taylor series

for taylor series do you assume that all y's are ao+a1x+a2x^2 etc?

Yes, for the purpose of this course you can assume that all y's can be written as $y(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dots$

Note that there are functions for which this is not possible, but this is a topic for another course.