Challanging Problem It is nessecary to solve quicly

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The discussion centers on a mathematical problem involving a car's motion along a real line, defined by a differentiable position function x(t) and its velocity v(t) = x’(t). The car starts and ends at a standstill, leading to the conclusion that the maximum velocity L must satisfy L > 1. The Mean Value Theorem (MVT) is utilized to demonstrate that there exists a point c in the interval [0,1] where the derivative of the position function equals 1, confirming that L > 0 and supporting the assertion that L > 1.

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  • Understanding of calculus, specifically the Mean Value Theorem.
  • Familiarity with differentiable functions and their properties.
  • Knowledge of basic kinematics and motion equations.
  • Ability to analyze velocity and acceleration in a mathematical context.
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  • Study the Mean Value Theorem in depth to understand its applications in motion problems.
  • Explore differentiable functions and their derivatives to strengthen calculus skills.
  • Investigate kinematic equations to relate position, velocity, and acceleration.
  • Practice solving similar problems involving motion on a real line with boundary conditions.
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Students of calculus, mathematicians, and anyone interested in the application of the Mean Value Theorem to real-world motion problems.

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Challanging Problem " It is nessecary to solve quicly"

I have a problem :
A car moves along the real line from x = 0 at t = 0 to x = 1 at t = 1, with
differentiable position function x(t) and differentiable velocity function v(t) = x’(t).
The car
begins and ends the trip at a standstill; that is v = 0 at both the beginning and the end of
the trip. Let L be the maximum velocity attained during the trip. Prove that at some time
between the beginning and end of the trip, l v’ l > L^2/(L-1).

Can you verify that L > 1 ?

Thankx
 
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You can show L > 1 using the Mean Value Theorem
 
By mean value theorem I can find that L>0
 
The MVT would say that there exists c in t = [0,1] such that
f'(c) = (x(1) - x(0)) / (t1 - t0) = (1-0) / (1-0) = 1.
 
That is my solution
How I can compleat it??
 

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You are from UOS
right?
 

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