Question 1-7

Application: 1. A certain radioactive substance decays exponentially. The percent, P, of the sub- stance left after t years is given by the formula P(z) = 100(1.2)-' . a. Determine the half-life of the substance. b. Find the instantaneous rate of decay at the end of the first half-life period. Mark Value: 4 2. The population of a town is decreasing at a rate of 1.8%/a. The current popula- tion of the town is 12 000. a. Write an equation that models the population of the town. b. Find the instantaneous rate of change in the population 10 years from now. c. Determine the instantaneous rate of change when the population is half its current population. Mark Value: 3 3. As a tornado moves, its speed increases. The function S(d) = 93logd + 65 relates the speed of the wind, S, in miles per hour, near the center of a tornado to the distance that the tornado has travelled, d, in miles. a. Calculate the average rate of change for the speed of the wind at the center of a tornado from mile 10 to mile 100. b. Estimate the rate at which the speed of the wind at the center of a tornado is changing at the moment it has travelled its 10" mile and it's 100" mile. Mark Value: 4 4. The height of a diver above the water is modeled by the function, h(t) = - 512 + 5t+ 10 , where t represents the time in seconds and h(t) represents the height in meters. Use the appropriate calculations for the rate of change in height to show that the diver reaches her maximum height at t = 0.5 s. Mark Value: 2 5. The demand function for snack cakes at a large bakery is given by the function 15 p(x) = 2x + 1 1x + 5 The x units are given in thousands of cakes, and the price per snack cake, p(x) is in dollars. a. Find the revenue function for the cakes. b. Estimate the marginal revenue for x = 0.75. What is the marginal revenue for x = 2? Mark Value: 2 6. State two intervals where the function y = 3 cos(4x) - 4 has an average rate of change that is; a. zero b. a negative value c. a positive value Mark Value: 3 7. Determine (f + g)(4) when f(x) = x2 - 3 and g(x) = - 6 x - 2 . For which value of x is (f + g)(x)) undefined? Explain why. What is the domain of (f + g)(x) and (f - g)(x)? Mark Value: 3