Quiz-1_Fall-2008-2009_Makdisi_Itani

Quiz-1_Fall-2008-2009_Makdisi_Itani - Math 201 — Fall...

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Unformatted text preview: Math 201 — Fall 2008—09 Calculus and Analytic Geometry III, sections 21—23 Quiz 1, November 4 — Duration: 1 hour GRADES: YOUR NAME: Q0£$kwxs & Sc\0¥°\5 YOUR AUB ID#: PLEASE CIRCLE YOUR SECTION: Section 21 Section 22 Section 23 Recitation M 11 Recitation M 9 Recitation M 1 Ms. Itani Professor Makdisi Ms. Itani INSTRUCTIONS: 1. Write your NAME and AUB ID number, and circle your SECTION above. 2. Solve the problems inside the booklet. Explain your steps precisely and clearly to ensure full credit. Partial solutions Will reCeive partial credit. 3. You may use the back of each page for scratchwork OR for solutions. There are three extra blank sheets at the end, for extra scratchwork or solutions. If you need to continue a solution on another page, INDICATE CLEARLY WHERE THE GRADER SHOULD CONTINUE READING. 4. Open book and notes. NO CALCULATORS ALLOWED. Turn OFF and put away any cell phones. GOOD LUCK! An overview of the exam problems. Take a minute to look at all the questions, THEN solve each problem on its corresponding page INSIDE the booklet. l. (4 pts for each part, total 8 pts) We consider the two series A=;(§>n, B=iln<n:l). n=1 a) Find the values 3N and m of the partial sums N n N 3N=Z(-§) , tN=Zln(n:L-1). 11:1 71:0 b) For each series A and B above, determine whether it converges 0r diverges; if it converges, find the value. 2. (7 pts for each part, total 14 pts) a) Find the second—order Taylor polynomial P2(a:) for the function f = \3/5 = 121/3 at the center a = 1. A b) We use P2 to approximate f for 0.7 g a: g 1.3. Use Taylor’s theorem to give a good estimate for the error. Please do not try to simplify your answer! 3. (4 pts for each part, total 20 pts) Determine whether each of the following series converges or diverges. Remember to justify your answer. 00 n! 0° 3+cos4n °° n77,31an a) 2—71 b) E T c) EH) 2.. n=0 n21 11:1 00 n+(lnn)2 OO 1 n n n=2 n=1 4. (5 pts for each part, total 15 pts) a) Find the first four nonzero terms of the Taylor series for 6"” and sin(10x2) centered at a = 0 (i.e., of the Maclaurin series). Use any method you like. b) Compute the limit lim 6‘3 — 1 +33 z—rO sin(10x2) ' 0.1 c) Find an approximation to L = / sin(10:c2) da: with an error less than 10‘9. (Use 0 the alternating series estimation theorem; do not bother to check the hypotheses.) 00 —l)”4”a:" 5. 5 ts for each art, total 15 ts Consider the ower series :3 = (—— ( p p p ) p f ( ) 1;; fl + 2 a) Find the open interval of convergence I °. b) Study the convergence of the series at each endpoint: does the series converge absolutely, converge conditionally, or diverge? 10 _1 11.471. 71. e) Let 310(22) = 7;) (“BE—+29: be the 10th partial sum. Estimate the error in using 510 to approximate f when so = —0.1. This means, give a good estimate for the error )slo(——0.1) —— f(—0.1)[. Caution: the series is not alternating when r = —0.1, so do NOT try to use the alternating series estimation theorem” ii 1. (4 pts for each part, total 8 pts) We consider the two series 00 17, OO 2 n + l A = — B = 1 . g3) ’ “( n ) 3.) Find the values SN and m of the partial sums N 5N=Z(§)n, tN=i1n<n:1). n=0 b) For each series A and B above, determine whether it converges or diverges; if it converges, find the value. n) 5“ z kis+gy+ = \ ‘ \%)L+_,+K1§)N = [Av-WU gnu cw‘ \Hfi-r‘y- qurN _ ~~\ \——r \~r {2N -. \A G) 4— “(3,1) \— M‘gb». hwy?) —~ N \ —\ \ = + N “95“ hr“; _ Jr) A \n ‘n F- 3 II I 2 l ,v- J ’ 9 A: \im 5” = :3 kcnvk Lfiyfl‘L—so dd."- “RF My (“In rnzc .Cu \(‘\<l) h—D‘ ’“Aifi “WSW “(£3)” Gunm‘u ' B: \KM'LN‘; \ZM “Ch/H) é“—‘ N\_c":h\' (‘6 N—d 4.”, know“) 4‘ a“) s- E “(Ag-é.) heels: H (anew-bun 96-“. Ed.“ (DJ-e M “WW culvvwixm Lax—I .r meni— Cr \WBL A, \~(“%‘ 2 0%.) 2. (7 pts for each part, total 14 pts) 3.) Find the second-order Taylor polynomial P2(:c) for the function f (:13) = {75 = 331/3 at the center a = l. b) We use P2 (ac) to approximate f (3:) for 0.7 5 :c S 1.3. Use Taylor’s theorem to give a good estimate for the error. Please do not try to simplify your answer! °~) 3. «E._>""Cx «mm D XVJ ‘ l 93 x-zr35 \,_5 Z "5“ x ’3 _?_fl‘ in i) mm (3604259 +611. wa\kx\=\¥@>vP1®l u an em um R1. = ‘9‘“(c’) c0” 5-” c, LAVJQA I X 3‘ ' 59 R’L : “9 ~W3 K“| . I 26C (Elijlémqlug buka ‘\B “k. we, M. 0v [and £6” <. «s lav.th \quX ""9 \C“\é P‘“l s (3 3 :3 Q” (C- < l “l5 \ c x I!) (SWH‘FKJ ., m cm.» < = ‘5‘") ' “11:??— S M 3:.) C us iguana? Gw- Quixth ® -==> (er-(5W3 2 [93 4/3 :. m €0.15ku W\\M.. '9' £8,315 (OrD‘e/l) mm. 5;“ (D M LugJ-l- QuscLLM. “\K .c \m MN. 6i sacs; Ufigr leakmul QFKEZX is is 0.3 (N aka 0N ADMQHQ\ hut 4"er i5§°.b°‘+3) 3. (4 pts for each part, total 20 pts) Determine whether each of the following series converges or diverges. Remember to justify your answer. oo 00 0° 3 80:02“!— b);3+;:s4n C)Zl(_1)nn21:n °° n+(lnn)2 00 l n n d); 723—1 e);(cos[<l—E>]> [cue-mun mm.» mu. rear) L c- 4. - D 5 at V“ ’5. 3+Q<.s5.\>,3_‘:v-L [Se «A: lthhx 2 Fl? =Q~I9.% n a”) 2% 3'“! es ' 9" Cir; wmwr \ 5: “Mai; : r! E r Q. N»: n=l n c 3 3 ) Herc- “ {nu/:“h -n=nl‘ <c-QCLOD" UM g” kg: ’\ [.Jlahuh (cL QAT3PH“ 0:“ R41 3mm» (cc “vol Sb CC, 1 n 3‘“ “'3 ' \%\ = :73.“ s (mm~ (my 27 - f «rum out“ “‘* raw-c Ma 2 C93“ h mag-4' “REC: mmm: gr: L13} G (OMB luu’k NJ “EARA “Hula, k) I.an abs-Net. C» (AQfl) Unfit»: A) as = “+U“)\ L? L t “'3‘ ) h h} F\ . a. n {I : ’Rx . E‘TH'GL USA “A; vial“ Cntvgwfi L z n \\ on NH “ML Gil: \#((\AA) A) \JMK‘EiLfi Ma. h K -> . t- t; ("'3 mm“ <‘ e-‘1 Wt «MLM- L: L L n [A a (q ac" I 1;.\ I ‘7‘,“ 4 L\ “kMfla “m ) SM ‘15 “NJ-“mhwuxwwifir (Q’Scfhs) raw) a ,‘L ‘ “3-! “3- «3,1 ' “L ’C" \W'XQ n1 4. (5 pts for each part, total 15 pts) a) Find the first four nonzero terms of the Taylor series for e—r and sin(10x2) centered at a = 0 (ie, of the Maclaurin series). Use any method you like. b) Compute the limit 11m fl. 95—.0 sin(10x2) 0.1 c) Find an approximation to L = / sin(10x2) d3: with an error less than 10—9. (Use 0 the alternating series estimation theorem; do not bother to check the hypotheses.) Q) m (5530:» hark-X '3 Hjumlu’fion'. “luau: ° 'L x 1 '> y‘- e = \‘VU +9 k0? *-'- c = >. —x -~ ?1_ C" 6* ——__,>su \ 7". ‘5‘ l. (u x) (___1)n4nwn 00 5. (5 pts for each part, total 15 pts) Consider the power series f = Z 12.20 a) Find the open interval of convergence I O. b) Study the convergence of the series at each endpoint: does the series converge absolutely, converge conditionally, or diverge? 10 (_l)n4nxn C) 310(33) —- 7;) W 310 to approximate f when a: = —0.1. This means, give a good estimate for the error I510(—0.1) — f(—0.1)|. Caution: the series is not alternating when ac = —0.1, so do NOT try to use the alternating series estimation theorem!! 9 u. m rake ks» (m M— M Mics “‘50: be the 10th partial sum. Estimate the error in using \GM\ 2 q“””"““_ ‘17 +1 _ ‘tbq. (in. g. A ~ A " ‘17: +7. +“ Na {KI—T +2. ’5‘ ._ 7. .. . l+ ‘ffi l ‘l‘ o z 09‘; .ch)=fi “i *r» *° “fizz”; «as m xer°a-a {0 «=9 “Hag N <t¢ L j EI'AQMVJ-D X;>‘,+ “4 mes Lao... git—WPM)“ °° H)“ 011‘, has Gm. 2 a f; “L 5 ‘J all-cruth 2043A“ HM“ UA~‘ __ 2° @anm‘fiwdmwkz t“ “+1.. l ®l~l- m: l‘ ‘ —O JMNA n“ U“ Jam” '\—-*.¢ ML“: m1 ~ m aim“ ‘°"“" S, ’\¥®mfi~¥ Kzlif ~ “€qu Alma A we stow ‘\ n t n # Z (.0 ‘r ( I" = L l G +1. Q “WWOK \n‘ ink (Lo-{4‘ ‘ (F0, 4—4.3“ Show gr I 6+L<6+fi -: Z‘I‘n ‘ he .L— '70:.) “(XV-“Mr \- \ 5 V: I" L (‘73 I 4 q X) 4.“ “LL SQN‘QS Unme.‘ lg Cnn‘lu‘kl .N 9.7 / Blank sheet. Ewe) cm casD-afl ,,.__ use. m fik—s’ca “Wav— \c., \-\h\"\\ ax,» a gm M Man» 5, WA _.....A. \cu 6‘ mg}: M 9:0». (.5 a ‘ any...“ ok U=..:\ W a“ + L-cr C. x: .\ f- M e Z 05:. =(= in s. \ Q: ‘cfliw < K Blank sheet. 2“ g C‘nsx‘txueci __.__A_\§ (\4' ' 7_ (RM ’Y‘M Mwa «\M is \ \ CQLMR q‘or hm ‘5 - '5 emu: (OH)“(\ N”: "\'.9.7~l° (“m )g,‘ Mo _ ‘1 (‘fii +7) “‘°"‘ S~3 ...
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This note was uploaded on 03/01/2010 for the course MATH 201 taught by Professor Variousteachers during the Spring '10 term at American University of Beirut.

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Quiz-1_Fall-2008-2009_Makdisi_Itani - Math 201 — Fall...

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