Constructing the "Tangent Parabola"

    The Newton polynomial  ,  has the form:

(i)        .

The coefficients are determined by forcing    to pass through three points  

        ,    and  .  

Here we have used the notation    and    for the second and third points.  
                    
Using the equation    and the above three points produces a lower-triangular linear system of equations:

          

These three equations can be simplified and written in lower-triangular form    

(ii)           

which is easily solved using forward elimination:  

           
and
           

Substitute into equation (i) and get  

(iii)        

Let   in equation (iii), the limit of the first difference quotient    is the derivative  ,
and the limit of the difference quotient    is the second derivative  .  
Therefore, the limit of the Newton Polynomial    is seen to be the Taylor polynomial    

    .  

Caveat.  This is the direction to take if polynomials of higher degree are to be studied.  However, since the three points are different than those of mentioned in the article this may be a little confusing.  Another important feature of this derivation is that the coefficients   are found by solving a lower-triangular system (which is easily solved by forward elimination, and this technique generalizes to higher order polynomials).  

Aside.  All the computations can be easily done letting Mathematica form and solve the equations.  The following Mathematica solution is proposed.

Construct the lower-triangular linear system to be solved.  Start with the equation

          

Substitute the three points into  eqn  and get three equations  eqn0  ,  eqn1  and  eqn2.  

Solve this lower-triangular system:

Substitute the solution values and obtain    which is the "secant parabola:"

We are done!

Aside.  We can take the limit of  .  This is just for fun!  

(c) John H. Mathews 2004