Assignment:
1. With a yearly rate of 3 percent, prices are described as P = P0 (1.03)^t, where P0 is the price in dollars when t = 0 and t is time in years. If P0 is 1.2, how fast are prices rising when t = 15?
2. The value of a car is falling 10 percent per year so that if C0 is the purchase price of the car in dollars, its value after t years is given by V(t) = C0(0.9)^t. At what rate is its value falling when it is driven out of the showroom? How fast has the car depreciated after year 1?
3. The world's population is abou f(t) = 6*e^(0.013t) billion, where t is the time in years since 1999. Find f(0), f'(0), f(100 and f'(10). Using units, interpret your answers in terms of population.
4. For f(x) = x^3 - 18x^2 - 10x + 6, find the inflection point algebraically. Graph the function and confirm your answer.
5. For the function, f, given in the graph, (a) sketch f'(x); (b) Where does f'(x) change its sign? (c) Where does f'(x) have local maxima or minima?
6. Using your answer to previous problem as a guide, write a short paragraph which describes the relationship between the following features of a function f:
(a) the local maxima and minima f. (b) the points at which the graph f changes concavity. (c) the sign changes of f', (d) the local maxima and minima of f'.
7. The function f has a derivative everywhere and has just one critical point, at x = 3. In parts (a)-(d), you are given additional conditions. In each case decide whether x = 3 is a local maximum, a local minimum, or neither. Explain your reasoning. Sketch possible graphs for all four cases.
8. An apple tree produces, on average, 400 kg of fruit each season. However, if more than 200 trees are planted per kim^2, crowding reduces the yield by 1 kg for each tree over 200.
(a) Express the total yield from one square kilometer as a function of the number of trees on it. Graph the function.
(b) How many trees should a farmer plant on each square kilometer to maximize yield?