MATH 251 Exam 2, Spring 2008

1 AMERICAN UNIVERSITY OF BEIRUT Faculty of Arts and Sciences ‐ Mathematics Department. Test II ­ MATH 251 Closed Book – 75 mn SPRING 2007 – 2008 STUDENT NAME . ID NUMBER . Problem 1 / 14 . Problem 2 / 14 . Problem 3 / 22 . TOTAL / 50 2 TABLE 1: Problems 1 and 2 Consider the following table of data for the function f(x). All computations shall be carried out with 8 significant figures. 3 # 1 ­ a – Use the Central Difference formula to approximate f ‘(0.5). Improve this result using Richardson’s extrapolation of the 1st and 2nd orders. For that purpose, give the formulae that provide , then fill out the table that follows. (10 points) 4 b –Using table 1, calculate the third derivative f “’(1.000) , using the Backward Difference Approximation. (4 points) f “’ (x) f ‘” (1.000) # 2 – a‐ Using table 1, write Newton’s interpolating polynomial of degree 3, that would best approximate f(0.4). Find f(0.4). (4 points) 5 b – Find the equation of the quadratic spline s(x), on the interval [0.125, 0.5], then draw its graph. Draw clearly the tangent to the curve at each node. (Let z1= s’ (x1) = 0) (10 points) 6 # 3 ­ Consider the following 4 by 4 square matrix: a­ Apply on this matrix Gauss Elimination with the scaled partial pivoting strategy, showing the status of the 4 by 4 matrix after each reduction. (Each pivot row and the corresponding multipliers should be identified and circled). Specify the index vector IV after each reduction. (10 points) Reduction 1: Modified matrix A including multipliers scales 7 8 b­ Specify the elements of the matrices P, L and U that satisfy the LU decomposition of the matrix A. (4 points) c­ Use the matrices P, L and U, to find the determinant of A. (2 points) 9 d­ Use the matrices P, L and U, to find the 2nd column (ONLY) of the inverse of A. (6 points)