Is quadratic usually sufficient for a good approximation? What situations require greater-than-quadratic approximation?
How quickly does the error term for the approximation shrink as we expand the Taylor series?
How strict is the tradeoff between number of operations to compute the approximation and how good the approximation is? At what point would you say "that's enough"?
Since the Taylor series have an error term at the very end, replacing complicated functions with approximation could be helpful but error prone. How can we determine whether the error term ignored would not have a negative influence.
Is there a function we can use to approximate the error for the Taylor series? There are definitely ways to generate upper bounds, but I feel like there are times when the upper bound is not tight enough.
How does quadratic approximation help with graphics algorithm?
in practice, how many terms of taylor expansion do we usually consider taking? for the tradeoff between accuracy and computational cost
How do we make linear approximations when the function is not differentiable/non-continuous?