A hill has the shape of the surface z=100-x^2-4y^2+xy, where x and y are measure

Question

A hill has the shape of the surface

z=100-x^2-4y^2+xy,

where x and y are measured in meters, east is the positive x-direction, and north is the positive y-direction. You are standing on the hill above the point (4, 1).

a.) For each of the directions, north, northeast, east, and southeast, determine whether you will move up, move down, or remain at the same height if you begin to move in this direction

b.) Find all direction u-> such thati f you begin to move in the direction u->, you remain at the same height. Express u-> as a two-dimensional vector.

Explanation / Answer

Gradient of z =    ((rac{partial z}{partial x},rac{partial z}{partial y})=(-2x+y,-8y+x)=(-7,-4)) at (x,y)=(4,1)

Directional gradient of z in any direction u is (-7,-4) = (-7,-4).u

North = (0,1)

directional gradient = -4 < 0

So, we will go down

So, we will go down

So, we will go up

We will go neither up nor down in a direction (u,v) if (-7,-4).(u,v) = 0

=> u = 4,v = -7 or u=-4,v=7

Therefore, the direction is (4,-7) or (-4,7).

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