What went wrong in calculating the curvature for r(t)=<t^2,lnt,tlnt>?

[bl4ke360]

Homework Statement

r(t)=<t^2,lnt,tlnt>

Homework Equations

k= |T '(t)| / |r '(t)|

The Attempt at a Solution

My professor's answer sheet solved the problem using the other method, k(t)=|r '(t) x r ''(t)| / |r '(t)|^3
and that answer ends up being 0.3, while mine is 0.4. I can't see where I made any mistakes, and why I got 9 points marked off. Can someone explain?

 

[Dick]

Homework Statement

r(t)=<t^2,lnt,tlnt>

Homework Equations

k= |T '(t)| / |r '(t)|

The Attempt at a Solution

My professor's answer sheet solved the problem using the other method, k(t)=|r '(t) x r ''(t)| / |r '(t)|^3
and that answer ends up being 0.3, while mine is 0.4. I can't see where I made any mistakes, and why I got 9 points marked off. Can someone explain?

Yes. T(t) is supposed to be a unit vector for any value of t. You normalized it so that |T(1)|=1. But it's not a unit vector for any t. So your T'(t) comes out wrong.

 

[phyzguy]

I think what you did wrong was when you calculated T(t), you plugged in |r'(1)| in the denominator, when you should have left it as |r'(t)| until after you differentiated T(t) to get T'(t). By doing this you lost the terms related to changes in |r'(t)|. Does this make sense?