Suppose that ? is the intersection of the plane ?+?+3?=4 and thesurface ?^2??^2=?^2?8 . Note that this intersection contains thepoint (3,4,?1) . Verify the assumptions of the implicit functiontheorem at this point; then if ?(?)=(?,?) ? ( x ) = ( y , z ) bethe function from ???2 R ? R 2 verifying the conclusion of theimplicit function theorem, compute ??(3) J ? ( 3 ) using thetheorem. Verify your conclusion by explicitly solving for (?,?) ( y, z ) in terms of ? x and differentiating.
