A bacterial culture grows according to

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

Problem

g(t)=e^{0.04t}

, where

t

is measured in minutes. A biologist wants a quick polynomial formula that estimates

g(t)

for

0\le t\le 4

with an error not exceeding

5\times10^{-5}

. Let

P_n(t)

be the

n

th-degree Maclaurin polynomial for

e^{0.04t}

, and let

R_n(t)=g(t)-P_n(t)

. For all

t\in[0,4]

, the

(n+1)

st derivative of

g

satisfies

|g^{(n+1)}(t)|\le 0.04^{n+1}e^{0.16}

. (a) Establish an inequality that bounds

|R_n(4)|

in terms of

n

. (b) Determine the smallest integer

n

for which

|R_n(4)|<5\times10^{-5}

. (c) Using the degree found in part (b), write the corresponding polynomial

P_n(t)

. Then approximate the population multiplier

g(3)

to six decimal places and state the maximum error associated with this approximation.