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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

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.

46. x4 + y4 + z4 = 3x2y2z2, (1, 1, 1)

(a)

To determine

To find: The equation of the tangent plane to the surface x4+y4+z4=3x2y2z2 at the point (1,1,1) .

Explanation

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

Calculation:

The given surface is, x4+y4+z4=3x2y2z2 .

Let the surface function be, F(x,y,z)=x4+y4+z43x2y2z2 . (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)=x(x4+y4+z43x2y2z2)=4x3+0+06xy2z2=4x36xy2z2

Find the value of Fx(x,y,z) at the point (1,1,1) .

Fx(1,1,1)=4(1)36(1)(1)2(1)2=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)=y(x4+y4+z43x2y2z2)=0+4y3+0+06x2yz2=4y36x2yz2

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

(b)

To determine

To find: The equation of the normal line to the surface x4+y4+z4=3x2y2z2 at the point (1,1,1) .

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Chapter 14 Solutions

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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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