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sample-final - MAT 21B-A Spring 2007 Prof Opmeer Thursday...

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Unformatted text preview: MAT 21B-A, Spring 2007, Prof. Opmeer Thursday June 14, 2007. Final NAME (print in CAPITAL LETTERS, last name first): ____ ID#: ____________________________ ._ Instructions: 0 Read each question carefully and answer in the space provided. If the space provided is not enough, please continue on the opposite (blank) side and CLEARLY INDICATE this. 0 YOU MUST GIVE CLEAR AND REASONABLY COMPLETE ANSWERS TO RE- CEIVE FULL CREDIT. Answers like ’yes’ or ’no’ or ’2’ will be awarded no credit. Proper notation and (mathematical) readability of your answers play a role in deter- mining credit. . Calculators, books, notes or similar things are not allowed. 0 The numbers between brackets refer to the number of points (out of 100) for that exercise. 1.[30] Compute the following indefinite integrals. Indicate the substitution that you make ( identity that you use (if you use one). (a)[6] f(a: + 1)2e‘” d1: if you make one) and state the trigonometric (W61 f m d“ __L__ e H + L Stu at 1'2, x" _ 3* $L¥~Z3 x = a“ ‘ . 29n\x-2\-’ Q“\‘”‘\]*C' £3 \ : ~93 '?>~— ~\‘ \ ~»/“:‘“‘ '6‘ 293:. (t 2' y) 1.[continued] (c)[6] f sin3 :1: cos‘ x dx ‘ : s ngkx (:0qu %“x dx u co x oh; 2 .—<a,\(\>< dx ’ -g ( \ycoSkfi) (95“ K 69¢»qu (d)[6] f “#353 da: : LS 9»! dx U: 121M 2 2 \ K (Lu 320R " _‘_\ udu 1 3 \ x): *C 7 7: MKB:*§€Q29 -\ l.[continued] )(h = Cast 8 (e)[6} f (£11747! d3 (ix: ch, 9 MM} d9 I gegé "an 9 A Er ” S x?“ wL‘fieceernB d9 (fay/$9 2.[12] Evaluate the following improper integrals. (a) [6] ff” e“ d2: :Qjm Sb 4 $9,”de ‘D'Wo \8 dx '3 “éK-kg : LPN ~éx\b (b)[5] fo°° (Viv; dx 3.[12] Consider the region bounded by the x-axis, the graph y = 3:1:4 and the lines a: = —l and x = 1. Set-up, but do not evaluate, integrals which represent the volume of the solid formed by revolving this region about (a)[3] the z-axis using the disc method. flog“ %o\id OéUv‘de \M ~. *wv‘h (b)[3] the y-axis using the shell method“ gag: W“ ' \Io\‘ 93V( ' ‘fi‘ *RWSS (d)[3] the line y = 3 using the disc method. AC3 NM» Lynda x=l5‘\ 4.[12] Consider the region bounded by the graph y = m2 + 1 and the line y = 5. Set-up, but do not evaluate, integrals which represent the volume of the solid formed by revolving this region about (a) [3] the y-axis using the disc method. (Ac/g N 60MB Ug‘x‘mCWC (b)[3] the x-axis using the shell method. 9 $‘- €06“ _ \lg n Qw- ‘(V‘K’RICXJW‘bS (c)[3] the line y = 5 using the disc method. gbhcl UAVM‘E‘ \lt\ «. “(w 2h * (d)[3] the line y = 5 using the shell method. HCS; énell V0\ " va-h flinch?“ 5.[6] Set-up, but do not evaluate, an integral which gives the length of the following curve. 27(t) = t3 - 6t”. W) = t3 + 6t”, 0 g t g 1. 3.x; 9>t-\2£) guw’c L: 8: it] 6.[6] Set-up, but do not evaluate, an integral which gives the area of the surface generated by revolving the curve y = Zfi, 1 S x 5 2 about the x-axis. 69‘ : 31¢. cadenfifa 7H2] (XJSMH‘) (a)[6] Set-up the area between the m—axis and the graph of the function 2:2 - 4x — 5 over the interval [—7, 7] in terms of integrals. Do not evaluate. Do not leave absolute value signs in your final answer. - S ‘7 A=S \(quxasB 6x + S \©—~ (xtapsfi Ax + 85(X1—4x's\ «ix -1 " (b) [6] Set-up the area of the region enclosed by the curves y = 23mm and y = sin 22: with 0 S a: _<_ 27r in terms of integrals. Do not evaluate. Do not leave absolute value signs in your final answer. :r 28km aemx = emZx Qsinx v <93an ((381 ngxm‘n {133M 2 0 cl («aim @on I \\ :« o CoSX—\ " O afix7\ X:- O‘TIQW x; 013'“ <~3\l\x 1 Q , 8.[10] (a)[5] Find f(4) if fox2 f(t)dt = zcos ms. at £03) ‘ 61K : Ccmw. r 'mugimrx z : C(3an r “mint Y 5% (70 ._..__._________.._9x 4‘00: Rf) =r COSQT witlg‘mlfi -W Z .2 - 9- ?Locb in K22 (b) [5] Use the definition of the natural logarithm as an integral to show that for all a and b with b > a > 0 the following holds: lnb—lna 1 “"_“—<-. 1< b b—a a X 6:9?" ,QnX : & €5th QCK) = /QX\X Q “<7 = -;-; I QemW W ‘ (‘n {1‘ ] QM? Commwds ; so)”, 530034013 mez 91”” dxgoemmw (M at - : J - 29» £00) ~§(a~3 T Q! %( \o a byé‘ b y (a), fl "0‘ Sm'fi Q {A (ago) 9100—- QJ‘O‘ r ('(c) I (9m some c in (0&7) / b’a QYXX : ‘>\? p =3} Q)n‘0~ kwfl 4 < i I 0~ ...
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sample-final - MAT 21B-A Spring 2007 Prof Opmeer Thursday...

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