4.5 Newton’s Method

1-3. Compute , the third approximation to the root of the given equation using Newton's method with the specified initial approximation .


1.

 

      

      

      

      


2. .

 

      

      

      


3. .

 

      

      

      


4. Calculate the second approximation to the root of the equation by using Newton's method with initial approximation .

 

      

      

      


5-6. Approximate the given number using Newton's method.


5.

 

      

      

      

      

      

      

      

      


6.

 

      

      

      

      

      

      

      

      


7.

 

      

      

      

      

      


8.

 

      

      

      

      

      


9.

 

      

      (ⅰ)

          

          

          

           

      (ⅱ)

          

          

          

          

          


10.

 

      

      (ⅰ)

          

      (ⅱ)

          

          

          


11. Apply Newton's method to the equation to derive the following reciprocal algorithm:

    . Hence compute .

 

      

      

      (ⅰ)

          

          

          

          

          

          


12. Use Newton's method to find the absolute minimum value of the function .