1. Let f(x)=sin(x) and apply Theorem 4.1. (a) Use x_(0)=0 and find P_(5)(x),P_(7)(x) and P_(9)(x) (b) Show that if vert xvert leqslant 1 , then the approximation sin(x)approx x-(x^3)/(3!)+(x^5)/(5!)-(x^7)/(7!)+(x^9)/(9!) has the error bound vert E_(9)(x)vert lt 1/10!leqslant 2.75574times 10^-7 (c) Use x_(0)^ast =pi /4 and find P_(5)(x) , which involves powers of (x-pi /4)

Question

Answer 1

(a) To find

,

, and

, we can use the Taylor polynomial approximation formula:

For

, we have

,

, and so on.

(b) To show that if

, then the approximation

has the error bound

, we can use the error bound formula for Taylor polynomials:

where

is the maximum value of

on

.In this case,

,

, and

. The maximum value of

on

is

, so we have:

Since

, we have

, so

.(c) To find

when

, we can use the same Taylor polynomial approximation formula as in part (a), but with

:$P_{5}(x) = \sin(\pi/4) + \cos(\pi/4)(x - \pi/4) + \frac

Soru