Tangent Plane to a Level Surface
1. Find the tangent plane to the surface x2 + 2y 2 + 3z 2 = 36 at the point P = (1, 2, 3).
Answer: In order to use gradients we introduce a new variable
w = x2 + 2y 2 + 3z 2 .
Our surface is then the the level surface w = 36. Therefore the normal to surface is
Vw = U2x, 4y, 6z).
At the point P we have Vw|P = U2, 8, 18). Using point normal form, the equation of the
tangent plane is
2(x − 1) + 8(y − 2) + 18(z − 3) = 0, or equivalently 2x + 8y + 18z = 72.
2. Use gradients and level surfaces to ﬁnd the normal to the tangent plane of the graph of
z = f (x, y) at P = (x0 , y0 , z0 ).
Answer: Introduce the new variable
w = f (x, y) − z.
The graph of z = f (x, y) is just the level surface w = 0. We compute the normal to the
surface to be
Vw = Ufx , fy , −1).
At the the point P the normal is Ufx (x0 , y0 ), fy (x0 , y0 ), −1), so the equation of the tangent
plane is
fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ) − (z − z0 ) = 0.
We can write this in a more compact form as
(z − z0 ) =
∂f
∂x
(x − x0 ) +
0
∂f
∂y
(y − y0 ),
0
which is exactly the formula we saw earlier for the tangent plane to a graph.