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Multivariable Calculus

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Find Curvature for Any Time

MVCALC-6VJZGS

Interpret:

$$\textbf{r}(t)=\frac{2}{3}(1+t)^{3/2}\,\textbf{i}+\frac{2}{3}(1-t)^{3/2}\,\textbf{j}+\sqrt{2}t\,\textbf{k}$$

...as the position of a moving object at time $t$. Find the curvature of $\textbf{r}(t)$ for any time $t$.

A

$\kappa=\frac{\sqrt{2}}{8\sqrt{1-t^2}}$

B

$\kappa=\frac{\sqrt{2}}{4\sqrt{1-t^2}}$

C

$\kappa=\frac{\sqrt{2}}{2\sqrt{1-t^2}}$

D

$\kappa=\frac{\sqrt{2}}{\sqrt{1-t^2}}$

E

$\kappa=\frac{\sqrt{2}}{4\sqrt{1+t^2}}$