The curvature $\kappa$ of the parabola $y = x^2$ at the origin $(0, 0)$ is:

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### 1. Strategy & Theoretical Principles The curvature $\kappa$ of a 2D plane curve $y = f(x)$ is given by: $$\kappa = \frac{|y''|}{\left(1 + (y')^2\right)^{3/2}}$$ ### 2. Analytical Formulation & Calculation For $y = x^2$: - $y' = 2x$ - $y'' = 2$ At the origin $(0, 0)$: - $x = 0 \implies y'(0) = 0$ - $y''(0) = 2$ Substitute into the curvature formula: $$\kappa = \frac{|2|}{\left(1 + 0^2\right)^{3/2}} = \frac{2}{1} = 2.0$$ (The corresponding radius of curvature is $\rho = 1/\kappa = 0.50$). ### 3. Distractor Forensics - Candidate evaluating radius of curvature reports $0.50$ instead of curvature $\kappa = 2.0$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Distinguish curvature $\kappa$ from radius of curvature $\rho = 1/\kappa$. - **Standard Reference:** Thomas, G.B., *Calculus*, Chapter 13.

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