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Sample Final Solutions - Full Name Section MATH 122 Sample...

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Unformatted text preview: Full Name: _ Section: MATH 122: Sample Final Exam Thursday, December 10, 2009 W Show all work and justify your answers. Your solutions should read nicely and be legible. They should not be composed of regurgitated fragments of your mind scattered throughout the page. If you run out of room for a problem on the front, then continue onto the back. Remember no calculators are allowed. 1. (a) Find the length of the curve defined parametrically by 39 = t and y 2 lnlcos(t)| for 0 S t S 1r/4. - [12 pts] 3t (b) Find all points of horizontal tangency on the curve defined parametrically by a: = 1 + 753’ 3t2 yk 1+t3' [13 pts] / 5‘3 €132"; (‘3!)3 I.“ li—iC—fimija I Jé’ecgl: : Sec-l: 5,, E02111] "S .Sec‘l: 0H: :lnlsect Hume] lo ._. }A(J3+,)-gn] O v4m2+1 2. E (a) valuate / $4 033: .. _ - = [15 pics] 6t b Bad t —— () v nae/(t_4){t+2)dt [10pt5]_ Max “M '--—-- ‘ 0‘) S 3 a Jim-«269 4| 9 - Seaae ( “' 3 ° 9 l le-I- 2x:- ‘Hm (9 6 : *‘fl-‘IQ ‘ 3 3! 56k 5 a Z".— 2 ‘3 S Mae ac; :2 50¢ a ”a 6’9 __ 35 case (99 Id“ U=$fn€9 .— "—'_ .1 ____--—-— 5-0 9 ' dos—“Casede rig/’2“ 9x \ b __‘3i°_._ (JG-HRH?) 6’ £ 4 a :::~m——: ____,..——-—""' q F—--— a 0° 1 3. (a) Determine if the integral [—2 m mine the value to which it converges. . [15 pts] alt converges or diverges. If it converges deter— (13) Use the error formula for. Simpson’s Rule to find the smallest number of ”subintervals 2 l necessary to approximate ln(2) = f Eda: with an error less than 10—6. Phrase your ' 1 answer as “let it be the least number of (even?) subintervals greater than . . . “ {10 pts] . b N l .L __ I‘M-x ' ‘H’: J ' a at a?) S +.“+H£+8 ($423?” 22-300 5 (ea) «:9 ~32 '" If“ "L Flay-Fl C 'fl—bv-‘Igo Q -3 -9 b m3] 2 gal-gm]: % (wave—rags a a . w ,3 . 5 "6 1:5 Eng—'5 %« 0"“) 4 10 h I 1... 4 '1 3 i We): ”—62: We) :3 r” .+ ”Us“ ‘ I :13 5!?) J- m 2% 7—3 m»; NW” x =— _ ___..._ ha P Cx)‘ X5 #nggl'l 'lJHL [.2 4' n saber“ 0Q QV-C”) fabu'nl-errle 96 I} ‘l' qooooc ' .3 4. (a) Find the unique solution of the differential equation hygg : a: + 1 such that y(2) : 1. [12 ptS] (b) A conical tank with a radius of 1 foot and a height of 2 feet is completely full of water. Water weighs 62.5 pounds per cubic foot. Find the work necessary to pump all the water out to a height of 3 feet above the top of the tank. {13 pts] an) 975‘] :633 d): = (“Sadr a) 527(9), 231+“; Ox '57 79:.- X+ln1X’-}C_ v09?! % \: 9434a4g «3.7 Cr—l..jh3 :7. Aflnlxi " *l-‘ina F05 b/c ycab Z O =3> y=+u+in1xl J-Jh 1) ( S: 'm IIc-f‘ Tr‘l xfi’KJIQflS-I 1 TI, 00 5. (a) Determine Whether the sequence {(1 — —) } converges or diverges. If it is conver- n =1 gent, determine to what value it converges. n [10 pts] - (b) Find the center of mass of the region bounded by flat) 2 m + l and 9(1) = \/$ 1. [15 pts] J’) “M M .. J, h . 1"" 45k 1"“8 K in(i"')' K350 32" 6&3 Ag“ (l-n) :xaoa (i'x ‘ ("€3.00 6 2 e (1536‘ 1.4, in 0- 3'5) L’H II». .121;— , (553) 1.2-, :_‘__'__, _ “I ' ' Xd’oa .3- : xa ,_—i—-,—*--'——‘ : gram ,,._J~ .... , 0‘ c» :2) " r n _ O In“ ({r%) :6 l COHJ‘QO'g—S ’____.....-.-.———-- OO 6. (a) Determine whether the series 2 rage-”J2 converges or diverges, State all tests that you use. 71:1 [10 pts] Do (—1)” 1110:) (1)) Determine Whether the series 2 T converges absolutely, conditionally or di— - 71:2 verges. State all tests that you use. _ [15 pts] n ’2 i .' h _ __._ 05 find "res’r? 1:00 9"” _ 6V3 4 l (OHM/jg; by Roz-3+ '76:} W 0" . in) As'r.‘ WKMN 139-53 ,3, fog + decreasb- ____._.. h , , , \ [In Ell—”L _ L»- J" 5" if 1"" '2' __ .4400 n * Xenon -X “" ’("m I “ O 09 _' V' in 60 P1 h 3? Z( 3‘ n Control/3:5 i} Z ELL-j (aflvtfjcig “-3 A . at: V1 Erika! Tm, , i=2: * ~ 1'? (X) “ I” ’" ’"" ' lei- pr 3:" 5:: ¥‘ (X)! X: 5' X‘L <’ C7 6n (6; an) 09 2— In rs beknvxts as S109 ink 1,.“ b In) - A ' 3. X 0X 7'" b‘wn *- dX 7"? X _ _ x ‘ s'acfi SM 1“ x a been a a, 3 m z X X diver» S “-9 I a: 0‘ £er e) ‘9‘ (QnV3f)C‘vS (unfifip‘dnxuy 8. (a) Let f(:1?) = / cos(t3)dt. Find the Taylor series for fix) about 0. 7 [10 pts] 0 (b) A bucket containing water is raised vertically at the rate of 2 feet per second Water is leaking out of the container at the rate .of— 2pound per second If the bucket weighs 1 pound and initially contains 20 pounds of water determine the amount of work W required to raise the bucket until its empty. g (—whdfisjy‘ a) (05 (£3 )Wf - ~ [15 pts] 7044 W}: 31+"); man's will be Ehfify 746.991: easier; :— 20 fat Swath“: gig-ngL—xélo ,—_~ 980 can; 0° 2nm2n+1 7. Let f (m) = 2 n=1 W (a) Find the interval of convergence for flat). State all tests that you use. . [13 pts] 1/4 (b) Use the given power series to find an approximation of (m)dm that has an error 0 less than 0.01. . [12 pts] Q é“? lxl 4 “1373\ i m "‘ am“; : 99 1 <——-> .7; asRoo+1es+.naoéJ) F X x 4 _,_ .—. $5“. 3 K” '2? 9" 5 Xr‘E/Q.’ QG‘QSZ j: (five/32$ bx, PPS-94?; “pi fl I ’ be «13/ = ' . (2—3/9, pm: 2 Ff" amoes 2», fame} ...‘ #J'f: «’3 30C “ C :3) 3 . VI— V" ’/ 9° n on“ M ” QM}? I 55) so 9C0dx ': 50 Z {:1 (ix _ ‘2 {3‘14335 D Z” 2‘“ n‘ l ‘7‘?”42 (264-3) (kl—V: ) ) 5+] RQi-H f2.— I00 1" (ROW) 437' - Q5” (Awe \[~ 7—1.7 )OOfl-l -J 394) 1535‘ 9.3+] '—' Y . . .4 (so/C “34H“; wager) I! Q a .‘1 I. ' 3; aka ,1 fl _.. n so Mad)! 2 ’7 79"“3‘}? ‘7‘? ‘2' ‘ 51:: Formulae K Error bound for Trapezoidal Rule: E: 3 fig; —— (1)3 Where KT = the maximum of | f”(w)i on [a,b]. Error bound for Simpson's Rule: E: g 1:73:14 {to —— (1)5 where KS 2 the maximum of Ef4(m)| on [a, b]. Taylor Remainder Formula' 7" (3:) 2 mm — 50”“ l n (n + 1)! 0° n a: e”: = Z '7? n=_0 00 - _ ("—1)” 2n+1 me e 2 (2n + was J. nix»). <53 yuqxw ...
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