(a) To determine Determine the equation of the tangent plane at point P 0 . Explanation Given information: The given surface is x 2 + y 2 + z 2 = 3 . Coordinates of point P 0 is ( 1 , 1 , 1 ) . Calculation: Consider the given surface as follows: f ( x , y , z ) = x 2 + y 2 + z 2 − 3 (1) Find the gradient ( ∇ f ) for the given surface: ∇ f = 2 x i + 2 y j + 2 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 , 1 ) for the given surface: ∇ f | P 0 = ( 2 x i + 2 y j + 2 z k ) | ( 1 , 1 , 1 ) = 2 ( 1 ) i + 2 ( 1 ) j + 2 ( 1 ) k = 2 i + 2 j + 2 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 ( 1 , 1 , 1 ) .

Thomas' Calculus: Early Transcendentals (14th Edition)

14th Edition

ISBN: 9780134439020

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 14.6, Problem 1E

(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(1,1,1).

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Chapter 14.5, Problem 44E

Chapter 14.6, Problem 2E

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Thomas' Calculus: Early Transcendentals (14th Edition)

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