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Sol-162FE-S2009 - ‘\x ‘21 MA 16200 FINAL EXAM 01...

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Unformatted text preview: ‘ \x. ‘21 MA 16200 FINAL EXAM 01 Name "Socuflows ten—digit Student ID number Lecture Time Recitation Instructor Section Number Instructions: 4‘va Spring 2009 1. Fill in all the information requested above. On the scantron sheet fill in your name, student ID number, and the section number of your recitation with an extra 0 at the left. See list below. Blacken the correct circles. QWF‘WP’E‘" Work only on the pages of this booklet, Books, notes, calculators are not to be used on this test. TA Lecture time Rec. time Sect. # TA Hyojung Lee 11:30 Kwangho Choiy 11:30 Sungmun Cho 11:30 Hyungyu Choo 11:30 Yean Su Kim 11:30 7:30 0022 Ritesh Nagpal 8:30 0001 9:30 0002 10:30 0003 Matthew Barrett 11:30 0019 12:30 0004 Jishnu J aganathan 1:30 0006 2:30 0007 Botong Wang 3:30 0008 4:30 0009 Young Su Kim Some Useful Formulas On the bottom under Test/ Quiz Number, write 01 and fill in the little circles. This booklet contains 25 problems, each worth 8 points. The maximum score is 200 points. At the end turn in your exam and scantron sheet to your recitation instructor. Lecture time Rec. time 2:30 2:30 2:30 2:30 For each problem mark your answer on the scantron sheet and also circle it in this booklet. 8:30 11:30 9:30 10:30 12:30 1:30 2:30 3:30 4:30 Sect. # 0010 0013 0011 0012 0023 0015 0016 0017 0018 MA 162 Final Exam 01 Spring 2009 1. The equation :32 + 41/2 ~ 21: — 4y 2; 7 in the plane describes A. a circle with radius 3 and a center (1,1) medg m 56f (Mr/ma B. a circle with radius 3 and center (1 , 2) C. a circle with radius 9 and center( (1,1) 61 ”2% +1 ) '1” Ll’ (‘92:! (M + 4;)“ “‘7 '1 D. a circle with radius 9 and center (1, 2) @otacircle a fiscal); vk LIL((9’:_L/)L a; a 2. Q» £51). + £3.41?) : 9] “I W M 0M alcogsg. 2. Determine whether the given pairs of vectors are orthogonal, parallel or neither |>—- 11—1 51 = (1,—1,1) 171—.— (1,1,1) C? = <416> 2 : <—6’ “9) a3 = —i’+ 25+ 5,; 1 03 || 03 s.1 + Q” | PT‘ {51,31 are neither, (1’2, 52 are orthogonal, (£3,133, are parallel. 31,51 are orthogonal, 6172,52 are parallel, 63, 33 are orthogonal. A. B. @3131 are neither, (1’2, 52 are parallel, 63, 53 are orthogonal. D . . E. —o 1,b1 are neither, 52, 52 are parallel, and 53, 33 are parallel. 0‘1 reorthogonal, fig, b2 are orthogonal, and a3, by, are parallel. g: 51:: [4+1 114:0 Mm amogw 53.1%]: >mrpm1a€ l WMWMMW ”M MA 162 Final Exam 01 Spring 2009 be three vectors in R3. Then co r' (D rt- 91 9‘1 01 ._, ((M3) x (25—19)) . (—5a+73+é) T: 1M, (M) >< (2,2453 & 13(5an 2,. (3x233 “(m5 mixflpflxg ewxzya =7. '5 e» 31%;?)«2ng— 3 D 7dx5)-é’ :— «3(Xm—E) E —3(c’i>< )-é’ 32mg «3(373) ‘9 (—§Z+7: +2) =-— .3 [(2%) {~53 + (3%:me 1L525) g5] : «@E 0 +* ' é) HEELS : “3(fog) 9/”: 4. The area between the curvesaczl—y2 andngfl—«lis l W ‘1 SW, [@éfiflj “(glfifldj E2 ’ij 4% H COIOO DDICD Cobb WIN w®cwe MA 162 Final Exam 01 Spring 2009 5. A spring has a natural length of 2111. If a force of 25 N is needed to keep it stretched to a length of 5m, how much work is required to stretch it from 2m to 4m? A.25J FNFfiXs jg: kBWnk-t: Q3 B. SM 0. 225 J D.%J mm: ‘(o 2’3”” ‘JX §9J @3 2a: 1‘3"” "$er 6 i 550%) 3: JO/o, :: 9%? MW“‘W\Q5iW§ g” 6. If the region bounded by y— — 3 + 233 — $2 and a: + y— — 3 is rotated about the y~axis, then the resulting solid Will have volume MA 162 Final Exam 01 Spring 2009 7. Evaluate the integral 5 (A‘ClV 4: UV ”" [V cit/t /0 ”MW La (A bA Mt- sA LA: mg. (oi we t ml 4v a sweat AAA. Ma AAA v: “gage T Chem 2: Mean gel? JO ’écwftit T A Gama: + if? my») U 5 flew-«(am : 1. g 8. Evaluate the integral /O7r/4tan2:cdm : , Ty 2:4 if max - , > w é a 97—23:" 3‘ CW)“ “" Mia 1—7r/4 31/23:. : {l”%l”(0“0l 2 :: ‘ ..._ :3. MA 162 Final Exam 01 Spring 2009 9. After the trigonometric substitution (13‘ = 4sin9, the integral /2\/§ 333 d X?“ ki-Si‘fkga 0 V16—m2 :3 ”(ix ‘T; Lia); @ 4,9 is transformed into the following integral: 2; m Wrijuowa r twig Gr 7r/3 @/ 43 sin3 9d6 O Q: éoA‘Yfl % 42 sin3 6 “i .3; B. /0 0080 de 8 (Di :7 9h \(03 J” “Tl” vr/6 : ’ \C éi *3 a» o. f 43316) we 9 (Zfi> 9 Lg» B 3 0 7r/6 2 - 3 3W 3 TI‘ \ D. / 4 sm 6 d6 fl 3 X w A ’ é (9% 8:56” 0 cos6 mgfif 74 — M” LILCMQAE} E / 42 s1n3 6d0 \( O Ll, CD 0 W3 4e : 29' (l V1 E0 Ll 10. Evaluate WJLW m -WM. a: + 2:1: + 5 d /_m2+1 55:76 XQ‘HBX: «r2174 +g A. a: + (2:1: + 4) tan‘1 a: + C ”Ly? :Drwm .V M“ .m+ln(m2+l)+4tan_1a3+0 l‘P+Lf C. (3:2 +2m+ 5) tan—193+C D. m+2m1n(a;2+1)+4tan-1m+o %:§G + EEL>A7Q L E. $+2lfl($2+1)+4tan_1m+0 X fi -: VF: if; MA 162 Final Exam 01 Spring 2009 11. Which of the following integrals converge? o , ' ‘3 Cl‘vw;3w(1)/:OO 2331—5 dw — &M_CP gfifigfi if: mfi—gn.(2fi”gilfi> 2 3—33 55“” (ED/0 w3+1dx (:11) 3/8”; gfiwwL/Ag =/[email protected]§T><) e) .A. Afl<fithmn €’%3 W . and (II) only : @315” (Igzgptyyk + ;\ :O.‘_2” EQGI) and( (111) 0n1y 5:3 - mm mwdml)[email protected]:) “fig Of\%$ 4! 9y oéxél E. none (ll/1m «(73%, | v.23» (4% < (:9 \ V’ f téf x 'eef x1 )4 flaeréé‘re lo fl chi, Cmvvxgm. 12. Let ('17, y) be the centr01d of the region bounded by the curves 3/ = 1/33, y = O, a: = 1, m = 2. Then the value of a: is given by i g A 3; 5 131 , $.7le 1: if: x j”; m “??ki ,0) ch 1 \j l R D'4m2 1 ’iWEQX 814 A [L 1 : t In IN I; X lg, .9” {‘1 E5 wgwmw. n MA 162 Final Exam 01 Spring 2009 2 13. If a = lim 003(2) and b: hm cos(—), then n—>oo 2 n-—>OO n A. aanndb=1 QM“ .1. n a B. a=1andb=0 {’1 had”? 663 (L> £565 M1 Wlfii C. a z 1 and I) does not exist ( 605th (g) aJchaffij W ,4 0,1”! l @a does not exist and b = 1 52/1 I+$ mfg WWA ”” 1%: mi) a9 > E. Neither a nor b exists. r [4 :m COSC‘%)7 wai t: 1 “—5 OD 14. Find the sum of the series f (cc) : Z :7 and find the set of values for which your 7121 answer is :ahda; f 3 3 ‘ :7: X y‘ E: We; W 2 ea 3: z es» B. 16(93): 9’ for —3 g :1: < 3 V‘" “2‘ ' 3— :13 1 ~35 3: . C.f(a:):3_$for—3<at<3 : WEEMW : 3 :z: X D. f(ac)=3_$for—3Sa:<3 3 E. flag): 1 forx7é3 /&( i3i 1 3—3: ‘7: va gt? 2 ‘31 I; m ae-W’imé 56“ 63 V15“! MA 162 Final Exam 01 Spring 2009 15. Z n3 + 1 1s +7 7% t A. Convergent by the integral test Edi: m VDD+ M WA” 0 5 I B. Convergent by the ratio test W Lflmfi’ 1 MA Wee. C. Divergent by the ratio test curt NUM QWIW 9 .Divergent by the limit comparison test E. Divergent by the root test ‘ V‘ L M W; t \WW. : i 7 o 00 _1 71—1 16. If we know that ln2 = Z L—j—, what is the least number of terms of the series n 7121 to use to be sure that we have approximated ln2 to within 10—2? . , "' 3‘” A. 9 \ng: l; i ' 1% t m _ .99 W ~ 7, r ,, "film . l M C‘ 999 301 55;“? ? \00 D. 9,999 we A, g; E. 999,999 . \00 too i a i 09 YM “(1.;an {Wire _r ggfi, é ‘ 911 V‘ Wt“! MA 162 Final Exam 01 Spring 2009 17. Find the interval of convergence of 2 1071393” 71:1 P \ V\ V\ 1/“ _ r"~ if)“: A ( 00,00) RM leak ”21% l g B (—10,10) ”QM,“ V‘ V‘ 1 1 :9: ' \0 i791“ : C (76%) WC” ”CW my”! 1 1 A L .L 13- [‘566) [ON 4| “3“ “ 10 X ”no 1 1 J 1 (39 V\ — —— —— " ~l @[10’1] A >2 ;—~L~% ‘3 (01, Q.) 0 @in— X 0 <3. 3 ’2 h? (,Mx/a/gxs I4 I M W“! 1L 90 [Dy 1.)”\ 479 \ 1 w 10 Z” «a X "I {(9 D 2— V1? 9‘: 2,“ W; WM?” WE? ha! 18. Find a power series representation for flan) = 2332:]; 1 and find its radius of conver_ gence R. 00 1”. _, . f 2, h A. 12(90):: (W - rm :1 M — x 2 am n=0 2 P2244? hrbo *' _ 00 _ n n 2n+1 __ ~_1__ \K 2% XZW+| 2 @f(x)_;( 1) 2 a: ,R—fi :Zeuz / (”ml 4| 00 “-20 _ _ n E _ z 1 __ ,__ ”L 00 m2n+1 “A; R ,1 \ D = —1 n , R: 2 v me) 7;} > 2 V2?” 00 n m2n+1 E f<w>=g<4> 2n ,R—fi MA 162 Final Exam 01 19. The first three terms of the McLaurin series of f (:12) av A1+1m+z fifm’; X’/l+('x2’)) fig 1 3 B 133.i 362 gm $4 a JD L +(i%J \ @$+—:—w3+gw 5 X/‘+6Q>KVX)+ :2! [x 1321*. 1 3 3 m5 D-—H§w+g : +iX3-P3wxgr—FM E.x+la:3—lw5 X ‘22. g 2 8 2751 MWS A» Spring 2009 2 33(1 — x2)"’% are $V§€§ «Wag, 7T 20. The Taylor series of f ( )—— — cos a: at a— — —2— is £17 __ _ 2n+1 (_1)n—1 ( 3—) (271+ 1)! (:3) M8 3 H o 021 M8 | C 2 g ”13 I 3w] § II o O M8 T C § A? [\3 3 + 5 § H o .5 M8 2 3 ”§ + Ji'i 3’ ”:0 (2712)! 00 n 1 (33+ §)2n+1 E' :0 (‘1) (2n+ 1) ~51, 4%ka 0” >4 WE) : O (3‘64 : mwax “g vii) '3' 4“”): Jam “PX 27:) m 0 gywzgw 1cm”) 2; Cos x <30 M4 a 2m! 2. :1; M) h:o (ZMIM ’ (3° vr'l : E: C") [Xx/3; HVQP Mm : B rePwesmL COS X :7 I93 / MAC {with ewes . MA 162 Final Exam 01 [ Spring 2009 NOV/x 3d b / K $2 :32 , , ’ e» EM » .. , “ 21. lime—(E11511: Am /§-'. 471+! at 6’” )A +3,- Z” 0 95 X» o F a m. ,, W amnaaw . th B. 1/4 1 L v 1% W ., on , . . . 3 C. 1/12 : >< 1+ ‘0 M 4» ® 1/24 X490 WWWMWeMLWNWn_W E. None of the above % 1’? 3 x W : ‘ J— ..n .,,..L ‘fl W 2% \ ( (5f (AM: /QK 4+?pr I M > 22. Find the points on the curve x=2t3+3t2 —12t, y=2t3+3t2+1 Where the tangent is horizontal. A :1, A. (20, —3) and (—7, 6) (£39; :: fl 3:, “V ”V G "7 .: o B. (*2,0) and (1,0) (11/54 $X/d$ [0-614- (of: "(7, C. (0,1) and (13,2) ,9 a; :1 D. (0,0) (01:0: I) t E. (0, —2) and (0,1) {:0 a x30 3 “”3 (MWMNJ 11 MA 162 Final Exam 01 Spring 2009 23. Identify the curve. Hint: Find a Cartesian equation for it. 7“ = 3 sin6 A. a circle of radius \/?_> centered at (0,0) F: 3 gm fi’ B. a parabola with vertex (0,0) -’5 fl; : Fé] k (9’ C. a half~line through (0,0) :2 :2, I . ~I>> x + , .,... 3 3» D. a cyc101d . . 3 2., 2" @a Cer18 of radius 3/2 centered at (0, -2—) «23’ x ,1, gr m 36’ 3O A XL+{'9.Z;BE7 i“ 21-) ; .0];— ‘A >54” [tat-W azeéij; 24. For which values of t is the curve concave upward? d CL At 2t, A. t < —2 (l/X Qifl/djf €622“! Z. B.ty<—2ort>2 \ ' C 1 \) H (0%, o. t>2 Z A ii (2) .21; «'1; =5" 2; ii ) - MA 162 Final Exam 01 Spring 2009 25. A part of the curve a: = 375, y = sin 215 is sketched below, where P is the highest point on the are shown, Then the length of the arc of the curve from P to Q is given by 4 A./7T V1+4 22tdt “I 7r/2 COS aJi’ Nfléffii“ @5wa 7r/2 ‘ @/ V9+4cos2 2t dt 6% (SM/MN; MW 7r4 m 7: @mfi‘fi) “’" L C./ V9+Cos22tdt w Zt‘i'ft 2 *2? 6:1? Lg, D. /2 V1+400322tdt 774‘: 7r E/ V9+cos22tdt 0 WW? simmer Arr : 5H7), /wé;f’i” (2 cos 2*? ) L My “Vt/44 13 ...
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