Chapter 3.1, Problem 26E

As the speed of a train increases, the amount of power needed to maintain that speed increases, and the rate of power increase also increases. Let P = f(v) be the power, in megawatts, needed for the train to maintain a speed of v kilometers per hour. (a) Sketch a possible graph of f. (b) For each of the functions f, f', and f", decide whether the function is positive or negative, and explain your answers. (c) What requires a greater increase in power: increasing the train's speed from 100 to 150 kilometers per hour, or increasing the train's speed from 200 to 250 kilometers per hour? Explain, based on your answer to part (b).