   Chapter 3.10, Problem 25E

Chapter
Section
Textbook Problem

Use a linear approximation (or differentials) to estimate the given number.25. 1001 3

To determine

To estimate: The value of 10013 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 10013 by using the linearization.

Since 1000 is near integer of the value 1001, choose the value a=1000 and the function f(x)=x3.

The linearization of the function f(x)=x3 at a=1000.

Consider f(x)=x3,

Differentiate with respect to x on both sides,

f(x)=ddx(x13)=13x131=13(x)23=13(x2)13

Substitute x=1000 in f(x) and simplify the terms as shown below.

f(1000)=13((1000)2)13=13(100×100×100)13=1300

Thus, the value of f(1000)=1300

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