James Stewart

ISBN: 9781305266643

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) .

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)(x−x0)+Fy(x0,y0,z0)(y−y0)+Fz(x0,y0,z0)(z−z0)=0 ”

“The normal to the level surface at the point P(x0,y0,z0) is defined as (x−x0)Fx(x0,y0,z0)=(y−y0)Fy(x0,y0,z0)=(z−z0)Fz(x0,y0,z0) ”.

Calculation:

Let the surface function be, F(x,y,z)=xy+yz+zx−3=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)(x−1)+Fy(1,1,1)(y−1)+Fz(1,1,1)(z−1)=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

Check out a sample textbook solution.

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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)

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Math
Calculus Multivariable Calculus 8th Edition

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)