These higher approximations are useful
in the following ways:
1. They tell us key information about f when all its lower
derivatives are 0 at x0.
2. They allow us to get bounds on the accuracy of lower approximations.
3. They can be used to deduce important facts (as in exercise
4. Being polynomials they are typically easier to manipulate
than f itself is.
The higher derivatives are sometimes of interest in themselves.
Thus the equations of motion of mechanics directly involve
acceleration, which is the second derivative of position.