MATH 210 Study Guide - Final Guide: Horse Length, Parallelogram, Directional Derivative
Document Summary
Consider the curve r (t) =(cid:0)t, t2, 2. 3 t3(cid:1). (1) find the arc length of r (t) from t = 0 to t = 1. (2) find the curvature at t = 1. Solution: (1) the derivative of r (t) is r (t) = h1, 2t, 2t2i. The magnitude of r (t) is computed and simpli ed as follows: We can now compute the arc length from t = 0 to t = 1. R (t)(cid:12)(cid:12)(cid:12)(cid:12) = 1 + 4t2 + 4t4 (cid:12)(cid:12)(cid:12)(cid:12) 3 (2) the curvature formula we will use is: The rst two derivatives of r (t) =(cid:0)t, t2, 2. We now evaluate the derivatives at t = 1. R (1) r (1) = (cid:12)(cid:12)(cid:12)(cid:12) R (1) r (1) = [(2)(4) (2)(2)] [(1)(4) (0)(2)] + k [(1)(2) (0)(2)] R (1) r (1) = 4 4 + 2 k. R (1) r (1) = h4, 4, 2i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
