Sol-166FE-S2007 - MA 166 FINAL EXAM Spring 2007 Page 1/11...

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Unformatted text preview: MA 166 FINAL EXAM Spring 2007 Page 1/11 NAME M— 10—DIGIT PUID RECITATION INSTRUCTOR RECITATION TIME LECTURER INSTRUCTIONS 1. There are 11 different test pages (including this cover page). Make sure you have a complete test. 2. Fill in the above items in print. Also write your name at the top of pages 2—11. 3. Do any necessary work for each problem on the space provided or on the back of the pages of this test booklet. Circle your answers in this test booklet. No partial credit will be given, but if you show your work on the test booklet, it may be used in borderline cases. 4. No books, notes, calculators, or any electronic devices may be used on this exam. 5. Each problem is worth 8 points. The maximum possible score is 200 points. 6. Using a £2 pencil, fill in each of the following items on your answer sheet: (a) On the top left side, write your name (last name, first name), and fill in the little circles. (b) On the bottom left side, under SECTION, write in your division and section number and fill in the little circles. (For example, for division 9 section 1, write 0901. For example, for division 38 section 2, write 3802). (c) On the bottom, under STUDENT IDENTIFICATION NUMBER, write in your Ill-digit PUID, and fill in the little circles. (d) Using a #2 pencil, put your answers to questions 1—25 on your answer sheet by filling in the circle of the letter of your response. Double check that you have filled in the circles you intended. If more than one circle is filled in for any question, your response will be considered incorrect. Use a #2 pencil. 7. After you have finished the exam, hand in your answer sheet and your test booklet to your recitation instructor. MA 166 FINAL EXAM Spring 2007 Name: ___— Page 2/11 1. For what value of c is the vector 2I— 3+ cl; perpendicular to the vector 05+ J + 1;? 2:)": +612)- (ct +F§+l~)'“ "“0 A. g 2.: ~1+C=O 3—1 C‘1 C. —2 -3— 1 D. E l v®§ 2. Which of the following statements are true for any three- dimensional vectors (1' and 6? (I) (6x6)-&’=O True: Zixb in its; :1 ML (II) ax x6: 6x a: Not true..- a x]: 2...be A (I) and (IV) only (III) Ia- 6| 3 Irina] T me; la . bi =\a\ M‘anse)® (I), (1“) and (IV) only (IV) ("i x (35) = 6 Trust. a W «3 a are. C. (II) and (III) only Parallel . D. (III) and (IV) only E. All 3. Find the value of k for which the graph of x2 + y2 + 22 —- 4 + 62: = k is a sphere of 3/ radiusz7. x2 +Kf; +3+4+E2+GE+72 k-l‘q'f? A.30 x 2.;(9- 99°“ +[2+3)—- =l<+l3 ‘ B. 16 36 k+1'f>" — 72” $25 :4. -1'3 =3C - k ? E. 54 E a 1L 4-./2xcosmdm= Xfithl?’ S; 5"”de 0 K 3X0“ +C 313: A. 71' UK: X Dru: m ~13; COS'X Z:_ du: Clx m:s£nx " 2 0 .2 1 «E— C. 1 _ ,2. 7r D. —2— 7r . E.. 5+1 MA 166 FINAL EXAM Spring 2007 Name: —— Page 3/11 1 IE 2 ,Q. 5. / sinzxcosswdx= J2 nah/fix (1.,emx) cosy/0(1): 0 1 0 A. 13 :f <s\n7"x__§m 4x?Cfl5xd:x_ B. l 7 3 LL: $\Yl‘;< ditt- Cnsx d} (“M/ @ —2— X = O —- LL': (9 ”MWWWM’ 15 y—JI ——a“;,\l,g,,:,iw““w D 0 v NZ ' 2 1 r $ 2 5» I n E — < 1 2. 4-) (L Li I L-” m ___:__ 5 *— ° " -‘ “W \ 0 , " ¥ X ' :5 6 / tanxsec xdm— S $.41; X (mcxtamx) all); Em 4x5 0 0 bk" MEX cl» :‘Bflcfxtam-xot‘x A T x:0 —-; Uzi / B 1/2 x ; g: M Mia . 5 — 3 4 2 _ "’ j w oh; 3 LL [r © 4 l 4" 1 D. .1. 4 5 :NE)__L'__._1_L:; D5” 4 4 4 4- . -4— d1: 7. For the inte ral / —— i choose a tri onometric substitution to sim lif the 3 mm, () g p y integral and (ii) give the resulting integral. 2 ‘ x X=25Me A. (i)x=28e00, (10/ law " 0d z , 7m: 0356 V _ sec {33; {4x 2 B (i):c— 2mg (19/ 2 anede l—i’bi/Ti $333” 3 ©(i)x=2sin0, (19/ . d0 x 4”,;(1' 25mg 2005.6 2sm6 I (Big D. (i):v=2sin0, (ii)/ mdfi 7‘ 2 5MB 1 E. (1) a: = 200s 0, (11) —-/ (:08ng MA 166 FINAL EXAM Spring 2007 Name: _——__ Page 4/11 2 5 8. / a: + dx is of the form (Where a, b, c are constants): x3+x gait ._—_ aim: .61.. .. thgm- X3 +X x(x2+i) 2" x244. JQX+ngX2 fl—M‘fgfixwirgM x3+x Y X?“ Yaw—t A. aln(:1:2 + 1) + blnlxl + C B. aln|x|+bln|x+1|+cln|x-1I+C’ C. atan‘1x+bln|:c| +0 D. alnlrzz3 +sc| +0 ® aln |x|+bln(x2+1)+ictan‘“1x+0 9. The region in the first quadrant bounded by the graph of y— - 1 + $2 ,the line y— - 5, and the y- axis is rotated about the y—axis to form a solid. The volume of that solid 1s A. /2 27r[52 — (1 — x2)2]dx 0 B. /2 27r$(1 + :02)d:c 0 C. /5 27r[5a:2 — (1 + $2)2]d:c .[02 27r:z:(4—x2)d:c E. /5 7r(1+:c2)2d$ 1 MA 166 FINAL EXAM Spring 2007 Name: _——_ Page 5/11 10. A solid sphere of radius 1 is divided into two parts by a plane perpendicular to a diameter, mid—way between the center and a tip of the diameter. Find the volume of the smaller part. Tl/Ul smug” raga 0W MSDLML SW A. in. Q o it! H (5.4:, ‘” ”ifLAiFL“ W” , [9% Va “ma 0 Uuu x-OVXV) CL“. firm», G. g” 1~>~<‘" D. %7r 5 fi’fl' X \ffliumx o i 1. l Lanai as» Mr «The 4* V:f Tr(l““"l’ial‘zeolf>!2.2T? L('*W2)d">‘ 4/2 2 1 '5 2': ”—231; :Tl(I—J3') "W i’%) 1' — %:'F(Ji“24)"" %*J2'L4~ 11. Consider the lamina bounded by the graph of y__ — 3:2 ,the :1:- -axis, and the line x— — 3, and with density p: 1. The x—coordinate a: of the center of mass of the lamina lS 9 '3 1 X33 B3 m:.1de:§'{a:9 '2' 03 0.2 m :fl-ixzalx D.§ ‘2 o 3 __ x4|3-‘3,‘- E; “3:” U ' 4 937:?“ §’_zr 14 MA 166 FINAL EXAM Spring 2007 Name: —_ Page 6/11 12. A tank is 10. ft. high and filled with water weighing 62.5 lbs/ft3. The cross-sectional area of the tank at 3/ ft. above its bottom is A(y). The work required to pump all the water to the top of the tank is 10 B. 62.5 /0 7r[A(y)]2(10——y)dy ” j :1‘ mm‘ 10 ‘d i gave ‘35 o. 62.5 (10—y‘)[10—A(y)]dy x , ,,«’ 0 ‘ of”? 10 u. L/ I D- 52-5 (10—y){102-[A(y)]2}dy x. O o 10 E. 62.5 / (10—y)2A(y)dy 0 13. Which of these improper integrals converge? 'I 00 d II 0O 3” 11 0/0 cosm$()/O 1+$2d$ (III)/0;dzc ' , t‘ (I) jmngohxgfiljprn [CDSXO\X:RXM [gi“i*]t A. Only (I) o . t-m 0 t'->W 0 B. Only (II) :L‘m si’n‘t DNE C. Only (III) ”90 hi) ft” X t if. Div D. Allofthem O ¢+X1 4.1x :%m It A“. M ® None ofthem t"°° o my?“ Ln .. t __ J. 2,; ~ gjwl-‘izmbrxfi—So ~{f3‘m 2. (”t meW ’01 ‘4: _ 1.1. ,' L—Lm —Qm't:co [ ) g0 de- atJ—ém-i- {txdx‘£4:;+{km;]tvt—90+( ) MA 166 FINAL EXAM Spring 2007 Name: _— Page 7/11 00 14. Suppose that 2 an 2 5 and .9n = a1 + a2 + - - - + an. Which one of these statements n=1 is true? _ ao ' _ . _ ‘ . _ :anzg _ A. ”lirnoan—5andnlggosn—OL ":1 W . . _ . _ th M Una. met ()1 UM B. “1310100,n — 0 and 111320 8n — 0 Sigma; 0" Four‘b‘OxL “my (3”? C. nlr’ngo (1n = 5 and ”11,1210 8n = 5 Of) Y) -) 0‘3 W _ __ o". 5 — ‘5 ”1:120 an— 0 and nlgrgo sn— 5 m " ' oo , ~ E. lim 3n = 5 but lim a,n cannot , ”‘9‘?“ . ' :- Q n—mo n—mo We‘ Also km“ W ‘11 n._, n be determined in new Marrawr> 0mm an : o In W no ' 15. Which of these series converge? 5, 1 5“ 4.7,“ ég+m DIv. Frjfim A. None Only (11) h+| (km) 1 . °° M c. (II) and (III) I (H) n n Ratio te9t' ”n+1M—a—ym < 1 g (”1) (m) ”mined 0‘ D. Only (III) °° n n2 , E. A11 (HI) 121(4) m Qfiman DNE “MW ." D‘vv MA 166 FINAL EXAM Spring 2007 . Name: —— Page 8/11 16. Which of these series converge absolutely? n n i 00 nil 00 n 1 (1) 2e 1) (7) (11) 2H) 73—; (III) ”2232(4) m (I) 21(6)) Wm'rm ”Vii" "WV 03“” on1y(I) <=<7 B. All ! ..... "-3 “X", :J’ i (I!) Z OW P L P 2‘6 C. (I) and (II) :I fins/1‘ a: , n'v, (m) Z .4...» div. mmfme with Eff—j $35; D. (II) and (III) W” “‘0'“ :5," E. (I) and (III) n 17. Find timimerval of convergence of the power series 2 Zn—flfi’; Raine 5: xn-H («NOR :J_ 71+! N"? 4;“! cm (7—590 [—2,2) \<n+2)'2"+'" x“ ” M B. (—oo,oo) mm; i} ”tum 0*“ '7—<x ‘2 C. (—2,2) °° - __l 1 WW an»? 1 $3212: wriv, bug Altawfieii D' ( 2’2] 2,; “w E. [-%a%) , “his“: aim p s—AN’ Q)»; \N Mm X2? . 2:, h+| 00 18. The radius of convergence of the power series 2 nkcn is n=1 ROS/t6.) tQSi 00 “'H ‘ (WM :(n+01x\ -—-+°° 0” ”’4‘" 1 W! X“ gov mitt? 2 MA 166 FINAL EXAM Spring 2007 Name: _— Page 9/11 19. Match the functions with their Maclaurin series. (1) e“6 ()0) (a) Zx",—1<$<1 n: 1 . x2 11:3 (2) 1_$ (0.) (b) 1+$+—2-!-+-§!-+...,—oo<x<oo . x2 m4 x6 (3) 811111? (CL) (C) 1_§!_+Z_E+H"_OO<$<OO 3 5 7 (4) cosx (C) (d) az—%+%—%+H.,—oo<x<oo 1 . (5) 1H2 (a) (e) 1—902 +324 —$6 +...,——1<a:<1 A. la,2b,3d,4c,5e 1b,2a,3d,4c,5e C. 1b,2e,3c,4d,5a. D. 1b,2c,3a,4e,5d E. 1a,2e,3d,4c,5b 2o <)(>(> cost—12w 4 A. 4—712 B. 6—142" G. 6-12: @0 2 E. ”—2 21. In the Taylor series of f (:12) —1— centered3 at a— —— 1, the coefficient of (:1: — 1)3is | (a) The 22:18)!)be 6/ (w) a. € (1) 1 7 3‘. A g §(X .1» 49.1%) :..'3" U) ’1 1 f (X\:~>< (a) 1 B- 7 £12} ‘ ”,3 8 CU _ ”-3. "*1 3. (x): 2% . 3/ ‘ '5] v C. 3 4,3) ”A a - (nan 25371 D. 1 CED-1 MA 166 FINAL EXAM Spring 2007 Name: _——_ Page 10/11 22. The length of the ellipse parametrized by :1: = 2 cos t, y :2 sin t, for 0 g t g 27r is given by a . in. MW 27" ' 2 27r ‘ O o M B./ \/3cos2t+1dt 0 : fl” ma} (most)? C. 27r(\/§cost+1)dt 6 ° W arr ' w D. 2 (x/Esint+1)dt :J 4 3,332: +0013» r all: E2/0‘ O V3sin2t+1dt 277 MAW :f \l '32 390“? l? fl.» 0 23. The lower half (3/ S 0) of the circle (:3 — 1)2 + y2 = 1 is described in polar coordinates C. r=2cos0, for —7r3630 D. r=cos€,for—%S€SO E. r=sin0,for0$0§7r MA 166 FINAL EXAM Spring 2007 Name: _—— Page 11/11 24. A point P has Cartesian coordinates (13,34) 2 (3, x/g). Polar coordinates (7‘, 0) for P are . (X) ‘3) 1(3! E) A (flag) B. («i-g) wwnmmwmy C (2x/gfi Z D. (Ni—g ~ 7. 12 «z; 1: 2‘5 V I t? '1’ 3 ‘f' @<2¢§,% 1 4.’ 25. The complex conjugate of the number + 7:is 3+27, 1+“ _ 1+4L 3~2i: A. 121.11.2- 3+gi ' 3+2i 3.4;: 13 13 ell—Eh 3-2£+\ZL+8 13 13 __ M 9+4 0. i—g—gi L '3 '10 10 _. 41 fig E g. 1’3 +13 _ ~ E'13+137’ . WM; 0. l+4k __ j). 19 :J-L... L’ L I —' |3 +‘3t 1'3 '3 ’5 +2: ...
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