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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 47 § 5.3 Applications of the Natural Logarithm Function to Economics

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 47 Relative Rates of Change Elasticity of Demand Section Outline

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 47 Relative Rate of Change DefinitionExample Relative Rate of Change: The quantity on either side of the equation is often called the relative rate of change of f (t) per unit change of t (a way of comparing rates of change for two different situations). An example will be given immediately hereafter.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 47 Relative Rate of ChangeEXAMPLE SOLUTION (Percentage Rate of Change) Suppose that the price of wheat per bushel at time t (in months) is approximated by What is the percentage rate of change of f (t) at t = 0? t = 1? t = 2? Since we see that

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 47 Relative Rate of Change So at t = 0 months, the price of wheat per bushel contracts at a relative rate of 0.22% per month; 1 month later, the price of wheat per bushel is still contracting, but more so, at a relative rate of 0.65%. One month after that (t = 2), the price of wheat per bushel is contracting, but much less so, at a relative rate of %. CONTINUED

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 47 Elasticity of Demand

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 47 Elasticity of DemandEXAMPLE SOLUTION (Elasticity of Demand) A subway charges 65 cents per person and has 10,000 riders each day. The demand function for the subway is (a) We must first determine E(p). (a) Is demand elastic or inelastic at p = 65? (b) Should the price of a ride be raised or lowered in order to increase the amount of money taken in by the subway?

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 47 Elasticity of Demand Now we will determine for what value of p E(p) = 1. CONTINUED Set E(p) = 1. Multiply by 180 – 2p. Add 2p to both sides. Divide both sides by 3. So, p = 60 is the point at which E(p) changes from elastic to inelastic, or visa versa.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 47 Elasticity of Demand Through simple inspection, which we could have done in the first place, we can determine whether the value of the function E(p) is greater than 1 (elastic) or less than 1 (inelastic) at p = 65. CONTINUED So, demand is elastic at p = 65. (b) Since demand is elastic when p = 65, this means that for revenue to increase, price should decrease.

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