1. ## Find the Curvature

Let k = curvature.

Find k given r (t) = t i + t^2 j + (1/2)t^2 k

I found ||T'(t)|| to be (5t + 3)/(5t^2 + 1)^(3/2).
Is this correct?

I found ||r'(t)|| to be sqrt {5t^2 + 1}. Is this correct?

I must find k = ||T'(t)||/||r'(t)||.

The answer is (sqrt {5})/(1 + 5t^2)^(3/2).

Can someone set up the correct fraction to find k?

2. ## Re: Find the Curvature

Originally Posted by USNAVY
Let k = curvature.
Find k given r (t) = t i + t^2 j + (1/2)t^2 k
Can someone set up the correct fraction to find k?
$\displaystyle \kappa (t) = \frac{{\left\| {R'(t) \times R''(t)} \right\|}}{{{{\left\| {R'(t)} \right\|}^3}}}$.
I think that I gave you this before. I have always advised students that if looking for say $\kappa(t_0)$
first evaluate $R'(t_0)~\&~R''(t_0)$ then use the kappa formula. That makes the calculations easier.

3. ## Re: Find the Curvature

Originally Posted by Plato
$\displaystyle \kappa (t) = \frac{{\left\| {R'(t) \times R''(t)} \right\|}}{{{{\left\| {R'(t)} \right\|}^3}}}$.
I think that I gave you this before. I have always advised students that if looking for say $\kappa(t_0)$
first evaluate $R'(t_0)~\&~R''(t_0)$ then use the kappa formula. That makes the calculations easier.
The kappa formula you suggested works best for polynomials or so I've read. But, what about for trigonometric functions mixed with e^t?

4. ## Re: Find the Curvature

Originally Posted by Plato
$\displaystyle \kappa (t) = \frac{{\left\| {R'(t) \times R''(t)} \right\|}}{{{{\left\| {R'(t)} \right\|}^3}}}$.
I think that I gave you this before. I have always advised students that if looking for say $\kappa(t_0)$
first evaluate $R'(t_0)~\&~R''(t_0)$ then use the kappa formula. That makes the calculations easier.
I did apply the formula you suggested and found the answer easily