Which of the following best explains why a fifth-degree Taylor polynomial centered at x=0 generally gives a better approximation to a smooth function f near x=0 than does a third-degree Taylor polynomial?
a) Because after truncating at degree 5 the error behaves like x5, whereas after degree 3 it behaves like x3, so the degree-5 error is smaller.
b) Higher-order derivatives give extra local information, so the degree-5 Taylor polynomial matches more terms of the series and leaves a remainder of higher order.
c) Because the factorial terms in the Taylor coefficients make every coefficient in a fifth-degree polynomial numerically smaller than in a third-degree polynomial, guaranteeing a smaller error.
d) Because once the Taylor polynomial degree reaches five for a smooth function, the remainder term becomes zero, so no error persists beyond that point.