Radius of Curvature Example

A curve  
\[y=f(x)\]
  has curvature given by  
\[\kappa = \frac{\frac{d^2 y}{dx^2}}{(1+(\frac{dy}{dx})^2)^{\frac{3}{2}}}\]
.
The radius of curvature is  
\[R= \frac{1}{\kappa}\]
.
Example:  
\[y= \sqrt{2x}= \sqrt{2} x^{\frac{1}{2}}\]
.
\[\frac{dy}{dx} =\sqrt{2} \frac{1}{2}x^{- \frac{1}{2}}= \frac{\sqrt{2}}{2}x^{- \frac{1}{2}}\]

\[\frac{d^2y}{dx^2} =\frac{- \sqrt{2}}{4}x^{- \frac{3}{2}}\]

\[\kappa=\frac{\frac{- \sqrt{2}}{4}x^{- \frac{3}{2}}}{(1+(\frac{\sqrt{2}}{2}x^{- \frac{1}{2}})^2)^{\frac{3}{2}}}\]

At  
\[x=2\]
,  
\[\kappa = \frac{\frac{- \sqrt{2}}{4}2^{- \frac{3}{2}}}{(1+(\frac{\sqrt{2}}{2}2^{- \frac{1}{2}})^2)^{\frac{3}{2}}} = \frac{\sqrt{5}}{4}\]
, and  
\[R= \frac{1}{ \kappa} = \frac{4 \sqrt{5}}{5}\]
..