   Chapter 3.10, Problem 23E

Chapter
Section
Textbook Problem

Use a linear approximation (or differentials) to estimate the given number.23. (1.999)4

To determine

To estimate: The value of (1.999)4 by using linear approximation.

Explanation

Result used:

The linear approximation of the function at x=a is L(x)=f(a)+f(a)(xa).

Calculation:

Obtain the value of (1.999)4 by using the linearization.

Since 2 is an integer near to the value 1.9999, choose the value a=2 and the function f(x)=x4.

The linearization of the function f(x)=x4 at a=2 is computed as follows,

Consider the function f(x)=x4,

Differentiate with respect to x,

f(x)=ddx(x4)=4x3

Substitute x=0,

f(2)=4(2)3=4(8)=32

Thus, the value is f(2)=32.

Substitute x=2 in f(x)=x4,

f(2)=(2)4=16

Thus, the value is f(2)=16

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