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Final_Fall-2004-2005_Yamani_Jurdak_Lyzzaik

# Final_Fall-2004-2005_Yamani_Jurdak_Lyzzaik - \L w> I aw...

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Unformatted text preview: .\ '.\L* w > - I aw FINAL EXAMINATION ;_ . ’ MATH 201 January 29, 2005; 3:00-5:00 PM. Name: Signature: Student number: Section number (Encircle): 17 18 19 20 Instructors (Encircle): Dr. H. Yamani Mrs. M. Jurdak Prof. A. Lyzzaik Instructions: 0 No calculators are allowed. 0 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: 0 10 points for each problem of PART I. o 5 points for each problem of PART II. o 0 point for no, wrong, or more than one answer of PART II. GRADE or PART 1/40: GRADE or PART 11/60: TOTAL GRADE/100: "‘ H uJHl'I i_il:i;.\:{Yl I l Part 1(1). Use the change of variables 1:; = :5 — y and v = a: + y to evaluate the integra} [/Rﬁv - y)2cosg[:c + y) dm 0531 over the square region R bounded by the lines so —- y = 1, 3: ~— y 2 —1, x+y=1,m+y=3. Part 1(2). Find the absolute maximum and minimum values of the function may) = 232 WW2- 72: an on the square region R : {(x, y) : 0 5 3,11} S 3}. ;“':‘j‘;"i‘;\*: :“igii'fl-r} i in- HIZIHPI -qp. .-—— Part I(3). Find the volume of the ”ice cream cone” C bounded by the cone do : 7r/6 and the sphere p : 2a coscf) of radius a, and tangent to the my—plane at the origin. Inn—HM...- gaurlrtlt; -\ h ['.\"l\' i-LIGSIT‘H 1.:BRARY E u!‘ III-IHH‘1 E “- Part 1(4). Use Lagrange multipliers to ﬁnd the point on the plane 231: + 3g + 42 = 12 at which the function ﬂay y, z) = 43:2 + y2 + 522 has its least value. -‘----— _...._._.._,.--.,_..,_. injumn,‘\ l .n'. hm'rn L l B R A R Y ”F H£JIRII'I Part II Part II(1). If f( x y) 2% for( (x y) :1: +\$y+y then (a) f is continuous at (0,0). (b)hm(x,y)—>( )f(\$ =“—"y) 1- __,(g g) f(a: y) does not exist. +(o,oj f(-’1‘7= U) = 2/3- (e) None of the above. Part II(2). The Maclaurin series of the integral 1': / \/1+t3 0!: is 0 (a) 1/2(1’2-r~1ll[2 2- -1£2 n+1\$3n+1 n: 0 (3n+1)!n (1212—112—2 (12— l (b) (UH 1(g 2-"(1 -n+!\$3n+1- 71:0 11! (3n+1) oo M22952—11g1§2—23---g1g2—n+12 3n+1 (C) 1120 (31144): 3: ‘ (d) (1K2 (—1/2—1!1/2—-2 11'2—11-l-llx 371+} ”=0 (Em—+1) (e) None of the above. Part II(3). The series "1)" End-“##— \f—(ln n)10 n=2\/— (3) converges absolutely. (b) converges conditionally. (c) diverges. (d) converges conditionally and absolutely. (e) None of the above. JHM Mun-Julrnx l'.‘d\'F-IE1SI'J it Linen” ; #(0,0) and f(0.0) = ‘xl (a. . “Mm.m_- Aﬂriﬁlrnq3 l'HJ'I'llfiSi'l 3‘: L I B R A R Y “F REHH"! "I- M5 Part 11(4). The value of the integral I 1 V17 / f 31’” d2: dy is 0 y (a) —1+e/2. (b) —1+e. (e) —1 + e/4. (d) -1+e/3. (e) None of the above. Part 11(5). The interval of convergence of the power series 00 “(E—2)” 2(4) ”4” is (a) [-236l. (b) ] — 2,6[. (C) [-276]- (d) l - 2: 6]- (e) None of the above. Part II(6). The value of the double integral 1 . m d2: d'y IS fol/“M (a) (alum/2. (b) 771112. (C) (7r 1n 2)/4. (d) (ﬁln2)/3. (e) None of the above. ._ ..—-—_..4......__..__..... _. 't . i LIBRARY Part 11(7). The polynomial that approximates the function “"“""“ ‘ "" HF HEIRL'1 3 x sint m) z j 7 it 0 throughout the interval [0, 1/2] with an error of magnitude less than 10—3 is (a) a: — 223/12. (b) :c — 333/6. (c) at —x3/18. (d) a: _ x3/24. (e) None of the above. Part 11(8). The series 2310:1671 converges if (a) an 2 1/nln2. (b) an = (1/71) 111(1 + 1/71). (c) an < bn and the series £30215“ converges. (d) on : nsin(1/n). (a) 11mm“? M 2 1. Part 11(9). Parametric equations for the line tangent to the curve of inter— section of the surfaces myz : 1 and 3:2 + 2y2 + 322 = 6 at the point P(1, l, 1) are (a):r=l+t=y=1-i-2t,z=1—i-1€1 —oo<t<oo. (b)m=1—t,y=1~2t,z=1+t,—oo<t<oo. (C)\$=1+t,y=1—2t,z=1—t,—Oo<t<oo. (d)3:=1+t,y-_~1—2t,z=1+t, —oo<t<oo. (e) None of the above. 9 . W._i.—-—.........._.... an... -- lnﬂlllilili“ ' ~ ‘ =-'~3""‘ L l U N A R "i' [ll- “KIRK"! gal—4 Part 11(10). A triple integral in cylindrical coordinates for the volume of the .41 solid cut out from the sphere :52 + y? + 22 = 4 by the cylinder I2 + yz : 2 is y (a) fclr [\$6059 _jE-;g 1'” dz oh" 056. (b) H IDES“ gig—:7 2,, 7" dz dr d6. (C) few/2 faking 42%; 7" dz d?" d6. (d) [0” jam" fom r dz dr :16. (e) None of the above. Part II(11). By using Green’s theorem, the value of the line integral fi(:c+y)dm+(y+x2)dy, where C is the positively—directed boundary of the region bounded by the circles \$2 + 2,12 = 1 and \$2 + y2 = 4, is (e) None of the above. Part 11(12). The only one TRUE statement of the following is (a) The ﬁeld (y sin z)i + (a: sin z)j + (cry cos z)k has potential function my sin 2 cos 2. (b) The ﬁeld (6“ cos y) i + (er sin y) j is conservative. (c) The differential 33:2 (is: + 2wy2 dy is exact. (d) The value of the line integral féﬂﬁ’llﬁz) 3; dx + :5 dy + 4 dz is 2. (e) If C is a simple closed curve, then ﬁcy dzc + :C dy aé O. ...
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