MATH 201 Exam 1 with Solution, Fall 2011

Math 201 - Fall 2011- 12 Calculus and Analytic Geometry 111, all sections · , Quiz 1 , October 22 - Duration: 90 minutes GRADES, GRADE TOTAL 7 (/10) 6 (/12) 5 (/14) 4 (/14) 3 (/17) 2 (/15) 1 (/18) I YOUR AUB ID#, PLEASE CIRCLE YOUR SECTION, Section 1 MWF 3, Kobeissi Recitation F 11 Section 5 MWF 10, Abi-Khuzam Recitation T 11 Section 9 MvVF 11, Brock Recitation T 12:30 Section 13 MWF 2, Nahlus Recitation Th 11 Section 17 MWF 8, Makdisi Recitation F 2 Section 21 M\VF I, Raji Recitation M 8 Section 24 M1"'-F 10, Egeileh Recitation F 11 INSTRUCTIONS, Section 2 M\VF 3, Kobeissi Recitation F 5 Section 6 MWF 10, Abi-Khuzam Recitation T 3:30 Section 10 MWF 11, Brock Recitation T 2 Section 14 M'YVF 2, Nahlus Recitation Th 3:30 Section 18 MWF 8, Makdisi Recitation Th 8 Section 22 MWF 1, R.ji Recitation M 9 Section 25 MWF 10, Egeileh Recitation F 2 Section 3 MvVF 3, Kobeissi Recitation F 4 Section 7 M\\lF 10, Ahi-Khuzam Recitation T 5 Section 11 MWF I I, Brock Recitation T 11 Section 15 MWF 2. Nahlus Recitation Th 8 Section 19 MWF 8, Makdisi Recitation Th 2 Section 23 MWF 1, Raji Recitation M 4 Section 26 MWF 10, Egeileh Reeitatioo F 3 Section 4 M\VF 3, I<obeissi Recitation F 10 Section 8 MWF 10, Abi-Khuzam Recitation T 2 Section 12 M\\lF 11, Brock Recitation T 3:30 Section 16 MWF 2, Nahlus Recitation Th 5 Section 20 MWF 8, Mokdisi Recitation Th 3:30 1. Write your NAME and AUB ID number , and circle your SECTION above. 2. Solve the problems inside t his w hite booklet. Expla in your steps precisely and clearly to ensure Full credit. Partial solutions wi ll receive part ial credit. 3. You may also use the back of a ny page for solu tions. Ir you need to continue a solu­ tion on another page, INDICATE C LEARLY WHERE THE GRADER SHOULD CONTINUE READING. 4. Closed book and notes. NO CALCULATORS ALLOWED. Turn OFF and put away any cell phones. G OOD LUCK! 1. (6 pts each part, 18 pts total) Fo I r find , 'With justification the limit I 01'1~ac 1 ~ t.he following sequences, exist. " or e se exp aln bnefiy why the limit. does not (a) n lO 2 9 lim ( - n)" ,,~oo n10 + n 9 (i- ~ ) ' e _2 ---;; (I f-+t (b) I hv>- '""' ~.~ :7" frr 3-;;~ ~ . =(" f- ,)I:: T3wf 3 "" ' J />'-> -, ,,-:::/(;J( _ (c) lim cosml' "-~ c=n7f ~ (:: i)" S 6 c.r ;2"" = I wt... l.e- w-? ~n -r I} i< ,,- f -rk "'-jI.-(M<.V- ~ ~ W"W~ 2. ~5 pts each. part, 15 pts total) For each of the series in (a) and (b), determme whe t.he ~ It. conver~es or diverges. for the series in part (c), find all val(~ of x for which the selles is convergent.. J ustify your answers. f _ L :z: A­ =w1APt"-> . I .J"N - d tll/n.u. ( (b) L(n'l" - 1) n =l L~" 1 k _ e.." - I - -'-.$..'"!- ~ ( ~).p" . " -,., lilt .., V\ - I We- '-<r7Wjf (vy,- ~. ~ w Ifi 2..g,.,.!!. . 1vm Jh,rr - 0 " - / ,,~ 1>1 w , L (2:)' (x - 3)" ~(,.;"~( J ~ ~, (> -J)" n_)", <"'1."'), 1>_11 . ~nl-l).('2")1 _" " ('2.").' ('4>/7) (~"r » C<fYJV~ wIo~, ~ ,().i1 <4')W~ 1v>­ ~ oJ ~I < .,. 3. (10 pts part (a), 7 pts part (b), 17 pts to tal) (a) Compute the nth. partial sum S." of the fonowing series, and use it to find, according to the value of c, the sum of the series. I L ",.;.1 -+ I C-t-I c.. i Jt" / _ o ) t<-nA ~'5" = J ~ O/C < ,) ~ "'--1- ( Of l c+f ~ )'>.-of ... ~. c.-~f ! =' I; ~ ~<;._-o, Jt c ;. ' ~ / 11--'1 "" __ 0 ­ ! ~ e-n,1 -- au c r-( C,4/ c > I} 4 1- ~ (b) Find the limit of the following sequence: - - 4. (14 pts) Lel. {{x) -- x2 _ 7x _ . I 8 ' Fmd the Taylor series expansion of f about a = 3, (; ., " centered at a = 3) , and use it to find j(n)(3) . 1/ - I X ):; --;-~--- ;x- 7x - 9 ( x-f) (?<'-f-I ) )/t; - I -=- -~? - ­ 0<-,);- 4 1-(!!..::2) 5 """'" L L ..l- _ ( -iJ~ . 1-] (, -3 )'" - .L-['tS '5~ 3{, 't" - r1 ;:. C) ..; V~J ~ tW I X_s/.< 4 ,., ("! (dL.~ :/- (J ) I I -~.-- '1~ -- )t. '15 5" rJ ' ~~ fir>vJ en) ( _I ) " )?) ( (-~ - q . ifn/"t - q. 5 "'I'I f 0) -­ ,,/t~W ~k;t. ~ 5. (7 pLS each part , 14 pta total) (0) Express the follo\\'ing funcLion as a Maclaurin series f(x) = 1" I- Mil dt x> O. , t ' .... "''' ~ " '" 1- .L. H) ~ ­ - .L (- I) T_ 1- =It ­ - ('11'1)1 17 =- 1 (£0) I • 1- w;fi- "" 1_J)"rl t"_1 - L t - t1 ~( (Ro) I I IY' ) .(b) For I = 0.1, find a specific pru-tial sum SA' of the series in part (8) fo whIch 11(0.1) _ sN I < 1O - ~. r 0t9 i').,r.J ."Yl 1 -ffo ./):= L (-I) _ ( 0. ) nco ( (1Jn)! -r> /lvV;"; VI ",~f/Ct.j u4tYlfla'~J ~/ "",' fh 1.0>"", ~1 m ",0~.r~ 1>-1 ~ ~J h r t'? )?~or . $ " ' j yY't IY><­ ~ ~ Wvu w-... J) .J-k ~ • . ' . 5 ('f :: I ..L. [0.1) /). I _ -'--1•. 0 1 1 1- Iffv., )_ )rJ I z: + { IJ' (I >-= ~ .(!.-.) S<:--'---xr.!..-f:.L. Ii; c_ 6. 3 )/60 /0 JI M" (/0 6> ft...c ,-- - rOT 6. (6 pts each part, total 12 pts) [' 1 (a) Let f(x) = Jo 1 + £3 dt. Find the Taylor series generated by f at a = O. l ~ p~ t~ II ~ (;):E -r .e:: I c~ __ .. ~ ... 0.:; '\013"'- I _ _ 2.J·,)(i) - Z.r· I! -t ­ / -f(3 - (/;:. / .I (b) Using the identity 1 ( l )n+ 1 n+ ' _=l _c+cl_ ... +{_l)nc"+ - c ) +c l +c prove tha.t the seri€S of part (a) converges to f(r) fur 0 5 x $ 1. ?' 11 l it 1_ 1: .;- -L _ ,_. +-(_1) -I: 1- ) -~-- = If -[ ) I /V/- "j'YA-ClMj !J.v~ o~XJ ,,~A~~ r ?< .... J~l+, ",.' 3n;-3 y (-I) C ,,{f I- ~. ,) J< -f ."Lfj.) = j -.!..-- ~ '" ',X ­ i ., nl'/ / 1(- t 1 o '1 " ift J h.f/.v. ;nr d "" f Y t J ft-f3 0# L /"t ] n",J;It- -:x / / f..{1/)</') f - I(-{ J - - '" 3",, 4- ?nA­ . - . ~ 1-t;;; Md ~ S J<.5{ ~" IZ-" /"J' / ::0 " n -> '" /'f'vl taJ ,;.m-,,-vCf' ;. ~ / fr " ~,A ~ / J!)-t->-t c- & ~ ') . , 7. (5 pts each part, 10 pts totaJ) (a) If 0 < an < 1, and Z::::l an converges, find all values of p for whi ch the series (b) If a n > 0, 2:':=<1 un diverges, and S" ::::: 2::=>'1Ok, prove that the series ~ L s? converges. n~ '2 1'\ n - l "'" a f­ leA T;, :;: :;1- SA-St. - I / - D - - - 5, 51 - - / "" . 0JWV~ trJuI A>5~ 2... Cl" '" . I 'Y 5,.. >"'_1 ~ , 5,