Question

Find the parametric equations for the tangent line to the curve with the given parametric equations at specified point.
x= e^t
y=te^t
z=te^(t^2)
(1,0,0)

Answer 1

Try a circle with parameter t=θ
x=cos(t)
y=sin(t)

dx/dt=-sin(t)
dy/dt=cos(t)

at t=0, (dx/dt,dy/dt)=(0,1)
at t=π/4, (dx/dt,dy/dt)=(-√2/2,√2/2)
...etc

Here:
x= e^t
y=te^t
z=te^(t^2)
(1,0,0)

Solve for t at the point (1,0,0) gives t=0 (the only way to get y=z=0).

dx/dt=e^t
dy/dt=(1+t)e^t
dz/dt=t(2+t)e^t

Substitute t=0 and find the equation of the line passing through (1,0,0) with the given slopes.