Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider f(t)=g(t)−h(t) , where and are the position functions of the two runners.

Question

Steacensen69 · Accepted Answer

Step 1f(t)=g(t)−h(t)&amp;nbsp;- the difference positions of two runners.g(t),&amp;nbsp;h(t)&amp;nbsp;are continuons and differentiable. So,&amp;nbsp;f(t)&amp;nbsp;is too continuons and differentiable.We want to show that there was no velocity difference between the runners at some point during the race.Find the first derivative of the equation&amp;nbsp;f(t)=g(t)−h(t).f′(t)=g′(t)−h′(t)Consider the intreval&amp;nbsp;(t0,t1), where is the start time&amp;nbsp;t0&amp;nbsp;and the finish time&amp;nbsp;t1.Is given that runners start a race at same time and finis in a tie.So,&amp;nbsp;g(t0)=h(t0)&amp;nbsp;and&amp;nbsp;g(t1)=h(t1).We have:&amp;nbsp;f(t0)=0&amp;nbsp;and&amp;nbsp;f(t1)=0We can apply the Rolle&#039;s theorem, because we have:&amp;nbsp;f(t0)=f(t1)&amp;nbsp;and&amp;nbsp;f(t)&amp;nbsp;is continuons and differentiable.So, there exists some cc that such&amp;nbsp;f′(c)=0f′(c)=g′(c)−h′(c)=0⇒g′(c)=h′(c)For&amp;nbsp;c=t,&amp;nbsp;t∈(t0,t1)⇒g′(t)=h′(t)AnswerApply the Rolle&#039;s theorem.

Two runners start a race at the same time and finish in a tie. Prove t

Answered question

Answer & Explanation

New Questions in Differential Calculus