   Chapter 2.9, Problem 25E

Chapter
Section
Textbook Problem

# 23-28 Use a linear approximation (or differentials) to estimate the given number. 1001 3

To determine

To estimate: The given number by using a linear approximation (or differentials).

Explanation

1) Formula:

i. Limit definition of differentiation:

f'x= limx0fx+x-fxx

ii. For small values of ∆x,

f'x fx+x-fxx

f'xx  fx+x-fx

fx+x   fx+ f'xx

iii. Power rule of derivative:

ddxxn=n xn-1

2) Given: 10013

3) Calculation:

Let fx=x3 ,  x = 1000 and x= 1

Substitute x = 1000 &x= 1 in  fx+x   fx+ f'xx

f1000+1   f1000+ f'1000(1)

f1001   f1000+ f'1000(1)

To find f (1000), plug x = 1000 in fx=x3

f1000=10003= 10

To find f’ (1000), find derivative of f(x)

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