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Math Help - Two functions with C^n continuity over an interval:Which one's the smoothest?

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    Senior Member x3bnm's Avatar
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    Two functions with C^n continuity over an interval:Which one's the smoothest?

    According to the definition of Smooth function - Wikipedia, the free encyclopedia:

    Functions that have derivatives of all orders are called smooth.


    Suppose I've two different functions f(x) = 2x^2 and g(x) = 4x^2 of continuity C^n over an interval [0,4]. Because both of these functions has derivatives of all order these two functions are smooth.

    What I like to know is which one of these two are smoothest?

    Is it possible to kindly help me detect which function( f(x) or g(x)) is the smoothest over the given interval?
    Last edited by x3bnm; July 6th 2012 at 02:39 PM.
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    Re: Two functions with C^n continuity over an interval:Which one's the smoothest?

    How are you defining "smoothest"? Some times texts will say that if f is differentable "n" times and g is differentiable "m" times, with n> m, then f is "smoother" than g. But you are using "smooth" to mean what those texts would call "infinitely smooth" so that does not apply here.
    Last edited by HallsofIvy; July 7th 2012 at 01:19 PM.
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    Senior Member x3bnm's Avatar
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    Re: Two functions with C^n continuity over an interval:Which one's the smoothest?

    Quote Originally Posted by HallsofIvy View Post
    How are you defining "smoothest"? Some times texts will say that if f is differentable "n" times and g is differentiable "m" times, with n> m, then f is "smoother" than g. But you are using "smooth" to mean what those texts would call "infinitely smooth" so that does not apply here.
    Thanks HallsofIvy for answering my question. Yes I meant f(x) = x^3 is smoother than g(x) = x^2 because x^3 is one time more differentiable than x^2. I found the answer. Originally my problem was like this:


    Suppose I've a acceleration vs. time graph where x means time in millisecond and y means acceleration in  \text{inch/ms}^2

    I've 2 sets of data with 4 points in each set. I've to find the approximated curve in each set that is smoother than the other(means one is more differentiable than other).

    I found the solution by the way. All I have to do is curve fit in each set using Linear Algebra. Thanks for helping me see the way.
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