Suppose you drive a car on a straight stretch of highway from toll booth to another toll booth {{MATH_91}} miles away in half of an hour. Must you have been driving at {{MATH_92}} miles per hour at some point? {{CMD_93}} Let {{MATH_94}} is the position of the car at time {{MATH_95}}, and {{MATH_96}} hours is the starting time with {{MATH_97}} hours being the final time, where {{MATH_98}}, and {{MATH_99}}. We can assume that {{MATH_100}} is continuous on {{MATH_101}} and differentiable on {{MATH_102}}. Now the Mean Value Theorem states that there is a time {{MATH_103}} such that {{MATH_104}} Since the derivative of position is velocity, this says that the car must have been driving at {{MATH_105}} miles per hour at some point. {{CMD_106}} {{CMD_107}}