MATH 201 Final, Fall 2003

FINAL EXAM.; MATH 201 ERIGAN CNEL STE § LIBRARY OF REOROE January 28, 2004 Name: Signature: Student number: Section number (Encircle): 17 (Miss Jaafar) 18 (Miss Jaafar) 19 (Mrs. Jurdak) 20 (Mr. Lyzzaik) Instructor: Prof. Abdailah Lyzzaik 1. Instructions: e Calculators are allowed. e There are two types of questions: PART I consists of six subjective questions, and PART II consists of seven multiple-choice questions of which each has exactly one correct answer. e GIVE DETAILED SOLUTIONS FOR THE PROBLEMS OF PART I IN THE PROVIDED SPACE AND CIRCLE THE“APPROPRIATE AN- SWER FOR EACH PROBLEM OF PART II. 2. Grading policy: e 12 points for each problem of PART I. * 4 points for each problem of PART II: 0.pejnt for no answer, -1 for a wrong answer or more than one answer of PART II, GRADE OF PART 1/72: GRADE OF PART II/28: TOTAL GRADE/100: Part I (1). Find the absolute maximum and minimum values attained by the function f(z,y) = zy — z — y + 3 on the triangular region A in the zy—plane with vertices (0,0), (2,0), and (0,4). {ivemtean UNIVERSEP J Liev ine Part I (2). Evaluate the integral 271 3 f yer daxdy. 0 Jy/2 OF tEVEey ERIC VS UN veasey LIBKaRry } Tn reer, Part I (3}. Set up a triple integral (without evaluating it) in cylindrical coordinates for the volume of the solid bounded by the ry—piane, the cylinder r? = cos 20, and the sphere 7 + y? +27 =1. Part I (4). Evaluate the integral | [isn (+2) dz dy, where F is the trapezoid in the zy~plane with vertices (1.1), (2,2), (4,0), and (2,0), by making the change of variables: u=y-g,v=y+or Part I (5). Evaluate the line integral ¢ 3ry dz + 22? dy, where C' is the boundary of the region R bounded above by the line y = z and below by the parabola y = x? — 2z. Interpret this integral in terms of vector fields. Part I (6). Find the interval of convergence of the power series State where the series converges absolutely and conditionally. Part II 1. If f(z, y) = Q27y/(r4 = y?}, then (a) Linge y@,0) f(z. y) = 0. (b) ime, y wo.) f(a, y) = 1. (c} limes, yy coo) F(z, y) = 2. (d) limyey) 0,0) {(x,y) does not exist. (e) None of the above. 2. An estimate of the integral 11 -—cosz [a 0 x with an error less than 1/(6!5) is (a) 1/2! + 1/(4!3). (b) 1/2! ~ 1/(413) + 1/(6!5). (c) -1/2! + 1/(4!3) — 1/(6!8). (d) 1/2! — 1/(4!3). (e} None of the above. 3. The function defined by (a) is continuous at (0,0). (b) kas no limit at (0,0). (c) has a limit at (0,0) but is discontinuous at (0, 0). (d) is bounded in the ry—plane. (e) None of the above. 4.1 w= f(x,y) where xs = e” cos@ and y = e” sin @, then (8) Wee + Wyy = Wee + wy /T + wee/t?. (b) Wee + Wyy = —Wrr + wr/T + Wee/T?. (C) Wer + Wyy = Wer + w,/T — wes/T?. (G) Woe + Wyy = Wer — W,/T + wee /T?. (e) None of the above. 5. An equation of the tangent plane to the surface with equation 2°+zz—-y? = 1 at the point (1,3, 2) is (a} 22 + 6y + 138z = 10. (b) 2x + Gy — 13z = 10. (c) 2x - 6y + 13z = -10 (d) 22 - 6y + 13z = 10. (e) None of the above. Lo, 6. The volume of the solid bounded by the cylinder y = 2? and the planes yt+2=4and z= 0 is given by the triple integral (a) Jo fo" Sq de dy de. (b) 2 Jf [25 fer Y de de dy (c) 2 J@ Jo°¥ de dy de. (a) J6 AE? ES, de dy ae (e) None of the above. 7. The function f(z,y) = 29 + 3ry — y® admits (a) a saddle point and a local minimum value. (b) no saddle point and no local minimum value (c) a local minimum value and no saddle point (d) a saddle point and no loca! minimum value (e) None of the above.