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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
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) .

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

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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