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User:Michael B. Porter/Teaching/Curvature problem

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Find the curvature of the curve .

There are two formulas we learned for curvature:


There was a lot more calculation involved in using the second formula. Here is the solution using the first formula:

so

And here is the solution using the second formula:

The derivatives are a little tricky:

so

and finally,

<math>\kappa=\frac{|\textbf{T}'|}{|\textbf{r}'|}=\frac{\frac{6t^2}{9t^4+4t^2}}{\sqrt{9t^4+4t^2}}=\frac{6t^2}{(9t^4+4t^2)^\frac{3}{2}}