ISPA - Taylor Polynomials and Approximations

Taylor Polynomials and Approximations

Taylor and Maclaurin Polynomials

Polynomials are often used to approximate other elementary functions. Recalling from earlier in this course, the equation of a tangent line approximates a curve at a point (c, f(c)). When finding polynomials to approximate curves, the graph of the approximating polynomial must pass through the point (c, f(c)) and have the same defined slope at that point as the original function.

Consider the polynomial P_n(x), where n represents the degree of the polynomial, and several of its derivatives expanded about c.

When x = c, P_n(c) = a₀, P_n'(c)= a₁, P_n''(c) = 2a₂,..., P_n⁽ⁿ⁾(c) = n!a_n. Because the value of the function and its first n derivatives must agree with the value of the polynomial and its first n derivatives at x = c, the following is true:

$f\left(c\right)=a_0,f^{\prime}\left(c\right)=a_1,\frac{f^{\prime\prime}\left(c\right)}{2!}=a_2,\ldots,\frac{f^n\left(c\right)}{n!}=a_n$

 

The next two definitions are based on these coefficients.

Let f be a function whose first n derivatives exist at x = c. Then the polynomial

The presentation below focuses on developing an intuitive understanding of how an arbitrary function can be approximated by a Maclaurin polynomial.

Using Maclaurin and Taylor polynomials to approximate functions that cannot be created by addition, subtraction, multiplication, or division is the theme of the next presentation.

Taylor Polynomials Remainder Term

Determining the accuracy of approximating a function f(x) with a Taylor polynomial is an important consideration. The exact value of the function is equal to the approximate value plus any error resulting from the approximation. The error is generally thought of as the remainder. This notion is expressed symbolically as $f\left(x\right)=P_n\left(x\right)+R_n\left(x\right)$. Solving for the remainder produces R_n(x) = f(x) - P_n(x). The error is defined as the absolute value of the remainder, or symbolically, Error=|R_n(x)| = |f(x) - P_n(x)|.

A general rule for estimating the remainder associated with a Taylor polynomial is Taylor's Theorem: If a function f and its first n derivatives f ', f '',..., f⁽ⁿ⁾

are continuous on the closed interval [a, b] and f⁽ⁿ⁾ is differentiable on the open interval (a, b), then there exists a number c between a and b such that

$f\left(b\right)=f\left(a\right)+f'\left(a\right)\left(b-a\right)+\frac{f^n\left(a\right)}{2!}\left(b-a\right)^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}\left(b-a\right)^n+\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(b-a\right)^{n+1}$ ,

where the remainder term $R_n\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(b-a\right)^{n+1}$

Taylor's Theorem is actually a generalization of the Mean Value Theorem.

When Taylor's Theorem is applied, it is customary to hold a fixed while treating b as the independent variable. When this occurs, Taylor's Theorem is referred to as Taylor's Formula, which is expressed as follows:

If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I,

$f\left(x\right)=f\left(a\right)+f'\left(a\right)\left(x-a\right)+\frac{f''\left(a\right)}{2!}\left(x-a\right)^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}\left(x-a\right)^n+R_n\left(x\right)$

where $R_n\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(b-a\right)^{n+1}$ for some c between a and x.

Lagrange Form of the Remainder and Lagrange Error Bound

Image Caption: Joseph-Louis Lagrange (1736-1813), was a French mathematician and astronomer of French and Italian descent. Before the age of 20 he was professor of geometry at the royal artillery school at Turin. Among his early successes were his method of solving isoperimetrical problems, on which the calculus of variations is based in part on his research on the nature and propagation of sound and on the vibration of strings as well as his studies on the libration of the moon and on the satellites of Jupiter. His contributions to the development of mathematics also include the application of differential calculus to the theory of probabilities and notable work on the solution of equations.'

The remainder in Taylor's formula, $R_n\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}\left(x-a\right)^{n+1}$, is called the Lagrange form of the remainder. The Lagrange Error Bound is the error associated with approximating a function value f(x) by a Taylor polynomial P_n that satisfies the inequality $\left|R_n\left(x\right)\right|\le\frac{\left(x-c\right)^{\left(n+1\right)}}{\left(n+1\right)!}\max\left|f^{\left(n+1\right)}\left(z\right)\right|$, where max|f ⁽ⁿ⁺¹⁾(z)| is the maximum value of f ⁽ⁿ⁺¹⁾(z) between x and c. The presentation below explores how to determine the error when approximating a function value with a Taylor polynomial.

To explore the Lagrange remainder using an interactive applet, CLICK HERE. Links to an external site.

Taylor Polynomials and Approximations Practice

Taylor Polynomials and Approximations: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of Taylor polynomials and approximations.

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