BC ONLYC4. 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 , 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
This approximation of e−0.25 is correct to four places.