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).

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.

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⁾(t_n).

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 R_n(x) between f(x) and the Taylor polynomial P_n(x). For example if

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

|R_n(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.