   Chapter 16.6, Problem 34E

Chapter
Section
Textbook Problem

Find an equation of the tangent plane to the given parametric surface at the specified point.34. x = u2 + 1, y = v3 + 1, z = u + v; (5, 2, 3)

To determine

To find: An equation of the tangent plane to the parametric surface x=u2+1,y=v3+1,z=u+v at the point (5,2,3) .

Explanation

Given data:

The equation of the parametric surface is x=u2+1,y=v3+1,z=u+v .

The specified point is (5,2,3) .

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 (u2+1) for x , (v3+1) for y , and (u+v) for z in equation (3),

ru=(u2+1)ui+(v3+1)uj+(u+v)uk=[u(u2+1)]i+[u(v3+1)]j+[u(u+v)]k=(2u+0)i+(0+0)j+(1+0)k=2u,0,1

Calculation of tangent vector rv :

Substitute (u2+1) for x , (v3+1) for y , and (u+v) for z in equation (4),

rv=(u2+1)vi+(v3+1)vj+(u+v)vk=[v(u2+1)]i+[v(v3+1)]j+[v(u+v

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