Find parametric equations for the tangent line to the curve

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = t, y = e^-3t, z = 4t - t^4; (0, 1, 0) (x(t), y(t), z(t)) =

Solution

Let r(t) = < x , y , z > = < t , e^-3t , 4t - t⁴ >

Then r\'(t) = < 1 , -3e^-3t , 4 - 4t³ >

The parameter corresponding to the point (0, 1, 0) is t = 0, so r\'(0) = < 1 , -3 , 4 >

So the parametric equations for the tangent line to the curve with parametric equations x = t , y = e^-3t and z = 4 - 4t³

at the point ( 0 , 1 , 0 ) is    x = t , y = 1- 3t and z = 4t