MATH 201 Final, Spring 2017

Math 201 - Final (Spring 17) T. Tlas ================================================ • Write the answers to questions 2, 3, 4 and 6 on their sheets. The other two questions have extra sheets for you to write your answers on them. Any part of your answers written on the wrong sheet will not be graded. Note that a sheet of paper has two sides, you can write on both of them. • There are 6 problems in total. Some questions have several parts to them. Make sure that you attempt them all. • This is a closed book exam and no calculators are allowed. ================================================ Name : ID # : Section : ================================================ Q1 Q2 Q3 Q4 Q5 Q6 TOTAL Problem 1 (6 points each) Which of the following series converge and which diverge? Those which converge, do they converge absolutely or conditionally? When possible find the sum of the series. i- ∞ � n=0 (2n)! 3nn! ii- ∞ � n=0 (−1)n 9n (2n)! iii- ∞ � n=0 1 (n + 1)(n + 2) iv- ∞ � n=0 n − 1 n2 + 3 ================================================ ADDITIONAL SHEET FOR PROBLEM 1 ANSWER Problem 2 (12 points) Evaluate the following integral � 0.1 0 e−x2dx with an error less than 10−8. Is your answer an over- or an under-estimate? ================================================ Problem 3 (20 points) Find the minimum value of the function f(x, y) = 4x2 + 9y2 subject to the constaint x2 − y2 = 1. ================================================ Problem 4 (10 points) Let f(x, y) = 1 √ x2+y2 . Integrate this function over the region R which consists of all points in the plane which are in the first quadrant, inside the circle x2 + y2 = 4 and above the line y = x. ================================================ Problem 5 (12 points each) Integrate the function f(x, y, z) = 1 √ x2+y2 over the following two regions: i- R1 is the region bounded from above by z = 2 and from below by the cone x2 + y2 − z2 = 0. ii- R2 is the region in the first octant inside the sphere x2 + y2 + z2 = 1 but above the cone x2 + y2 − z2 = 0. ================================================ ADDITIONAL SHEET FOR PROBLEM 5 ANSWER Problem 6 (10 points) 1 + 1 1 + 1 1+ 1 1+... = ? ================================================