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Unformatted text preview: Name 50w? 0N5 March 18, 2008 Math 133 — Hourly Exam 3A Do all problems (Total = 100 points). Show all work. Calculators are not allowed. (1) (a) (6 points) Write out the partial fractions decomposition of the following expression
but DO NOT SOLVE for the coefﬁcients A, B, . . . _$:1_. : A * 3 + c + ”D
(90 — 3)2(9L“2 — 1) x— 2; (>03? 'x\ XH.
3“”
‘5‘;ng Gk}
(x “\(xu)
2 __
(b) (16 points) Use partial fractions to ﬁnd I = / u—Zﬂf day. Edna“ 7' “(76+Ll) MA Xl4u\ lfrcévcﬁhlt. Su‘nu. bL'LlQG : O‘lL<O Wﬁ C 32%“ X " 8 _ A B¥+C ,— pAi‘lfﬂi) 4' (By“v x
X3+va ' 7? * X‘H . "T, XNHX
(Lamas ¢wanh
M3\—(,\\ “W’t3\W5 ‘ 2x2“? X'8 = (A*5) X1 3r Cx 4 ‘lA
Mater 045 1» 'Ansal
% Czl So
HA=’8
bl ”1)ng \ ‘ Ukr X71 L‘
‘0“ I = x l WW A“ 30‘3”" \§ ama,‘ ax. ’ dm
: “Q. QMl’Xl + 8 x31 (1* ‘ X+H .L . .‘ 25
'“ *QQMM +19% * lmexLL) " \A ~= :1 QMM 1» l J'Mx‘vﬂ '4 1; maizvdg *0 OR =\ (xiii: “ 3‘: main. (EA * C, \‘ 0° d
(2) (10 points) Evaluate the improper integral I = /0 ﬂ ”Do "Be. {AA€/%V\\\‘C. imie~ra\ '. Lei u=l+xL du=élxebsc iku=xé2x
"x44 _ i‘c'iu _; .4ch =—1U{‘ __~\ =—\
(\m‘)" ‘ 3‘ UL” ‘ 9 0‘ ‘9' u 94m?) (3) Determine whether the following integrals converge or diverge. DO NOT EVALUATE.
Justify your answers using the Comparison Test (CT) and Limit Comparision Test (LCT). a : °° dx _ ‘ . _ _\ __\_
()(8pts) I /1 W Csmvam, "FE/J: m Lm‘e‘ glﬂm—X “ pm 2 ‘ ._'S_{L—’_«“x‘3_
° 3:2: 3;?) 2 ET?” \ixzuﬂ % ._ Ex; Mint) ‘ Erm<xzilx "\ (4) (14 points) Solve the differential equation \/_dy d—— — y2 with the initial condition y(0)=
:5 5}}?ka VCuchbkL ‘. '35— da = (11—; d’X 1. :9
3.393%th ivCA’ial Cencl‘s‘lﬁ‘sw ‘. X=O) ¢6=l ‘4) “l = 25+C” Q i ‘\ 4“: ‘
501% (w 25‘ ”A 21—; \ (5) (8 points each) Calculate the limit of the following sequences. Show all steps clearly. (a)7lLIlgo[ln(n+3)— —lnn] : [MM Q (Liz) “>00 ﬂVxVL (Lﬂ\ ) ‘—
(b) lim V2n—1 = 50AM (9“‘3/n : LN“ elimULnJ) Y\
n—>oo uD “300
_ [W Quit—1)
. 6 new 00
a 2‘43” QAAliX“) g 456% 5 sin n (c) If an 2 n2 What is nil—n; an? 5“an “i S $a‘n n 5 \
, _\ \
bat Mug 3;; 5 50m“? 5 T“. g, a“ h.
i \ ‘ w \
Bah” Lle *x : O CMAA lUV‘A 4:. 3 D So LUV“ 1% : 0
“W h “Q00 h ) Y\ \>% "\‘kc Sanéwi‘c/k meta“, (6) (8 pts) (a) Does the inﬁnite geometric series 3+1+1+1+
4 4 12 36 converge or diverge? If it converges, What is the sum? Justify your responses.
1"; i , ~  3‘ '—
iS V) Us SaM‘L, sari cs Loriﬁ A ' A MA T: 5
Sinuz. i‘r\ 4 \ ‘Hnis Series 0 Convaées . We Show 3% A 39"!” :: —§9~_;:v :é;:€L>
\r: w Ms L: /3 tr:— 8 (b) (8 pts) Write the repeating decimal 0.37 as a geometric series. Then write the sum as a
fraction. 0.37 ‘3 .57 57 57 “1 23.3 4 7307; 4 7;}: Jr" . . . _\_
W?) V: C\ aﬁom'i'ﬁc, aria; \m‘i‘e‘ A: 3%00 ““9“ V ‘1 loo c ”(ng '— A 37 ‘ .31. 23,29.
037 .: 7:;=7;‘1—};u I00 33 *' Q Ii ...
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