Let . A second–degree Taylor polynomial is centered at and used to approximate near…

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

Problem

Let

f(x)=e^{x^{2}}

. A second–degree Taylor polynomial

P_2

is centered at

x=0

and used to approximate

f

near

x=0.50

. (a) Determine

P_2(x)

. (b) Compute

f'''(x)

in closed form. (c) Find a constant

M

such that

|f'''(z)|\le M

for every

z

in the closed interval

[0,0.50]

. Clearly indicate how you obtained

M

. (d) Using the value of

M

from part (c), establish a numerical upper bound for

|f(0.50)-P_2(0.50)|

. Present your result to six significant digits.