   Chapter 14, Problem 50RE

Chapter
Section
Textbook Problem

Find parametric equations of the tangent line at the point (−2, 2, 4) to the curve of intersection of the surface z = 2x2 − y2 and the plane z = 4.

To determine

To find: The parametric equation for the tangent line to the curves of intersections of the surface z=2x2y2 and the plane z=4 at the point (2,2,4).

Explanation

Given:

The equation of surface is,z=2x2y2.

The equation of plane is,z=4.

Equation used:

F(x0,y0,z0)r(t0)=0 where r(t0) is a tangent line of the function F(x,y,z).”

Calculation:

Let the functions be f(x,y,z)=2x2+y2+z and g(x,y,z)=z4.

Compute the gradient vector f(x,y,z) at the point (2,2,4) as follows.

f(x,y,z)=fx,fy,fz=fx(2x2+y2+z),fy(2x2+y2+z),fz(2x2+y2+z)=4x,2y,1f(2,2,4)=8,4,1

Thus, the value of f(x,y,z) at the point (2,2,4) is f(2,2,4)=8,4,1.

Compute the gradient vector g(x,y,z) at the point (2,2,4) as follows

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