Math 2224 Common Final Exam Fall 2012 FORM C INSTRUCTIONS: Please enter your NAME, ID NUMBER, FORM DESIGNATION, and CRN on the op-scan sheet. The CRN should be written in the box labeled 'COURSE'. Do not include the course number. Darken the appropriate circles below your ID number and below the Form designation letter. Use a # 2 pencil. Machine grading may ignore faintly marked circles. Mark your answers to the test questions in rows 1-14 of the op-scan sheet. Your score on this exam will be the number of correct answers. You have one hour to complete this portion of the exam. Turn in the op-scan sheet with your answers and the question sheets, including this cover sheet, at the end of this part of the final exam. Exam Policies: You may not use a book, notes, formula sheet, or a calculator or computer. Giving or receiving unauthorized aid is an Honor Code Violation. Signature Name (printed) Student ID # [1] Consider the series S = \u0013 \u0012 X 3 n n=1 2 . Which of the following statements is true? \u0012 \u001350 1) The 50-th partial sum equals 3) S = 3 3 5 2 3 5 2) S = 3 5 4) The 50-th partial sum diverges [2] The interval of convergence of 1) (3, 5) 2 5 2) x = 1 only X n! (x 1)n equals 4n n=1 3) (1, 1) 1 4) (0, 2) [3] The value of 0 1) e2 1 x 2) [4] Evaluate ey dy dx is given by y2 Z 4Z 2 1 2 e 1 2 3) e2 4) 1 2 e 1 4 x2 + y 2 x 4 or state that the limit does not exist (x,y)(0,0) 2(x2 + y 2 ) lim 1) 0 2) The limit does not exist [5] The sequence an = 3) 1 2 4) 1 4 n2 + 4n n2 + 1 1) converges to 2 2) converges to 1 3) diverges 4) converges to 0 [6] If z = x3 y, where x = tet and y = sin(t), then dz equals dt 1) 3t2 e3t (t + 1) cos(t) 2) 3x2 et (t + 1) + cos(t) 3) 3t2 e3t + cos(t) 4) 3x2 et cos(t) [7] The volume of the pyramid with vertices (0, 0, 0), (1, 0, 0), (1, 2, 0), (0, 2, 0), and (1, 2, 3) is represented by which of the following integrals? 1) Z 3 Z 2z/3 Z z/3 0 3) 0 0 2) 0 Z 2 Z 1 Z 3y/2 0 dx dy dz 0 Z 2Z 1Z 3 0 dz dx dy 4) 0 Z 3Z 2 0 dz dx dy 0 2z/3 Z 1 z/3 2 dx dy dz [8] Which of the following surfaces matches the level curves below? 1) z = x2 y 2 + 6 3) z = q 2) z 2 = x + y + 6 x2 + y 2 + 6 4) z = q x2 + y 2 4 3 2 y 1 0 1 2 3 4 4 3 2 1 0 1 2 3 4 x [9] The Maclaurin series of f (x) = e2x equals 1) X (1)n (2x)2n (2n)! n=0 X xn 3) n=0 n! 2) 4) X (1)n (2x)2n+1 (2n + 1)! n=0 X (2x)n n! n=0 [10] At the point (0, 2), the function f (x, y) = x3 + x2 y + y 2 4y + 2 has 1) not a critical point 2) a saddle point 3) a local minimum 4) a local maximum 3 2 2 [11] The volume of the region q that is above the xyplane, inside the surface x + y = 4 and beneath the surface z = x2 + y 2 is represented by which of the following integrals in spherical coordinates? 1) Z 2 Z /4 Z 2/ sin 0 3) 0 /4 sin d d d 2) Z 2 Z /2 Z 2/ sin 0 0 Z 2 Z /2 Z 2 0 2 2 sin d d d 4) 0 0 2 sin d d d 0 Z 2 Z /4 Z 2 0 [12] The series /4 2 sin d d d 0 X (1)2n converges by the 2 n=1 n + 1 1) direct comparison test, using bn = 1/n2 2) ratio test 3) n-th term test since n lim an = 0 4) alternating series test 1 dA, where D is the region in the first quadrant bounded D 1 + x2 + y 2 by y = x, x = 0, and x2 + y 2 = 4, is given by [13] The double integral 1) tan1 (2) 4 Z Z 2) tan1 (4) 4 3) ln 5 4 [14] Find the directional derivative of the function f (x, y, z) = direction of the vector v = 4i 2j + 4k 1) 3 2 2) 2 3) 1 3 4) 4 1 4 4) ln 5 8 xyz at the point P (2, 1, 2) in the