Barrons AP Calculus - David Bock

C4. Approximating Functions with Taylor and Maclaurin

Polynomials.

The function f (x) at the point x = a is approximated by a Taylor polynomial Pn (x) of order n:

The Taylor polynomial Pn (x) and its first n derivatives all agree at a with f and its first n
derivatives. The order of a Taylor polynomial is the order of the highest derivative, which is also the
polynomial’s last term.
In the special case where a = 0, the Maclaurin polynomial of order n that approximates f (x) is

The Taylor polynomial P 1 (x) at x = 0 is the tangent-line approximation to f (x) near zero given by

f (x) P 1 (x) = f (0) + f ′(0)x.

It is the “best” linear approximation to f at 0, discussed at length in Chapter 4 §L.

A NOTE ON ORDER AND DEGREE
A Taylor polynomial has degree n if it has powers of (x − a) up through the nth. If f (n) (a) = 0,
then the degree of Pn (x) is less than n. Note, for instance, in Example 45, that the second-order
polynomial P 2 (x) for the function sin x (which is identical with P 1 (x)) is or just x, which
has degree 1, not 2.

EXAMPLE 44

Find the Taylor polynomial of order 4 at 0 for f (x) = e−x. Use this to approximate f (0.25).

SOLUTIONS: The first four derivatives are −e−x, e−x, −e−x and e−x ; at a = 0, these equal −1, 1,

−1, and 1 respectively. The approximating Taylor polynomial of order 4 is therefore

With x = 0.25 we have