A car travels along a straight highway from the town of Arborville at time hours to…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A car travels along a straight highway from the town of Arborville at time t=0 hours to the city of Brookton at time t=3 hours. The position of the car, measured in miles east of Arborville, is a twice–differentiable function s of time t (in hours). When the trip begins the odometer reads 12,480 miles, and when the car arrives in Brookton the odometer reads 12,660 miles. Throughout the interval 0≤t≤3 the car remains on the same highway, where the legal speed limit is 60 miles per hour.
(a) Calculate the average speed of the car for the interval 0≤t≤3. Express your answer with appropriate units.
(b) State the conditions on the position function s that allow a conclusion about the car’s instantaneous speed during the trip, and verify that these conditions are satisfied on the interval [0,3].
(c) Justify that there exists a time c with 0<c<3 at which the car’s instantaneous speed s′(c) is equal to the average speed found in part (a). Interpret this result in the context of the speed limit on the highway.
(d) Suppose a patrol officer’s radar recorded the car’s highest observed speed during the trip to be 70 miles per hour. Based on your work in parts (a)–(c), discuss whether the available information is sufficient to conclude that the driver necessarily exceeded the speed limit at some time. Fully explain your reasoning.