Curvature of Natural Log

1. Sep 29, 2014

lovexmango

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. Sep 30, 2014

HallsofIvy

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

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