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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 linear approximation of the function f(x, y) = 1 − xy cos πy at (1, 1) and use it to approximate f(1.02, 0.97). Illustrate by graphing f and the tangent plane.

To determine

To find: The linearization of the function f(x,y)=1xycosπy at the point (1,1) and use it approximate the value of f(1.02,0.97) ;illustrate using the graph and the tangent plane.

Explanation

Given:

The function is, f(x,y)=1xycosπy .

The point is P(a,b)=(1,1) .

Theorem used:

“If the partial derivatives fx and fy exist near (a,b) and are continuous at (a,b) , then f is differentiable at (a,b) .”

Result used:

“Linear approximation of the function f(x,y) at (a,b) is f(x,y)f(a,b)+fx(a,b)(xa)+fy(a,b)(yb)

“Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z=f(x,y) at the point P(x0,y0,z0) is zz0=fx(x0,y0)(xx0)+fy(x0,y0)(yy0)

Calculation:

Take partial derivativewith respect to x in the function f(x,y)=1xycosπy ,

fx(x,y)=0(1)ycosπy=ycosπy

Take partial derivative with respect to y in the function f(x,y)=1xycosπy ,

fy(x,y)=0x[y(sinπy)(π)+(cosπy)(1)][y(uv)=u'v+uv']=πxysinπyxcosπy

Obatin the partial derivative of x at the point (1, 1),

fx(a,b)=fx(1,1)=1(cosπ.1)=1(1)[cos(π)=1]=1

Obtain the partial derivative of y at the point (1,1) ,

fy(a,b)=fy(1,1)=π(1)(1)sin(π.1)1cos(π

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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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