Evaluating a polynomial function

First of all I'd like to say thanks for all the help with this assignment so far, theres no way I'd be even close to as far as I am currently without all the help that I've received.

All I need to do now is to evaluate the function that I've got for smoothness, and length. Now length is easy, i can just plug it into a program and have it break up my polynomial into lost of short linear equations and approximate the length off all of these added together (if theres a way to do this differently eg. a formula or something please let me know). But I'm stuck on how to evaluate the smoothness of my polynomial function, if anyone knows of any methods to achieve this please let me know.