How Should I Calculate Curvature for Standard and Vector-Valued Functions?

[member 508213]

Homework Statement

For the first problem I am asked to find the curvature for y=cosx

We are studying vector value functions so I tried to rewrite this as a vector valued function so I can find the curvature. I just chose r(t)= <t,cost,0>. I found rI(t)=<1,-sint,0> and rII(t)=<0,-cost,0> and used the curvature formula of lv(t)xa(t)l/(lv(t)l^3 to find the curvature...but then I thought that if I had used a different vector valued function such as <t^3,cost^3,0> it would not work out the same so I am not sure how I should tackle this problem now.Second Problem is similar...It is asking "at what point does y= e^x have the maximum curvature" so I tried doing a similar approach as I did in the previous problem (even though I suspect my method is faulty) and got imaginary roots when solving for a point which makes me think the maximum curvature might be as t->infinity but I'm not sure.

I'm having difficulty with the concept of curvature here I think, especially because the function is not already written as a vector valued function (maybe their is a way to deal with this without vector valued functions, I'm not sure.)

Homework Equations

The Attempt at a Solution

Sorry kind of attempted solution in the 1st part...

Thank you[/B]

 

[Dick]

Homework Statement

For the first problem I am asked to find the curvature for y=cosx

We are studying vector value functions so I tried to rewrite this as a vector valued function so I can find the curvature. I just chose r(t)= <t,cost,0>. I found rI(t)=<1,-sint,0> and rII(t)=<0,-cost,0> and used the curvature formula of lv(t)xa(t)l/(lv(t)l^3 to find the curvature...but then I thought that if I had used a different vector valued function such as <t^3,cost^3,0> it would not work out the same so I am not sure how I should tackle this problem now.Second Problem is similar...It is asking "at what point does y= e^x have the maximum curvature" so I tried doing a similar approach as I did in the previous problem (even though I suspect my method is faulty) and got imaginary roots when solving for a point which makes me think the maximum curvature might be as t->infinity but I'm not sure.

I'm having difficulty with the concept of curvature here I think, especially because the function is not already written as a vector valued function (maybe their is a way to deal with this without vector valued functions, I'm not sure.)

Homework Equations

The Attempt at a Solution

Sorry kind of attempted solution in the 1st part...

Thank you[/B]

You don't have to promote the problem to a three dimensional vector problem. Look at equation 14) here. http://mathworld.wolfram.com/Curvature.html

 

[member 508213]

You don't have to promote the problem to a three dimensional vector problem. Look at equation 14) here. http://mathworld.wolfram.com/Curvature.html

Oh, thanks! Didn't realize there was such a formula. We've just been studying vector valued functions so putting it into 3d and using that formula was all i knew