Explanation Given: The plane sin ( x + y ) + tan ( y + z ) = 1 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 sin ( x + y ) + tan ( y + z ) = 1 at ( π 4 , π 4 , − π 4 ) First, let f ( x , y , z ) = sin ( x + y ) + tan ( y + z ) Now, find the gradient vector of this function at the point ( π 4 , π 4 , − π 4 ) , 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 ( π 4 , π 4 , − π 4 ) , f y ( π 4 , π 4 , − π 4 ) , and f z ( π 4 , π 4 , − π 4 ) as f x ( x , y , z ) = ∂ ∂ x ( sin ( x + y ) + tan ( y + z ) ) = ∂ ∂ x sin ( x + y ) + ∂ ∂ x tan ( y + z ) = cos ( x + y ) ∂ ∂ x ( x + y ) = cos ( x + y ) , f y ( x , y , z ) = ∂ ∂ y ( sin ( x + y ) + tan ( y + z ) ) = ∂ ∂ y sin ( x + y ) + ∂ ∂ y tan ( y + z ) = cos ( x + y ) ∂ ∂ y ( x + y ) + sec 2 ( y + z ) ∂ ∂ y ( y + z ) = cos ( x + y ) + sec 2 ( y + z ) and f z ( x , y , z ) = ∂ ∂ z ( sin ( x + y

6th Edition

ISBN: 9781465208880

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

Publisher: Kendall Hunt Publishing

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Chapter 11.6, Problem 20PS

To determine

To find a unit vector that is normal to the surface

sin(x+y)+tan(y+z)=1 at (π4,π4,−π4)

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 19PS

Chapter 11.6, Problem 21PS

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Chapter 11 Solutions

Calculus

Ch. 11.1 - Prob. 1PS Ch. 11.1 - Prob. 2PS Ch. 11.1 - Prob. 3PS Ch. 11.1 - Prob. 4PS Ch. 11.1 - Prob. 5PS Ch. 11.1 - Prob. 6PS Ch. 11.1 - Prob. 7PS Ch. 11.1 - Prob. 8PS Ch. 11.1 - Prob. 9PS Ch. 11.1 - Prob. 10PS

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To find a unit vector that is normal to the surface sin ( x + y ) + tan ( y + z ) = 1 at ( π 4 , π 4 , − π 4 ) 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

sin(x+y)+tan(y+z)=1 at (π4,π4,−π4)

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 sin ( x + y ) + tan ( y + z ) = 1 at ( π 4 , π 4 , − π 4 ) 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 sin(x+y)+tan(y+z)=1 at (π4,π4,−π4) 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

sin(x+y)+tan(y+z)=1 at (π4,π4,−π4)

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