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Final_Fall-1997-1998_Abu-Khuzam_Lyzzaik

# Final_Fall-1997-1998_Abu-Khuzam_Lyzzaik - W l 3‘ ‘ men...

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Unformatted text preview: W l 3‘ ‘ men: E'NWEH . ‘ L l B K A R Y (W 311mm - v FIN AL EXAM; MATH 201 w P 9P1.“ February 6, 1998; 8:00—10:00 A.M. Name: Signature: Student number: V Section number (Encircle): 3 10 11 12 Instructors (Encircle): Prof. H. Abu-Khuzam Prof. A. Lyzzaik 1. Instructions: 0 No calculators are allowed. 0 There are two types of questions: PART I consisting of four subjective II consisting of twelve multiple—choice questions of questions, and PART h has exactly one correct anSWer. which eac o GIVE DETAILED SOLUTIONS FOR THE PROBLEMS OF PART D SPACE AND CIRCLE THE APPROPIATE AN- 1 IN THE PROVIDE SWERS FOR THE PROBLEMS OF PART II. 2. Grading policy: problem of PART I. I 10 points for each 0 5 points for each problem of PART II. 0 0 point for no answer, wrong answer, or more than one answer of PART 11. GRADE OF PART 1/40: GRADE OF PART 11/60: TOTAL GRADE / 100: Part 1(1). Find the absolute maximum and minimum values of the function f (as, y) z 3:3 + 33y —— y3 on the triangular region R with vertices (1,2), (1, —2), and (—1,—2). Part 1(2). Evaluate the integral 1 x/E / f ez/ydydrc. U a: Part 1(3). Set up a triple integral (without evaluating it)in cylindrical coordinates for the volume of the solid bounded by the my—plane, the cylinder r:1+sin6,andtheplane\$+y+z=2 Part I(4). Find the interval of convergence of the power series 00 1 n 11:1 State Where the series converges absolutely and conditionally. Part II 1. The area of the region lying outside the circle 7“ = 3 and inside the cardioid r 2 2(1 + cost?) is (a) — 71'. (b) + 7r. (0) 9x/§ + Tr / 2. (d) Qx/g — 7r / 2. (e) None of the above. 2. The slope of the tangent line to the curve 1" = 8c0536 at the point of the graph corresponding to 6 = 7r/4 is (e) None of the above. 3. If f(:r,y) = for (m) a: (0,0) and f(0,0) = 0, then (a) lim(x.y)~<o,o> f (m, y) = 1/ 2. ('0) f is discontinuous at (0,0). (c) limwiypmjo) f(a:,y) = 0. (d) 1im(z,y)—’(l,—1)f(-Tay) = 2- (6) None of the above. 4. An estimate to four decimal plaCes of the value of the integral 0.1 2 2 f a: 6—:c d3." is U (a) 10—4. (b) 2 x 10‘4. (c) 5 x 10—4. (4) 3 x 104‘. (e) None of the above. 5. The Maclaurin series of the integral / v31+t2dt is 0 1 1 1 ( )< ~1)---( —n+1) b m (_1111—1)-~(1—n+1) 1 1 1 1 m < )1 a )-~( —n+1) 2m ((33:— n=1 n.‘(§n+1) E ' 1 1 1 0° ( )1 —1)---( —n+1) 2 +1 (d)\$+anl%—xn . 211+ l 172111-1- (e) None of the above. 6. If an 2 + \$- and bn = 712(61/“2 — 1), then (a) the sequences {an} and {bu} diverge. (b) the sequences {an} and {bn} converge. (c) the sequence {an} diverges and {bu} converges. (d) the sequence {an} converges and {bu} diverges. (6) None of the above. 7. The sum of the series (a) 15/4. WHM (C) 7/4 (d) 13/4 (e) None of the above. 3 3 8. The function deﬁned by f(33,y) = cos for (my) #5 (0,0), and f(0, 0) = 1 (a) has no limit at (0,0). (b) has a limit at (O, 0) but is not continuous at (0,0). (c) is continuous at (0, 0). (d) is unbounded. (e) None of the above. 9. If 2 = f(\$,y) where a: = 6"6059 and y : ersinﬁ, then (8) f3 - f3 = 64°11“?- f3)- (b) ff + f3 = 64112? + f3)- (01224—132: 62TH: " fag)- (d) if + f: 2 6W? + f3)- (e) None of the above. 10. An equation of the tangent plane to the ellipsoid 2—552 + 3312 + 22 = 12 at the point P(2,1,\/5) is (a) 393 — 6g + 2E2: : 12. (b) By — 62: + N6: = 3. (c) 3y + 61: + 2x/Ez = 27. (d) 39; + 6y + was = 24. (6) None of the above. 11. If the directional derivatives of f (3:, y) at the point P(1 , 2) in the direction of the vector i+j is 2%? and in the direction of the vector —2j is —3, then f increases most rapidly at P in the direction of the vector (a) 3i — j. (b) 3i + j. (c) i — 3j. (d) i + 3j. (e) None of the above. 12. The function f(3:,y) = ‘%’ 3 + %y3 — x2 - 31' — 4y — 3 admits (a) a local maximum value 4 / 3. (b) a local minimum value —42/ 3. (c) a saddle point (3,1,f(3, 1)). (d) an absolute maximum value f(1, 1). (e) None of the above. ...
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Final_Fall-1997-1998_Abu-Khuzam_Lyzzaik - W l 3‘ ‘ men...

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