A state patrol officer saw a car start from rest at a highway on-ramp. She radio

Question

A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to another officer 30 mi along the highway. When the car reached the location of the second officer 26 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 drive exceeded the speed limit?
A? -the officer can conclude the drive exceeded the speed limit becuase by the extreme value theorem, at some time c, the car was travelling at a speed of ? miles per hour.

B? -the officer can conclude the drive exceeded the speed limit because, by Rolle's theorem, at some time c, the car was travelling at a speed of ? miles per hour.

C? -the officer can conclude the driver exceeded the speed limit because by the mean value theorem,at some time c, the car was travelling at a speed of ? miles per hour.

Explanation / Answer

The correct answer is C. Consider the car's position (miles) as a function of time (hours). f(0)=0, f(26/60)=30. The slope of the secant between these two points is 30/(26/60) ~ 69 mph f(t) is a continuous and differentiable function, so by the mean value theorem at some point of time t0, the tangent line had slope f'(t0)=69 mph. But f'(t) is the speed of the car, so at some time the car was going 69 mph.

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