   Chapter 14, Problem 33RE

Chapter
Section
Textbook Problem

Find the linear approximation of the function f ( x , y , z ) = x 3 y 2 + z 2 at the point (2, 3, 4) and use it to estimate the number ( 1.98 ) 3 ( 3.01 ) 2 + ( 3.97 ) 2 .

To determine

To find: The linear approximation of the function f(x,y,z)=x3y2+z2 at the point (2,3,4);use this linear approximation and approximate the value of (1.98)3(3.01)2+(3.97)2 .

Explanation

Given:

The function is, f(x,y,z)=x3y2+z2 (1)

Calculation:

The linear approximation of the function f(x,y,z) at (3,2,6) is defined as,

f(2,3,4)f(2,3,4)+fx(2,3,4)(x2)+fy(2,3,4)(y3)+fz(2,3,4)(z4) (2)

Find the partial derivative of the equation (1)with respect to x ,

fx(x,y,z)=3x2(y2+z2)=3x2y2+z2

Substitute (x,y,z)=(2,3,4) and find the value of fx(2,3,4) .

fx(2,3,4)=3(2)2(3)2+(4)2=3(4)9+16=12(5)=60

Find the partial derivative of the equation (1) with respect to y ,

fy(x,y,z)=x3(12)(y2+z2)121(2y)=x3(y2+z2)12(y)=x3yy2+z2

Substitute (x,y,z)=(2,3,4) and find the value of fy(2,3,4)

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