   Chapter 3.8, Problem 11E

Chapter
Section
Textbook Problem

# 11-12 Use Newton’s method to approximate the given number correct to eight decimal places. 75 4

To determine

To use:

TheNewton’s method to approximation the given number

Explanation

1) Concept:

Use Newton’s formula to find for nth approximation

2) Formula:

i. Newton’s formula for nth approximation is xn+1=xn-fxnf'xn for n=1,2,3,

ii. Power rule of differentiation ddxxn=nxn-1

iii. ddxconstant=0

3) Calculation:

Assume x=754

Taking 4th power on both sides,

x4=75

x4-75=0

So assume, f(x)=x4-75

So calculating 754 is similar to solving x4-75=0

Differentiate f(x)=x4-75 using power rule,

f'x=4·x4-1-0

=4x3

So Newton’s formula for nth approximation becomes

xn+1=xn-xn4-754xn3

Since 34=81 close to 75, so x1=3 is good choice for initial approximation

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