Taylor Series Expansion

Explained: derivative approximation gradient approximation gradient-Hessian approximation Taylor polynomial Taylor series expansion

Polynomials are frequently used to locally approximate functions. There are various ways this may be done. This article discusses several forms of differential approximation, culminating with Taylor series.

Univariate Approximations
Consider a function f: that is differentiable in an open interval about some point . The linear polynomial

is called a derivative approximation. It provides a good approximation for f, at least in a small interval about . This is because:

equals f at , and

has the same first derivative as f at .

If f is twice differentiable in an open interval about , we can improve the approximation with a quadratic polynomial

[2]

Consider the function

[3]

which has first and second derivatives

	[4]
	[5]

on . Let’s construct a linear polynomial approximation for f about the point = 0. Applying [1], we obtain

	[6]
	[7]

This is graphed in Exhibit 1

Example: Linear Approximation Exhibit 1

Linear approximation [7] fit to function [3].

We can improve the approximation, at least for values x close to 0, with a quadratic polynomial. Applying [2] at = 0, we obtain

	[8]
	[9]

This is graphed in Exhibit 2.

Example: Quadratic Approximation Exhibit 2

Quadratic approximation [9] fit to function [3].

Multivariate Approximations
Polynomial approximations [1] and [2] generalize to multiple dimensions. For f: , gradients replace first derivatives, and Hessians replace second derivatives, so linear polynomial [1] and quadratic polynomial [2] become

	[10]
	[11]

with primes '  now indicating transposition, as opposed to differentiation. We call these gradient approximations and gradient-Hessian approximations, respectively.

Consider the function

[12]

which has gradient and Hessian

	[13]
	[14]

Let’s construct a gradient-Hessian approximation about the point (0,1). Applying [11], we obtain

	[15]
	[16]

Taylor Series
The linear and quadratic polynomial approximations discussed in this section are examples of a more general concept called Taylor polynomials. Consider a function f: whose first m derivatives exist in an open interval about a point . The polynomial

[17]

is called the -order Taylor polynomial of f about the point . It provides a good approximation for f, at least in a small interval about . If all derivatives exist for f in an open interval about , we may consider the limiting polynomial as m approaches infinity. This is called the Taylor series expansion of f about the point . In some—but not all!—cases, a function equals its Taylor series expansions on . For example, functions and sin(x) both equal their Taylor series expansions about the point = 0:

	[18]
	[19]

Taylor polynomials and Taylor series generalize to higher dimensions.

Exercises
Apply [2] to construct a quadratic polynomial approximation for the function f(x) = xex about the point x[0] = 0. [solution] Apply [11] to construct a quadratic polynomial approximation for the function about the point x[0] = (0,0). [solution] Construct the Taylor series expansion for the function log(1+x) about the point x[0] = 0. [solution]
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cubic spline interpolation Any of several methods of interpolating with cubic splines. gradient, Hessian, Jacobian Multidimensional generalizations of first and second derivatives. interpolation Any procedure for fitting a function to a set of points in such a manner that the function intercepts each of the points. linear and quadratic polynomials Two basic forms of polynomials. method of least squares Any of several techniques for fitting a curve to data so as to minimize the sum of squared differences between the curve and data points.
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