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 10.3).
4. Being polynomials they are typically easier to manipulate than f itself is.
5. 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.
6. Finally, they extend the distance from the expansion point over which they are accurate, when compared to a lower approximation.