You are standing above the point (5,5)(5,5) on the surface z=15−(2x2+2y2)z=15−(2x2+2y2). (a) in which direction should you walk to descend fastest? (give your answer as a unit 2-vector.)

Since the surface is the top part of a sphere centred on the origin, the directions of steepest descent are any vector originating from the origin (0,0).

At (5,5), the direction would be dictated by the line joining (5,5) and the origin (0,0), i.e. along <5,5> or equivalently along <1,1>.

You can also go through vector calculus to arrive at the same result.

So the first thing we do is to multiply the top equation by 3. the reason i want to do this is i notice there will be a 6y on both equations and can possibly add/subtract them. so we get 6y = -3x + 27. adding that and the second equation we get 6x = 27-15 which is 12. divide by 6 and we get 2. ∴ x = 2. substituting 2 into the first equation we get 2y = -2 + 9 which is 7. divide by 2 and we get 7/2. ∴ y = 7/2, x = 2.

12