The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Mean Value Theorem and Taylors Formula

We can easily find the Taylor polynomials of f(x) by differentiating. Let us now try to find a formula for the Taylor remainders. The Mean Value Theorem gives a formula for the Taylor remainder, R0(x).

MEAN VALUE THEOREM (Repeated)

Suppose f(t) is differentiate at all t between c and d. Then

09_infinite_series-689.gif

for some point t0 strictly between c and d.

When we replace d by x, this gives the formula

f(x) = f(c) + R0(x). R0(x) = f'(t0)(x - c).

Taylor's Formula is a generalization of the Mean Value Theorem which gives the nth Taylor remainder.

TAYLOR'S FORMULA

Suppose the (n + l)st derivative f(n+1)(t) exists for all t between c and x. Then

f(x) = f(c) + f'(c)(x - c) + 09_infinite_series-690.gif(x - c)2 + ... +09_infinite_series-691.gif(x - c)n + Rn(x)

where

Rn(x) = 09_infinite_series-692.gif(x - c)n+1

for some point tn strictly between c and x.

Notice that the remainder term looks just like the (n + 1)st term of a Taylor polynomial except that f(n+1)(c) is replaced by f(n+1)(tn).

When r = 0 Taylor's Formula is sometimes called MacLaurin's Formula.

Taylor's Formula can be used to get an estimate of the error Rn(x) between f(x) and the Taylor polynomial Pn(x). For example if

|f(n+1)(t)| ≤ Mn+1 for all t between c and x. then we obtain the error estimate

|Rn(x)| ≤ 09_infinite_series-693.gif |x - c|(n+1).

Taylor polynomials with the error estimate are of great practical value in obtaining approximations. In the next example we use Taylor's Formula to approximate the value of e.


Last Update: 2006-11-08