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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.

36. r(u, v) = sin u i + cos u sin v j + sin v ku = π/6, v = π/6

To determine

To find: An equation of the tangent plane to parametric surface r(u,v)=sinui+cosusinvj+sinvk at the point u=π6,v=π6.

Explanation

Given data:

The vector function is given as follows.

r(u,v)=sinui+cosusinvj+sinvk;u=π6,v=π6

Formula used:

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) is,

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

The expression to find normal vector from the tangent vectors of the parametric surface is,

n=|ijka1b1c1a2b2c2| (2)

Where, 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.

The expression to find the tangent vector ru of the parametric surface is,

ru=xui+yuj+zuk (3)

The expression to find the tangent vector rv of the parametric surface is,

rv=xvi+yvj+zvk (4)

Calculation:

The parametric surface r(u,v)=sinui+cosusinvj+sinvk at the point u=π6,v=π6.

Let x=sinu,y=cosusinv,z=sinv.

Write the expression to find point (x0,y0,z0) in the surface.

(x0,y0,z0)=(sinu,cosusinv,sinv)

Substitute π6 for u and π6 for v,

(x0,y0,z0)=(sin(π6),cos(π6)sin(π6),sin(π6))=(12,(32)(12),12)=(12,34,12)

Thus, (x0,y0,z0)=(12,34,12).

Calculation of tangent vector ru is as follows.

Substitute sinu for x, cosusinv for y, and sinv for z in equation (3),

ru=(sinu)ui+(cosusinv)uj+(sinv)uk=[u(sinu)]i+[u(cosusinv)]j+[u(sinv)]k=(cosu)i+[sinvu(cosu)]j+(0)k=(cosu)i+(sinusinv)j+(0)k

Modify the equation as follows.

ru=cosu,sinusinv,0

Substitute π6 for u and π6 for v,

ru=cos(π6),sin(π6)sin(π6),0=32,(12)(12),0=32,14,0

Thus, ru=32,14,0.

Calculation of tangent vector rv is as follows.

Substitute sinu for x, cosusinv for y, and sinv for z in equation (4),

rv=(sinu)vi+(cosusinv)vj+(sinv)vk=[v(sinu)]i+[v(cosusinv)]j+[v(sinv)]k=(0)i+[cosuv(sinv)]j+(cosv)k=(0)i+(cosucosv)j+(cosv)k

Modify the equation as follows

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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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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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