Evaluate, thusIs 27.aun jeun 8uI5) Calculate the Taylor polynomials T2 (a) and T3 (x) centered at a = a of the function f(x) = sin a for the givenvalue of a.(a) a = 0T(b) a =« PreviousNextMacBook AirF5F7%&*5664

Answered: Evaluate, thus Is 27- 5) Calculate the…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Evaluate, thus
Is 27.
aun jeun 8uI
5) Calculate the Taylor polynomials T2 (a) and T3 (x) centered at a = a of the function f(x) = sin a for the given
value of a.
(a) a = 0
T
(b) a =
« Previous
Next
MacBook Air
F5
F7
%
&
*
5
6
64

Expert Solution

Step 1

The given function is $f (x) = \sin x$ .

Calculate the Taylor polynomials $T_{2} (x) and T_{3} (x)$ at $a = 0$ and $a = \frac{π}{2}$ .

Taylor polynomial is given as $T_{n} (x) = \sum [\frac{f^{n} (a)}{n!}] \times {(x - a)}^{n}$ , where n ranges from 0 to ∞ and a is the value at which polynomial function is to be determined.

^{$\begin{array}{rcl} T_{2} (x) & = & \frac{f (a)}{0!} {(x - a)}^{0} + \frac{f' (a)}{1!} {(x - a)}^{1} + \frac{f'' (a)}{2!} {(x - a)}^{2} \\ = & f (a) + f' (a) (x - a) + \frac{f'' (a)}{2} {(x - a)}^{2} \end{array}$}

$\begin{array}{rcl} T_{3} (x) & = & \frac{f (a)}{0!} {(x - a)}^{0} + \frac{f' (a)}{1!} {(x - a)}^{1} + \frac{f'' (a)}{2!} {(x - a)}^{2} + \frac{f''' (a)}{3!} {(x - a)}^{3} \\ = & f (a) + f' (a) (x - a) + \frac{f'' (a)}{2} {(x - a)}^{2} + \frac{f''' (a)}{6} {(x - a)}^{3} \end{array}$