Explanation Given: The plane x 4 + y 4 + z 4 = 3 Concept Used: Gradient of a function f ( x , y , z ) at a point P ( x 0 , y 0 , z 0 ) is: ∇ f = f x ( x 0 , y 0 , z 0 ) i + f y ( x 0 , y 0 , z 0 ) j + f z ( x 0 , y 0 , z 0 ) k The standard equation of tangent plane to a surface S with equation f ( x , y , z ) = C at point P ( x 0 , y 0 , z 0 ) is f x ( x 0 , y 0 , z 0 ) ( x − x 0 ) + f y ( x 0 , y 0 , z 0 ) ( y − y 0 ) + f z ( x 0 , y 0 , z 0 ) ( z − z 0 ) = 0 Calculation: To find a unit vector, normal to the surface x 4 + y 4 + z 4 = 3 at ( 1 , − 1 , − 1 ) First, let f ( x , y , z ) = x 4 + y 4 + z 4 Now, find the gradient vector of this function at the point ( 1 , − 1 , − 1 ) , since gradient vector at any point is perpendicular to the surface at that point. Gradient of a function f ( x , y , z ) is ∇ f = f x ( x 0 , y 0 , z 0 ) i + f y ( x 0 , y 0 , z 0 ) j + f z ( x 0 , y 0 , z 0 ) k Thus, first find f x ( 1 , − 1 , − 1 ) , f y ( 1 , − 1 , − 1 ) , and f z ( 1 , − 1 , − 1 ) as f x ( x , y , z ) = ∂ ∂ x ( x 4 + y 4 + z 4 ) = 4 x 3 , f y ( x , y , z ) = ∂ ∂ y ( x 4 + y 4 + z 4 ) = 4 y 3 and f z ( x , y , z ) = ∂ ∂ z ( x 4 + y 4 + z 4 ) = 4 z 3 Thus, f x ( 1 , − 1 , − 1 ) = 4 ( 1 ) 3 = 4 , f y ( 1 , − 1 , − 1 )

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 11.6, Problem 18PS

To determine

To find a unit vector that is normal to the surface

x4+y4+z4=3 at (1,−1,−1)

And to find the standard form of the equation of tangent plane to this surface at above point

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Chapter 11.6, Problem 17PS

Chapter 11.6, Problem 19PS

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Chapter 11 Solutions

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To find a unit vector that is normal to the surface x 4 + y 4 + z 4 = 3 at ( 1 , − 1 , − 1 ) And to find the standard form of the equation of tangent plane to this surface at above point

Solution Summary: The author explains how to find a unit vector that is normal to the surface and the standard form of the equation of tangent plane to this surface.

Videos

To find a unit vector that is normal to the surface

x4+y4+z4=3 at (1,−1,−1)

And to find the standard form of the equation of tangent plane to this surface at above point

Want to see the full answer?

Chapter 11 Solutions

To find a unit vector that is normal to the surface x 4 + y 4 + z 4 = 3 at ( 1 , − 1 , − 1 ) And to find the standard form of the equation of tangent plane to this surface at above point

Solution Summary: The author explains how to find a unit vector that is normal to the surface and the standard form of the equation of tangent plane to this surface.

Videos

To find a unit vector that is normal to the surface x4+y4+z4=3 at (1,−1,−1) And to find the standard form of the equation of tangent plane to this surface at above point

Want to see the full answer?

Chapter 11 Solutions

To find a unit vector that is normal to the surface

x4+y4+z4=3 at (1,−1,−1)

And to find the standard form of the equation of tangent plane to this surface at above point