Mean Value Theorem, Rolle's Theorem

In summary: I am understanding it now.In summary, the problem asks to prove that at some point during a race, two runners will have the same speed if they start and finish at the same time. This can be proven by applying the Mean Value Theorem and Rolle's Theorem, which state that if a function is continuous and differentiable on an interval and has the same values at the endpoints, there will be a point within the interval where the derivative is equal to 0. In this case, the function represents the difference in position between the two runners, and the derivative being equal to 0 means that at that point, the runners have the same speed.
  • #1
phillyolly
157
0

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.
 
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  • #2
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.
 
  • #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?
 
  • #4
phillyolly said:
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.
 
  • #5
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?
 
  • #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.
 
  • #7
Thank you a lot for your support.
 

1. What is the Mean Value Theorem?

The Mean Value Theorem is a mathematical theorem that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point within the interval where the slope of the tangent line is equal to the average rate of change of the function.

2. What is the significance of the Mean Value Theorem?

The Mean Value Theorem is significant because it provides a way to link the instantaneous rate of change of a function (represented by the derivative) to the average rate of change over a given interval. It is also used to prove other important theorems in calculus.

3. What is Rolle's Theorem?

Rolle's Theorem is a special case of the Mean Value Theorem that states that if a function is continuous on a closed interval, differentiable on the open interval, and has the same values at the endpoints of the interval, then there exists at least one point within the interval where the derivative is equal to zero.

4. How is Rolle's Theorem used in calculus?

Rolle's Theorem is used to prove that a function has a critical point (where the derivative is equal to zero) within a given interval. It is also used to prove the Intermediate Value Theorem and the Second Derivative Test.

5. Can Rolle's Theorem be applied to all functions?

No, Rolle's Theorem can only be applied to continuous functions on a closed interval that are differentiable on the open interval. Some functions, such as those with discontinuities or sharp turns, do not meet these criteria and therefore cannot be analyzed using Rolle's Theorem.

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