MATH 201 Final, Fall 2004

FINAL EXAM.; MATH 201 January 28, 2004 Name: Signature: Student number: Section number (Encircle): 17 (Miss Jaafar) 18 (Miss Jaafar) 19 (Mrs. Jurdak) 20 (Mr. Lyzzaik) Instructor: Prof. Abdallah Lyzzaik 1. Instructions: • Calculators are allowed. • 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. • 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: • 12 points for each problem of PART I. • 4 points for each problem of PART II: 0 point for no answer, -1 for a wrong answer or more than one answer of PART II. GRADE OF PART I/72: GRADE OF PART II/28: TOTAL GRADE/100: 1 www.amal-aub.org Part I (1). Find the absolute maximum and minimum values attained by the function f(x, y) = xy − x − y + 3 on the triangular region R in the xy−plane with vertices (0, 0), (2, 0), and (0, 4). 2 www.amal-aub.org Part I (2). Evaluate the integral � 2 0 � 1 y/2 y ex3 dx dy. 3 www.amal-aub.org Part I (3). Set up a triple integral (without evaluating it) in cylindrical coordinates for the volume of the solid bounded by the xy−plane, the cylinder r2 = cos 2θ, and the sphere x2 + y2 + z2 = 1. 4 www.amal-aub.org Part I (4). Evaluate the integral � � R sin �y − x y + x � dx dy, where R is the trapezoid in the xy−plane with vertices (1, 1), (2, 2), (4, 0), and (2, 0), by making the change of variables: u = y − x, v = y + x. 5 www.amal-aub.org Part I (5). Evaluate the line integral � C 3xy dx + 2x2 dy, where C is the boundary of the region R bounded above by the line y = x and below by the parabola y = x2 − 2x. Interpret this integral in terms of vector fields. 6 www.amal-aub.org Part I (6). Find the interval of convergence of the power series ∞ � n=2 (2x − 1)n ln n . State where the series converges absolutely and conditionally. 7 www.amal-aub.org Part II 1. If f(x, y) = 2x2y/(x4 + y2), then (a) lim(x,y)→(0,0) f(x, y) = 0. (b) lim(x,y)→(0,0) f(x, y) = 1. (d) lim(x,y)→(0,0) f(x, y) does not exist. (c) lim(x,y)→(0,0) f(x, y) = 2. (e) None of the above. 2. An estimate of the integral � 1 0 1 − cos x x2 dx with an error less than 1/(6!5) is (a) 1/2! + 1/(4!3). (b) 1/2! − 1/(4!3). (c) 1/2! − 1/(4!3) + 1/(6!5). (d) −1/2! + 1/(4!3) − 1/(6!5). (e) None of the above. 3. The function defined by f(x, y) = tan �x3 − y3 x2 + y2 � for (x, y) ̸= (0, 0), and f(0, 0) = 0 (a) has no limit at (0, 0). (b) has a limit at (0, 0) but is discontinuous at (0, 0). (c) is continuous at (0, 0). (d) is bounded in the xy−plane. (e) None of the above. 8 www.amal-aub.org 4. If w = f(x, y) where x = er cos θ and y = er sin θ, then (a) wxx + wyy = −wrr + wr/r + wθθ/r2. (b) wxx + wyy = wrr + wr/r − wθθ/r2. (c) wxx + wyy = wrr − wr/r + wθθ/r2. (d) wxx + wyy = wrr + wr/r + wθθ/r2. (e) None of the above. 5. An equation of the tangent plane to the surface with equation z3+xz−y2 = 1 at the point (1, 3, 2) is (a) 2x − 6y + 13z = 10. (b) 2x + 6y + 13z = 10. (c) 2x + 6y − 13z = 10. (d) 2x − 6y + 13z = −10. (e) None of the above. 6. The volume of the solid bounded by the cylinder y = x2 and the planes y + z = 4 and z = 0 is given by the triple integral (a) � 4 0 � 4−y 0 � √y −√y dx dy dz. (b) � 4 0 � 4−z 0 � √y −√y dx dy dz. (c) 2 � 4 0 � 2√y � 4−y 0 dz dx dy. (d) � 2 −2 � x2 0 � 4−y 0 dz dy dx. (e) None of the above. 9 www.amal-aub.org 7. The function f(x, y) = x3 + 3xy − y3 admits (a) a saddle point and no local minimum value. (b) a local minimum value and no saddle point. (c) a saddle point and a local minimum value. (d) no saddle point and no local minimum value. (e) None of the above. 10 www.amal-aub.org