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Find an x value for which the instantaneous velocity equals the average velocity

  1. Jun 18, 2012 #1
    1. The problem statement, all variables and given/known data
    if d(x)=x+2cosx represents the distance traveled by a particle from x=0 to x= 2pi.Find an x value in (0,2pi) for which the instantaneous velocity equals the average velocity.






    3. The attempt at a solution

    This question is a direct application of the mean value theorem (LMVT)
    D(x) = x + 2cos(x)
    ∴ D(0) = 0 + 2cos(0) = 2
    ∴ D(2pi) = 2π + 2cos(2π) = 2(π + 1)
    ∴ Average velocity = Total distance/Total time
    = [D(2π) - D(0)]/(2π - 0)
    = (2 + 2π - 2)/(2π)
    = 1 m/s
    Instantaneous velocity =D ' (x)
    ∴ v(t) = D'(x) = 1 - 2sin(x)
    Given that, to find 'c' such that D ' (c) = avg velocity = 1
    ∴ 1 - 2sin(c) = 1
    ∴ sin(c) = 0
    ∴ c = 0 or π
    But 'c' needs to lie in (0,2π)
    ∴ At c = π, the average velocity = instantanoeus velocity = 1

    ive been at this and im just wondering if it's right?
     
  2. jcsd
  3. Jun 18, 2012 #2

    tiny-tim

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    Science Advisor
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    hi indigo1! :smile:
    looks ok :smile:
     
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