# 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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