(a) Explanation Given information: The given surface is x ln y + y ln z = x . Coordinates of point P 0 is ( 1 , 1 , e ) . Calculation: . Consider the given surface as follows: f ( x , y , z ) = x ln y + y ln z − x (1) Find the gradient ( ∇ f ) for the given surface: ∇ f = ( ln y − 1 ) i + ( x y + ln z ) j + ( y z ) k Tangent plane is the plane through the given point P 0 perpendicular to the gradient of f at P 0 . Find the gradient ∇ f | P 0 at point P 0 ( 1 , 1 , e ) for the given surface: ∇ f | P 0 = ( ( ln y − 1 ) i + ( x y + ln z ) j + ( y z ) k ) | ( 1 , 1 , e ) = ( ln 1 − 1 ) i + ( 1 1 + ln ( e ) ) j + ( 1 e ) k = − i + 2 j + 1 e k The equation of tangent plane to the surface f ( x , y , z (b) To determine Determine the equation of the normal line at point P 0 .

14th Edition

ISBN: 9780136208013

Author: Hass

Publisher: PEARSON

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Chapter 14.6, Problem 9E

(a)

To determine

Determine the equation of the tangent plane at point P0.

(b)

To determine

Determine the equation of the normal line at point P0.

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Chapter 14.6, Problem 8E

Chapter 14.6, Problem 10E

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Chapter 14 Solutions

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Determine the equation of the tangent plane at point P 0 .

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Determine the equation of the tangent plane at point P0.

Determine the equation of the normal line at point P0.

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