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Understanding Arc Length

  1. Dec 19, 2011 #1
    1. The problem statement, all variables and given/known data

    This is probably very simple, but I'm teaching myself arc length via Paul's Online Calculus Notes and there's a simplification on the page:

    eq0013MP.gif

    I was wondering why the first [itex]Δx^2[/itex] was simplified to 1? I understand the other [itex]Δx^2[/itex] came out of the square root.
     
  2. jcsd
  3. Dec 19, 2011 #2

    vela

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    What you're thinking is akin to [itex]\sqrt{a^2+b^2} = \sqrt{a^2+1}\ b[/itex], which isn't correct. You can only pull something out of a radical if it's a factor. What Paul is doing is this:
    [tex]\sqrt{\Delta x^2 + [f'(x_i^*)]^2 \Delta x^2} = \sqrt{\Delta x^2(1 + [f'(x_i^*)]^2)}[/tex]Now because Δx2 is a factor under the radical, you can say
    [tex]\sqrt{\Delta x^2(1 + [f'(x_i^*)]^2)} = \sqrt{\Delta x^2}\sqrt{1 + [f'(x_i^*)]^2}= \Delta x \sqrt{1 + [f'(x_i^*)]^2}[/tex]
     
  4. Dec 19, 2011 #3
    Oh you are completely right, sorry for such a silly question and thank you for explaining.
     
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