..Ex1 Am stuck

1. During the winter, a smart thermostat is set to adjust the temperature on a 24-hour cycle. It is programmed to reach a maximum of 68"F at 2PM, and a minimum of 60"F at 2 AM. Let T(t) be the thermostat setting t hours after midnight. (a) Find a possible formula for T(t). (b) Find a value of t so that the temperature at that time is 67"F. 2. A sample of a radioactive element decays at a rate of 12% per year. At f = 0 (measured in years) the sample has a mass of 50 kg. (a) Find the formula for the function f, where M = f(t) is the mass (in kg) of the sample at time t. (b) How long does it take for the mass of the sample to reduce by 80%? 3. On January 1, 2000, the population of a certain city was 2 million people. For the next twenty years, the population increased at a constant rate of 0.5 million people per year. Starting on January 1, 2020, the population began decreasing at a rate of 20% per year. (a) Find a formula for P(t), the population of the city (in millions) t years after January 1, 2000. (b) When will the population be 1 million? 4. Let P = f(t) be the population of the United States, in millions, t years after January 1, 1800. Assume f is modeled by an increasing function. Write a sentence that explains the practical meaning of each mathematical statement. (a) f(100) = 76 (b) f-(10) = 21 (c) f(10) = f(0) +6 5. Determine the domain of the function f(x) = In(15r + 2| -3). Express your answer in interval notation. 6. Calculate the average rate of change of f(x) = x] + 3x over the interval [0, 3]. 7. The graph y = f(x) is given below. Determine all values of a such that lim f (x) does NOT exist.