Mean Value Theorem, Rolle's Theorem

[phillyolly]

Homework Statement

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.]

Homework Equations

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

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.

 

[Dick]

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.

 

[phillyolly]

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?

 

[Office_Shredder]

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

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

 

[Dick]

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?

 

[phillyolly]

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.

 

[phillyolly]

Thank you a lot for your support.