Answer
\[ a_N = \frac{18}{{{[1 + 4x^2]}}^{3/2}} \]
Work Step by Step
Step 1 Given \[ y = x^2, \quad \text{with fixed speed } 3 \] Step 2 Since \[ \kappa = \frac{ \left| \frac{d^2y}{dx^2} \right|}{\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2}} = \frac{|2|}{{{[1 + 4x^2]}}^{3/2}} \] Then \[ a_N = \kappa\left(\frac{ds}{dt}\right)^2 = \frac{18}{{{[1 + 4x^2]}}^{3/2}} \] Result \[ a_N = \frac{18}{{{[1 + 4x^2]}}^{3/2}} \]
