Constant approximation

f(x) = f(a)

error = f ' (c)(x - a) for some c between a and x.

Linear approximation

f(x) = f(a) + f '(a)(x - a)

Quadratic approximation

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