A cubic function is technically a "better approximation" than a quadratic function, but at the point it seems to be more of a quadratic shape, whereas a cubic function has to peel of to negative infiniti (and another to positive infiniti) right? Isn't that a poorer approximation?
keenan
Taylor's theorem is a purely local statement—the idea to think about here is that one can always find a sufficiently small neighborhood around the point $x_0$ where a 4th-order approximation is better than a 3rd-order approximation. What you're essentially thinking about is global, variational approximation, rather than local approximation. E.g., over some interval, minimize the difference between the original function and a polynomial approximation of fixed degree (with respect to some norm). But even in this case, one likely has nonzero coefficients for all terms in the polynomial.
A cubic function is technically a "better approximation" than a quadratic function, but at the point it seems to be more of a quadratic shape, whereas a cubic function has to peel of to negative infiniti (and another to positive infiniti) right? Isn't that a poorer approximation?
Taylor's theorem is a purely local statement—the idea to think about here is that one can always find a sufficiently small neighborhood around the point $x_0$ where a 4th-order approximation is better than a 3rd-order approximation. What you're essentially thinking about is global, variational approximation, rather than local approximation. E.g., over some interval, minimize the difference between the original function and a polynomial approximation of fixed degree (with respect to some norm). But even in this case, one likely has nonzero coefficients for all terms in the polynomial.