MATH 210 Final, Spring 2004

LIBRARY OF BEIRUT Math 210 FINAL EXAMINATION, JUNE 4, 2004 1. (10 pts.) A function fis defined by f(x) = x3 sin(->), for x + 0, and f(0) = 0. Prove that f is differentiable and uniformly continuous on R. 2.(10 pts.) A sequence a, is defined by dnsi = ay + +, and a; = 1. Obtain the asymptotic behavior of a, and prove your answer. 3. (10 pts.) Letf': K + R be a continuous function, where K is a closed and bounded interval. Fora ¢ R put E = wek : fx) = a}. Prove that E is a compact subset of R. 4, (10 pts.) Letf: R> Rhea differentiable function, and assume that if (x)| < A < 1 forall x © R. A sequence x, is defined by Xn+1 = f(frn)). Prove that x, is a convergent sequence irrespective of the value of x, suet 5. (10 pts.) Let {be given by f(x) = eo oe Prove that fis defined and continuous on -1 <x < 1. Prove also that fis differentiable on (-1, 1) and find its derivative. 6. (10 pts.) Consider the infinite series 0 Dlnxe™ — (n= 1)xee "7. n=] If S, is the n" partial sum, compute Sp, limy..5,, and determine whether or not the series converges uniformly in a nbhd (-6,6) of 0. 7. (10 pts.) Evaluate the following limits and prove your answers: ln pnt (a) lim f Ke + X)"e2dy, (6) lim nm modo 1+t at. 8.( 10 pts.) State without proof four “ deeper” Properties of continuous functions.Prove that if fis continuous at a, and f(a) + 0, then 1/fis continuous at a. 9. (10 pts.) Define X+1 fa) = f sin(t3)dt. Show that fe) = 1 ¢ 20st") _ cos[(x + "Ly _ 2 a. COSU gy, 2 x xt (+1)? 438 and conclude that \/(x)| < C/x? , where Cis a constant; and that Ie sin(t? )dt converges. 10. (10 pts.) IfAx) = D0", 8 , prove that f(x) = 2%, =8m= for all x (0,27). n=! 11. (10 pts.) Prove that Sy eS dx is a convergent integral, which is not absolutely I+x? convergent. 12. (15 pts.) Let ¢ and So be continuous on / : 45x b. Define a sequence S,, of functions on J by Sil) = a+ * $()Sn-i(Odt,n > 1,a fixed in I (a) If S, converges uniformly to a function S on J, prove that S is differentiable and that S'(x) = O(x)S(x). (b) Show that<{S,,} is uniformly convergent on J. 1. (a) Use mathematical induction to prove that 1.2+2.3+4---n(n4 1) = n(n + 1)(n + 2)/3. (6 points) (b) Show that »? 4 2 (mod 6). € Z. State whether vour proof is direct or indirect? (6 points) 2. Let = Rx (-s,9). B= (Rx R) \ {(0.0)}. and f: A B be the function defined by fir. y) = (e" cos y.e" sin y). (2) Graph the sets 4 and B in a cartesian system. (3 points) (b) Show that f is a one-to-one correspondence. (6 points) (c) Find the inverse fiction of f. (2 points) 3. (@) There are four large groups of people. each with 1000 members, Any two of these eranps lave TAO members in common. An three of these groups have 10 members in common. Aud there is 1 person conumon to all four groups. All together. how many people are in these groups? (6 points) (b) There ave 10 questions in a Discrete Mathematics final examination. How mauy ways are there to assign scores to the problems if the sum of the scores is LOO and each question is worth at least 5 points? (@ points) (c) How inauy different words can be made from “SUCCESS” given that the first and last lerrors umst both be “S'” (@ points)