1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Mean Value Theorem, Rolle's Theorem

  1. Jul 6, 2010 #1
    1. The problem statement, all variables and given/known data

    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 g and h are
    the position functions of the two runners.]

    2. Relevant equations

    If this is ever helpful:
    Mean Value Theorem:
    f ' (c)= [f(b)-f(a)]/[b-a]


    3. The attempt at a solution

    We know that if a function f is continuous on an interval [a,b] and differentiable on (a,b), and f(a) = f(b) = 0, then there is some point c in (a,b) such that f'(c) = 0.

    The velocity equation is
    f'(t)=g'(t)-h'(t)
    g^' (t)-h(t)=0
    g'(t)=h'(t)


    I am not at all sure I am doing it correctly - my solution is too simple.
     
  2. jcsd
  3. Jul 6, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You've got it right and too simple but you aren't expressing yourself very well either. The MVT tells you there is a c such that f'(c)=0. It doesn't tell you f'(t)=0 for all t. Explain that to me again.
     
  4. Jul 6, 2010 #3
    Hi, I don't know how to explain your question. Because I take Calculus I as an online class, I have nobody to explain this to me. May you please help me out to understand this problem?
     
  5. Jul 6, 2010 #4

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    He's talking about here, where you seem to assert the derivative is zero for every value of t.
     
  6. Jul 6, 2010 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Apply the mean value theorem to the problem. If the start time is a and the finish time is b, what does the mean value theorem tell you? Start with the basics, why are f(a)=0 and f(b)=0?
     
  7. Jul 6, 2010 #6
    Hi! I got the answer!

    At time t=0, f(start_time) = 0 because at the starting point, both runners are at the same spot. Similarly, at the finishing line, f(finish_time) = 0 because in the end the runners finish tied. Knowing that in the Rolle's theorem at some time c between 0 and the finish time, f'(c) = 0, we can conclude that at some time c, the difference in their velocities is 0, which means that at time c, they essentially have the same speed.
     
  8. Jul 6, 2010 #7
    Thank you a lot for your support.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Mean Value Theorem, Rolle's Theorem
  1. Mean Value Theorem (Replies: 2)

Loading...