   Chapter 14, Problem 25RE

Chapter
Section
Textbook Problem

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.25. z = 3x2 − y2 + 2x, (1, −2, 1)

(a)

To determine

To find: The equation of the tangent planeto the surface z=3x2y2+2x at the point (1,2,1) .

Explanation

Given:

The surface is, z=3x2y2+2x .

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)=3x2y2+2xz (1)

The equation of the tangent plane to the given surface at the point (3,1,1) is defined by,

Fx(1,2,1)(x1)+Fy(1,2,1)(y+2)+Fz(1,2,1)(z1)=0 (2)

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

Fx(x,y,z)=x(3x2y2+2xz)=6x0+20=6x+2Fx(1,2,1)=8

Thus, the value of Fx(1,2,1)=8

(b)

To determine

To find: The equation of the normal line to the surface z=3x2y2+2x at the point (1,2,1) .

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