(a) Explanation The given function is, f ( x , y , z ) = 8 − x y z . The gradient of f ( x , y , z ) is computed as follows. ∇ f ( x , y , z ) = ∂ f ∂ x ⋅ i + ∂ f ∂ y ⋅ j + ∂ f ∂ z ⋅ k = [ ∂ ∂ x ( 8 − x y z ) ⋅ i + ∂ ∂ y ( 8 − x y z ) ⋅ j + ∂ ∂ z ( 8 − x y z ) ⋅ k ] = [ ( 0 − ( 1 ) y z ) i + ( 0 − x ( 1 ) z ) j + ( 0 − x y ( 1 ) ) k ] = − y z i − x z j − x y k Thus, the gradient of f ( x , y , z ) is ∇ f ( x , y , z ) = 〈 − y z , − x z , − x y 〉 (b) To determine To find: The equation of the plane such that the set of all vectors orthogonal to the gradient with their tails at P ( 2 , 2 , 2 ) form a plane.

2nd Edition

ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Chapter 12.6, Problem 72E

(a)

To determine

To find: The gradient of f(x,y,z)=8−xyz and use it to evaluate at the point P(2,2,2) .

(b)

To determine

To find: The equation of the plane such that the set of all vectors orthogonal to the gradient with their tails at P(2,2,2) form a plane.

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Chapter 12.6, Problem 71E

Chapter 12.6, Problem 73E

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The gradient of f ( x , y , z ) = 8 − x y z and use it to evaluate at the point P ( 2 , 2 , 2 ) .

Solution Summary: The author explains how the gradient of f(x,y,z)=8-xyz is computed as follows.