Let be a non-zero real constant and define

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

Problem

Let

k

be a non-zero real constant and define

f_k(x)=e^{kx}

. A general Taylor polynomial of degree

n

about

x=0

is sought. (a) Derive an explicit formula for

f_k^{(m)}(x)

for a non-negative integer

m

. Your answer should be expressed entirely in terms of

k

m

, and

x

. Justify your result. (b) Using part (a), write the degree-

n

Taylor polynomial

P_n(x)

for

f_k(x)

centered at

x=0

in closed form. (c) Specialize your formula from part (b) to

n=4

and write

P_4(x)

explicitly. (d) For fixed

n

, discuss qualitatively how the magnitude of

k

influences the accuracy of

P_n(x)

as an approximation to

f_k(x)

for values of

x

near

0

. Support your discussion with mathematical reasoning.