Answer
$$\frac{6}{5 \sqrt{10}}$$
Work Step by Step
We find:
$\kappa=\frac{\left| y^{\prime \prime}x^{\prime} - x^{\prime \prime}y^{\prime}\right|}{\left(y^{\prime 2}+x^{\prime 2}\right)^{3 / 2}} $
$\begin{array}{ll}3 e^{3 t}= x^{\prime}(t)& y^{\prime}(t)=-e^{-t} \\ 9 e^{3 t}= x^{\prime \prime}(t)& e^{-t}=y^{\prime \prime}(t) \\ 3=x^{\prime}(0) \quad -1=y^{\prime}(0) & \\ 9=x^{\prime \prime}(0) \quad 1=y^{\prime \prime}(0) & \\ \kappa=\frac{|-(-1)(9)+3(1)|}{(1+9)^{3 / 2}}=\frac{12}{10 \sqrt{10}}\end{array}$
