Fall 2005 Calc 3- Final

# Fall 2005 Calc 3- Final - MA 261 FINAL EXAM Name Student...

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Unformatted text preview: MA 261 FINAL EXAM December 13, 2005 Name: Student I.D. #: Lecturer: Recitation Instructor: Instructions: 1. This exam contains 22 problems worth 9 points each. 2. Please supply a_ll information requested above and on the mark—sense sheet. 3. Work only in the space provided, or on the backside of the pages. Mark your answers clearly on the scantron. Also circle your choice for each problem in this booklet. 4. No books, notes, or calculator, please. key. lac/as MM 12:50 AcaM be ﬁﬁbe Name: MA 261 Final Exam 1. Compute the angle 6 between a = i + k and b 2 2i —j + k 7T A.— 4 7r B.— 6 7r 0-3 D. cos‘1 ( E. cos‘1 ( 2. Find the equation of the plane that contains the line 1‘ = i+ 2] + 3k + t(i —j + k) and that is parallel to the vector v : —i + 2j + 3k. A. 2x—3y+z:—1 B. 3x—y+z=4 C. 5x+4y—z:10 D. 4m—y—z2—1 E. m+y—2z=—3 mla ME 2 3. If the acceleration of a moving particle is a(t) : (2t + 1)i + 2th — 4tk and its initial velocity is v(0) = —i + k, then its velocity when t : 1 is: A.L+§—k Bi+§+% C.4+§+k D.—L+§+2k Ei+§—% 4. A solid region E is deﬁned by O < a: < y, 0 < z < «3:2 +212, 3:2 +212 + 22 < 4‘ In spherical coordinates E is deﬁned by: A.0<9<Ego<¢<§30<p<2 7T7T 71' B. —~ — 2 O<6<4,4<¢<2,0<p< C O<9<E,O<¢<E,O<p<4 7T 7T 7T D.—0— — 4 4< <2,0<¢<4,0<p< 7T 7T7T 7T .— —ﬂ ~ 2 E 4<0<2,4<¢<2,0<p< 3 5. The arclength of the curve: 2 2 \$=1—2t2,y=4t,z=3+2t2,ogtg2, A'fomthFZdt is given by the integral: 2 B./ V8t4+4t2dt 0 2 C./ \/8t+4dt 0 2 D./ 4V2t2+1dt 0 2 4Vt2+1dt 0 6. A vector parallel to the tangent vector to the curve 4 :3 = 4x/t, y : t2 — 2, z = I at the point P(4, —1,4) 0n the curve is: A. 4i—9j+4k B.&+m+k C. 2i+2j—4k D. 4i+2j+4k E. 3i+2j+6k 4 7. The plane tangent to the graph of f(a:,y) :3my2—m2—4y wherem=2 and yzlis: A. \$+2y—z=4 B. a:—8y+z=—8 C. 2m—2y+z=0 D. —\$+4y—z:0 E. 2m—8y—22—4 3 2 2:1: 8. Evaluate lim m. (w,y)+<o,o) 2:2 +112 A' 0 B. 1 C. e D. 62 E. The limit does not exist 9. Find the Slope of the curve 1:2 — my — 2y3 2 0 at the point (13,31) : (2,1). 2 A. —3 4 B. 3 3 0. —§ 3 D- g E. _§ 10. Let u be a unit vector that points in the same direction as v # 3i + 4j. If f(:c, y) = x2 + my — y2 compute the directional derivative Duf(2, 1). mcow I 6 11. Which vector is perpendicular to the tangent plane of the surface 3:23; — 23:2 + 2y 2 —6 at the point (2,1, 3)? 12. Find the absolute maximum of f(:1:, y) = 2:1: — 2y — 3:2 — yz. A.4i—j+3k B.i+j—2k C.m—j+k D.i—j+2k E.—L+m—2k A.3 B.4 0.0 D.2 7 13. If E is the solid in the ﬁrst octant that is bounded on the side by the surface 3:2 + y2 = 4, and on the top by the surface 3:2 + y2 + z : 4 represent the volume of E as an integral. 2 “4—3/2 4—z2—y2 A. f f / ldzdyda: 0 0 0 4 VAL—2:2 4—2 B./ / / ldzdyda: 0 0 0 2 VAL—2:2 4—z2—y2 C./ / / ldzdyda: 0 0 0 2 4—1-2 1:2+y2 D. / f / ldzdyda: 0 0 0 2 W 4—x2—y2 E. / / / ldzdyda: 0 0 0 14. If R is the planar region deﬁned by 1 < 3:2 + y2 S 9 and a: Z 0, compute f/R WdA. _ 267T A. — 3 137r B. — 3 C. 97r D. 47r E. 137r 8 15. If S is the part of the surface z I ﬂy + 2:2 that lies above the rectangle 0§x_<_1,0§yg2,compute//a:d5. S A. §(5\F5—1) B. §(2\F2—1) o. %(5\/5—1) D. §(2\F2—1) E. §(2\/§—1) 16. Compute de, where E is the intersection of the ball 3:2 + y2 + 22 < 1 E with the ﬁrst octant. 3—156 0.1712 77' D.—6— 3 12% 9 17. The vector ﬁeld F(m, y) = (3/3 + 3m2)i + (331% + 1)j is conservative. Find f so that v f = F. A. f(m,y) : avg/3 + m3 +y B. f(m,y) = 6m + 63/2 + my 1 C. ﬂay) = y“ + \$3 +y3\$+m D. f(m, y) = \$314 + 3m3y 1 3 E- f(\$,y) = Ey4+3\$2y+§\$2y2+\$ 18. Evaluate / ydm + cos ydy Where C' is the polygonal path from (O, 0) to (1,0) to (1, 2) t0 (0,02) to (0,0). A. —1 B. 0 C. —2 3 D. —— 2 E3 3 10 19. Evaluate / 4dm + 3dy where C is given by A_ 7 m:t2,y:Ct3,0§t§1. B. 12 C. 8 D. 4 E. 10 20. If F = i +j + 2k and Sis the intersection of the solid cylinder :32 + y2 g 1 with the plane 2:3 + y — z : 1, compute F - 11 (15 (using an upward pointing n). S A. —7r 37r _—2_ C. 27r B. 11 21. If S is the cone z : V502 +y2, 0 g z 3 2, that is upward oriented, and if F : —yi + :cj + zk, compute curl F - dS. S 47r 167r —47r 871' — 167r racew? 22. Let F = 423i — zj + :ck. Compute F - dS, where S is the union of the S hemisphere :02 + y2 + z2 = 1, z 2 0, and the base given by :02 + y2 S 1, z : 0. (Use the outward—pointing normal.) 271' A. —— 3 167r B. — 3 871' C. _ 3 D. 4—” 3 E. i” ...
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Fall 2005 Calc 3- Final - MA 261 FINAL EXAM Name Student...

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