Find the curvature?

Question

Find the curvature of r(t)=<e^-2t, 2t, 4>.

r'(t)=<-2e^-2t, 2, 0>

What to do next? Show your steps.

Answer 1

r'(t)=<-2e^-2t, 2, 0>

r''(t)=<4e^-2t, 0, 0>

|r'(t) x r''(t)| = |<-2e^-2t, 2, 0> x <4e^-2t, 0, 0>| = 8e^-2t

K = |r'(t) x r''(t)|/|r'(t)|^3 = 8e^-2t/(4e^-4t + 4)^(3/2) = e^-2t/(e^-4t + 1)^(3/2) <==Answer