Background  

    The limit of "the secant line" is "the tangent line."  We recast this in the context of polynomial approximation.  The secant line is the Newton polynomial  ,  and the tangent line as the Taylor polynomial  ,  where each has degree  .    

The Newton polynomial  ,  has the form:

(i)        .

The coefficients and are determined by forcing    to pass through two points   and .  Here we have used the notation   for the second point.  
                    
Using the equation    and the two points produces a lower-triangular linear system of equations:

          

which simplifies to be

(ii)          

which is easily solved using forward elimination:  ,  and  .  Substitute   and into equation (i) and get

(iii)        

     As h approaches 0 in equation (iii), the limit of the difference quotient    is the derivative    and the limit of the Newton Polynomial    is seen to be the Taylor polynomial  :

        .  

This background is the motivation to investigate the idea of polynomial approximations and their limits.

    The secant line which is used to approximate    it is based on two points and .  What if we used three points?  Then we could determine a polynomial of degree , which could also be used to approximate  .  Thus, we have the concept of  "the secant parabola" with interpolation points , , and .  

  
        Figure 1. The secant parabola approximating  ,  at    using  

When the interpolation points are moved closer to the middle point   the "the secant parabola" approaches a limiting position.

  

        Figure 2. The secant parabola approximating  ,  at    using  

The limiting position of the secant parabola is called the "tangent parabola."  The following figure shows this case when h goes to 0 and all three nodes coincide.

  
        Figure 3. The tangent parabola approximating  ,  at    where  .  

Can you guess what the tangent parabola will turn out to be?  We will reveal this pleasant surprise at the end of the article.

The Secant Parabola     

    A precise discussion of "the secant parabola" is now presented.  Recall that a polynomial of degree    expanded about    can be written in the form

(1)        ,  

where the coefficients are to be determined.  Since    is to be the interpolating polynomial for   , it must pass through the three points  , , and .  Using the first point we obtain the relation  

        ,  

which implies that
    
        .   

Proceeding, we solve for the two coefficients by first making the substitution in equation (1) and writing

(2)        .    

Then make substitutions for the two points , and , respectively,  in equation (2) and obtain two relations

          
    and  
        .  

Simplification  produces the following two equations which will be used to solve for .

        ,  
(3)  
        .  

Subtract the first equation in (3) from the second and then divide by  2h  and get  

        .

Add the equations in (3), subtract and then divide by and get

        .

The Secant Parabola Formula

    The secant parabola for which passes through , , and involves the variable x and parameters and has the form

(4)        .  

An Alternative Construction & Proof   Tangent Parabola  Tangent Parabola  

A Numerical Experiment

    Formulas for the above graphs of the secant parabola   for   are constructed using formula (4).  The computations are centered at the point using the step sizes  .

Example 1.  Use in formula (4) and compute the coefficients of for the secant parabola  .
Solution 1.

Example 2.  In a similar fashion, use in formula (4) and compute the coefficients of for the secant parabola  .
Solution 2.

Finding the Limit Numerically

    The limit of the secant polynomials is found by evaluating formula (4) using decreasing step sizes  .
The numerical results are summarized in Table 1.  

        
        
        Table 1. The secant parabola approximating  ,  at    where .  
Details

Finding the Limit Symbolically

    The entries in the table show that the coefficients of    are tending to a limit as  .  Thus the "tangent parabola" is  

(5)          .  

The first limit in (5) is well known, it is  

        .

The second limit in (5) is studied in numerical analysis, and is known to be , which can be verified by applying L'hopital's rule using the variable h as follows

         .  
Exploration 1.

Therefore, we have shown the limit of the "secant parabolas" to be

(6)        .  

Therefore, the "tangent parabola" in (5) is revealed to be the Taylor polynomial of degree .  
Exploration 2.

For our example with  ,  and  ,  we have

          
    
          
Exploration 3.

The Lagrange Connection

    In numerical analysis, the Lagrange interpolation polynomial is constructed, and it can be shown to be equivalent to the formula in (4), however the hand computations are messy.  If a computer algebra system, such as Mathematica is used, then it is easy to verify that the two forms are equivalent.  First, enter the formula for the Lagrange polynomial

Then enter formula (4)

The above two formulas can be expanded and shown to be equal.
Details

Therefore, formula (4) is equivalent to Lagrange interpolation, hence the Lagrange form of the remainder applies too.

The Remainder Term

    In numerical analysis, the remainder term for a Lagrange interpolation polynomial of degree  ,  is known to be

        ,

where depends on and lies somewhere between  .  

Proof  Lagrange Polynomials  Lagrange Polynomials  

    When we take the limit of     as    it is plain to see that we get

        ,

which is the remainder term for the Taylor polynomial of degree  .  This cinches the fact that the limit of the secant polynomial is the tangent polynomial.

Proof  Maclaurin and Taylor Polynomials  Maclaurin and Taylor Polynomials  

Conclusion

    The purpose of this article has been to show that the Taylor polynomial is the limiting case of a sequence of interpolating polynomials.  The development has been to first show graphical convergence, which is quite rapid.  This can be illustrated in the classroom by using graphical calculators or with computer software such as Mathematica or Maple.  Then a selected set of interpolating polynomials is tabulated, which is a new twist to the idea of limit, it involves the concept of convergence of a sequence of functions.   Finally, the power of calculus is illustrated by discovering that the limiting coefficients are and .  Then one recognizes that the "tangent polynomial" is a Taylor polynomial approximation.  Moreover, we have motivated the "what if" exploration by showing what happens to "the secant parabola" with interpolation points , , and when the points "collide" at the single point .  Thus the mystery behind the Taylor polynomial being based on a single point is revealed.  It is hoped that teachers reading this article will gain insight to how to use technology in teaching mathematics.  Higher degree polynomials have been investigated by the authors in the article "Investigation of Tangent Polynomials with a Computer Algebra System ", and some of the ideas are given below.

The Tangent Cubic Polynomial

    A natural question to ask now is: "What about polynomial approximation of higher degrees?"  Exploration of the Newton polynomials involves complicated symbolic manipulations and is prone to error when carried out with hand computations.  These derivations can become instructive and enjoyable when they are performed with computer algebra software.  Let be the Newton polynomial that passes through the four points    for  .  It may be shown that the Taylor polynomial is the limit of    as  .  We shall use the power of Mathematica to assist us with this derivation.  Begin by setting    equal to the general form of a Newton polynomial of degree n by issuing the following Mathematica commands:

Now form the set of four equations that force the polynomial to pass through the four equally-spaced points.  Notice that this is a lower-triangular system of linear equations.

Then solve this lower triangular linear system, and construct the polynomial  ,  and store it as the function  .  

Finally, compute the limit to verify that our conjecture was correct:

Eureka!  The limiting case of    as    is the Taylor polynomial .  Observe that the option    must be used in Mathematica's limit procedure.  This is a mathematicians way to tell the computer that    is "sufficiently differentiable."

Example 3.  Use the function    and the point    and draw the graphs of Newton cubic polynomials with and compare them with the Taylor polynomial.  
Solution 3.

Various Scenarios and Animations for the Tangent Parabola.

Example 4.  Use the function    and the point    and plot a sequence of Newton cubic polynomials converging to the Taylor polynomial.  
Solution 4.

Tangent Polynomials of Higher Degree

    Consider the seven points   for  .  We can construct the Newton polynomial of degree 6 and take the limit to obtain the Taylor polynomial.

Now form the set of seven equations that force the polynomial to pass through the four equally-spaced points.  

Then solve this lower triangular system, and construct the polynomial  ,  and store it as the function  .  

Finally, compute the limit and see that it is the Taylor polynomial.

Example 5.  Use the function    and the point    and plot a sequence of Newton polynomials of degree 6 converging to the Taylor polynomial.  
Solution 5.

References

1.  Russell Howell and John Mathews, "The Tangent Parabola," The AMATYC Review, Vol. 23, No. 1, Fall 2001, pp. 25-32.
2.  Russell Howell and John Mathews, "Investigation of Tangent Polynomials with a Computer Algebra System, " The AMATYC Review, Vol. 14, No. 1, Fall 1992, pp. 20-27.
3.  Kurtis Fink and John Mathews, Numerical Methods Using Matlab, 4th Edition, ISBN 0-13-065248-2, Prentice-Hall Pub. Inc., Upper Saddle River, NJ, 2004.  

Research Experience for Undergraduates

Lagrange Polynomials  Lagrange Polynomials  Internet hyperlinks to web sites and a bibliography of articles.  

Newton Polynomials  Newton Polynomials  Internet hyperlinks to web sites and a bibliography of articles.  

Maclaurin and Taylor Polynomials  Maclaurin and Taylor Polynomials  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook The Tangent Parabola

(c) John H. Mathews 2004