Download Math 1502 K, Calculus II Final Exam December 12th 2008 - Prof. Jean Bellissard and more Exams Calculus in PDF only on Docsity! Math 1502 K, Final Exam December 12th 2008, Name:.......................... 1 Georgia Tech School of Mathematics Math 1502 Calculus II Final Exam : 2 hours & 50 minutes December 12th 2008, Section K First Name : โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ Last Name : โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ Students : please do not write anything in the box below ! ! I.1 II.1 III.1 2 pts 2 pts 2 pts I.2 II.2 III.2 2 pts 2 pts 2 pts I.3 II.3 III.3 2 pts 2 pts 6 pts I.4 II.4 III.4 3 pts 4 pts 6 pts I.5 II.5 3 pts 8 pts I.6 II.6 8 pts 6 pts Math 1502 K, Final Exam December 12th 2008, Name:.......................... 2 WARNING : Put your name on the top of each page. Read carefully, read the comments in italic, take your time, do not panic and double check what you write. Write the result cleanly and use the blank pages for your calculations. Take the time to write in plain English the arguments used to get or justify the answer. Math 1502 K, Final Exam December 12th 2008, Name:.......................... 5 5. Is the following series converging (Indicate the test used to conclude) โ (k!)2 (2k)! Converges 2 Diverges 2 Test used : 6. Give the interval of convergence of (Hint : analyze the convergence inside and on the boundary of the interval) โโ k=0 xk (2k + 1)1/3 Interval of convergence = Math 1502 K, Final Exam December 12th 2008, Name:.......................... 6 (Use this page for your calculation) Math 1502 K, Final Exam December 12th 2008, Name:.......................... 7 II- Linear Algebra 1. Indicate whether the following function is linear or not and why g ([ x y ]) = ๏ฃฎ๏ฃฐ y + x0 xโ y ๏ฃน๏ฃป Linear ? YES 2 NO 2 Why? 2. Find all solutions (a, b, c) of the equation A2 = B whenever A = [ a b 0 c ] B = [ 1 3 0 4 ] Math 1502 K, Final Exam December 12th 2008, Name:.......................... 10 6. Compute the inverse and the determinant of B below (Hint : use the same row reduction method for both the inverse and the determinant) B = ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฐ 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 ๏ฃน๏ฃบ๏ฃบ๏ฃป det B = Bโ1 = Check : B ยทBโ1 = Math 1502 K, Final Exam December 12th 2008, Name:.......................... 11 (Use this page for your calculation) Math 1502 K, Final Exam December 12th 2008, Name:.......................... 12 III- Problem In this problem A denotes the matrix below and it will be asked to com- pute its double QR-factorization. It is advised to use row reduction on the question 1 and to check carefully the result in order to use it in the next questions. A = ๏ฃฎ๏ฃฐ 1 1 โ1 1โ1 0 0 1 1 2 โ2 3 ๏ฃน๏ฃป 1. Give a basis of Im(A). (Hint : row reduce A and conclude) (Use back page for your calculation) Basis of Im(A) 2. Give a basis of Ker(A). (Use back page for your calculation) Basis of Ker(A) Math 1502 K, Final Exam December 12th 2008, Name:.......................... 15 4. Give the QR-factorization of Rt (see 3) in the form R = TQtr. Check that T is invertible (Hint : keep track of the size of both matrices Qr and T ) Qr = T = Math 1502 K, Final Exam December 12th 2008, Name:.......................... 16 (Use this page for your calculation)
