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Minimum distance of functions in a metric space

  1. Jun 17, 2010 #1
    1. The problem statement, all variables and given/known data
    A metric on C[0,1] is defined by:

    [tex]d(f,g) = ( \int_0^1 \! (f(x) - g_t(x))^2 \, dx )^{1/2}[/tex]

    Find t e R such that the distance between the functions [tex]f(x) = e^x - 1[/tex] and [tex]g_t(x) = t * x[/tex] is minimal.


    2. Relevant equations
    Given above


    3. The attempt at a solution
    The first thing I did was multiply the inner part of the integral out then evaluate it:
    [tex]( \int_0^1 \! (e^x-1) - (tx))^2 dx )^{1/2}[/tex]

    [tex]= ( \int_0^1 \! e^{2x} - 2e^x - 2te^xx + 1 + 2tx + t^2x^2 dx )^{1/2}[/tex]

    [tex]= (\frac{1}{2}e^2-\frac{1}{2} - (2e - 2) - 2t(e) + 1 + 2t(\frac{1}{2}) + t^2(\frac{1}{3}) )^{1/2}[/tex]

    [tex]= ( \frac{1}{2}e^2 + \frac{5}{2} - 2et + t + \frac{1}{3}t^2)^{1/2}[/tex]

    But I'm not sure I did that right, because now I don't know where to go from here. Any tips?
     
  2. jcsd
  3. Jun 17, 2010 #2

    Office_Shredder

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    So the distance between f and g depends on t. You might even say it's a function of t.... do you know how to minimize a function that has only a single variable?
     
  4. Jun 17, 2010 #3
    I would need to find the first derivative, find the roots and then check their sign in the second derivative. Can I do this without the square root around the entire function? Because with it the derivative is enormous and I'm not sure how I'd go about solving it for 0.
     
  5. Jun 17, 2010 #4

    Office_Shredder

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    It's a pretty common technique to use that [tex]\sqrt{a}<\sqrt{b}[/tex] is true exactly when [tex]a<b[/tex] (assuming everything is positive here), so to minimize a function [tex]\sqrt{f(x)}[/tex] it's enough to minimize [tex]f(x)[/tex] itself (the same thing holds true for maximizing)

    Even without using that though, by the chain rule the only difference is that you would have the function again in the denominator, and you can clear that out when you try to solve for the derivative equal to zero
     
  6. Jun 17, 2010 #5

    HallsofIvy

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    You should not need to integrate- you find the derivative of a vector by differentiating and by the "Fundamental Theorem of Calculus", the derivative of [itex]\int_a^x f(t)dt[/itex] is just f(x) itself.
     
  7. Jun 17, 2010 #6

    Office_Shredder

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    This isn't really a fundamental theorem of calculus situation
     
  8. Jun 17, 2010 #7
    Ok thanks, in the end I have

    [tex]\frac{2}{3}t - 2e + 1 = 0[/tex]

    [tex]t = 3e - \frac{3}{2}[/tex]
     
  9. Jun 17, 2010 #8

    Dick

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    Double check that. It looks like you are integrating x*e^x to get e. That's not right.
     
  10. Jun 17, 2010 #9
    Woops! You're right, I had x*e^x instead of x*e^x-e^x. Well that got rid of the nasty e now I'm left with t=3/2. Thanks!
     
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