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Test 3 Spring 2008 Answers - Section Color EF Math 122 Test...

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Unformatted text preview: Section Color: EF: Math 122 Test 3 April 8, 2008 Name k E Y “\ICDO‘li-DWMH Style Points l Total Directions: 1. No books, notes or thinking that Robots In Case is funny (although I don’t need to tell you that). You may use a calculator to do routine arithmetic computations. You may not use your calculator to store notes or formulas. You may not share a calculator with anyone. 2. You should show your work, and explain how you arrived at your an- swers. A correct answer with no work shown (except on problems which are completely trivial) will receive no credit. If you are not sure whether you have written enough, please ask. 3. You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or you will be penalized. 4. You may leave as soon as you are finished, but once you leave the exam, you may not make any changes to your exam. U? 1. (10 points) (a) Compute the limit of the following sequence: 00 n3 _ 5 sin n 1/3 877,3 + 7n + 3 cos n n: 1 W 3 Nﬂz’». 63m} N “L H_ i gNa-rﬁnarzmsn % «l N~=>OO (b) A ball is dropped from a height of 6 feet and begins bouncing. The height of each bounce is three—fourths the height of the pre— vious bounce. Find the total vertical distance traveled by the ball. (Hint: consider separately the distance traveled up and the distance traveled down. 00 3 u (0 -.-.- §DQ(;\ -.- “-3; 2H 00 q MD A rt»: .2 II ii 83 2. (20 points) For each of the following series, determine if it converges or diverges. For each test you use, you must name the test, perform the test, and state the conclusion you reached from that test. * en + 3n, 0C N (aw-r 3 no.1“ f ﬁg)” “QM 9H?“ {L N~=aoo -———;—-—--~ .4 Smog f (EFF) CON‘L‘ 0° 1 b t < ) ”22:2 nxd/lnn 00 0:9,.“ x E X \ MAO Ty'vreeam “F95"w 5 on 0‘0””de g “) ... 1 X MK ’2 '2. 00 \$60 “=7 13.55 -= 23—: (.mei’5 \a: DN’E. (ON) x5 3. DW gown \/ DEC. V (C) ZNW” 71:1 90 ‘ u \C 2 g ”L. < \ 0% a " 26%;“) “WW“ ‘5- : ”We N:\ K633 Vb‘ COM“ 00 3+cosn n <d> Z< l l) 1121 ’ N COSN N r: L (z r” ‘ (3:4 3 O \ ”VEST “\$00 3. (20 points) Determine if the following series converge absolutely, con~ verge conditionally, or diverge. For each test you use) you must name the test, perform the test, and state the conclusion you reached from that test. “0 1 (a) (“HRH n 2 1+ (f0) N-‘V’R‘T‘EQM JQNW‘ \ vs :1 4’ #0 N900 W (59-53 71:2 1 l “L com (P—TE‘ST Come “we; 4 ”Z” N‘ TEST (NZ‘H) N 00 1 __ n+1 (C) ”25 1) lnﬁn) +1 \ A l i L C. \ ,_.. .QmN i M14 N400 \ ‘ l QnN'“ l \ M = O V” 5;”? 4 New DnNHOO I NH +3 71, n. (CUE; 1)1 3-5-~-(2n-1) jam N900 >4 (5+3 ﬂ=O~~3 11. (15 points) Consider the power series: ‘30 (LU — a)" :1 n 3" n: (a) Where is the power series centered? Xuq QR qu (b) Find the radius of convergence. Qk—osﬁtﬁﬁﬁ 1.. 13:0 \4\ \x-—O\\<3 (W) a“ “(x—o3“ (c) Find the interval of convergence. [Hintz Check the endpoints.] '4 m 0% (OWZVOQ _, f “L. Dnl. " N N '5'é ,. N=\ “' N ,3 00 g 00 - 2 (0‘5 9‘) 1: g L9. Com! com; N 5““ N) N5 9:) 5. (15 points) The Maclaurin series for ﬁx) 2 111(1 +5132) is 111(1 + (1:2) 2 :1;2 m 1: CC :6 (CL) Write the Maciaurin series for LCZf(—{L‘) . ’2— x“ x9 __ x8 \$04) =3 ”4 "‘ ~— “‘ -~ ..... A 1 3 '1 g, e 10 2? , x 41m)»:- x“~—L +22... “2:.” :2, 2 H (b) Wiite the Madaumi bLTle fol m ’3 5 1F / G x '3’}: :" 'F‘(x—) ‘1': ax - H...:_ “'4' m3 "" gr‘vi’“ ' ’ ‘ ‘ \"f‘X (c) Find P4(x) the fourth degree Mackiurin polynomial for ﬁx). L} ’l K PH (’0 ‘“ X ‘1'” (d) Use 134(55) to approximate f(1) (e) Find a bound on the error if 134(23) is used to approximate f (1) Without using 111(2). mﬁOR \e No mom; mum MESKV T921“ \ BZRGK 4 ~—-— 3 6. (5 points) Below is the graph of y : f(:E) 4~ /~-\ / / ,// // 3 / ,/ // /2/ ‘\\\~\\m _,_,.1r/ /’ i / ,/ 1‘ f / / l i /i i x 1 i i J l I I I i | | l x Ivﬁihllinhl.” —O.5 / - 0.5 L0 1.5 2.0 Which of the following could be the second degree Taylor polynomial for f(:c) at a =1. (e) 172(33) = 2 ~ (:5 — 1) + 3(3; ~ 1)2 7. (10 points) Indicate Whether the following statements are true or false by circling the appropriate letter. A statement which is sometimes true and sometimes false should be marked false. 8) “31111320 an diverges and ”lingC b” cllverges then “131010 (aw/+51”) T ® diverges. 0‘9 a If ambn > 0 and 2 an converges, and Hinge}; = 00, b) ”:1 “ ® F ()0 then 2 bn converges. n21 0CX OO - . . ‘ . If the series 5 an diverges and the series E bn diverges, C) n=l 77:1 T (9 ()0 then the series Em” + bn) diverges. n: 1 d) If lirn an : 0 then 2 an converges. T ® n—>oo 71:1 00 If an > O, and Z (171 converges, then e) 00 ”=1 ® F E 1 sin anl converges. 72:1 ...
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