midterm151a.01f

# midterm151a.01f - proximate the Fxed point, starting with p...

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UCLA MATH 151A/3, Monday October 29, 2001 MIDTERM EXAM NAME STUDENT ID # This is a closed-book and closed-note examination. Please show all your work. Partial credit will be given to partial answers. There are 4 problems of total 100 points. PROBLEM 1 2 3 4 TOTAL SCORE 1. (a) Do three iterations (by hand) of the bisection method, applied to f ( x ) = x 3 - 2, using a = 0 and b = 2 (Fnd p 0 , p 1 and p 2 ). (b) How many iterations does the theory predict it will take to achieve 10 - 5 accuracy, to ap- proximate the root of x 3 - 2 = 0 by the bisection method on the interval [0 , 2] ?

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2. (a) By a theorem from the course, show that the function g ( x ) = 1 + e - x has a unique Fxed point on [1 , 2] (given values: e - 1 = 0 . 3679, e - 2 = 0 . 1353). (b) ±or p 0 = 1, compute p 1 using the Fxed-point iteration. (c) How many iterations does the theory predict it will take to achieve 10 - 5 accuracy, to ap-

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Unformatted text preview: proximate the Fxed point, starting with p = 1 ? 3. (a) Write down Newtons iteration as applied to the function f ( x ) = x 3-2. Simplify the formula as much as possible. (b) Write down the Secant iteration as applied to the same function f ( x ) = x 3-2. 4. (a) Construct the Lagrange polynomial interpolating the points x = 0 and x 1 = h , to approxi-mate the function f ( x ) = cos x , where h &gt; 0 is a small parameter. (b) Write down the error formula as applied to the above interpolation. Find an upper bound for the error on the interval [0 , h ], function of the parameter h . (c) Construct the Lagrange polynomial interpolating the points x = 0, x 1 = h , and x 2 = 2 h to approximate the function f ( x ) = cos x ....
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## This note was uploaded on 10/28/2010 for the course MATH 151a taught by Professor Staff during the Spring '08 term at UCLA.

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midterm151a.01f - proximate the Fxed point, starting with p...

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