Question

You are standing above the point (2,5) on the surface z=30−(2x2+2y2). (a) in which direction should you walk to descend fastest? (give your answer as a unit 2-vector.) direction = (b) if you start to move in this direction, what is the slope of your path? slope =

Answer 1

(a) Use u = -∇z(2, 5)/||∇f(2, 5)|| 

= -<-4x, -4y>/||<-4x, -4y>|| {at (2, 5)} 
= <8, 20>/√((-8)^2 + (-20)^2) 
= <2, 5>/√29

(b) Du f(2, 5) = ∇z(2, 5) · u 
= ∇z(2, 5) · -∇z(2, 5)/||∇f(2, 5)|| 
= -||∇z(2, 5)||^2 / ||∇f(2, 5)|| 
= -||∇z(2, 5)|| 
= -4√29