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Mean Value Theorm

  1. Nov 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Let f be a function defined on an interval [a,b]. min f' <= f(b) - f(a) / b -a <= max f' where min f' and max f' refer to the mininmum and maximum values of f' on [a,b]

    Then the question is
    Using that inequality; estimate f(.1) if f'(x) = 1 / (1 + x^4 cos x) for 0 <= x <= .1 and f(0) = 1


    3. The attempt at a solution

    I really don't know where to start with this, I have min f'(x) <= f'(c) <= max f'.
    The book says to use a calculator too.
     
  2. jcsd
  3. Nov 16, 2009 #2

    Mark44

    Staff: Mentor

    For your interval, a = 0 and b = .1. You are given that f'(x) = 1/(1 + x^4 cos x). Clearly f'(0) = 1, and this will be the maximum value of f' on the interval in question. Use your calculator to get an approximate value for f'(.1).

    In the following inequality, min f' and max f' refer to the minimum and maximum of f' on the interval [0, .1].
    min f' <= (f(.1) - f(0))/.1 <= max f'
    ==> .1*(min f') <= f(.1) - f(0) <= .1*(max f')

    Now add f(0), which is given, to all members of this inequality to get a lower bound and an upper bound on f(.1).
     
  4. Nov 16, 2009 #3

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Hint: State the mean value theorem if a = 0 and b = .1 and use what you are given.
     
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