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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Suppose you need to know an equation of the tangent plane to a surface S at the point P(2, 1,3). You don't have an equation for S but you know that the curves

r1(t) = ⟨2 + 3t, 1 –t2, 3 – 4t + t2

r2(u) = ⟨1 + u2, 2u3 – 1, 2u + 1⟩

both lie on S. Find an equation of the tangent plane at P.

To determine

To find: The equation of the tangent plane to a surface S at the point P(2,1,3) .

Explanation

Given:

Let S be the surface and the curves are,

r1(t)=2+3t,1t2,34t+t2r2(u)=1+u2,2u31,2u+1

Calculation:

Let r1(t)=2+3t,1t2,34t+t2 (1)

Differentiate the equation (1) with respect to t at the point P(2,1,3) and obtain the tangent vector r1'(t) ,

r1'(t)=0+3,0(2t),04(1)+(2t)=3,2t,4+2t

Let r2(u)=1+u2,2u31,2u+1 (2)

Differentiate the equation (2) with respect to u at the point P(2,1,3) and obtain the tangent vector r2'(u) ,

r2'(u)=0+(2u),2(3u2)0,2+0=2u,6u2,2

To find the curve passes through the point P(2,1,3) or not. By assuming the value of t=0 and u=1 .

Assume t=0 and u=1 in the equation (1) and (2),

r1(0)=2+3(0),1(0)2,34(0)+(0)2=2,1,3

Assume u=1 in the equation (2),

r2(1)=1+(1)2,2(1)31,2(1)+1=1+1,21,2+1=2,1,3

From this, it is clear that the curve satisfies the point P(2,1,3) when t=0 and u=1

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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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