   Chapter 14, Problem 29RE

Chapter
Section
Textbook Problem

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.29. sin(xyz) = x + 2y + 3z, (2, −1, 0)

(a)

To determine

To find: The equation of the tangent planeto the surface sin(xyz)=x+2y+3z at the point (2,1,0) .

Explanation

Given:

The surface is, sin(xyz)=x+2y+3z .

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:

Let the surface function be, F(x,y,z)=x+2y+3zsin(xyz) (1)

The equation of the tangent plane to the given surface at the point (2,1,0) is defined by,

Fx(2,1,0)(x2)+Fy(2,1,0)(y+1)+Fz(2,1,0)(z0)=0 (2)

Take partial derivative with respect to x at the point (2,1,0) in the equation (1),

Fx(x,y,z)=x(x+2y+3zsin(xyz))=1+0+0yzsin(xyz)=1yzcos(xyz)

The value of Fx(x,y,z) at the point (2,1,0) is,

Fx(2,1,0)=1(1)(0)cos(2(1)0)=10=1

Thus, the value of Fx(2,1,0)=1

(b)

To determine

To find: The equation of the normal line to the surface sin(xyz)=x+2y+3z at the point (2,1,0) .

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