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Math Help - please help mean value

  1. #1
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    please help mean value

    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) )
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by bobby77 View Post
    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
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