Suppose

f(t)

can be written in a Taylor series representation centered at a number

a

as\

f(t)=f(a)+f^(')(a)(t-a)+(1)/(2)f^('')(a)(t-a)^(2)+(f^(''')(a))/(6)(t-a)^(3)+cdots

\ Suppose

h>0

below.\ a. Substitute

t=x+h

and

a=x

above and write the result.\ b. Substitute

t=x-h

and

a=x

above and write the result.\ c. Combine the information from parts

a

and

b

to show that

f^('')(a)

can be written in the form\

f^('')(a)=N_(1)(h)+K_(2)h^(2)+K_(4)h^(4)+cdots

\ where

K_(2),K_(4)

, etc. are independent of

h

, and give an explicit formula for

N_(1)(h)

.\ d. Explain why

N_(1)(h)

is an

O(h^(2))

approximation of

f^('')(a)

. Do you recognize

N_(1)(h)

?\ e. Use

N_(1)(h)

and

N_(1)((h)/(2))

to create an

O(h^(4))

approximation

N_(2)(h)

of

f^('')(a)

.

f(t)=f(a)+f(a)(ta)+21f(a)(ta)2+6f(a)(ta)3+ Suppose h>0 below. a. Substitute t=x+h and a=x above and write the result. b. Substitute t=xh and a=x above and write the result. c. Combine the information from parts a and b to show that f(a) can be written in the form f(a)=N1(h)+K2h2+K4h4+ where K2,K4, etc. are independent of h, and give an explicit formula for N1(h). d. Explain why N1(h) is an O(h2) approximation of f(a). Do you recognize N1(h) ? e. Use N1(h) and N1(2h) to create an O(h4) approximation N2(h) of f(a)