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Very basic question

  1. Dec 15, 2013 #1
    h(t) = [itex]\sqrt{t - 1}[/itex]
    h(t + Δt) = [itex]\sqrt{t + Δt - 1}[/itex]
    h(t + Δt) - h(t) = [itex]\sqrt{t + Δt - 1}[/itex] - [itex]\sqrt{t - 1}[/itex]

    So far so good. This is where I get confused:

    [itex]\frac{h(t + Δt) - h(t)}{Δt}[/itex] = [itex]\frac{1}{\sqrt{t + Δt - 1} - \sqrt{t - 1}
    }[/itex]

    I don't understand why dividing both sides by Δt allows for this statement to be true. Can someone explain this to me? It would be very much appreciated.

    Thanks a lot.
     
  2. jcsd
  3. Dec 15, 2013 #2

    lurflurf

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    Homework Helper

    That should be
    [itex]\frac{h(t + Δt) - h(t)}{Δt}[/itex] = [itex]\frac{1}{\sqrt{t + Δt - 1} + \sqrt{t - 1}
    }[/itex]
    to show it observe that
    $$\frac{h(t + Δt) - h(t)}{Δt} = \frac{\sqrt{t + Δt - 1} - \sqrt{t - 1}
    }{Δ t} \cdot \frac{\sqrt{t + Δt - 1} + \sqrt{t - 1}}{\sqrt{t + Δt - 1} + \sqrt{t - 1}} =\frac{1}{\sqrt{t + Δt - 1} + \sqrt{t - 1}
    }$$
    That is often called rationalizing the numerator, but in this case we don't really care that the numerator is rational we want to have a sum instead of a difference.
     
  4. Dec 15, 2013 #3
    Thank you very much for your response, sorry for the typo.
     
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