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Curvature of Natural Log

  1. Sep 29, 2014 #1
    Let k(x) be the curvature of y=ln(x) at x. Find the limit as x approaches to the positive infinity of k(x). At what point does the curve have maximum curvature?

    You're supposed to parametrize the graph of ln(x), which I found to be x(t)=(t,ln(t)). And you're not allowed to use the formula with the second derivative, only k(t)=magnitude T'(t)/ magnitude v'(t).
    I have problem simplifying the formula for T'(t) and k(t).
  2. jcsd
  3. Sep 30, 2014 #2


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    If x= (t, ln(t)) then v= (1, 1/t) so |v| is [itex]\sqrt{1+ 1/t^2}[/itex] and [itex]T= (\frac{1}{\sqrt{1+ 1/t^2}}, \frac{1}{t^2\sqrt{1+ 1/t^2}})[/itex].

    To simplify [itex]t^2\sqrt{1+ 1/t^2}[/itex], take one of the "[itex]t^2[/itex]" inside the square root: [tex]t\sqrt{t^2+ 1}[/tex]. You can simplify the first component by multiplying both numerator and denominator by t: [itex]\sqrt{1}{\sqrt{1+ 1/t^2}}= \frac{t}{t\sqrt{1+ 1/t^2}}= \frac{t}{\sqrt{t^2+ 1}}[/itex]. So [itex]T= (\frac{t}{\sqrt{t^2+ 1}}, \frac{1}{t\sqrt{t^2+ 1}})[/itex]
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