CMPS 251 Final, Spring 2001

AMERICAN UNIVERSITY OF BEIRUT VIATH 251. Final Exam June 15. 2001 ‘Time = 1 Honr and 50 Minnites Part I: Multiple choice (each question catrics a weieht of 10 points) (VIP: Checking more than oue auswer 18 considered as a wrong answer!) 1. Given a frmetion (7) which las derivative of infinite order. What does the following expression, al —h) a . Se 2h), estiinate? (a) fr) (b) f"Gr) (e) fila) + £" Cr) (ad) Ce) = f(r) (e) none of the above 2. The integral [! 2"dr is to be approximated by using Romberg method. What is the value of nsnch that [R(n, 2) — RU.n — 1)| < 0.005. (a) 1 (b) 2 (c) 3 (d) 4 (e) none of the above 3. Yon are to obtain an approximate solution to the equation: c?—4e+1 = 0. One iteration by the Newton's method is performed, using the initial approximation vy = 0. resulting in a first approximation, ©. One iteration of the Secaut method is then performed. using 2o and 7, as the initial approximations. proditcing au approximate solution, y. Determine :r2 (aj 3/8 (b) 31/60 (c) -1/2 (a) 1 (e) none of the above . Given the followings table of data: | 4 7 [3 a {| 1 |: 3 fo) J2yrfi 4 [2 Using Lagrange interpolating polynomial for the above data, what is the value of Iq(-2)° (a) 1 (b) 5 (ce) 1% (a) 35 (e) none of the above 1 + Refer to the data of the previous qnestion. You are to approximate f(s) 2 by a quadratic spline, Q;(r). for 4 = 0.1.2.3. What is the coefficient of we? in gr)? (a) -7 (b) -6 (¢) 5 (d) T (e) none of the above Refer to the data in previons question (4). The Least Squares method ix used fo approximate a liuear relationship to f(r). that is, a + be. Using this approximation. what is the value of ((@.5)" (a) -0.65 (bb) 0.15 (c) 1.54 (d) 2.65 (e) none of the above Part Il: Written Questions (Please show all work!) Cousider the following [nitial Value Problem (1\P): (0) (ai Find the approximate value of vr(1) usiug Enler’s method and hes 0.4. (pts) (hb) Pind the approximate valne of (1) using Runge-Kutta method of order 2 aud fy = 0.5. (5 pts) (oy Tis known that the exact soliitiou solution to the above [VP ts OD of the form rsa + dle Jetermine the values of a and § and then compure the absolute errov for the above approximations. respectively. (LO pts) X. You are given the following svstem of eqnatious where @ is a real munber. (a) Use the Naive Ganssiau Elimination method to solve the above system. (10 pts) (bo) Disesue the sobttion(s) af the above system when @ = 1 and a =