Section 3.6 Let f (t) = t 2 . Find the relative rate of change of this function.

1. Section 3.6 • Let f (t) = t 2 . Find the relative rate of change of this function. • a. The relative rate of change RRC = f’ (t)/f (t). • RRC = 2t/t 2 = 2/t. • b. Evaluate the relative rate of change when t = 1. • RRC (1) = 2/1 = 2 • c. Evaluate the relative rate of change when t = 10. • RRC (1) = 2/10 = 0.2 ln x x 2

2. 2. Let Find the relative rate of change of this function. • a. The relative rate of change RRC = f’ (t)/f (t). • b. Evaluate the relative rate of change when t = 10. • RRC (10) = 2 10 = 20

3. Let f (t) = 25 (t – 1) . • a. Find the relative rate of change of this function. • This function is f (t) = 25 (t – 1) 1/2 and the relative rate of change RRC = f’ (t)/f (t). • b. Evaluate the relative rate of change when t = 6. • RRC (6) = 1/10 = 0.1

4. 4. ECONOMICS: National Debt. If the national debt of a country (in trillions of dollars) t years from now is given by the following function, find the relative rate of change of the debt 10 years from now. N (t) = 0.5 + 1.1 e 0.01t The relative rate of change RRC = f’ (t)/f (t). OR use your calculator. Graph the function and find RRC = f ’ (10)/f (10) RRC(10) = 0.01215688/1.715688 = 0.0071 = 0.71%

5. 5. GENERAL: Population. The population (in millions) of a city t years from now is given by • P (t) = 4 + 1.3 e 0.04t . • Find the relative rate of change of the population 8 years from now. • b. Will the relative rate of change ever reach 1.5%? The relative rate of change RRC = f’ (t)/f (t). You may use your calculator for this. See problem 4. Graph it and look. In about 15.3 years.

6. For the demand function, D (p) = 200 – 5p; • Find the elasticity of demand E (p) • Determine whether the demand is elastic, inelastic, or unit elastic at a price of p = 10. Demand is inelastic.

7. 7. For the demand function, D (p) = 300 – p 2; • Find the elasticity of demand E (p) • Determine whether the demand is elastic, inelastic, or unit elastic at a price of p = 10. Demand is unitary.

8. 8. For the demand function, D (p) = 300/p; • Find the elasticity of demand E (p) • Determine whether the demand is elastic, inelastic, or unit elastic at a price of p = 4. NOTE: D (p) = 300 p – 1 Demand is unitary.

9. 9. For the demand function, D (p) = 100/p 2 ; • Find the elasticity of demand E (p) • Determine whether the demand is elastic, inelastic, or unit elastic at a price of p = 10. NOTE: D (p) = 100 p – 2 Demand is elastic.

10. 10. AUTOMOBILE SALES - An AUTOMOBILE DEALER IS SELLING CARS AT A PRICE OF $12,000. The demand function is D(P) = 2(15 – 0.001P)2, where p is the price of a car. Should the dealer raise or lower the price to increase the revenue? Demand is elastic, lower the price.

11. 11. CITY BUS REVENUES – The manager of a city bus line estimates the demand function to be D (p) = 150,000 (1.75 – p) ½, where p is the fare in dollars. The bus line currently charges a fare of $1.25, and it plans to raise the fare to increase its revenues. Will the strategy succeed? Demand is elastic, the strategy will not work.

12. 12. OIL PRICES – A European oil-producing country estimates that the demand for its oil (in millions of barrels per day) is D (p) = 3.5 e – 0.06p, where p is the price of a barrel of oil. To raise its revenues, should it raise or lower its price from its current level of $120 per barrel? Demand is elastic, lower the price.