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midterm_F

# midterm_F - 0625 3 = 0 0208 Knowing that f(1 5 = 0 3535...

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UCLA MATH 151A, WINTER 2000, MIDTERM, FRIDAY, FEBRUARY 9 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 2 problems of total 20 points. PROBLEM 1 2 3 4 5 TOTAL SCORE Let f ( x ) = 1 2 x and x 0 = 0, x 1 = 1, x 2 = 2. 1. Determine P 2 , the Lagrange interpolating polynomial, of degree at most 2, which agrees with f at x 0 , x 1 , x 2 . 2. Use the following table to estimate P 2 (1 . 5) with a two-digits with a two-digits chopping arithmetic. x x 2 1 8 5 8 1 . 5 2 . 25 0 . 125 0 . 625 3. Use the Theorem of the course to find a bound of the absolute error between f (1 . 5) and P 2 (1 . 5). (You can use the fact that ( ln (2)) 3 1 3 , 1 . 5 24
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Unformatted text preview: . 0625 3 = 0 . 0208.) Knowing that f (1 . 5) = 0 . 3535, compare to the results you obtained and conclude. Let f ( x ) = x-2 cos ( x ) 1. Proove that the equation f ( x ) = 0 admits a solution p in the interval [0 , π 2 ]. 2. Use the following table to determine the Frst steps of the bisection method when starting with the x π 16 2 π 16 3 π 16 4 π 16 5 π 16 6 π 16 7 π 16 8 π 16 f ( x )-2-1 . 76-1 . 45-1 . 07-. 62-. 12 . 41 . 98 1 . 57 points 0 and π 2 . You will sumarize your results in this table n a n b n p n f ( p n ) π 2 1 2 3 ... ... 3....
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