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Math Help - When is instantaneous and average rate of change the same?

  1. #1
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    When is instantaneous and average rate of change the same?

    Given the function f(x)= (x-2) / (x-5), determine an interval and a point where the ave. R.O.C and the instantaneous R.O.C are equal.


    IROC = [ f(x+h) - f(x) ] /h
    AROC = f(x2) - f(x1) / x2 - x1

    I know that in order to satisfy this, the x's must satisfy both equations when using the AROC and IROC formulas.
    I'm not sure where to start, though.
    I could do guess and check, but I need help for an algebraic method.
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  2. #2
    Senior Member abhishekkgp's Avatar
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    This can be seen from Lagrange's Mean Value Theorem.
    in fact there should exist infinitely many such intervals.
    Choose any interval [a,b] which satisfies continuity of the function in [a,b] and differentiability in (a,b) then you will always get a c \epsilon (a,b) such that:
    f'(c)=[f(b)-f(a)]/(b-a)

    eg: a=6, b=7
    AROC=(2.5-4)/(1)=-1.5
    f'(x)= (-3)/[(x-5)^2]=-1.5
    then we have (x-5)^2=2
    so x=5+sqrt(2) which of course belongs to [6,7]
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    Senior Member abhishekkgp's Avatar
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    deleted post.
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