Find the curvature of the function y = 5/x at x = -4. Find...

Find the curvature of the function y = 5/x at x = -4. Find the radius of curvature at x = -4. If the radius is infinite, enter 'oo' for ?. Find the center of curvature at x = -4. If the center of curvature does not exist, enter "DNE". Hint if needed: Multiply the normal vector by the radius of curvature, and then add this vector quantity to the position vector. The components of the result will be the coordinates of the center of curvature. :) Note: For very large or very small numbers you will probably wish to use scientific notation to enter your answers.

Transcript

00:04 I hear this question you are given a function, right? and you asked to find the curvature function at this point, right? now the curvature of function i'm going to call the kappa, which is a definition given by the second order derivative of this function, divided by 1 plus the first order derivative squared and take the part of 3 to 2, right? so, you know, the set, we can find the second, the first, let's find the first, order derivative of this function, that's of course given by minus 5 over x squared, right? and the second order derivative is given by actually 10 times x to the minus 3.

00:46 So what we need to do is just to plug in the values, right, for these two quantities and to this expression of a curvature.

00:56 So if you plug in the values, you found this to be given by, and i found this to be about minus 0 .16.

01:08 Well, this is about, that's given by, of course, minus 5 divided by 16.

01:14 16, that would actually give us, so it's minus 5, minus 5 divided by 16, and that's given by minus 0 .3 1.

01:34 So that would be this.

01:35 And you plug this values into this expression, you found this to begin by minus 0 .16, right? and then divided by minus 0 .16 rate and divided by 1 plus 1 prime squared.

02:04 Oh, actually, i'm a mistake.

02:06 So i need to do it more careful.

02:09 I think that's, yeah, the second order is minus 0 .6.

02:15 31, it's not, it's not like minus 0 .31, right? that's what we found divided by 1 plus 0 .16 squared, right? and then to the power of 1 .5, right? and i would find this actually to begin by minus 0 .298 basically, right? so that's minus 0 .298.

02:47 The radius curvature, that's basically the inverse of this.

02:50 So the universe of this is going by 1 divided by 0 .298, which actually gives you 3 .356.

03:03 I found the center of the curvature at this point.

03:07 But the center of this, you know, it has a, it has a, it has curvature of this, right? so basically what's happening is there, suppose this, you look at this curve, right, and it has a curvature on this side, basically, right? and it has a radius.

03:25 This point actually, of course, has minus four.

03:28 What is the wire point? the wire, of course, minus five over fours and minus 1 .25, right? and then the curvature will be somewhere here...

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