Given the vector r(t) = t i + t^2 j find T(t), T(1) a dn then N (1).
After this I am to sketch the plane curve and graph the vectors T(1) and N(1) at t = 1.
r(t) = t i + t^2 j
First we find the unit tangent vector
T(t) = (i+2tj)/ sqrt(1+4t^2)
Now use the quotient rule to find T '(t)
T'(t) = (1+4t^2)^1/2 (2j) - (i+2tj) 4t (1+4t^2)^ -1/2
Unit tangent and unit normal vectors are investigated. The solution is detailed and well presented.