




f(x) = f(a)
error = f ' (c)(x  a) for some c between a and x.
f(x) = f(a) + f '(a)(x  a)
Cubic approximation
Therefore, bounds on these derivatives f ', f '', f ''', f '''' in the interval [x,a] give bounds on these errors.
We can give bounds on the error in each approximation in terms of the minimum and maximum values_{ }of the derivative in the first term left out._{ }
Let x_{ }> a._{ } _{ }
In each case we assume that the derivative thatappears explicitly in the approximation formula are continuous between x and a.
Then,
_{Proofs}