Answer
$\kappa = 2$
Work Step by Step
Step 1 To find the curvature at the given point, which is the origin $(0,0)$, we will use the relation between the osculating circle's radius $\rho$ and the curvature $\kappa$ at a point given by: \[ \kappa = \frac{1}{\rho} \] Step 2 From the figure, we see the diameter of the circle is one, which means the radius is $0.5$. To find the curvature, we will just plug in $\rho = 0.5$ into the formula: \[ \kappa = \frac{1}{\rho} \Rightarrow \kappa = \frac{1}{0.5} = 2 \] The curvature at the origin is $\kappa = 2$. Result \[ \kappa = 2 \]
