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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

Find the point on the plane x − 2y + 3z = 6 that is closest to the point (0, 1, 1).

To determine

To find: The point on the plane x2y+3z=6 .which is closest to the point (0,1,1)

Explanation

The given plane equation can be expressed as,

3z=6x+2yz=6x+2y3z=213x+23y

The distance d between (0,1,1) and any point (x,y,z) is,

d=(x0)2+(y1)2+(z1)2d2=x2+(y1)2+(z1)2

Substitute the value of z in the above distance equation,

d2=x2+(y1)2+[(213x+23y)1]2=x2+(y1)2+(113x+23y)2

Thus, f(x,y)=x2+(y1)2+(113x+23y)2 .

Take the partial derivative of f(x,y) with respect x and obtain fx .

fx=x(x2+(y1)2+(113x+23y)2)=x(x2)+x(y1)2+x(113x+23y)2=2x+0+2(113x+23y)(13)=2x23(113x+23y)

Simplify further as follows.

fx=2x23(113x+23y)=2x23+29x49y=209x49y23

Hence, fx=209x49y23 (1)

Take the partial derivative of f(x,y) with respect y and obtain fy

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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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