I thought I understood tangent planes until I saw tangent hyperplane and 3-d contour here. So a little bit of help will be most welcome. (i) and (ii) is no problem and agreed to the textbook answer. I am a bit stuck on (iii).

Question (Calculus, Binmore and Davies, 3.9(#5*))

Let

i. Find the gradient of f at the point

ii. Find the tangent hyperplane to the hypersurface

where

.

iii. Find the normal and the tangent plane to the contour

at

.

Answer.

i.

ii. To find tangent hyperplane, I want to use the formula

iii. To me, the equation for the countour

looks exactly the same as the equation to the surface in (ii) as u=0 in both cases.