Newton-Raphson Method (part 2)

 
The Newton-Raphson method does not work always.  If  f(x) is twice differentiable and  f(x) * f " (x) > 0 in the

open interval I joining c and x1 ,  the sequence of approximations x1 , x2 , ... will converge to the root c.
 

 
Application #2:   Let f(x) = 2* x^3 - 3*x^2 - 1.  The function has a root in the interval (1,2).
 

    1. Show that the Newton-Raphson Method  with x1 = 1   fails to generate values that aproach the root in (1,2).
    2. Estimate the root by starting at  x1 = 2 .  Determine x4 rounded off to five decimal places, and evaluate f(x4).
     
(Step by Step Solution -- ) :
 
  1.  define(F(x), 2* x^3 - 3*x^2 - 1);     returns RESULT
  2.  define(F1(x), diff(F(x),x));     returns RESULT
  3.  define(F2(x), diff(F1(x),x));     returns RESULT
  4.  x[1] : 1;     returns RESULT
  5.  x[1]:block(if(F1(x[1])=0)then x[1]:2,x[1]);     returns RESULT
  6.  x[2] : x[1] - F(x[1]) / F1(x[1]);     returns RESULT
  7.  x[n] := x[n-1] - F(x[n-1])/F1(x[n-1]);     returns RESULT
  8.  x[4], numer = true;     returns  RESULT
  9.   Check the approximation by computing f(x[4]),numer=true;     returns  RESULT
 

Exercise:  Salas & Hille's Calculus (3.9 #39)

 


 
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