Question

in what direction from the point (1, 3, 2) is the directional derivative of Φ=2xz-y^2 a maximum? what is the magnitude of this maximum ? ​

Answer 1

Answer:

in what direction from the point (1, 3, 2) is the directional derivative of Φ=2xz-y^2 a maximum? what is the magnitude of this maximum ?

Answer 2

Answer:

Step-by-step explanation:

Gradient: For a scalar function ϕ(x, y, z), the gradient is the maximum rate of change which is given by- g r a d ϕ = ∇ ϕ ⇒ ( ^ i ∂ ∂ x + ^ j ∂ ∂ y + ^ k ∂ ∂ z ) ϕ gradϕ=∇ϕ⇒(i^∂∂x+j^∂∂y+k^∂∂z)ϕ Directional derivative: It gives the rate of change of scalar point function in a particular direction. The maximum magnitude of the directional derivative is the magnitude of the gradient. The directional derivative of ϕ is maximum along normal to the surface. a̅  = ∇ϕ  Calcaulation:  Given: ϕ = 2xz - y2 g r a d ϕ = ∇ ϕ ⇒ ( ^ i ∂ ∂ x + ^ j ∂ ∂ y + ^ k ∂ ∂ z ) ϕ gradϕ=∇ϕ⇒(i^∂∂x+j^∂∂y+k^∂∂z)ϕ ∇ ϕ = ( ^ i ∂ ∂ x + ^ j ∂ ∂ y + ^ k ∂ ∂ z ) ( 2 x z   −   y 2 ) ∇ϕ=(i^∂∂x+j^∂∂y+k^∂∂z)(2xz − y2) ∇ϕ = 2zî - 2yĵ + 2xk̂ ∇ϕ(1,3, 2) = 4î - 6ĵ + 2k̂Read more on Sarthaks.com - https://www.sarthaks.com/2674044/directional-derivative-of-2xz-y2-at-the-point-1-3-2-becomes-maximum-in-the-direction-of