10.1 The Quadratic Approximation

The linear approximation to f is exactly true if f ' is constant for that means that f is linear. The inaccuracy of the linear approximation to f at x₀ at argument x arises from the changes to f ' between arguments x₀ and x.

If f ' is differentiable in the interval between x₀ and x we can get a better approximation to f at x by making a linear approximation to f ' and using it to estimate the change to f in the interval.

In short if f ' is differentiable in that interval we can compute its derivative, called the second derivative of f with respect to x and written as f "(x) or as or sometimes as and use it to improve the estimate of f.

All of our standard functions have differentiable derivatives and even differentiable second derivatives, etc on forever wherever they are defined, except perhaps at specific singular points.

They are said to be "infinitely differentiable" because we can keep differentiating them as long as we like. We may therefore compute second derivatives, and also third and higher derivatives and generate a sequence of better and better approximations to any such function.