maximum weekly revenue and find this maximum revenue. Remember -- if E = 1 then revenue is maximized! Max revenue

Income Elasticity of Demand: Live Drama Waner pg. 344, #15 The likelihood that a child will attend a live theatrical performance can be modeled by q = 0.01(0.0078x2 + 1.5x + 4.1) (15  x  100). Here, q is the fraction of children with annual household income x thousand dollars who will attend a live dramatic performance at a theater during the year. Compute the income elasticity of demand at an income level of $20,000 and interpret the result. (Round your answer to two significant digits.)

Income Elasticity of Demand: Live Drama Waner pg. 344, #15

Income Elasticity of Demand: Live Drama Waner pg. 344, #15 Your interpretation: At a family income level of $20,000, the fraction of children attending a live theatrical performance is increasing by 0.77% per 1% increase in household income.