   Chapter 16.6, Problem 33E

Chapter
Section
Textbook Problem

Find an equation of the tangent plane to the given parametric surface at the specified point.33. x = u + v, y = 3u2, z = u − v; (2, 3, 0)

To determine

To find: An equation of the tangent plane to the parametric surface x=u+v,y=3u2,z=uv at the point (2,3,0) .

Explanation

Given data:

The equation of the parametric surface is x=u+v,y=3u2,z=uv .

The specified point is (2,3,0) .

Formula used:

Write the expression to find tangent plane to the parametric surface with the normal vector n=a,b,c at the specified point (x0,y0,z0) .

a(xx0)+b(yy0)+c(zz0)=0 (1)

Write the expression to find normal vector from the tangent vectors of the parametric surface.

n=|ijka1b1c1a2b2c2| (2)

Here,

The vector a1,b1,c1 is a tangent vector ru of the parametric surface and

The vector a2,b2,c2 is a tangent vector rv of the parametric surface.

Write the expression to find the tangent vector ru of the parametric surface.

ru=xui+yuj+zuk (3)

Write the expression to find the tangent vector rv of the parametric surface.

rv=xvi+yvj+zvk (4)

Calculation of tangent vector ru :

Substitute (u+v) for x , 3u2 for y , and (uv) for z in equation (3),

ru=(u+v)ui+(3u2)uj+(uv)uk=[u(u+v)]i+[u(3u2)]j+[u(uv)]k=(1+0)i+(6u)j+(10)k=1,6u,1

Calculation of tangent vector rv :

Substitute (u+v) for x , 3u2 for y , and (uv) for z in equation (4),

rv=(u+v)vi+(3u2)vj+(uv)vk=[v(u+v)]i+[v(3u2)]j+[v(u

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