Mean Value Theorem and the police A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to a patrol officer 30 mi along the highway. When the car reached the location of the second officer 28 min later, it was clocked going 60 mi/hr. The driver of the car was given a ticket for exceeding the 60-mi/hr speed limit. Why can the officer conclude that the driver exceeded the speed limit?

Verified step by step guidance

1. The Mean Value Theorem (MVT) states that for a continuous function on a closed interval [a, b] that is differentiable on the open interval (a, b), there exists at least one point c in (a, b) where the instantaneous rate of change (derivative) is equal to the average rate of change over [a, b].
2. In this scenario, the car's position as a function of time, s(t), is continuous and differentiable. The interval is from t = 0 to t = 28 minutes (or 28/60 hours).
3. The average speed of the car over the 30-mile journey is calculated by dividing the total distance by the total time: \( \text{Average speed} = \frac{30 \text{ miles}}{28/60 \text{ hours}} \).
4. Convert the average speed to miles per hour to compare it with the speed limit. If this average speed exceeds 60 mi/hr, then by the MVT, there must be at least one point in time where the car's instantaneous speed was greater than 60 mi/hr.
5. Since the average speed over the interval is greater than 60 mi/hr, the Mean Value Theorem guarantees that the car must have exceeded the speed limit at some point during the trip, justifying the officer's conclusion.

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) states that if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the instantaneous rate of change (derivative) equals the average rate of change over that interval. In this scenario, it implies that the car must have exceeded the average speed over the distance traveled.

Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. In this case, the car traveled 30 miles in 28 minutes, which can be converted to hours to find the average speed. This average speed is crucial for determining whether the car exceeded the speed limit of 60 mi/hr.

Speed Limit Enforcement

Speed limit enforcement relies on the principle that drivers must not exceed the posted speed limit at any point during their travel. Given the average speed calculated from the distance and time, if the average speed exceeds the speed limit, the law implies that the driver must have exceeded the limit at some point, justifying the ticket issued by the officer.