Sol-166FE-S2005

Sol-166FE-S2005 - MA 166 FINAL EXAM Spring 2005 Page 1/11...

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Unformatted text preview: MA 166 FINAL EXAM Spring 2005 Page 1/11 NAME $OLUTION$ 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 or calculators 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 student 9 digit ID number (probably your social security number), and fill in the little circles. ((1) 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 2005 Name: _.____.__ Page 2/11 1. Whichflof the following statements are always true for any three—dimensional vectors 6 and b? (I) Ia-Elslana A- (Donly (II) 16x13] glam B. (II) only (111) a x a = 6 C. (I) and (II) only (I) 62-17 —_\a|r)mse _ _. D. (II) and (III) only \a'glflEllEl lwsel é lallk‘ tm @ all (IL) \ax'E) = )aH-Glsine 3mm mm Crud, El 91 x 91 \I 01 2. Le 173 = {+jand 172 = —;+;+ Find the values of a and I) such that the vector 17 = 21' + aj + bk is perpendicular to both 171 and 172. 4 ___, —o __ _ fl¢mi 1qu.'UL-: 2+0. :0 A. a— 4,b_4 —-> d —’ "’ — B. a=2, b=—1 at 1:112 -. ruwuzz —2+CL“’"-° C 1 b .a=—, =2 —#—“2. 13:4- CL" ’ a=—2, b=4 E. none of the above 3. Find the total area of the finite regions bounded by the curves 3/ 2 4(233 — x) and = 0. L3 ': 4X (xZ-A.) \&:0 Wm x20) .13 1 ® 2 MA 166 FINAL EXAM Spring 2005 Name: —___— Page 3/11 4. Let R be the region in the first quadrant bounded by the curves 3/ = 3V5, y : 3, and x = 0. Find the volume of the solid obtained by rotating R about the y—aXis. P9 .U @ .w SIS” fil>1 Oil‘s”? “DIS” “M Methool o}, oUsks Volunu. o’l Daniel oUSlA: AV=TV(%%‘>QM( __ 3 4 ’5 5' “iii-“Mtflilfifl 3" W H ‘81 5 "3: MQU’M‘A 0/. S’laells'. Valum 5% 9W; AV: 2wx<373fi)dx 4 . 1 '- - -— £3.13 \l:f0 21w (3 ray/“)dx :61? 500’ {/3011 :07 [35:2, E37561) “mm/2 3/9)- g 5. Let R be the region in the first quadrant bounded by the curves y = x, y 2 v4 — x2, and a: = 0. The volume of the solid generated by rotating R about the y—axis is given f \ inn/(seal NW A. 7r/ 2[(4 — x2) — x2] dac / W = x 0 qng“ x1 x/i 2 ” 27r an 4—1: —:c dzL' m6. @ /0 (V ) x \/§ C. 27r/ (y2 — y)dy 0 Memo! 0; shells D, 2W/2u—x/Z—Tadx \lo\ume a; tweak stew: fl 2 AV : 2 fix (W._.X)AX E. 27r/fi(y_y2) dy Vi \/=§ Qnu<mf_¢()&y O MA 166 FINAL EXAM Spring 2005 Name: _—_ Page 4/11 6. The bottom half of a spherical tank of radius 10 ft is filled with water. It is known that water weighs 62.5 lbs / ft3. If the y—axis is vertically upwards with y = 0 at the center of the tank, the work required to pump all the water to the top of the tank is given by V ,1 0 WQFN a} MeYeéwtht . . 0 N t 3.1000 “3) GE's-WOOD A?) a“) 2 7. / :czlnx da: 2 1 Twiegvnl’iow M TawE: quqj-‘zu'u-vJIUCLLL A. 4ln2—1 81n2 1:; 3 a, 1 B. 3 J4. Kw”?de tibia—g” \1. S4, 3 c. 41n2+l u—me (Lu-:xzec; 8132 37’ n AM’z licbr q):% ' a 3 _g I :Iltm) 5.3 X3 )1 3 3122—1 3 * 7 1 _ ‘ e 3, dim) E ‘7 9] "("12) MA 166 FINAL EXAM Spring 2005 Name: 8/0 l 4 tan4 m sec4 :5 dac = fl 4 4- '2 7. T: tunxsecxgechx —- ‘— 0 1 D "I a t¢n4s< (L +toqu> 323x 3.7-: 2 u:tQ~nm mLLAZ$ve¢ ’Xd~y x:o .——» u:0 Va-E") “L: 5‘ 9. For the integral / find the resulting integral. 1 wzx/x2 + 4 Page 5/11 7 18 12 35 3 11 16 41 5 21 A. C. D. E. dw, choose the right trigonometric substitution and 1 c036 . x321: -71" 641'." = -_. x cum? 4 2 m 2tan0, 4/Sin26d9 (Lx =Z$u SOUS cos0 B.:c=2tan0;/ .2 d0 \1x2‘+4. :25{Ce Sln 6 1 cos0 C. $=2se00; —/ d0 f ___L__.——-—- (ix __ #9236133 4 sin26 2' Z ’ ’2. ’ s a X \‘X +4 4fom9 '2 ea D. $Z2Sin9; 60520 d6 4 Sin 0 __._L J 5246 d3 a 4‘ tmla’ E. x=tan0;/ (30820 (16 sm 0 L j L039 dye, 4 50719 2’ z Bx-rC Z 1+4 3 ” 3%. + X714 + x 7—9—7- x2+4 dx: X A. 1n5 3 9: +213 x'a’q _: A(x2+7_) +§x+€>>¢ B tan x2+’+‘>@+8)x +CY+2A ©3 A+B -.-.'1. C=0) 2A=4 . 51112 c A>2, 3:4, C=O D. 4 x1+4 CL 7' h - '1 Y: Q— __ X I E —31n2 Kit—7.x J, (T x1k17’o\x ZFQ’W' "iLfi(y2*2> , a :: yaw-Ailing; -(.1i52,,.3.) =2“? +7; 3 :ZLMZ—élna =22tna MA 166 FINAL EXAM Spring 2005 Name: _—______ Page 6/11 3 d i 11‘/2 53:31:! A. 1n2—ln3 _ i X_ 9’ 1 B. 1n2+1n3 L t g, __4=,_. CL“ = LAM f _g___ A7 @ The integral is divergent —2 x‘1 t—fl‘ '2 “’5’ D_ 0 t- [EQ/n’X'il—l E“ 00 t—u- ‘2 =2 E knit-1"; —£m3]-=...eo T—bl‘ 'g 9‘ .4223... = '2 X”! 12. Thelength ofthe curvey: 2+2)3/2, 03mg 1, is A. V3 . L L=§\(i.gy—M 13.4 .0 y y C. 00 7. 0% T Jé 3-2 (xi-r9.) 22x13 X(X2+*°-> D. tan—1(3) w 4 1+(&)2 i 4— X2(><1+2):x4+2x7‘+i @g 1 L: So {(Xz’r'fL OLX 7' g. (Xv—+4.) 4.x 0 (3+x0 3 - '- 13. Find the center of mass (mg) of the semicircular lamina bounded by the curves y : V1; 302 and y = 0, and with density p 2: 1. MA 166 FINAL EXAM Spring 2005 Name: ___—— Page 7/11 5 2n+1 271—1 + 371—1 00 14. Find the sum of the series 2 [ J if it is convergent. n=l 00 E E + 2WH A_ % W2! Zia-I ego-1. 0° vpm B. 5 hrs. (7; L: 5 +4” 2 3 C' 7 h=l ‘73! ® 22 1 E. Th ' 'd' t = 5 ’1. 1’ + 4. 7...;— e serles 1s 1vergen - v ' '2' 7. :3. {LO -\-— 4 '3 '-"— 9‘2 . 00 n , . . 15. The series 2—31 W13 0:1vergent 1f 09 A. p< *1 n— L 1 “ ._._—— COMM “MI-L :- —-——-—=v-: 2P4. B p<——— W11, ‘17—? n; n 2 oo oo 00 None 5n+1 n2+2 271 a I II ~— III —— ( ) 77122311,—1 ( ) n21 /n5 +—‘n ( ) 7;”? h+, B F ’ I L 5% CL) (kn/my» heme Q0 _, £300 3",; C. onl}; ) no «2 0° D. II and III only ’ ' v in - :L l r W ° W .74 - ._/ at) (b ‘3’“ ' UMP mu” n/z mm" E. (III) only me-‘fl oumw)ow\du/>L limit wmwa‘wn fest l 2 , Z Rdio IeSr: 2"+'L:fld-m ill”. : an - 11,900 MA 166 FINAL EXAM Spring 2005 Name: _______4 Page 8/11 17. Which of the following series are absolutely convergent? (I) 2 Sin” (11) 2(4)“ n31” (111) Z (—1)"; n2 n__1 (1) glighfi minnoan A'None h:’ n n3, n7, W B. All bxav \Nlfl" E '17“- C. (III) only M‘ng Q» Convenaapmf, D: (I) only G) ~V) was. Lon v, @ (I) and (II) only w p) h .— L ‘ H (D hank; “3+” “we ‘ ma WWW}, witL We W “Mari -‘. {,0 Okibsi “fivm our) ‘"‘ n . m i x, 3’1") 121;: t V’ o“ “W 173' mm) c, m aJas- WV- 0—"9 £1 hzi 18. The interval of convergence of the power series Z T m” nzl 5 n is Efitlo A ( 5 5) 14+ mi *1 ' ’_7 _ Emit-L“ rim 2 ix 5” 2 2 YI-eaD (in “flat! SW'GHJ) 2“)" __§ 5 b 2 2’ 2 :: m __ J... x = 2-. w—aoo 5 “+1” ‘5)“ C- (—2? \'\ A , 2 axles mm]. \l. ?‘x\¢g_g D_ (_OO,OO) 0"“ \X\ < i E. The series converges only for a: :0 6T -%< x ‘52-: Why. x:—§ -, i 2“ GE)”: £6451? WW x:_25,‘ I ‘3: 2 lo MA 166 FINAL EXAM Spring 2005 Name: —_______ Page 9/11 1 19. Use the Maclaurin series for 6%”2 to approximate the integral / 6“”2 day. The small— 0 est number of terms needed to approximate the integral with error < 0.01 is Z '2 ex 1: L+-x'r1%+—lfi-+rr' A-1 ’L 2. 3‘ 4 B 2 '1. i Z '3 . -X .7. 5 € &X=‘S <1+éx9fifii2+£12+égzwc,3 0 0 2! 3g 44 y _._j1 2 4 Q 8 ®4 —— (1—-x +'2L._ 2&n+.é_._.u)cix E 5 0 2. 6 4 i ' 2 L x? —' 1 '3 D ’2. 23.6 i : Jgi O ‘ 4 WWSWWUYM I 3’53 > 03’ 2's ‘0' ‘ ‘ n+‘texms A 20. In the Taylor series for f : 1 :1; about a = 1, the coefficient of (x — 1)2 is T La Tofi’bsr Hide/2 o&, 4 633.0001 03:1 A‘ 1 3%) 4- £10m ( L) 4' {(2%) (x 1)}? B' _1 i x, .______..a — I " ' 4! 2/ C.—1 (X) : x 1 g 1+) 2 D'_Z (17 \+)< - 5‘ " 1 g _ a: ’L‘l’x) __ x .. xx 1 ® 2 v? f (X) = v 9% MO (2 —3 2,. 86):-2M2) =—4 {lz’m _ __ J. 2! ' ‘3 21. The graph of the parametric curve :1: = 3 cost, 3; = 2 sin2 t is part of a(n) $- A. circle 2; 3— - ’3) + (2 " i B. ellipse gt ... 2 1%.)(2 Wotan @ parabola D. hyperbola E. line MA 166 FINAL EXAM Spring 2005 Name: —— Page 10/11 22. An equation of the tangent line to the parametric curve as = t4 + t2 + 1, y = t2 —- t at the point corresponding to t = —1 is d” %1 Qty—1 A. 2$+y—8:0 : ———————~: --—-—"""" ch 9.. 413 +213 B. 2m—y—4=0 {Lt ' 2 1—0 Wirmm 12—1: fl: “2'1 1‘12};- ©$— y+ — X: L+1+1=3 E. :r+y—5::0 23. A point P has Cartesian coordinates (33,31) : (—1, Polar coordinates of P are MA 166 FINAL EXAM Spring 2005 Name: ___.____ Page 11/11 24. Th h fth l t. e grap o e po ar equa 10n ® ellipse 7.2 2 1 B. line 2c0320+331n26 , C. c1rcle is a(n) D. parabola Tl n 2 E. h b 1 w <2ws19+3$4m 9) :1. yperoa Q. flange 4—“: rusinze ; L 2x2+3u32=1 ellipse M34 x/5+z' «an. =2[m>(—L;)+is?n(-E)] A. %(cos 5§+¢sin 5—5) \FS +‘L :2[m>(T—') +{shn(r)] ®COS%+iSin 5?” C 6 , 2 _ C. fl (cos §+¢sin g) “2;? 1 E [Cg5(’;‘£‘g) D. cos g+isin g +l$\h(—%—Bj E. cos %+isin % = ilw>(— %)+?S"n 1%)] :. 00%;?) + L sm(§§) Q: Fifi : (fi,£)(fi-+;) ’ 3.45-113 41.: Q - £25 €+i \fi +£)({3-L) f ’3 +0. 4 . . ~ ' JI— I: 1i (,1 __ 0175‘) :éa. [cos (-§)+IS‘“< 3)] —- 5.1? l 5”“ —’— .— CJOS 3 'l' 3 25. The polar form of the complex number with argument between 0 and 277 is ...
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Sol-166FE-S2005 - MA 166 FINAL EXAM Spring 2005 Page 1/11...

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