Question

Find the curvature of y=sec x.

Answer 1

\[k(x)=\frac{ \left| y'' \right| }{ [1+(y')^2]^{3/2} }\]

Answer 2

\[y'=\sec x \tan x \]

Answer 3

\[y''=\sec x(\sec^2 x+\tan^2 x)\]

Answer 4

\[k(x)=\frac{ \left| \sec x(\sec^2 x+\tan^2 x) \right| }{ (1+\sec^2 x \tan^2 x)^{3/2} }\]

Answer 5

Is ^ that the correct answer? Do I need to simplify it more? If so, how?

Answer 6

@mathstudent55 @Nurali @Destinymasha @robtobey @ganeshie8 @SolomonZelman @freckles

Answer 7

@zepdrix

Answer 8

so far it looks good to me
let me think on the simplifying part

Answer 9

i honestly see no other way of writing it that would make it prettier

Answer 10

http://mathworld.wolfram.com/Curvature.html

Answer 11

http://math.kennesaw.edu/~plaval/math2203/curvature.pdf using theorem 156 on page 91

Answer 12

oh @robtobey I looked at that site and i didn't even see that formula on line 14
so I went to find it somewhere else :p

Answer 13

but I wonder if we need |y''| as the site I found suggests on top or y'' as mathworld suggests
but either way I do not see a way to simplify what @Idealist10 as posted

Answer 14

we could write everything in terms of sec(x)
and replace sec(x) with y :p

Answer 15

\[k=\frac{|y(y^2+(y^2-1))|}{(1+y^2(y^2-1))^\frac{3}{2}} \\ k=\frac{|2y^3-y|}{(y^4-y^2+1)^\frac{3}{2}}\]

Answer 16

does it say you are to write the curvature as a function of x ?
if that isn't given then I don't see why we couldn't write it in terms of y

Answer 17

@SithsAndGiggles

Answer 18

Refer to the attachment from Mathematica 9.

Answer 19

@ganeshie8 @thomaster @SolomonZelman @e.mccormick

Answer 20

@SithsAndGiggles

Answer 21

Seems like that as simple as it will get, though you can remove the absolute value bars from \(\sec^2x+\tan^2x\) since this factor is non-negative.

Answer 22

So what's the final answer @SithsAndGiggles ?

Answer 23

I would say you already have it.

Answer 24

So the answer is \[k(x)=\frac{ \left| \sec x(\sec^2 x+\tan^2 x) \right| }{ (1+\sec^2 x \tan^2 x)^{3/2} }\]

Answer 25

?

Answer 26

@SithsAndGiggles

Answer 27

Yes. Equivalent to that, you can also write
\[k(x)=\frac{ | \sec x|(\sec^2 x+\tan^2 x) }{ (1+\sec^2 x \tan^2 x)^{3/2} }\]
Same difference.

Answer 28

Thank you so much!