Define for all real by s(x)= cases x x , & x 0, 1, & x=0

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

Problem

Define

s(x)

for all real

x

s(x)=\begin{cases}\dfrac{\sin x}{x}, & x\ne0,\\ 1, & x=0. \end{cases}

Selected derivatives of

s

x=0

are listed below.

s(0)=1,\quad s''(0)=-\dfrac16,\quad s^{(4)}(0)=\dfrac1{120},\quad s^{(6)}(0)=-\dfrac1{5040}

(1) Part 1: (a) Using the data above, write the degree–6 polynomial

R_6(x)

that matches the value and the listed even–order derivatives of

s

x=0

. Express

R_6

in standard form. (2) Part 2: (b) Use

R_6

to estimate

s(0.3)

. Report your result to 6 decimal places. (3) Part 3: (c) The calculator value of

s(0.3)

0.985067

(rounded). Compute the percentage error of your estimate from part (b), correct to three significant figures. (4) Part 4: (d) Explain qualitatively how adding the

x^{8}

term that would match the eighth–order derivative of

s

x=0

is expected to affect the magnitude of the error found in part (c).

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