Question

Find the tangent line of the curve defined by the vector equation at the point t=0. u(t)=(1+t^3) i+te^(-t) j+e^(2t) k

Answer 1

You can take the derivative of u(t) and evaluate it at 0 and divide it by the magnitude of u'(0). From here, you have vector. With this vector and a point (that is t=0), so solve for (x,y,z), you have a vector parallel to the line and a point. This is all you need to parameterize a line!

Answer 2

u'(t) is ?

Answer 3

The derivative of the vector valued function that you provided^^