For the given position vectors rt compute the unit tangent v

For the given position vectors r(t) compute the unit tangent vector T(t) for the given value of t. If r(t) = (cos 4t, sin 4t) then T(pi/4)= if r(t) = (t2, t3) then T(1)= if r(t) = e4ti + e-tj + tk. then T(-1)=

Solution

r\' = 4e^4t i + -1e^-t j + 1k

Ir\'I = (16e^8t + e^-2 + 1)

T(t) = 4e^4t/(16e^8t + e^-2 + 1)i + -1e^-t/(16e^8t + e^-2 + 1) + 1/(16e^8t + e^-2 + 1)k

T(1) = 4e^-4/(16e^-8 + e² + 1) i + -1e¹/(16e^-8 + e² + 1)j + 1/(16e^-8 + e² + 1) k