MA303test1SP12soln

MA303test1SP12soln - Math 303 Midterm Exam 1 fjfilatbi. 35...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 303 Midterm Exam 1 fjfilatbi. 35 a LC,ka Section: Name: Instructions: 0 TO RECEIVE CREDIT YOU MUST SHOW ALL YOUR WORK and WRITE NEATLY. In other words, any problem without any scratch work won't receive any credit. Be sure to explain your reasoning. - The format will be closed book, closed notes and no cell phones, iphone, blackberries, calculators or other such devices are allowed while taking this test. o it is your responsibility to keep your paper safe. Do not let anybody see your answers or solutions. a You are not allowed to use your own scratch paper, a blank paper for scratch work is to be provided if necessary. Make sure that you are NOT * having access to or consulting notes or books during the exam. 7k looking at or copying from another student's paper. * enabling another student to copy from one's paper. * talking or otherwise communicating with another student during the exam. * disturbing other students during the exam. Time Allowed: qo minutes 12 problems on 12 pages. 1(5 points) C I - 2(5 pomts) n I:- . Problems 3(5 points) 4(5 points) , 5(5 points) 6(5 points) 7(5 points) 8(5 points) 9(5 points) - 10(5 points) 11(5 points) n 12(5 points) Total(60 pts) - Cfi 2 Math 303 Midterm Exam 1, February 14, 2012 Problem 1. (3 points) Which of the following statements must be true? I. The radius of convergence of i 37x2” IS3 $5233.; n=0 H. The interval of convergence of; (sac-$4)" is (—2,—4) Fake? ||| Th T l ‘ ' f ' ' 0° —1"——x2n_3 Thai» . e ayorsenes expansmno smxlsngz( ) (2n_3)! A. Onlyl “Jr . Onlyll QVASL KM fl 3“ < i Only III 1;, \w a m D. H, m mm 3"“ w- X E. I, II, III 9 w rug g3<\ <9 mdi -> f 3 $> i3 ’—p\%.9~a% \\ ' wfifl Y\ \gxiH\< \ <-> / \\\N\ W ’ h Mm m @me <3» #5 < 3% <~°2 ._ :2 < x < \ < "’1‘? a i ,me .31 ex. “ fl Wmfliw 2% Q": Tm? ==> L :2— e Y‘L “:10 L Mi «:33 A n [014? \xfi\ »~r0\9\‘t”5’ \pJ ‘01:”: \ ‘fi ,_ TWO/Ova m \inrtvual 9Q convifgfl‘vu ‘5 125‘ ) ! >5 1 i3 m \A ad“ \ ’" if; X m g T (Cy) X : if“ a .‘N\ 2 (x/ - ’ x (they Gm mi) , V\ 2% —3 A 2n~3 //rx \\ “U (A? 7‘ __ m en 23—... Sigma we 7 1 \i \ \ x it ~— at—m (Wu 1 @«m x ~ V\':_ 2, \‘ \i‘. .,,....»r-’/ hf; 7/ Math 303 Midterm Exam 1, February 14, 2012 3 Problem 2. (3 points) Let p denote the radius of convergence of the series in question. Which of the following is true about the Taylor series of 1 i x about the point x0 = 2? A' 1 1 x Z i (*1)n(x —— 2)" and p z: 2 Se‘UA‘CL’Q/‘i ‘ r WWmmMMflwmmfiawgwwm _ nzo _7 , .V {q .__..._i,_ I °° 4 "“ A29 Pol < i . \ 1 ’ -—-"‘"‘ na— B- 1cx=;<—1>n<x+2>nandp=1 H - Z is m n—O \ Y\'=U 2.» 1 _ °° _ n _ n _ *uiwwflw fi,-_.,e.,.___s... A. m ‘ c.1_x_rg)( 1) n!(x 2) andp~1 A t w x , 1 00 n n M : M —. l ‘- 1-x=g(—1)+1(x—2) andp=1 (PX «~(¥_2}’2_ “Pix 1} 1 __ 0° _ n+1(x‘_2)rl __ CK) bi E. 1_x 1;} 1) ———~——~m andp—.oo ‘4 "A v (A) Z (“SHAW \"F(X-’l\l a ‘iw “‘0 L 00 l” w : g , Gflflfl CW1“? - \xtfl< & '7 ’ hf'U 2YV§K 29,90 3 i V LUV ~‘c 2 x—z Tim le—‘r’t ! w in ‘\ i ,_ ___3__.__ w _::L. w w‘» ,C—2) :99 «xx - A——(14r-’c -—\--“C fi’fE in {wec‘x} n:0 m m cm «#1 ‘f‘: T /. ‘3 & V‘xa Q fizkl VA 4\ é:> \xn'i,\4’\ => f"“ M * ' ' n h vr 40: G“) G 4) ‘ M 0 0.2 3} 30;) 3.129”! Col/\E‘stfic—e, an -: Y” P 708 p wail n \ w _. mi vPrv:.L* 52;, £42); 1—:4 ~> 020-— i f ' ’ * w 3"» z. w- : a z : ..—-— 5+: % (X); “Liz, ‘79 i (93‘ saw?” 4 > W‘ 13, («In 1 Q , 2" U( :L “m t. L- s». «2 We a JW aw 'i X}: f :2) «v.1! '5 a j 2 2%. ’l 5 é‘k ! 9‘“ Q Dix-H “‘ , {‘2 ' x 3 V n s n» s92~scwwn a a: (g "z; n '- <:> é — mu 1 i X I"\ “a l w m "* {gua\ Y‘L‘ , “a \ \ k “ \ ~_} t: (Q *\~ -_- s“ {\‘lf n 5 \ “Ea L 3; \ (.43 ‘ CX—Z\ VQTio 1:” $3 : g AMK ~ / a I L f \fi30 Q 4 Math 303 Midterm Exam 1, February 14. 2012 Problem 3. Suppose following must be true. i. (n +3)(n +2)(n + 1)an+3 = 11,, for n = 0,1,2, 00 x31; 00 x3n+1 00 “' y(x)=a0 (1-hng1 ))+l11 (X+n=1—(m>+a2 (39—bit; (311 ! Hi. If y(0) = 2, y’(0) = 3, y”(0) = 6, then the coefficient x3n+2 (3n + ). as of x5 is that we would like to learn the solution y(x) = 220 anx” of y’” — y = 0 around x = 0. Which of the 90 0’3 , i 'r' (, (“*2 n La '1 n ,4, an: A. Onlyl : anx J: 2 map :4 9 «e/ “ 9 B. Onlyll 7 j ‘ 5 - A643 H_/} ":1 I, II \ n .l,|ll 9:: W3 31 ( +0) X E. I,l|,i|| > gym, Zhoflxhflanx : LCn+3>(n+'2ln., 01M? €33 n “Lg: O g5 Z i@+%)(n+25 (am an???” Gin] X 2: Q G I ' ncoL r l _ £7 @+‘53U\+2§(v\—i~\\> awe) : a“ , he 041)?) r ~ ' 01% 04 (x q __ CR12.— __:_ la; ' 9‘9" 0m, 3' 787%) 2 “H, ‘3 5 5H ’5 5 l p I at, a; \ 0‘6 “ 2e; d, __ I 2 I“ Q r; "“ W a _ (13 z _3‘_ x Ox-jr— key) j," ‘X 8 3.},(3 23‘ é * (95¢; 6; ‘5 \ i \ \ ( v G“ mi a?n : CM 1‘»): Qfivwz * @MQ—W 0%” {any .+\ (awry r-I Wfi ‘33 i 9!“ if? Vi n y‘gfi ‘5 gm ‘3 " 3. ova . 3093* 2 Om '94“ z: } 0?)“ X “J” L QBWH ‘4‘“ L "WA-72;. \ SQ ‘ V330 W: \ “3‘3 9‘0" X?fl+2—\ ‘¥—\\ , ‘ 7 0L 9:954. 1. . \ - 5Q +0» 3 w“ 1*?” " if“??? ? “rm-3‘ " S» g ( K v i‘ 0 O L) \“- viii j “\ / hm 3m: 9w é’fio}:3} 5%0‘): {o I \ \- 7. i; \y H g / 00:1) &\:q>) 0‘3: j V . " “ ’* M Q \ a ‘5" - a r. f»— : I 05, X 3::- Cigyx "'—> ‘5 5 l. \20 ‘ 1““ (3‘. u x 0 fig»); allr’0\)_. i p we; L~ ’» :0 :71. __ = w * : k ‘ 3:6 ex 2% ' Q: 3 3 ’ \ 395% c H u; @551: 5% "—17 Q5 ‘” 3 2,0 Math 303 Midterm Exam 1. February 14, 2012 S Problem 4. Assume that 201 —2)an_2x"”3 + E 2an+5xn+5 = 0. What is the Taylor series y(x) = Z anxn'w‘n‘E-n QO:\; ? n: 11wa 11:0 A. W) = er ‘W—w’ ' «~er “Ext”: m n s»: a C. y x =ex ‘ “49‘ X ‘3 r .. D_ y(x) ___ e—x/Z ‘3 ofin-‘e-x 4‘7 L 2 X k/O E. y(x) 2 ex “:9 “:3 a) / , m " —~ Q { {Q3933 QWAV’ZQH X W \ Q m 2“ avg “A > O 2> \A,%\ W m fipifli a Z, _. a ;_O :‘> W :m w j 9 {1kg ._e Z '35 4” m \i Z _, ~4— . f f, OK : "‘ '-_"" QQ‘ ——-—~ k ’ A70 3 f C A / m ‘3 («2.3 a \ ~— - "‘ _,_, 2 \ Gym. Q9; 4-K egg/No”de ) QR «w 0L0 wi> am A; \’\ \I I {323 M V\ 16517: 3:. /> CE“ K — ‘ ‘ w a p 6 Math 303 Midterm Exam 1, February 14, 2012 Problem 5. Consider the initial value problem (iVP): y” — xy’ — y = O with y(0) = 11 and y’(0) = 5. Which of the following is the one including the first five nonzero terms in the solution y(x) of NP above? [Hintz you can differentiate the differential equation to find the coefficients of xk for k = 3, 4.] (KO —_; 3(0): ii ) 0K1; %‘(0\:5 11 5 5 A.y=11+5x+—x2+—x3+—x4+~- , i (, __ “(3.. \i 4 3 8 i _ , o e. a ._ (o ,—.-. o _. m - \ . 52 113 54 ,3 (OW 3‘) :3 B. y=11+5x+—x +——x +—x +--' %1 2 411 711 4 “(03 ii C.y=11+5x+—2—x +Zx3+§x 02‘ L :: ‘— ‘ 9d D-y=11+5x+%x2+§x3+;x4+m ‘ 2‘ _ 112 53 114 @y—11+5x+2x +3x +8x + m :3 3’- r:—— -M 4) ‘ qfifi‘tgwiiah agifxngsfgo rm obvi/ 35m — ‘66:}, >03! \, iv 0 ( a: \g m 3(\ 0. “(QB '- Matt?“ .2; § ;\> a I“ 0“) :33 g(0‘5: Ekiri‘m 34:0 3 wt XML ué M e E3 to — a G m g (0“; w fig 3 E); z: a3”; ”‘ 31, 5 "3 ' it V 31% v r so t e ii , mm» m w 0 D‘ \fGAS‘inSiM; U7 W i” (m G L” f ‘. . .‘ fl _, 3 {0) t: Mikey $220? L f)?“ a i g [Wm I 32 Z, J, M ‘ Lit 2"“ 8 90 ~ ’5 “t L n ‘ 5X “55-” *” ' 1/669) : EGAN/4“ “f Q®4794$+02kafl$fi "MAL? V\:O L E3 3 Math 303 Midterm Exam 1, February 14, 2012 Problem 6. Which of the following statements must be true? Taxi” I. y(x) = 01x4 + 62x41nx is the general solution of xzy” — 7xy’ + 163/ 2 O for x > O. ‘i‘f‘iAfé g9;ng ll. y(x) = c1x5 + czx3 is the general solution of 2x2y” + 3xy’ — 15y = 0 for x > O. ((336, Ill. y(x) = clx“1 cosh/(311m) + czx"1 sin(\/§h1x) is the general solution of xzy” + 3xy’ + 4y = O for x > 0. . - , i“ A. Onll .‘ t 2., iii 5‘ i i ‘_ J: » B. I, My Raw W63“: 1" L3 ‘3‘ mion "1% figj W Q jig” x G) I. m » ii; D. n, m _.__ \J E. l,ll,i|i {/ in {aqua 4v (4?“ +371: :C: "XLUflwEWX/‘g “‘3” “L2” '1: 25>. r(i“-fl «% {:13} i“? if? '3: {-3 iii-WE) -< rezi ca: We“; ,( a“: 2 ix?) ’6 :7 '3 199i iii ‘75:; «A.» C; x X j I fin 3‘3 35> viva“; 4???) Mr “V ’5“ Q ‘ m / ’ ,fl. 1“}! 3* «w: “Var *H mm ‘f‘QJt‘ lt“‘”l§-§f (his if «H \; 1+3 8 Math 303 Midterm Exam 1, February 14, 2012 Problem 7. Supposing that we would like to find a solution in the form of y = x’ {22:0 anx", which of the following must be true about the differential equation 2x2y” — xy’ + (1 + x)y = O ? ' ' ' ' . K ('35 vi ‘FMS’B I. The indicial equation is 2r2Q3r+1=0. Var—"M x “a 7‘ fiwyz/ II. an[r+n—1][2(r+n)—1]+an_1=0. X4") 1 fix ’ _\ Pal/SM”. One ofthe solution isy1(x) =x1/2 1+ i ——-———xn-————- qo : WM x (3;; ' 2 n=1m[1.3-5~-(2n—1)] ' W , . ; O Onlyll r 00- Xn a “° Xn+r~ :5 r(v~—\3+i>of*‘l° B. I,” 306: X OM ” Z @n' rz’rg _\,r+_\_ =0 C. Li” M10 M/o 2‘ 2‘ i , D. II III . 1‘” “564‘s?— E I’” I” I 00. " ‘n—i—rfl WA -~ fiat: Zanmr—U >< .- (fer/{fl “90 iM-r'z. =3? 17-1/9, i 2- H on 3 Over—fl X " la (XV 2 Z ‘M 6‘” mac ESL ,L n i +X : o=2xgwa+0 “W 2; l For r1 ’ /Z grn<n_)§:3 Math 303 Midterm Exam 1, February 14, 2012 Problem 8. What is solution of the initial value problem y” + y = cos(2t), y(0) = 1, y’(0) = 1 ? A. y = 1m?” +sin(t) + mil”) 51‘7/(53 ~e xi + ‘llsi : 6 cos (2t) , 45in (2t) 51+ Ll“ B.y= 3‘ —sm(t)+ 3 [Z 5 \ . . 2‘ ‘ _ 5 C. y: #45120) +COS (t)+2511;(t) (S A? W (3 ._ {El—Hp + + D. y:c05(2t) Asm(t)_4sin(2t) \ ) 3 3 E g 3%3 @y=-C053(2t)+sin(t)+:1£9:£)~ \/l$\: ‘ ""l ( SQWQE w‘ f“; 39“? ‘i S _‘_ As+ E + {is-4:3) 3 =1 (#er B3CSY+LD+ ($491)} (filo g : {Mm $34? (Prim 31+ GP“ (33$ + (“3* D» \ Mic;o >z> A23, (3:23; 3+?) 2: 3 RN , \me :7, : Z O Liairbzo NJ _ \ r: A- 5 ~ L S "V a ‘JT‘ H 3 37+ \ 3 SZ+L+ 31—472 Sari ‘x \/(§\ :1 Li S '— ‘L g 0. “Jr I 3 51H 3 5&2 57%” kfil‘fif‘gf; ‘3, CogH:\ _. if; .47., 10 Math 303 Midterm Exam 1, February 14, 2012 50 7 Problem 9. What is f(t) so that L{f(t)} = A. f(t) =e—2t [6cos(2t)+25in(2t)]+5t+4 =3" [4cos(2t) ~§sin(2t)] +10t—4 c. f(t) =e’t [65in(2t)——2cos(2t)]+5t—4 D. f(t) 252* [2cos(2t)+6sin(2t)] +10t+4 E. f(t) =e—t [4sin(2t)—-2cos(2t)]+4t—1O a % C§+’D 6c: M iv??? A;- Z 51<sl+23+5\1 m g ‘3‘”23‘5 2. '3'.an \ (ZS—FD 3 60 :_ faig{gl.§2_s+5\) +‘E‘si3 “95’5"? C B ~ g 2 (affilms + ‘53 L30: @+C)§ “‘7 (ZUR-kgybws + :0 Fig—J50 $> 3 ‘ wag—:0 si;> 5A=~10 2% Rig": 2 2k~k%+®=0 :5» $519940 "‘5 ”" A—srczo ==> C:”A=>C‘Lf .1750 W m :_L_Jr__ £977 A? Mfg—2‘ M 3‘“ (gums ‘3 5 3" S? +1345 L; :2. AM :2. We LN : 5 z % E; “3/7": g”: 25+ 5 (9—K? 1°” (9% +1 [sum +2 J. —-" ’1 ~": r-‘t \\ 3+ ,— /\ \ .. r C ‘90 *HJdot JrLVP/ECQAZM ’33 9‘4: 3 Math 303 Midterm Exam 1, February 14. 2012 Problem 10. Which of the following statements are true? _ sint, ifogt<2 _1+se‘25 " Iff(t)_{sint+cos(t—2), ift22, ’thenqflmfi s2+1 ' _ t if0§t<3 _1+e—35 II. If g(t) — {3 ift Z 3, ,then L{g(t)} —— 52 . III. lfh(t) = e—Z‘cosh(3t), then L{h(t)} = A. Only I“ B. I, ll \ c. l,||| ‘ \ 4 « * ” JV” .H,I|l L gm‘ 39"“ "Luzm 00% 9‘) E. I, H, m 2 ( ~ 6 ‘5 : w, L . “M S~H r ‘ 3‘5 \ ’bt \ \«QJ 32%“? “T” 3 fl : “f” S 51 S 4‘ ' “*WZEW» :> the} rim. T 5+2?) «fl 12 Math 303 Midterm Exam 1, February 14, 2012 Problem 11. What is the solution of the initial value problem y” +2y’ +5y = (4 — t) 6(t — 3), y(0) = O, y’(0) = O ? 1 _ A. y(t) = flaw-(F3) (4—19) cos(t—3) - i . B. y(t) = %u3(t)et’3(4— t) sin(2(t—3)) ($3 3% 23 +53 if * 3‘ SW > } 1 _ ~ ~ ~35 C. y(t) = —u3(t)e 2“ 3)cos(t—3) I . ~ 5 “F33 e W) = game-(F3) smog—3)) 31%) *1; flm 5mg < I E. y(t) :: iu3(t)ez(t‘3) COSQU -3» / 0V5) :0) 3‘ (93:0 ' / ’35 (5% 29%3 \qu = Q ~3$ a \/ L9) : 3142.935 j( 2:35 ’\ fl: (Sagan)? ,, 2. W L e 35 fl,” 9: 1 (SA—l)?“er ' r-CKC‘E ’C) - ,. _c9 t-C «my g. 3 6 mm} ~ Meet» 6 P‘fl 3 ’ “CL a, W LN?— K “N _ «ire L 3 5m <9. 07"3-33 Math 303 Midterm Exam 1, February 14, 2012 13 Problem 12. Find the solution y(t) of the initial value problem 0, ifogt<3 y"+4y= (t~—3)/3 if3§t<6, y(0)=0,y’(0)=0. 1 ift26, A W) = u3(t) [¥ — 81mg _ 3)] + u6(t) [% _ mag - 6)] B W) = use) [11‘8— ~ “8:23;; ‘ 3:] + u6(t) [% _ mfg; — 6)) i—3 sm2t—3 t— sm2t—6 C y“) = “30) - zlzt 3 ] ~u6(t) [F6~ 6] D ya) = u (t) — Cfs(1(8‘ )] ~u6(t) [% _ cos(1(8— )] @m = use) [3113’- — Smazfi‘ 3)] — uea) 11:6. _ sm(22(:— 6)] L3“‘\'L\-:j-— (‘43 "‘46 C'E\3 Jr UKQUCX .— A \ ‘ 5W «2 :3) w — (it; “W 3W0) :9 «Lo “HQ, LQPKQCQI Wnsvgém M 6,2};— -—33 ~$3 ._ 9&3 ‘ ‘— e’ “V’ = 3 Q. ._ €— 9, 32 __ SS; ' l L ~35” £63 3 22> V83 2 3 e 52- “SQ—PD « -— f; + B 85—h c r\ — As (Sam + BC: w.) + (Cab) 37— 31(514‘0 t’> $1 5 -+‘-( A»\-C:O) 941920) L‘Avrrg) 43:4 ...
View Full Document

Page1 / 13

MA303test1SP12soln - Math 303 Midterm Exam 1 fjfilatbi. 35...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online