2 You are given \(\mathrm { g } ( x , y , z ) = 6 x z - ( x + 2 y + 3 z ) ^ { 2 }\).
- Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 125\).
- Find the equation of the normal line to \(S\) at the point \(\mathrm { P } ( 7 , - 7.5,3 )\).
- The point Q is on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = 125 + h\), where \(h\) is small. Find the vector \(\mathbf { n }\) such that \(\overrightarrow { \mathrm { PQ } } = h \mathbf { n }\) approximately.
- Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
- Find the two points on \(S\) at which the tangent plane is parallel to \(x + 5 y = 0\).