Answer
$\frac{\sqrt{2}}{3}$
Work Step by Step
See Theorem $12.5 .2 b$
\[
\begin{array}{l}
\kappa(t)=\frac{\left\|\mathrm{r}^{\prime}(t) \times \mathrm{r}^{\prime \prime}(t)\right\|}{\left\|\mathrm{r}^{\prime}(t)\right\|^{3}} =\qquad \kappa(0)\\
\begin{array}{l}
\mathrm{r}^{\prime}(t)=\left\langle 2,2 \mathrm{e}^{2 t},-2 e^{-2 t}\right\rangle, \quad \mathrm{r}^{\prime}(0)=\{2,2,-2\rangle \\
\mathrm{r}^{\prime \prime}(t)=\left\langle 0,4 c^{2 t}, 4 c^{-2 t}\right\rangle, \quad \mathrm{r}^{\prime \prime}(0)=\langle 0,4,4\rangle \\
\mathrm{r}^{\prime}(0) \times \mathrm{r}^{\prime \prime}(0)=\left|\begin{array}{ccc}
i & j & k \\
2 & 2 & -2 \\
0 & 4 & 4
\end{array}\right|=(8+8) \mathrm{i}-8 j+8 \mathrm{k} \\
\frac{\|\langle 16,-8,8\rangle\|}{(12)^{3 / 2}}=\frac{8 \sqrt{6}}{12 \sqrt{12}}=\frac{\sqrt{2}}{3}=\kappa(0)
\end{array}
\end{array}
\]
