Test 2 Solutions - EJBH'IMECICIB 15:0 IFAK...

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Unformatted text preview: EJBH'IMECICIB 15:0? IFAK scannerlflmail.math.mcmaster.ca + Bonnie flosterlin CICI'IKCICIE MATHEMATICS 1AA3 TEST 2 Day Class Dr. D. Haskell Dr. D. Ghioca Duration of Examination: 60 minutes Dr. 0. Unlu McMaeter University 13 March 2006 NAME(PLEASE PRINT): _30 L— 35? l D ’U 3 Student No.: Tutorial No.: THIS TEST HAS 3 PAGESAND 6 QUESTIONS. YOU ARE RESPONSIBLE FOR EN— SURING THAT YOUR COPY OF THE PAPER IS COMPLETE. Attempt all questions. Total number of points is 50. Marks are indicated next to the problem number. Any Casio 5:991 calculator is allowed. USE PEN TO WRITE YOUR TEST. IF YOU USE A PENCIL YOUR TEST WILL NOT BE ACCEPTED FOR REMARKING (IF NEEDED) Write your answers in the space provided. Good luck. Continued on next page 03x’14f2008 15:0? IFAK scannerlflmsil.math.mcmsster.cs + Bonnie flosterlin 002E008 MATH 1AA3 Test 2 Name: Student No: _______________—_____ Table of Formulas 1) sin(2:c) = 25in(:t) eos(:r) 2) oos2(:r) = $0 + oos(2:r)) 3) sin2(:r.) = %(1 — cos(22:)) 4) feeds) dz: = In | sec(:r) + tsn(a:)| + C 5) / sec3(:r) da: = ésedz) ten(:r) + éln | sec(:c) + tan($)| + C 1 6) f (is: = areten(z) + G 7) The Taylor, series for the function f(;r) centered at a is given by 69 Hz) = z airmen: — a)“. 11:0 If ]f("+1)(r)l 5 M for 3.11 In: — el 1: :1, then the nth remainder term satisfies IRn(:r:)| E M'Im — uln'H. (11+ 1)! 8) e” = _ —:r”; converges for all I. 55211. ; converges for all :r. 10) sin(:r) = E: ——22“+1; converges for all 3:. “=0 (2n +1)! r W __1 _2 ._ 11) (1+ 1) = E iHMr—Bll—(r-flmn; converges for [ml <'. 1. __ . Continued on next page + Bunnie flusterlin UGSIUQS Student No: Name: MATH 1AA3 Test 2 03f14x’2008 15:0? IFAK scannerlflmail.math.mcmaster.ca e e h h .r. .r. .F... ..I m 0 ll r C m a L... . m .m. e r e .m E .T.. C. n .1.. w d J .m {x “HUM...” HHHHH Iii... 3.... 7... I 2 Ill-,frdr/rl [flirt/(I‘le m - n i glib”... Rd... .H II. . m E. I. J. In... fill/1...}! [III/f!!! .IIILIIr N \H x. 1.. Hi]. “ D V .15... JHIIHU... “Hf {xxx/#15. //....r////.r I'll/k \x.\\.... 1...... r . . H... .1 __ l\\ ....5..../ x 1111' 1.555.... 5,555. lllrlr; .x\\\.__. W1! ..m C _ a Eu mu... 3 fr.......................... ___..rfixzsrr [It'll/.1. K539... /.rf u H d _d '11!!! \\.\\.k 4.... um q .S [if \.\._\..\..~ 1.1.]. .1. e \I... I fill. m e "u 1 O \L. a h rlx \\\\.._ {If n i t $33.2... 1.2.122... II]... \\\b. 2.? V. MW t m 5.2.5:}... 3555...... EIHH MN... .I. :J m xxx/t5?! //x//../// if... \Nxx... [H D. r. 0 {/Jrlrfr... xiii/1?}... Ill... \\ a. p «m Em a.“ III/filial. trill/III}: \. r 1111111.... 1552?!!! .r I... h 1M4 1d m! [flint/trim: Ill-[III]! HIEI y 1 a m o "m. T E. w t. . . m m m B D r- .1.. H. t d. B e. W e h b D .H LL. ‘0 w d e I f If If}... \\. \\ b m e M 5H5...” “thunk E...” a... “HHHM AA”... _ x C h .m {Ir/Ira}... ___.Q\\\\\\ ___... .1.... a... .. III/1.5....) xw ___ .. If! \\.\.\\\ t. {fr/z... ._____\ if»... 2.... .............:.....I.../ .5... .1.. IT 1% :1. C ffffiffxr a..._\.._.\\\.\\ .__ ... I... J. .3... .... fill/III... .. _.. ___. a.“ 1.- o o 1 any... .. a“... :3 h E 5.6.5.... ___ bx 12...... 3.... III/{xxx}. .....Q..\ = W 1% a CF... .___...r.r... xx... Ill/fli/# ___..uxxk... h 3 II\ a .21.}... xx... [riff/xxx; .....C.\ I. \I: If!!! m P e d U... Ill. 1 B B. #33:. ........_..._..__. I... ......\\...\.\\.\.\\. 1 xxx: 2 .._._ T H“. .W fi _ \Exflhum ___.rrrfzrf if..." # x\\\\\\.\.\ .0 \\._\\.\.\_._.._ {Zr/Jill... ... ___. \\.\\\\\\\ t t E 1 ._ Mar ___...__.. a... fix .. \\.\.\\\\..\\ B. \\\\_\N._...___.. ___ xix/11.1.... ___... a... e. L“ x p :al\. .\.\\.\\.\.\..._..__ sag/Jill!!! ._._... HR We. _r ___ ___. \\.\.\.\.\\\\\\ L .1.-W. m y \\\\\\\......_..n.....“. ##HHH/rrfl HHHHiJI _. u; m \\\\.\\ 1.11.11: r... \..\.\.__..._.. . .. .. \ \\\\|\ 3 0 4 W. y___.m \\\\\\\\.. $214.11.. .. ., 5 .3. 1.. 5.4.3:. 3. __ ._ .mmwuux HH\H\\\\ \I... r... a .W .IL I d )( m m. m ) . . .l. l tlx. .C (U. A C Continued on next page 03f14f2008 15:9? IFAH scannerliimail.mathmcmasterta + Bunnie Dusterlin 004E008 MATH lAAB That 2 Name: Student Nu; m 3" — 1 2)(a) [4] Calculate the series 2 “+1 . 11.23 4 m I'D 1w _ \ m Em A. 2 _. t S“ m. — w _ I‘m-H L~---—- ‘1- "\ *1: A -_ '5 n1“; 0-0 DO — 2W)" - 2 m“ n L” "t 'T “:0 W n | EL [1‘- — [2: tea -(-=.+~r+++(a)1’] Hum : L” t J. (a? _‘ .4— \ u \JIH, H- M.) "H “t 1—th " 7t Tb“ (b) [3] Show that y = sin(z) is a solution of the differential equation _ E: ‘ \SL ' cos(:n)y’” + sin(:z:)y = — 003(23). : an {n} (:tnfnfl 1- ahfilnhm \Im; —m{x) :: ~¢n1{131'§1\1‘{1) : ._ m arm-9&5 Q\?) GJthn-b/~ Continued on next page 03x’14f2008 15:0? IFAK eeannerlflmail.math.mcmaeter.ea + Bunnie Dusterlin 005E008 MATH 1AA3 Test 2 Name: Student No.: 3) a) [4] Find the power series expansion centered at 0 for f :5:12 dm. 3k 190 L m P _ *1 7L 1“ .w «new an a '2 “E ‘A\DLV\ (ml ("'5’ 1" 4L”. a J“; Z ‘3“ J Jung n‘ m\ A». = in gm 9a mic.) m val ; 2:135:90 JoC Continued on next page 03f14f2008 15:0? IFAK scannerlflmail.math.mcmaster.ca + Bunnie flusterlin UGBIUQS MATH lAAB Test 2 Name: Student No.: 4 12 Find 11 01's ri nere r a: = 1 A \‘H’V )[ ]a) t eTayl a‘e es ee t datOfe fl: (3_I)2. SELAHH QM: {rs—DJ fim : 1‘ —.~_ g as swam-s "I \ _ I - Q Wk ewruflbu 1c,“ ; -1 (w) M W0 - g “L ‘rwt.’ MM ‘v- Ive- ll 31' '1 . :— HV “H L ? m Lew—2.3 mm) H" 10 (a); “‘3: MM :1 LL... 3 3 m 2 1 lf 0k hd‘f‘“? . g 'm : Hyena) (3-36.) can 1“ (e a *- 3'5.‘ "Wm. M h. 1 Lbkflu‘".{ Mu! * otwlw'Hu-q “94‘ M 4-“ e w» m. .f m :-L—1)l*‘s)--~ (-1—th (H) W __ If“ H {H W _ (jflfi = put-3L. me (m (m 5 ° * 3M h m . \ H's] : :L: (“*Xffil 0L b) How many terms of the Taylor series of flat) do we need to compute so that the approx- imation to f(;r) is within 10'2 on [—1,1]? ' \Rntma ~‘— m“ , LAM n7, 4‘ "m 10” (am, QA-‘t I)“ 4-H —\ fwm : {maxi (m a We} K}; '— Cog—:qfifil J \ffifi‘hfl f WWWL‘W (mm! 6M 1M“ ll 3"“ \ Kh-rl‘ W-kIZ. fins \Rn‘1\\5(mm J --‘;13 1 37?; fl 1 Continued on next page 1 Vt": '- F\\J "1 h an”? 41 5 III.)2 ‘L °\ “:1. $15 1 in : 0-00? C1 o-m 03f14x’2008 15:0? IFAK soannerlflmail.math.memaster.ea + Bonnie flosterlin OWENS MATH 1AA3 Test 2 Name: Student No.: 5) a) [E] Find the power series solution to the differential equation 3;” — y : 0’ Mt \ALO “It i It: “3 7' — Z “(k-W C‘s“ J Fr:- L m1) “P1. cm “W” 2: W (“I- ‘X CH1 fi -' Z C111" : O __ E k 1.“ “7-1 «:0 ‘1 h C" Wufl‘q'x H m K Sinking +c\ “Q 'J-d .. .. I E (“firolmfl {Kalil Z CW“ ‘ 0 10-. Ca 2 Ca “1:. “Fania- nzo H:¢. Clo“, = L DR "a flea: Cl CMLXMQ _ C. H c. ca“ '- an‘ | CT. “Ht — (lwflu b) [j Find the solution 19(3) to the above differential equation which satisfies the initial conditions y(0) = 1, y’(0) = 1. Justify your answer. m " fl (1‘ Jr LW-tl fir!) fitu '{ X C hi “ l“-\ C i '2. t] z“ X) : arm :1. 't is \j ‘5' “1‘ (Rhys “n10 {ahfily Continued on next page 03:14:2003 15:0? IFAK scannerlflmail.math.mcmaster.ca + Bunnie flusterlin CICISKCICIE MATH 1AA3 Test 2 Name: Student No.: _____ ———-—__.____________ . 6) Consider the following differential equation: :1! E3” =(100—yxy —10). a) [31 On the axes given, sketch the solutions with initial values ( E9 ym) 100, 1 ( THE END ...
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