## Taylor polynomials

9.10 TAYLOR's FORMULA

If we wish to express f(x) as a power series in x - c, we need two things:

(1)    A sequence of polynomials which approximate f(x) near x = c,

(1)

a0, a0 + a1(x - c), ...,a0 + a1(x - c) + ... + an(x - c)n,....

(2)     An estimate for the error En between f(x) and the nth polynomial,

(2)

f(x) = a0 + a1(x - c) + ... + an(x - c)n + En.

In the last section the formula

was used to obtain power series approximations. A much more general formula of this type is Taylor's Formula. In Taylor's Formula the nth polynomial Pn(x) is chosen so that its value and first n derivatives agree with

f(x) at x = c.

The tangent line at x = c,

P1(x) = f(c) + f'(c)(x - c),

has the same value and first derivative as

f(x) at x = c.

A polynomial of degree two with the same value and first two derivatives as f(x) at c is

P1(x) and P2(x) are the first and second Taylor polynomials of f(x) (see Figure 9.10.1).

Figure 9.10.1 First and Second Taylor Polynomials

To continue the procedure we need a formula for the nth derivative of a polynomial.

LEMMA 1

Let P(x) be a polynomial in x - c of degree n.

P(x) = a0 + a1(x - c) + a2(x - c) + ... + an(x - c)n.

For each m ≤ n, the mth derivative of P(x) at x = c divided by m! is equal to the coefficient am,

PROOF

Consider one term ak(x - c)k. Its mth derivative is

 k(k - 1)... (k - m + l) ak(x - c)k-m if m < k, m! am if m = k, 0 if m > k.

At x = c, the mth derivative of ak(x - c)f is:

0 if in < k, m! am if in = k, 0 if in > k.

It follows that

P(m)(c) = m! am.

This lemma shows us how to find a polynomial P(x) whose value and first n derivatives agree with f(x) at x = c. The mth coefficient of P(x) must be

DEFINITION

Let f(x) have derivatives of all orders at x = c. The nth Taylor polynomial of f(x) at x = c is the polynomial

By Lemma 1, Pn(x) is the unique polynomial of degree n whose value and first n derivatives at x = c agree with f(x),

Pn(c) = f(c), Pn'(c) = f'(c),.... Pn(n)(c) = f(n)(c).

The difference between f(x) and the nth Taylor polynomial is called the nth Taylor remainder,

Rn(x) = f(x) - Pn(x). Thus

f(x) = f(c) + f'(c)(x - c) +(x - c)2 + ... + (x - c)n + Rn(x).

Last Update: 2006-11-08