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Prove that the difference between two expressions is always smaller than some amount

  1. Jun 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Firstly, I'd just like to point out that this is not actually a course related question. I have been trying to teach myself mathematics, and have been grappling with this for a couple of days. The book has no answer at the back for this particular question.

    Variables:
    [itex]0<x<1[/itex]
    [itex]0<b<a[/itex]

    Show that [itex](1-\frac{1}{2}x^{2})^{2} < 1-x < (1-\frac{1}{2}x)^{2}[/itex]. Hence show that if [itex]0<b<a[/itex], the error in taking [itex] a-\frac{b^{2}}{2a}[/itex] as an approximation to [itex]\sqrt{a^{2}-b^{2}}[/itex] is positive and less than [itex]\frac{b^{4}}{2a^{3}}[/itex].

    2. Relevant equations

    N/A

    3. The attempt at a solution
    The first part is relatively easy:

    Expansion of the inequality involving x gives:

    [itex]1-x-\frac{3}{4}x^{2}+\frac{1}{2}x^{3}+\frac{1}{4}x^{4}<1-x<1-x+\frac{1}{4}x^{2}[/itex]​

    Due to the fact fact that

    [itex]0<x<1[/itex]​

    The following is true:

    [itex]x^{n}>x^{n+1}[/itex]​

    This concept can be used to prove that

    [itex]\frac{3}{4}x^{2}>\frac{1}{2}x^{3}+\frac{1}{4}x^{4}[/itex]​

    The last part is more straightforward, it is simply due to the fact that:

    [itex]\frac{1}{4}x^{2}>0[/itex]​

    I have no idea how to connect the statement involving [itex]a[/itex] and [itex]b[/itex] to this set of inequalities, however from what I understand the initial statement is:

    [itex]0<a-\frac{b^{2}}{2a}-\sqrt{a^{2}-b^{2}}<\frac{b^4}{2a^3}[/itex]​

    I have attempted a bit of algebra jiggling, which gives:

    [itex]2a^{2}+b^{2}<3a^{4}[/itex]​

    Evidently, this is only true when [itex]a>1[/itex]

    Any help would be much appreciated! I would really love to put this to rest, so that I can move beyond page 34... there are about 450 more to go.
     
    Last edited: Jun 29, 2012
  2. jcsd
  3. Jul 1, 2012 #2
    Re: Prove that the difference between two expressions is always smaller than some amo

    Hint: Let [tex]x = \left( \frac{b}{a} \right) ^2[/tex]
     
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