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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 f is a differentiable function of one variable. Show that all tangent planes to the surface z = xf(y/x) intersect in a common point.

To determine

To show: All the tangent planes to the surface z=xf(yx) intersects in a common point.

Explanation

Given:

The equation of the surface is z=xf(yx) .

Calculation:

Let, g(x,y)=xf(yx) (1)

Differentiate equation (1) with respect to x using the product rule ddx(u.v)=dudxv+dvdxu .

gx(x,y)=f(yx)+xf'(yx)(yx2)=f(yx)x(yx2)f'(yx)gx(x,y)=f(yx)(yx)f'(yx)

Differentiate the equation (1) with respect to y .

gy(x,y)=(0)f(yx)+xf'(yx)(1x)=xf'(yx)(1x)=x(1x)f'(yx)gy(x,y)=f'(yx)

The general equation of the tangent plane at the point (x0,y0,z0) is, zz0=fx(x0,y0)(xx0)+fy(x0,y0)(yy0) .

Here, fx(x0,y0)=gx(x,y) and fy(x0,y0)=gy(x,y) .

Rearrange the general equation of tangent plane

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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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