A twice-differentiable function has an th–degree Taylor polynomial about

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

A twice-differentiable function

f

has an

n

th–degree Taylor polynomial

T_n

about

x=a

. The coefficients of

T_n

are determined by evaluating derivatives of

f

x=a

. Assume

n\ge 2

. (a) Explain, in complete sentences, why matching the first

k

derivatives of

T_n

to the corresponding derivatives of

f

x=a

guarantees that the coefficients of the first

k

non-constant terms of

T_n

are uniquely determined. (b) Describe how increasing the degree

n

of the Taylor polynomial affects the local accuracy of

T_n

as an approximation to

f

near

x=a

. In your answer, explicitly connect the role of higher-order derivatives to the observed improvement.