For the curve given by rt 4 sin t t cos t cos t t sint 1

For the curve given by r(t) = [4 sin (t) - t cos (t) ,cos (t) + t sin(t) - 1t2 + 8], Find the unit tangent T(t) = [, , ] Find the unit normal N(t) = [, , ] Find the curvature k(t) =

Solution

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

r\'(t)=(t.sin(t) , t.cos(t) ,-2t)

||r\'(t)||=5.t

T(t)=(1/5.sin(t), 1/5.cos(t),-2/5)

N(t)=T\'(t)/||T\'(t)||

T\'(t)=(1/5.cos(t), -1/5.sin(t),0)

||T\'(t)||=(2/5)

N(t)=(5/2).cos(t), -(5/2).sin(t), 0)

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

=(2/5)/(5.t

=2/5t