Test 2 Solutions

# 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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