Background.

    When a Taylor series is truncated to a finite number of terms the result is a Taylor polynomial.  A Taylor series expanded about , is called a Maclarin series.  These Taylor (and Maclaurin) polynomials are used to numerically approximate functions.  We attribute much of the founding theory to Brook Taylor (1685-1731), Colin Maclaurin (1698-1746) and Joseph-Louis Lagrange (1736-1813).

Theorem (Taylor Polynomial Approximation).  Assume that  ,  then

    ,
    
where is a polynomial that can be used to approximate  , and we write  

    .

The remainder term has the form

    ,

for some value that lies between .  The formula is referred to as the Lagrange form of the remainder.

Proof  Maclaurin and Taylor Polynomials  Maclaurin and Taylor Polynomials  

Corollary 1.  Assume that  , and that the Taylor polynomial of degree for is , then

       for  .

Proof  Maclaurin and Taylor Polynomials  Maclaurin and Taylor Polynomials  

Corollary 2.  Assume that  , and that the Taylor polynomial of degree for is , then

    ,

where  .  

Proof  Maclaurin and Taylor Polynomials  Maclaurin and Taylor Polynomials  

Animations (Maclaurin and Taylor Polynomials  Maclaurin and Taylor Polynomials).  Internet hyperlinks to animations.

Computer Programs  Maclaurin and Taylor Polynomials  Maclaurin and Taylor Polynomials  

Example 1.  Consider the function  .  
1 (a).  Find the terms up to    in the Maclaurin series for  f[x].
1 (b).  Investigate the error term for the Maclaurin polynomial of degree n = 10 over the interval  [-0.5, 0.5].  
1 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].
Solution 1 (a).
Solution 1 (b).
Solution 1 (c).

Example 2.  Consider the function  .  
2 (a).  Find the terms up to    in the Maclaurin series for  f[x].
2 (b).  Investigate the error term for the Maclaurin polynomial of degree n = 10 over the interval  [-2.0, 2.0].  
2 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].
Solution 2 (a).
Solution 2 (b).
Solution 2 (c).

Mathematical notation.  Mathematica has adopted the notation    for the natural logarithm.  This can be illustrated by using either differentiation or integration.  Since    starts with the upper case letter  L,  the word    is a "reserved word."

Example 3.  Consider the function  .  
3 (a).  Find the terms up to    in the Maclaurin series for  f[x].
3 (b).  Investigate the error in the approximation over the interval [-0.5, 0.5].
3 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].
Solution 3 (a).
Solution 3 (b).
Solution 3 (c).

Example 4.  Consider the function  .  
4 (a).  Find the terms up to    in the Maclaurin series for  f[x].
4 (b).  Investigate the error term for the Maclaurin polynomial of degree n = 10 over the interval  [-2.0, 2.0].  
4 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].
Solution 4 (a).
Solution 4 (b).
Solution 4 (c).

Example 5.  Consider the function  .  
5 (a).  Find the terms up to    in the Maclaurin series for  f[x].
5 (b).  Investigate the error term for the Maclaurin polynomial of degree n = 20 over the interval  [-2.0, 2.0].  
5 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].
5 (d).  The relationship to the standard normal distribution.
Solution 5 (a).
Solution 5 (b).
Solution 5 (c).
Solution 5 (d).

Various Scenarios and Animations for the Taylor and Maclaurin polynomials.

Example 6.  Find the Taylor polynomial for   expanded about  , , and .  
Solution 6 (a).
Solution 6 (b).
Solution 6 (c).

Example 7.  Find the Maclaurin polynomial for    expanded about  .  
Solution 7.

Example 8.  Find the Maclaurin polynomial for   expanded about  .  
Solution 8.

Example 9.  Find the Taylor polynomial for   expanded about  , and .  
Solution 9 (a).
Solution 9 (b).

Example 10.  Find the Taylor polynomial for   expanded about  , and .  
Solution 10 (a).
Solution 10 (b).

Example 11.  Find the Taylor polynomial for   expanded about  , and .  
Solution 11 (a).
Solution 11 (b).

Example 12.  Find the Maclaurin polynomial for   expanded about  .  
Solution 12.

Example 13.  Find the Maclaurin polynomial for   expanded about  .  
Solution 13.

Example 14.  Find the Maclaurin polynomial for   expanded about  .  
Solution 14.

Example 15.  Find the Maclaurin polynomial for   expanded about  .  
Solution 15.

Example 16.  Find the Maclaurin polynomial for   expanded about  .  
Solution 16.

Example 17.  Find the Maclaurin polynomial for   expanded about  .  
Solution 17.

Example 18.  Find the Maclaurin polynomial for   expanded about  .  
Solution 18.

Example 19.  Find the Maclaurin polynomial for   expanded about  .  
Solution 19.

Animations (Taylor and Maclaurin Polynomial Approximation  Taylor and Maclaurin Polynomial Approximation).  Internet hyperlinks to animations.

Old Lab Project (Maclaurin Polynomials  Maclaurin Polynomials).  Internet hyperlinks to an old lab project.  

Old Lab Project (Taylor Polynomials  Taylor Polynomials).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

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

Download this Mathematica Notebook Maclaurin and Taylor Polynomials

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(c) John H. Mathews 2004