midterm151a.01

Midterm151a.01 - g x = π 0 5 sin x 2 has a unique Fxed point on[0 2 π(b ±or p = π compute p 1(c How many iterations are necessary to achieve

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UCLA MATH 151A/2, Friday February 9, 2001 NAME STUDENT ID # This is a closed-book and closed-note examination. No calculators are allowed. Please show all your work. Partial credit will be given to partial answers. There are 4 problems of total 20 points. PROBLEM 1 2 3 4 TOTAL SCORE I. Let f ( x ) = πx - cos ( πx ). (a) Prove that the equation f ( x ) = 0 has at least a solution p in the interval [0 , 1]. (b) By the Bisection method, fnd p n , n 2 on [0 , 1]. Use the table x 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1 f ( x ) -1 -0.297 0.179 0.702 1.253 1.783 2.242 2.581 2.772 and write your answers in the next table. n a n b n p n f ( p n ) 0 0 1 1 2 (c) How many iterations are necessary to solve πx - cos ( πx ) = 0 with accuracy 10 - 5 on [0 , 1] ?

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II. (a) Use the Theorem from the course to prove that

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Unformatted text preview: g ( x ) = π + 0 . 5 sin x 2 has a unique Fxed point on [0 , 2 π ]. (b) ±or p = π , compute p 1 . (c) How many iterations are necessary to achieve the accuracy 10-2 ? III. Let f ( x ) = x 2-6. (a) Prove that f has a positive zero p . (b) With p = 1, fnd p 2 by Newton’s iterations. IV. Let f ( x ) = cos x , x = 0, x 1 = 0 . 6 and x 2 = 0 . 9. (a) Construct an interpolation polynomial of degree at most two to approximate f . Use cos (0 . 6) = 0 . 8 and cos (0 . 9) = 0 . 6. (b) Use the Theorem of the course to Fnd an error bound for the approximation....
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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.01 - g x = π 0 5 sin x 2 has a unique Fxed point on[0 2 π(b ±or p = π compute p 1(c How many iterations are necessary to achieve

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