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Unformatted text preview: MA2213 Lecture 12 REVIEW 1.1 on page 10 1. Compute , 1 ) ( lim â‰ âˆ’ â‰¡ â†’ x x e x g x x 2. Compute , ) 1 log( ) ( lim â‰ + â‰¡ â†’ x x x x g x ompute quadratic Taylor polynomials for ) ( lim ) ( f ; ), ( ) ( f x g x x g x x â†’ â‰¡ â‰ â‰¡ 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 [ âˆ’ 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 + + + âˆ’ = on page 11 and take ) ( p 3 x to be the third degree Taylor polynomial about . = a uggestion Study â€˜bounding the errorâ€™ 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 + â‰¡ about . = a 4. (b) Use the method developed on page 14 to obtain a Taylor series with remainder for ). ( tan ) ( 1 x x g âˆ’ â‰¡ Suggestion: Use part (a) âˆ« + = âˆ’ 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 âˆ’ . (a) 1 1 3 âˆ’ + x . (d) x e e x x 2 âˆ’ âˆ’ . (c) 3.2 on page 88 eview Newtonâ€™s method for finding roots, in articular Example 3.2.2 on pages 8183 that xplains the importance of choosing a sufficiently closeâ€™ 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 Î± with order of onvergence 1 â‰¥ p if there exists a constant â‰¥ c uch that . ,     1 â‰¥ âˆ’ â‰¤ âˆ’ + n x c x p n n Î± Î± 3 , 2 , 1 = linear, quadratic, cubic convergence. 3.4 on page 96 . Experimentally confirm the error estimate 13 . 1 â‰ˆ Î± where and is a sequence that converges n x to Î± obtained by the secant method. ,     62 . 1 1 â‰¥ âˆ’ â‰ˆ âˆ’ + n x c x n n Î± Î± is the unique positive root of 1 6 âˆ’ âˆ’ x x uestion What is the order of convergence of he secant method for this problem ? uestion What is the order of convergence of ewtonâ€™s method, the secant method, for finding he root of the function 2 ) ( f x x â‰¡ ? 4.1 on pages 134135 6. As a generalized interpolation problem, find the quadratic polynomial for which . 4 ) 1 ( , 1 )...
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 Fall '07
 Michael
 Linear Algebra, Polynomials, Numerical Analysis, Elementary algebra, elementary row operations

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