Answer and Step-by-step explanation:
SOLUTION:
Given function:
h (x, y, z) = ln (x2 + y2 – 1) + y +6z, p0(1, 1, 0)
Derivatives of the function:
∂h / ∂x = (1 / x2 + y2 -1). (2x)
= 2x / x2 + y2 -1
∂h/∂y = (1 / x2 + y2 – 1) (2y + 1)
= (2y / x2 + y2 -1) + 1
∂h / ∂z = 6
Now compute the partial derivatives:
∂h (1, 1, 0) / ∂x = 2(1) / 1 +1 -1
= 2
∂h(1, 1, 0) / ∂y =( 2(1) / 1 +1 -1) + 1
= 3
∂h (1, 1, 0) / ∂z = 6
Δh (1,1,0) = ( 2, 3, 6)
The definition of gradient is:
Δf (x, y, z) = (∂f / ∂x , ∂f / ∂y)
Put the coordinate values to arrive at solution:
Umax = Δh (1,1,0) / | Δh (1,1,0) |
= (2, 3, 6) / |2, 3, 6|
= (2, 3, 6) / √(22+33+62)
= (2, 3, 6) / √49
= (2, 3, 6) / 7
= (2/7 , 3/7, 6/7)
Umin = - U max = ( -2/7, -3/7, -6/7)
Compute direction by finding unit vector:
Du h max = Δh (1,1,0) . u max
= (2, 3, 6) . (2/7, 3/7, 6/7)
= 2(2/7) + 3(3/7) + 6(6/7)
= 4/7 + 9/7 +36/ 7
= 49 / 7
= 7
Du h min = -Du h max
= -7