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

In summary, the conversation discusses finding the curvature for y=cosx and using a vector valued function to solve it. However, it is discovered that there is a formula (equation 14) that can be used instead of promoting the problem to a three dimensional vector problem.
  • #1
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]
 
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  • #2
Austin said:

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
 
  • #3
Dick said:
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
 

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

What is curvature of a function?

Curvature of a function is a measure of how much the curve of the function deviates from a straight line at a given point. It represents the amount of bending or turning of the curve at that point.

How is curvature calculated?

The curvature of a function at a point can be calculated by finding the second derivative of the function at that point and then dividing it by the square of the first derivative. This can also be expressed as the ratio of the radius of the osculating circle to the arc length of the curve.

How is curvature used in real life?

Curvature is used in various fields such as physics, engineering, and computer graphics. It is used to analyze and design curved structures, determine the trajectory of moving objects, and create smooth and realistic shapes in computer graphics.

What is the difference between positive and negative curvature?

A function has positive curvature at a point if the curve is bending in a concave manner, meaning it is curving towards the inside. Negative curvature, on the other hand, occurs when the curve is bending in a convex manner, curving away from the inside. A function can also have zero curvature, which means it is a straight line.

Can a function have varying curvatures?

Yes, a function can have varying curvatures at different points along the curve. This is because the curvature depends on the second derivative of the function, which can change as the function changes. A function can have positive curvature at one point, zero curvature at another point, and negative curvature at yet another point.

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