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Use the result of Exercise 79 to approximate the square roots in Exercises 80–83. In each case, start with x0=1and stop when xk+1xk<0.001.

82.4

Short Answer

Expert verified

The approximate value of the root of4is2

Step by step solution

01

Step 1. Given datax

The given term is 4and x0=1.

Here, we have to find the root of the functions.

02

Step 2. Finding the value of x1

Let us consider the functionf(x)=x2-4

We have the equation xk+1=xk-fxkf'xk.......Equation (1)

Therefore,

f(xk)=xk24f(xk)=2xk

Substituting the values in equation (1)

xk+1=xkxk242xk

Now to find the value of x1, substitute k=0in equation (2)

x0+1=x0x0242x0x1=x0x0242x0

Substitute x0=1

x1=(1)(1)242(1)=1142=132=1+32=42=2

Therefore,x1=2

03

Step 3. Finding the value of x2

Now to find the value of x2, substitute k=1in equation (2)

x2=x1x1242x1

Substitutex1=2

x2=(2)(2)242(2)=2444=204=2

Thereforex2=2

04

Step 4. Finding the value of x3

Now to find the value of x3, substitute localid="1649345412906" k=2in equation (2)

x3=x2x2242x2

Substitutex2=2

x3=(2)(2)242(2)=2444=204=2

Therefore,x3=2

05

Step 5. Finding the value of x4

Now to find the value of x4, substitute localid="1649345423154" k=3in equation (2)

x4=x3x3242x3

Substitutex3=2

x4=(2)(2)242(2)=2444=204=2

Therefore,x4=2

06

Step 6. Finding the root of the function  

Here,

|x4x3|=|22|=|0|

Since, role="math" localid="1649335839464" |x4x3|<0.001let us stop the iteration.

Therefore, the approximate value of the root of 4is 2

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