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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane.

38. r(u, v) = (1 − u2 − v2) iv j − u k; (−1, −1, −1)

To determine

To find: An equation of the tangent plane to the vector function r(u,v)=(1u2v2)ivjuk at the point (1,1,1) .

Explanation

Given data:

The vector function is given as follows.

r(u,v)=(1u2v2)ivjuk

The specified point is (1,1,1) .

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)

Write the vector function as follows.

r(u,v)=(1u2v2)ivjuk

Calculation u and v at the point (1,1,1) :

Equate the x-, y-, and z-coordinates of vector function r(u,v) to the corresponding coordinates of the point (1,1,1) as follows.

1u2v2=1v=1u=1

From the equations, it is clear that the point (1,1,1) corresponds to u=1 and v=1 .

Therefore, u=1 and v=1 .

Calculation of tangent vector ru :

Substitute (1u2v2) for x , (v) for y , and (u) for z in equation (3),

ru=(1u2v2)ui+(v)uj+(u)uk=[u(1u2v2)]i+[u(v)]j+[u(u)]k=(02u0)i+(0)j+(1)k=2u,0,1

Substitute 1 for u ,

ru=2(1),0,1=2,0,1

Calculation of tangent vector rv :

Substitute (1u2v2) for x , (v) for y , and (u) for z in equation (4),

rv=(1u2v2)cvi+(v)vj+

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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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