Suppose c ∈ I, an open interval and that a function f, defined on I has n + 1
derivatives. Then for each m ≤ n the following formula holds for x ∈ I.
where y is some point between x and c. Fix c,x in I. Let K be a number, depending on
c,x such that
Now the idea is to find K. To do this, let
Then F =
F = 0
. Therefore, by Rolle’s theorem there exists y between c and x
such that F′ = 0
. Do the differentiation and solve for K. This is the main result on
Taylor polynomials approximating a function f. The term f
the Lagrange form of the remainder.