Explanation Given: Thesurface x 3 + 2 x y 2 − 7 x 3 + 3 y + 1 = 0 and a point P 0 ( 1 , 1 , 1 ) Concept Used: For a function that defines a surface f ( x , y , z ) differentiated at a point ( x 0 , y 0 , z 0 ) . The tangent plane to f ( x , y , z ) at ( x 0 , y 0 , z 0 ) is the plane with normal vector ∇ f ( x 0 , y 0 , z 0 ) that passes through the point ( x 0 , y 0 , z 0 ) . The equation of the tangent plane is given by ∇ f ( x 0 , y 0 , z 0 ) · 〈 x − x 0 , y − y 0 , z − z 0 〉 = 0 Where ∇ f is the gradient of the function f which is ∇ f = 〈 f x , f y , f z 〉 . And, the equation of the normal line is given by x ( t ) = x 0 + f x ( x 0 , y 0 , z 0 ) t y ( t ) = y 0 + f y ( x 0 , y 0 , z 0 ) t z ( t ) = z 0 + f z ( x 0 , y 0 , z 0 ) t Calculation: For the givensurface x 3 + 2 x y 2 − 7 x 3 + 3 y + 1 = 0 and a point P 0 ( 1 , 1 , 1 ) Consider f ( x , y , z ) = x 3 + 2 x y 2 − 7 x 3 + 3 y + 1 f ( x , y , z ) = 2 x y 2 − 6 x 3 + 3 y + 1 The gradient of the function f is ∇ f = 〈 f x , f y , f z 〉 that is ∇ f = 〈 2 y 2 − 18 x 2 , 4 x y + 3 , 0 〉 At P 0 ( 1 , 1 , 1 ) ∇ f ( 1 , 1

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

See similar textbooks

Videos

Question

Chapter 11, Problem 37SP

To determine

To find:The equation for the tangent plane and normal line to the surface x3+2xy2−7x3+3y+1=0 at P0(1,1,1) .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11, Problem 36SP

Chapter 11, Problem 38SP

Students have asked these similar questions

An object of mass 4 kg is given an initial downward velocity of 60 m/sec and then allowed to fall under the influence of gravity. Assume that the force in newtons due to air resistance is - 8v, where v is the velocity of the object in m/sec. Determine the equation of motion of the object. If the object is initially 500 m above the ground, determine when the object will strike the ground. Assume that the acceleration due to gravity is 9.81 m/sec² and let x(t) represent the distance the object has fallen in t seconds. Determine the equation of motion of the object. x(t) = (Use integers or decimals for any numbers in the expression. Round to two decimal places as needed.)

Early Monday morning, the temperature in the lecture hall has fallen to 40°F, the same as the temperature outside. At 7:00 A.M., the janitor turns on the furnace with the thermostat set at 72°F. The time constant for the building is = 3 hr and that for the building along with its heating system is 1 K A.M.? When will the temperature inside the hall reach 71°F? 1 = 1 hr. Assuming that the outside temperature remains constant, what will be the temperature inside the lecture hall at 8:30 2 At 8:30 A.M., the temperature inside the lecture hall will be about (Round to the nearest tenth as needed.) 1°F.

Find the maximum volume of a rectangular box whose surface area is 1500 cm² and whose total edge length is 200 cm. cm³

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

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The equation for the tangent plane and normal line to the surface x 3 + 2 x y 2 − 7 x 3 + 3 y + 1 = 0 at P 0 ( 1 , 1 , 1 ) .

Solution Summary: The author explains the equation for the tangent plane and normal line to the surface p_0(1,1,1).

Videos

To find:The equation for the tangent plane and normal line to the surface x3+2xy2−7x3+3y+1=0 at P0(1,1,1) .

Want to see the full answer?

Chapter 11 Solutions