MATH 201 Final, Fall 2004

FINAL EXAMINATION \. MATH 201 January 29, 2005; 3:00-5:00 P.M. Name: Signature: Student number: Section number (Encircle): 17 18 19 20 Instructors (Encircle): Dr. H. Yamani Mrs. M. Jurdak Prof. A. Lyzzaik Instructions: e No calculators are allowed. e There are two types of questions: PART I consists of four work-out problems. Give a detailed solution for each of these problems. PART II consists of twelve multiple-choice questions each with exactly one correct answer. Circle the appropriate answer for each of these prob- lems. Grading policy: e 10 points for each problem of PART I. 5 points for each problem of PART II. e 0 point for no, wrong, or more than one answer of PART II. GRADE OF PART 1/40: GRADE OF PART II/60: TOTAL GRADE/100: aed rer | LERRARY } \ | on WIRE Part I(1). Use the change of variables u = z—y andv = 2+ y to evaluate the integral [Le — y)? cos*(x + y) dx dy over the square region A bounded by the lines c ~ y = 1, r- y = —1, ety=lety=3. Part 1(2). Find the absolute maximum and minimum values of the function f(z,y) = 22? -ayty’ - 7x on the square region R = {(z,y):0<2,y < 3}. Pe sey | | oF RETIRE Part 1(3). Find the volume of the ice cream cone” C bounded by the cone @ = 7/6 and the sphere p = 2acos@ of radius a and tangent to the xry—plane at the origin. eee JAMEKICAN UNIVEKSIT LIBRARY ! OF BEIROT i Part 1(4). Use Lagrange multipliers to find the point on the plane 2x + 3y + 4z = 12 at which the function f(z, y,z) = 42? + y? + 52” has its least value. PAMRICAN UE SIVENSITe® LIBKARY OF BEIRUT Tee memento tipennitsieren Part II Part II(1). If f(z, y) =v" | for (z, y) then e+ aetyty (a) f is continuous at (0,0). (b) lime.) (0,0) f(@,y) = 1. (c) lime y)-+(0,0) f(a, y) does not exist. (d) limee,y)-»(0,0) f(@, ) = 2/3. {e) None of the above. Part II(2). The Maclaurin series of the integral xo | V1+8 dt is 0 (1/2)(1/2 -1)(1/2 —2)---(1/2 —n-+1) (a) Dio (3n+1)!n me nth (b) E2 (2/2) /2 = 1) (0/2 ~2)--(1/2 =n+1) 7 3n+1 n=0 n! (8n+1) (€) Co OR ee, (d) oo = (1/2)(1/2 -1)(1/2 —2).--(1/2 =n) 3nd n=0 (3n+1) (e) None of the above. Part II(3). The series -1)" loa} tan Inn)? fan Ve (a) converges absolutely. (b) converges conditionally. (c) diverges. (d) converges conditionally and absolutely. (e) None of the above. — i nnn AMeHIC aN ¢ MIVEDSTS a LIBRARY OF Berry # (0,0) and f(0,0) = 0, 5 i { Se eet etree ne AMERICAN UNIVERS TOR Part II(4). The value of the integral Lope [ | * evle dx dy is 0 Jy LIBRARY QF REIRG YD Tt cere enced (a) -1 + e/2. (b) -I+e. (c) -l+e/4. (d) ~1+e/3. (e) None of the above. Part II(5). The interval of convergence of the power series St(-1 (2 = = is (a) [-2, 6]. (b) | — 2,6]. (c) [-2, 6]. (d) } — 2,6]. (e) None of the above. Part II(6). The value of the double integral CI” ap ee (a) (wn 2)/2. (b) mIn2. (c) (win 2)/4. (d) (n In 2)/3. (e) None of the above. Part II(7). The polynomial that approximates the function Memtnes tsi poets LIBRARY | Of BEIRUT } = sint , ia) = [Sa —— o ¢ throughout the interval [0, 1/2] with an error of magnitude less than 107° is (a) 2 — 23/12. (b) « -— 23/6. (c) «— «3/18. (d) 2 — 23/24, {e) None of the above. Part II(8). The series 0°: , a, converges if (a) ay, = 1/niP?. (b) an = (1/n)In(1 + 1/n). {c) Qn < bp and the series °°, 6, converges. n=1 (d) @n = nsin(1/n). (e) limp soo flan! >i. Part II(9). Parametric equations for the line tangent to the curve of inter- section of the surfaces ryz = 1 and x? + 2y? + 3z? = 6 at the point P(1,1,1) are (a)xc=lit, y=14+2t,z=14+t,-w<t<c. (b)e=1—-t,y=1—2t,z=1+t, -w<t<oo. (c)m=1lt+t,y=1-2t,2=1-t,-w<t<oo. (d)v=1l4+t,y=1-2t,z=14+t, -wo<t<o. {e) None of the above. 9 JAMIA © SSE LIBRARY OF BEIRET ‘ 3 nm Part II(10). A triple integral in cylindrical coordinates for the volume of the solid cut out from the sphere 1? + y? + 2? = 4 by the cylinder z? + y* = 2y is (a) fo goose pes r dz dr dé. (b) Jo Jon? fy ~; 7 dzdr dd. (c) far? seein ve r dz dr dé. d) ft fo5"8 [Mr dz dr dO. 0 Jo 0 e) None of the above. (e) Part II(11). By using Green’s theorem, the value of the line integral fer) dz + (y+ 27) dy, where C is the positively-directed boundary of the region bounded by the circles c? + y?=1 and 2? +y? = 4, is (a) —7. (e) None of the above. Part II(12). The only one TRUE statcment of the following is (a) The field (ysin z)i+ (xsin z)j + (zy cos z)k has potential function ry sin z COs Z. (b) The field (e* cos y) i + (e* sin y) j is conservative. (c) The differential 32° dx + 2zy? dy is exact. (d) The value of the line integral Seer ydr+a2dy+4 dz is 2. (e) If C is a simple closed curve, then fo y dz + x dy # 0.