   Chapter 14.6, Problem 46E

Chapter
Section
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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