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MA241_test_1_soln

# MA241_test_1_soln - .4(When you are ﬁnished fold the test...

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Unformatted text preview: {.4 (When you are ﬁnished: fold the test over vertically and write. your name. on the. l'Ja‘ok) MAQL’ll—OlQ, Fall 2008 Test 1 No notes allowed. Dedicated calculators which are not. capable. of symbolic manipulation are allowed (i.e.. uo cell phones, PDA’S, laptops: or TIAQ‘S or Tl-QQ's or equivalent). You must. Show all work and sufﬁciently justify all steps in order to receive credit for a probleru! \V 11:11 the exception of problem '2, all answers must be exact (i.e., no calculator approximations or decimal answers). ll" possible‘ answer questions in the space provided; you may continue your work on the back page if needed. but clearly indicate which problem the work its for. 1. (50 pm. total) Evaluate the following integrals: ‘ ‘ l ':: 3 L636 ‘5‘ = (Y) r ,, 33,4 r I; f : will (Eli) (8 13m) j—l nil?” (in: ; a » 1"" {3' ' 5 [A {It (If .L '4“ : x— _,4x 1 (I; " L" lu(.r) Lot ,, ___I/l._ 1 ﬁlly: H? 4\ '- lb) (8 Pts-l 3 (int i :r r: r , zILj'i Y .1 {/7 ‘44-; .22 v2 ix. . _ i Li: I ‘ If 1 5m : l l ‘" ("1"; ' ‘l i — I‘ ' r # yr ; l ,v {r J: x‘ \ A— 3 K: A \ '1}, \ r r '- — — I» ~ 4 is; o R, 1L, ; _ of r l ,12 l r? If: , L- 7 .1 5.”, ’1 VJ (Li / x ,r 1~ A — :1 ‘r i —‘ m , — .‘r l " j , # («.2 Ma r a ‘ , 1”] ﬂ / L "(T r l N 'v A : If")! i l '5‘ — ;'_ «w; : ~ " + C ‘ 1" 5” a“ l n - " 1 _ H 1 7 ~ —1 041—3 53x , .1 553233: d‘T‘ éhf a '3 J 50 ;%X3£-* 4’1. 12—), 5x -j XdK’T-“k’ : ‘ A“ xn, We"! * W I,”- Izarlt“: m-_ﬁ—_ﬁ/_r_‘ﬁ_ -_ k_‘ - if 1 "' :f r 1 x K 9K C 9 ‘ y 5:" ‘Scix./v): iXﬂ -X5 + an: .«éuwiJV “j—_r *— ‘_f ﬂag/x L/=(fY Lah- éef M: X { #1..— gx = Q 42/44 : .2_.4 r u" H \/121L 1' (15“: i X 0A, with“ Q“: a; 026/“ :. X 57")“ ik'g —€7_ x1) I -u, I g + c = {2 < 5 + .9 1r 5f {3&1}: 4:”? a 5 L? x3+z +x—f f i X££'/»y ' X 5‘9) Allkc; C 3+0: .7.) D A =/ ’4 8; —J' ( 8 3’ 1 _ x+ +xf 50/ x (11¢ Xi(x11’-r') d3; (Note: the factor “at” in the denominator has multiplicity 2). 2 a} X51"D<R +X"? -“- A wry/+1) “ﬁg/Xx“) 7L (GHQ-)x WM; 3 ' - 1 :qu1F-A-x +Bx‘1f'8 7‘ C-r‘f 1L0)? :(ﬂ+C)X?+(ﬂ+DI)x1 L/tLiirX'+(6-) .L--'A+ <1_._5ﬂfx :J;/H/X/+_.L +lzﬂnﬁéﬂﬂﬁx}+cl X X X11")! / 7 X n [O ‘2. (13 pus. total) In this problem: you will approximate the deﬁnite integral -1.6 _/0 using both the Trapezoid Rule and Simpson‘s Rule with 11:4. \/:L: + 8”: (LI / 6 ~63) '/ (1):: (a) (7 pts) Filst complete the follewing‘ table (round to 5 decimal places): (b) (6 pts.) Now use your values fmm part. to approxinmte the integral using the two methods (Show your use of each formula): 1. Trapezoid Rule: W ' 4. ,m._-l+.:,2":z.lz_“)+ we” Tﬂwm mm (_ ) ii. Simpson's Rule: @<//+‘HM‘?_ M20722- lJ/EF’ r’ “+3333 5079(3’975 3 n . Fl , 9 ,X' ‘3 '1! 1’ +3“ X(Xf => ’1‘» x -.~.1' l :x."x ‘3 ll""‘*’ 4. (9 pts. each,_ 18 p133. total) Determine whether the following improper integrals are convergent 01' divergent. 06 6’ If f ‘ _ , I a \$idm : my. 13 : [1,44 --— 2 AW? - —' 4— -—-3I ( ) 2 + X + X + g f o 1‘ LL—acp é 45—36 £_ 6"‘[email protected] i? W 7 r - g - J ‘ 50 J] "’- ng f5 51*I/s71~-L.”w- ‘ x2 do CO _ _. [— “fir”, 002+Si112(\$) (b) [2 eh d2: (4 3 .« J «1+ 9"”“1 < —— ’ - J3"- r 7m.» . Sm‘fx) <{I <6 2+ YMMV) S j‘ T \ a“ t 9”" 5’ ’5: :15’ﬂ’195) (‘3 m .00 1‘ :3+ 5.64%?“ < 3,. (1x AH f ‘3’“,jx _. f5: 5 Ax \ | 1g. _ 35 .J- : (5' a; 3 a} _ H ‘ ‘" I, I ’- f n _'____ x gamer/5M”, .: I’HI J 8‘3” ’2 {NW ("2/4. {496 ) F g I 5 .2 63?): M _ #900 I; m r 2/ #1 21+ 5’” ’XJ a! 4/? cont/Efﬁ’"? my»: ‘ H 'F ‘M ‘ (23" 1’ MM) AL; ﬁr:— w/ a? - ppm. J v 5. (6 pts. each: 12 pts. total) For each part;1 set up the integral to ﬁnd the arc—length 0f the function over the given interval. You do not need to evaluate the integral. (b) L:ct+ot,y=572t 0ng3 ff (Ha {Y " ‘ ; 1- _ 5‘ L; 1 eff : —I: 1:) Z— : fa; (/ f f_ Ii f.) Tif— l“ x V. L Bonus Completely evaluate either of the integrals from the lust problem (4 ptsr each). CT! ...
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MA241_test_1_soln - .4(When you are ﬁnished fold the test...

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