   Chapter 14, Problem 30RE

Chapter
Section
Textbook Problem

Use a computer to graph the surface z = x2 + y4 and its tangent plane and normal line at (1, 1, 2) on the same screen. Choose the domain and viewpoint so that you get a good view of all three objects.

To determine

To graph: Thetangent plane, normal line and the surface on the same screen for the function z=x2+y4 at the point (1,1,2) .

Explanation

Given:

The level surface is, z=x2+y4 .

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, x2+y4z=0 (1)

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

Fx(1,1,2)(x1)+Fy(1,1,2)(y1)+Fz(1,1,2)(z2)=0 (2)

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

Fx(x,y,z)=x2+y4z=2x+00=2x

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

Fx(1,1,2)=2(1)=2

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

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

Fy(x,y,z)=x2+y4z=0+4y30=4y3

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

Fy(1,1,2)=4(1)=4

Thus, the value of Fy(1,1,2)=4

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