MATH 201 Exam 1, Fall 2004

Math 201 — Fall 2004-05 Calculus and Analytic Geometry III, sections 5-8 Quiz 1, October 28 — Duration: 1 hour — LAM ay GRADES (each problem is worth 12 points): AYVNG 1 4 SUSMALLING: NYO 1 2 3 4 |-5 6 TOTAL/72 YOUR NAME: YOUR AUB ID#: PLEASE CIRCLE YOUR SECTION: Section 5 Section 6 Section 7 Section 8 Recitation M 1 Recitation Tu 12:30 Recitation Tu 2 Recitation Tu 3:30 Professor Makdisi Mr. Khatchadourian Mr. Khatchadourian Mr. Khatchadourian 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. Each problem is worth 12 points. 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. No calculators, books, or notes allowed. Turn OFF and put away any cell phones. GOOD LUCK! wt cjafesse Malecisy FER ar REP ee i ERICAN UNIVEKSIT®, - [ize ARY pecause- Y acc oO” fL, Q pecec 2” S Abis is ow altertak Serres ak converges ly a\ter nox 8. ded. fs , act coc ry qos), S af GS claimed, Ru hous oh, oN Ge) 5s ews Geyveeces C Mis. direct Cupar ded " j Linkin o 2. (4 pts for each part, total 12 pts) Find the limit of each sequence below: 10” sinn a) lim b) lira cos™+(n?(0.1)") n—-0o ni NO inne —\o" a) |s n(<\ S A Buk a —9 ( expo gaeth c<< Gon al goth) y -0e \s-0-0 ) se boy as conlwich Macoter, Coane limi) a — cinoit meng as © LIBR ARY OF BEIRUT ~ S se Tayler Apangen aC) =X me asxa\- fe a C \+ Za) = Za ant AMERICAN GNTVERSIT LIBRARY OF BPIRUT ERICAN UNIVERSIT* LIBRARY OF BEIRUT a 3. (3 pts for each of a-d, total 12 pts) a, b, and c} Find ONLY the radius of convergence R for the following three series. Do NOT test the endpoints. (32 —2 VG = 2\r >> oa DS " 9X45) x n=0 d) Find the sum of the series (a) in terms of x. What is the interval of convergence? x ar| ar | 5) uur be ay Ss 2G -3y Lene Ant\} = (x “2) ) Nn Bat ° 3\.-%\ Mi rhe ruven Lit BKARY | OF RETELE \ ne 4. (6 pts for each part, total 12 pts) a) Find the second-ordér Taylor polynomial P2(z) of the function f(z) = ./Z, at the center a = 100. b) For 90 < x < 110, estimate the error |f(x) — P2(x)|. Your answer should have the form |f{x) — P2(x)| < a, where a is an explicit constant. You do not have to simplify your expression for a. a) EG) = ace = fF ((oc) =>\o . \ - CO = a a Vo =) a (\ee) = S ra > Be C°a = -\ yore => "(bey = Ne be hee CNG = 3x5 (oseh\ Sr rr b) (x- os Pe Go = Flos) RCW) Cavs) + EMC) “ = \|o 4 Lp Cx- loo) om | Cx -\00f- Cacke the) ) Zooe = 1s ale corre As whe PC lee #DX) = =lo-r by O* ~ Ler | Lb) \oe, ~\o < x-loe <= jo go |x < lo, ~ nm Mere. Ae \eons| is \ 6O \ = £ (<) (ax an lehoeon 2 <a \po avd x, ° so in partulrn \R w\ -|%o? ° [rescs ve S _ 3 3 = \ x \ (0) aS < acke Aoec = Wo “Otis 36x lyon Sf és Sl \ ~ =3 oo < cn” (ite) que \e ACL oPaS Ove A = _ oo Sy. 5 LL rege Yo Sx eld roe * Qwy??e “MSe cals O9 _) peeRiC AN UNIVERST LIBRARY OF BEIRUT 4 ERICA CNIVERSIT LIBK, RY 5. (6 pts each part, total 12 pts) oF beret a) Using power series, express the integral L= —_ tan~ as a certain alternating series. For full credit, the answer should be written using D notation. You can get nearly full credit for just writing out the first four (nonzero) terms in the series. b) Find (with justification, of course) a specific partial sum s, for which the error satisfies |s, — L| < 1071}. Note: in parts (a) and (b), you may use without proof the fact that your series satisfies the conditions of the alternating series estimation theorem. oo kel ei x herd ; Cyr 9 2 Sex = . okrl eat oh Qe et) aon \ ~ > Ss De eC") ") 6 =o Cert) C4EHD) b) we mk d& sondweh [betwen Sk + c-F CN wee * CZEV) (Ue eZ) ont Fey > Sy Leby ab The Mporion cathe the Re Reus Seong L Leben. Me See Mish ur Gn sorduich - G ©) ond ey — 6! Tt ° m \ per ©. Jie S tof Ne Tener a OO) ope \) TELL CeCe ae Ra Yo since \L-si\< See i~>\ = . (4 pts for part a, 8 pts for part b, total 12 pts) oo Qn a) Show that the series dX Taga b) Estimate the error that we obtain if we use the partial sum ss to approximate the value of the series. converges. ee — 2° ee Zo, covery menpore ult, “Yd 3% Nhe. geomsitic Se res Ss (2) (germeddc. wilh reZel) ‘ dase as 6 | S Linwany | = 2 5 en 2 | Ce ee : 13! 138 Te whic, be comes 1 \9 S? C, lorge nN Xe ecror is? <2: (%) Sma experertally Risk,