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Unformatted text preview: MA2213 Lecture 12 REVIEW 1.1 on page 10 1. Compute , 1 ) ( lim ≠ − ≡ → x x e x g x x 2. Compute , ) 1 log( ) ( lim ≠ + ≡ → x x x x g x ompute quadratic Taylor polynomials for ) ( lim ) ( f ; ), ( ) ( f x g x x g x x → ≡ ≠ ≡ here g is the function in problems 11 and 12. 1.2 on page 18 . Bound the error in using on the interval x e ]. 1 , 1 [ − to approximate Use the remainder ) ( p 3 x x a c c n a x x n n n and between ), ( f )! 1 ( ) ( ) ( ) 1 ( 1 + + + − = on page 11 and take ) ( p 3 x to be the third degree Taylor polynomial about . = a uggestion Study ‘bounding the error’ on page 15 eview power series and formuli on page 17 1.2 on page 19 4. (a) Obtain the Taylor polynomial for 2 1 1 ) ( f t t + ≡ about . = a 4. (b) Use the method developed on page 14 to obtain a Taylor series with remainder for ). ( tan ) ( 1 x x g − ≡ Suggestion: Use part (a) ∫ + = − x t dt x 2 1 1 ) ( tan and 2.2 on page 54 eview the elementary concepts in Chapter 2 bout errors, particularly the sources of error n pages 4547, and lossofsignificance errors and how to reduce such losses) on pages 4750 2 cos 1 x x − . (a) 1 1 3 − + x . (d) x e e x x 2 − − . (c) 3.2 on page 88 eview Newton’s method for finding roots, in articular Example 3.2.2 on pages 8183 that xplains the importance of choosing a sufficiently close’ initial estimate, and the error analysis on ages 8386 . m a a m , , > a positive integer ead about order of convergence on page 101 : sequence n x converges to α with order of onvergence 1 ≥ p if there exists a constant ≥ c uch that . ,     1 ≥ − ≤ − + n x c x p n n α α 3 , 2 , 1 = linear, quadratic, cubic convergence. 3.4 on page 96 . Experimentally confirm the error estimate 13 . 1 ≈ α where and is a sequence that converges n x to α obtained by the secant method. ,     62 . 1 1 ≥ − ≈ − + n x c x n n α α is the unique positive root of 1 6 − − x x uestion What is the order of convergence of he secant method for this problem ? uestion What is the order of convergence of ewton’s method, the secant method, for finding he root of the function 2 ) ( f x x ≡ ? 4.1 on pages 134135 6. As a generalized interpolation problem, find the quadratic polynomial for which . 4 ) 1 ( , 1 )...
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This note was uploaded on 01/06/2012 for the course MA 2213 taught by Professor Michael during the Fall '07 term at National University of Singapore.
 Fall '07
 Michael
 Polynomials, Numerical Analysis

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