We will describe here a method for locating a root of an equation f(x) = 0.  This method is called the

Newton-Raphson Method.  Assume that the solution exists and is denoted c.  Start at a point x₁ .  The tangent

line  (x₁,f(x₁ )) intersects the x-axis at a point x₂ ,  which is closer to  c  than x₁ .  Determine a new point x₃ :  the

intersection of the line  (x₂ , f(x₂)) and the x-axis.  Continuing, we obtain better and better approximations x₄, x₅,

. . . , x_n   to the root   c.  Hence
 
 
 

f ' (x[n-1]) =  -f(x[n-1]) / (x[n] -  x[n-1])

Or,
 
 

x[n] =  x[n-1]   - f (x[n-1]) /  f ' (x[n] )

Application #1:   Let f(x) = x^2 - 24.  Use the Newton-Raphson Method  estimate a root of the equation f(x) = 0 

starting at x₁ = 5.  Find x₄  and evaluate  f(x₄ ).
 

(Step by Step Solution -- ) :
 

1.   define(F(x), x^2 -24);     returns RESULT
2.  define(F1(x), diff(F(x),x));     returns RESULT
3.  define(F2(x), diff(F1(x),x));     returns RESULT
4.  kill(x); x[1] : 5;     returns RESULT
5.  x[2] : x[1] - F(x[1]) / F1(x[1]);     returns RESULT
6.  x[n] := x[n-1] - F(x[n-1])/F1(x[n-1]);     returns RESULT
7.  x[4];     returns  RESULT
8.   Check the approximation by computing F(x[4]);     returns  RESULT