 Polynomials are frequently used to locally approximate functions. There are various ways this may be done. We consider here several forms of differential approximation.

###### 2.3.1 Univariate Approximations

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

[2.16] provides a good approximation for f, at least in a small interval about x. This is because:

• p1 equals f at x, and
• p1 has the same first derivative as f at x.

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

[2.17] Consider the function

[2.18] which has first and second derivatives

[2.19] [2.20] on . Let’s construct a linear polynomial approximation for f about the point x = 0. Applying [2.16], we obtain

[2.21] [2.22] This is graphed in Exhibit 2.2. Exhibit 2.2: Comparison of function f and linear polynomial p1.

We can improve the approximation, at least for values of x close to 0, with a quadratic polynomial. Applying [2.17] at x = 0, we obtain

[2.23] [2.24] This is graphed in Exhibit 2.3. Exhibit 2.3: Comparison of function f and linear polynomial p2.
###### 2.3.2 Multivariate Approximations

Polynomial approximations [2.16] and [2.17] generalize to multiple dimensions. For f : n → , gradients replace first derivatives and Hessians replace second derivatives, so linear polynomial [2.16] and quadratic polynomial [2.17] become

[2.25] [2.26] Consider the function

[2.27] [2.28] [2.29] Let’s construct a gradient-Hessian approximation about the point (0, 1). Applying [2.26], we obtain

[2.30] [2.31] ###### 2.3.3 Taylor Polynomials

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 x  . The polynomial

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

[2.33] [2.34] Taylor polynomials and Taylor series generalize to higher dimensions.

###### Exercises
2.1

Apply [2.17] to construct a quadratic polynomial approximation for the function f(x) = xex about the point x = 0.

2.2

Apply [2.26] to construct a quadratic polynomial approximation for the function f(x1, x2) = x1exp(x2) about the point x = (0, 0).

2.3

Construct the Taylor series expansion for the function log(1 + x) about the point x = 0.