Two students Devyn and Riley raced to class down a straight hall. They start at the same time and finish in a tie. Was there a point during the race that Devyn and Riley were running at exactly the same velocity? {{CMD_161}} Let {{MATH_162}} represent Devyn's position with respect to time, and let {{MATH_163}} represent Riley's position with respect to time. Let {{MATH_164}} be the starting time of the race, and {{MATH_165}} be the end of the race. Set {{MATH_166}} Note, we may assume that {{MATH_167}} and {{MATH_168}} are continuous on {{MATH_169}} and that they are differentiable on {{MATH_170}}. Hence the same is true for {{MATH_171}}. Since both runners start and finish at the same place, {{ENV_172}} In fact, this shows us that the average rate of change of {{MATH_173}} is {{MATH_174}}. Hence by the mean value theorem, there is a point {{MATH_175}} {{MATH_176}} with {{MATH_177}}. However, {{MATH_178}} Hence at {{MATH_179}}, {{MATH_180}} this means that there was a time when they were running at exactly the same velocity. {{CMD_181}} {{CMD_182}}