Sol-162E3-S2010

# Sol-162E3-S2010 - MATH 162 — SPRING 2010 ~ THIRD EXAM —...

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Unformatted text preview: MATH 162 — SPRING 2010 ~ THIRD EXAM — APRIL 13, 2010 VERSION 01 MARK TEST NUMBER 01 ON YOUR SCANTRON STUDENT NAME gt? (“U Tl 0M5 STUDENT ID RECITATION INSTRUCTOR INSTRUCTOR RECITATION TIME INSTRUCTIONS 1. Fill in all the information requested above and the version number of the test on your scantron sheet. 2. This booklet contains 13 problems. Problem 1 is worth 4 points. The others are worth 8 points each. The maximum score is 100 points. 3. For each problem mark your answer on the scantron sheet and also circle it is this booklet. 4. Work only on the pages of this booklet. 5. Books, notes and calculators are not allowed. 6. At the end turn in your exam and scantron sheet to your recitation instructor. Useful Formulas i m2n i $2n+1 cosac = (—1)” , sina: : (—1)” “:0 (2n)' n20 (2n + 1)! 1 2 00 1)(4 points) For the series 2 (—1)“n2, the partial sum 34 equals 71:1 A)2. i L q, ENC-‘3“; 2’l +2 "L 2 L “’32 +3fo “5;! GHQ =—\+Lt-»‘i-H(a D)-2 : [0 E)30 2) (8 points) Which of the following statements are true? 00 d’ (I) If 11m an : 0, then Zan converges. gniée, . Z 1:? CL ngﬂ (II) If 2 Ian} converges, then Z an converges. Twang” ‘ MSDMR Cmvé/ﬁm 75> jim‘t n21 n21 tyrrwwrwzw _ WHWﬂﬁ \qw ___ra,“anwcﬁmh,-,,_7,,_,,_,mmw~r» 'r ~~~**‘ 00 an+1 00 i. Q \ 111 If n . 11:1 m _: gm 4 H ; ( ) Z an converges, then Za [ an W > “999 a I 0 n=1 n=1 W“ | #9 go Vt oo 00 > .’ (IV) If 0 S (17L 3 I)” and 2 bn diverges, then 2 an diverges. \ 2 66“ erigrm by Y n=1 ‘ 11:1 ‘Zﬂ’ho Taft. (V) If an — 2, then 2a,, convergf; \‘5 i 4 J, | E: L A A 1 H) d III 1 n_ ‘ W‘ \‘x t “L m H (W , a on . \ )()( n< ) y (4%“ \\ mt if; canwa b \N and only hug-5 «17¢ a \‘\ ﬂ I‘M, [A ~ 9“ CII,IVandVon1. , \ ﬁt A ~— V‘ ><><) <> y ,ﬂVWT.««—»>0MZ[£) cmverﬁm. Mag”? 5 a: 1' b 11), (III) and (V) only. n 33 E)‘(II), (III) and (IV) only. “‘> 2_, an Wrangler Lia WH‘ CMWKTSSK 725%. DO 3)(8 points) Which of the following alternatives is true about the series 2 (1—1)2—? n ogn n=2 n=1 m °° 1 A) It converges by the comparison test with Z 30" Get) xix-‘2 I 4% . X 00 1 B) It diverges by the comparison test with Z 71:1 : jt 32 Cl)“ 00 1 f C) It converges by the comparison test with Z t )9? 71:1 00 ’ 47 1 — ‘— l D) It diverges by the comparison test with E ’f M ( W 3 71:1 E) t converges by the integral test. r ,‘Lﬂéap 4)(8 points) Which of the following series diverge? l a; ‘h’ “7‘ - 00 L (1):”:1 DM. “ago? guano ﬂ‘l7owinblw n=1 n hm A r 00:“ > M \ a» (III) 712:; Inn \ D “#500 M22,“ W 1 1% 0 ' mbw‘ﬁlfjwm- A) only. \ MﬂpwrdwvmwWinnW w~~~W~~rw~w B)(II) only. )DN. gm 4 n WWWI . g0 CIndIInl. ~—/>_,L. A,» ’L‘ . )(ML ()Oy ﬁn ‘? h QMA 2h clNUyw D) (II) and (111) only. (a [mpaﬁsm \851‘ @411 of them‘ 4 5) (8 points) Which statement is true about the following series? (7" h (I) Z (—1)n Z 6% emf gal/lei +63le 3 I 5M” 44:7: div.) «fan‘ch 7— ): </ 00 or M, W \rm (III)Z(—1)wﬁ dim?“ \ n2 (H)§:(_1)n Mamie/l3 mvww 8t 11:1 2 My? Avawxgag) \>’§€N\¢E§JF117I A)All are conditionally convergent.\> f + 0 W A Naﬁwée Y\ ' ‘ . wan l B)All are divergent. @(I) is conditionally convergent; (II) is absolutely convergent. D)(I) is absolutely convergent; (II) is conditionally convergent. and (II) are conditionally convergent; (III) is divergent. 6)(8 points) Let S : i Find the smallest integer N such that we “:1 1 N em“ can be sure that ISN —S[ < m, Where 5N = n22; m i l A 8. WW , > “‘ (8‘10 , Blg' Altair; trim @10 3 (W7), w > “’0 i ' ’ I 2 ,_,.L at D) 11‘ 51 may g0 > (00 l a a, l l E 12. \-/ a” M W. ) to (H 56‘? >) 1% 7 no» WW?” in I I M a 4 4,. I ‘24) ' tit~ 7i Mi Ibo :35 “Wafﬂe S/thlﬁylb M 3 Ax: W WM it mumgmm, x (1,51 W40 tactile QR CWJ-éijﬂaw; i; 7) (8 points) The radius and interval of convergence of the power series Z n n=1 satisfy W i genial? lest“, Em Elm/a) A) The radius is equal to 1 and the interval is (—1, 1). “@‘f’ an C “Vth Ml B) The radius is equal to 2 and the interval is (0, 4). -: WW}: NH C) The radius is equal to 1 and the interval is (1, 3). The radius is equal to 1 and the interval is (1, 3]. k—ﬁ 90 max E) The radius is equal to 1 and the interval is [1, gal/125 Cﬂmmﬁ/ﬁ (6r bkzl 1" 3 S7" In vxo" w_‘¢X-;4(w‘sl4><4 (l) X \ > 2, Y\+l 2%! wkucﬁ clhrexfgw {MW} WPQQM m“) g is“) so A X: 3 ==> 2% MAMA cmmgm (Meme/.5 MH is“ (See ab“) 8) (8 points) Which of the following is a power series representation of the function :c — 2 m") ”$2 — 433 + 5? X76 L m ii X“ - X” 3'“ A) :2; $35 — 2)”. / x81 L04 in g (XL-9X41?) i“ (l l: (waif) oo _1)n I (7° 09 B) (I: _ 2y». a,» r_ V\ , ﬁrm 22:0 Wall M 2 NM“ C) 2&7 — 2)”+1. ' h :0 ":00 2m H 9: WW I D) Emma — 2W. : >, (-70“ (Krill 710:0 R “30 E) Z (;_:)1)(w — 2W1 1 6 1 (4 — 56)3 (Hint: Start with the power series of (4 — x)”1 and differentiate it enough times.) °°n(n—1)n-2 —'l'-:~l-(~—Jm):.i§ XmgiwL-s 2(4"+1) m . Lf‘x 4 (“(1%) Li? Fiﬁ-o i120 Ltw 9) (8 points) The Maclaurin series of the functon f = is B) x71“. ' /’—L‘— v“ f Y‘ XWM’ °° —1“nn—1 2 ' i Z ‘— ‘/ °° —1“nn—1 2 go _ wiele—w X OH); EH) “:2, WM" ~ 9° mm"; 22:) ‘ X p Chic); “:3; 2(L‘fﬂﬁ) 10) (8 points) The Maclaurin series of f = (cos a3)2 is equal to l (Hint: Use that (cosm)2 = 5(1 +008 on n 2m 1 °° (—1)“ m” Cw >< : *I if” A)2+§ nl ' h:o( ) K 1 0° (—1)“4" 2n 9° ._+Z x M h 2V\ (70 3 3338!” 493M : :5 (,1) (a), Zen“? Xe (2m (W 1 °° (—1)”4" n 9" h D) —+ r I V‘ 2 % 2(22)! JZ “P (: 4056174 5:; J): 4, Z XZM E) é+2ﬁju> 564” “:0 2C2“) L 11)(8 points) Let f = Z ~ 2)". We can say that the ﬁfth derivative of f at the point 2 is equal to “:0 g (9) C 5“ g A>f<5)<2>=10- 1.. : .2, a We; : a .2: .32 B)ﬂWm=t4 EL 5» 'A “>2=32 f ( ) W‘ W t ‘ at P D)ﬂWm=2L 29-: g; , bi me»: n n I M ED.f“M2)-100. ‘ ' . _ m1.a5u-@n—n n 12) (8 pomts) If we use that 1 _ m — 1 + “2:31 2n n! :1; , and that d . 1 . . . . ZZZ aresm a: —— m, we conelude that the Maclaurm senes 0f arcsma: 18 equal to 7/ " °° 1..3 -5- --(27L- 1) 2n+1 5%” +7; 2”(2n+1)n! x ' l a. : l +E l‘g‘nglt/r‘lm X2“ B)m+°° 1.3.5...(2n~1)x2n+1 {\—12’ I L“ m! W1 WHQn+Dl ' W °°1-3-5---(2n—1) n 90 C)“: 2n<2n+1>nl x' j ‘ t : x +— j in“ n=l n=1 w1-a5~-2n—1 71:1 ' 8 13) (8 points) Let be a function deﬁned on [1,00) such that > 1 for all :1: and lim az—>oo 1B A)$1 and \$2 diverge. B) 5'1 converges and 52 diverges. @91 diverges and 32 converges. D) 51 and 52 converge. E) Nothing can be said about the convergence of the series. as”? m um— me/‘W “(Est 9322. WE ——— = 1. What can we say about the convergence of the series \k/ a ‘ \ V N §W| /" W .1 m 23 , (AL)? , ) (W E: i g!) >__ 9% Cmukmga‘ ...
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## Sol-162E3-S2010 - MATH 162 — SPRING 2010 ~ THIRD EXAM —...

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