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## 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

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) + (x - c)2 + ... +(x - c)n + Rn(x)

where

Rn(x) = (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)| ≤ |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