   Chapter 14.6, Problem 47E

Chapter
Section
Textbook Problem

Use a computer to graph the surface, the tangent plane, and the normal line on the same screen. Choose the domain carefully so that you avoid extraneous vertical planes. Choose the viewpoint so that you get a good view of all three objects.47. xy + yz + zx = 3, (1, 1, 1)

To determine

To graph: The tangent plane, normal line and the surface on the same screen for the function xy+yz+zx=3 at the point (1,1,1) .

Explanation

Given:

The surface is, xy+yz+zx=3 .

Result used:

“The tangent plane to the level surface at the point P(x0,y0,z0) is defined as Fx(x0,y0,z0)(xx0)+Fy(x0,y0,z0)(yy0)+Fz(x0,y0,z0)(zz0)=0

“The normal to the level surface at the point P(x0,y0,z0) is defined as (xx0)Fx(x0,y0,z0)=(yy0)Fy(x0,y0,z0)=(zz0)Fz(x0,y0,z0) ”.

Calculation:

Let the surface function be, F(x,y,z)=xy+yz+zx3=0 . (1)

The equation of the tangent plane to the given surface at the point (1,1,1) is defined by,

Fx(1,1,1)(x1)+Fy(1,1,1)(y1)+Fz(1,1,1)(z1)=0 . (2)

Take partial derivative with respect to x at the point (1,1,1) in the equation (1),

Fx(x,y,z)=xy+yz+zx=y+z

The value of Fx(x,y,z) at the point (1,1,1) is,

Fx(1,1,1)=1+1=2

Thus, the value of Fx(1,1,1)=2 .

Take partial derivative with respect to y at the point (1,1,1) in the equation (1),

Fy(x,y,z)=xy+yz+zx=x+z

The value of Fy(x,y,z) at the point (1,1,1) is,

Fy(1,1,1)=1+1=2

Thus, the value of Fy(1,1,1)=2

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Which property of real numbers is illustrated in each part of problem 3-6? 5. a) b)

Mathematical Applications for the Management, Life, and Social Sciences

In Exercises 21 to 32, solve each equation. 3y=214y

Elementary Geometry For College Students, 7e

In Exercises 14, find the slope of the line shown in each figure.

Finite Mathematics for the Managerial, Life, and Social Sciences

The graph of x = cos t, y = sin2 t is:

Study Guide for Stewart's Multivariable Calculus, 8th

Find f(x) if f(x)=1x2 and f(2) = 0. a) 1x+12 b) 1x12 c) 3x3+38 d) 3x338

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 