Tangent plane parallel to given direction

A question is this type if and only if it asks to find points on a surface where the tangent plane or normal line has a specified direction (e.g., parallel to an axis or vector).

1 questions · Challenging +1.3

8.05d Partial differentiation: first and second order, mixed derivatives8.05f Nature of stationary points: classify using Hessian matrix
Sort by: Default | Easiest first | Hardest first
OCR MEI FP3 2007 June Q2
24 marks Challenging +1.3
2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
  3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
  4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.