Calculus parametric

Question

Find parametric equations x,y,x for the line that is tangent to the given curve at the given parameter value

r(t)=(3 sin t) i+(t^4−3 cos t) j+(4 e^2t) k​, t = 0

                            

Answer 1

In a nutshell

r(t)=(3 sin t) i+(t^4−3 cos t) j+(4 e^2t) k​, t = 0

r'(t)=(3 cos t) i+(4t³ + 3 sin t) j+(8e^2t) k​,

The directing vector of the tangent line at t = 0 is

u = r'(0) = < 3 , 0 ,8 >

and therefore the parametric equations of the tangent line at the point P (0 , -3, 4 ) are

x = 3t

y = -3

z = 4 +8 t