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Best expansion point Suppose you wish to approximate cos

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett ISBN: 9780321570567 2

Solution for problem 82E Chapter 9.1

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 82E

Best expansion point Suppose you wish to approximate cos (π/12) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or ? Use a calculator for numerical experiments and check for consistency with Theorem 9.2. Does the answer depend on the order of the polynomial?

Step-by-Step Solution:

Solution 82E

Step 1:

Given function  is  f(x) = cos(x)

General  expression of Taylor series of  a function f(x) centered at ‘a’ is ;

                                  f(a) +(x-a) +++.............

Consider , f(x) = cos(x) , then f(0) = cos(0) = 1

                 = -sin(x) , then     = -sin(0)  =0

                    = - cos(x) , then     = -cos(0)  = -1

                      = sin(x) , then     = sin(0)  = 0

  =  cos(x) , then     = cos(0)  = 1……………..

Therefore , the Taylor polynomial  of the function f(x) = cos(x)with center zero is ;

                            cos(x) = 1 - + -+.............

Take x =  , then  we get ,

                           cos() = 1 - + -+.............

                                         1 - 0.0342694597+1.95732644986-.........

                                       0.9659262729

Step 2:

The Taylor polynomial of the function f(x) with center  is;

         f(x)=  f( ) +(x- ) +++.............

Consider , f(x) = cos(x) , then f() = cos() =  

                 = -sin(x) , then     = -sin()  =

                    = - cos(x) , then     = -cos()  =

                      = sin(x) , then     = sin()  =

  =  cos() , then     = cos()  =  ……………..

Therefore , the Taylor polynomial  of the function f(x) = cos(x)with center  is ;

                cos(x) =  +(x- ) +++.............

                 cos(x)  = -(x- ) -++.............

Take , x = , then  we get ,

                 cos()  = -(- ) -++.............

                                = +( ) -++.............

                                 0.86602540378 +0.1308996939-0.02967822269+...............

                                0.96724687499

Step 3 of 3

Chapter 9.1, Problem 82E is Solved
Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

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