How twisty is it?

Calculus Level 3

Evaluate the curvature of the point \((0,0,0)\) on a twisted cubic curve \(\mathbf{r}(t) = t\ \mathbf{i} + t^2\ \mathbf{j} + t^3\ \mathbf{k}\).

Details and assumptions

The curvature of a curve given by a vector function r can be found as follows: \[\kappa(t) = \frac{|\mathbf{r}^{(1)}(t)\times\mathbf{r}^{(2)}(t)|}{|\mathbf{r}^{(1)}(t)|^3}\]

where \(\mathbf{r}^{(1)}(t)\) denotes the first derivative of \(\mathbf{r}(t)\), \(\mathbf{r}^{(2)}(t)\) denotes the second derivative of \(\mathbf{r}(t)\) and \(\mathbf{a} \times \mathbf{b}\) denotes the cross product of vectors \(\mathbf{a}\) and \(\mathbf{b}\).


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