   Chapter 14.4, Problem 31E

Chapter
Section
Textbook Problem

If z = 5x2 + y2 and (x, y) changes from (1, 2) to (1.05, 2.1), compare the values of Δz and dz.

To determine

To compare: The values of Δzanddz if the function z=5x2+y2 and (x,y) changes from (1,2) to (1.05,2.1).

Explanation

The given function is, z(x,y)=5x2+y2 . (1)

Let the changes from P(a,b)=(1,2) to P(c,d)=(1.05,2.1) .

The derivative, dz is defined as, dz=zxdx+zydy .

Thus, for the given data, dz=zx|(1,2)dx+zy|(1,2)dy (2)

The increment of z , Δz is defined as, Δz=f(c,d)f(a,b) .

Thus, for the given data, Δz=f(1.05,2.1)f(1,2) (3)

Take the partial derivative of the function with respect to x at the point (1, 2) of the equation (1),

zx=x(5x2+y2)=5(2x)+0=10xzx|(1,2)=10

Take the partial derivative of the function with respect to y at the point (1, 2) of the equation (1),

zy=y(5x2+y2)=0+2y=2yzy|(1,2)=4

The value of increment at x is,

dx=Δx=1

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