Calculus Arc length

Question

Find the curvature of the curve defined by→r(t) = ⟨5cos(t),7sin(t)⟩ at t=2π/3.

Answer 1

First, compute the tangent vector r'(t) = 〈-5 sin(t), 7 cos(t)〉. You'll want to compute the magnitude of the tangent vector, ||r'(t)||, so you can write the unit tangent vector T(t) where

T(t) = r'(t)/||r'(t)||

Next, you'll compute the normal vector T'(t) and its magnitude ||T'(t)||.

Finally, you'll compute the curvature as

κ = ||T'(t)|| / ||r'(t)||