• Dec 8th 2006, 05:45 AM
bobby77
Two runners start a race at same time and finish in a tie.use mean value theorem to show that at some time during the race they have the same speed
(Hint:Let s1(t) be the distance function of first runner and let s2(t) be the distance function for second runner.consider f(t)=s1(t) -s2(t) )
• Dec 8th 2006, 06:43 AM
CaptainBlack
Quote:

Originally Posted by bobby77
Two runners start a race at same time and finish in a tie.use mean value theorem to show that at some time during the race they have the same speed
(Hint:Let s1(t) be the distance function of first runner and let s2(t) be the distance function for second runner.consider f(t)=s1(t) -s2(t) )

Take the hint, then f(t) is continuous on [0,T] (where T is the time they finish),
and assume that s1 and s2 are differentiable on (0,T), then so is f and by the
mean value theorem there is some point c in (0,T) such that:

f'(c)=(f(0)-f(T))/(0-T)=0

(as f(0)=f(T)=0).

But f'(t)=s1'(t)-s2'(t), so at some point c in (0,T) s1'(c)=s2'(c) as required.

RonL