Math 1B - Spring 2003 - Reshetikhin - Final

Math 1B - Spring 2003 - Reshetikhin - Final - Math 1B Final...

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Unformatted text preview: Math 1B, Final Exmnination Section 1, 11—125mm, N.Reshetikhin, May 19, 2003 Student's Name: TA’s name: Student’s i.d. number: 1.10 puts Evaluate the integral 1 / 33v 1 + Igdm ‘ o 09/23/2003 15:46 FAX 510 642 9454 .002 2.10 puts Evaluate the integral [:62 111(1 + 1:)dx 09/23/2003 15:46 FAX 510 642 9454 .003 3.15 pnts Indicate which of the following statements are true and which are false. Do not show your work. 2: 0° 1 °° sin2 . 1. f 2 d3: converges by comparison test w1th / Edam 1 55 1 m sin2 °° 1 2. f xix diverges by comparison test with / —dz. 1 5'3 1 x 2 dm , . . 3, ./0‘ main: IS a convergent unproper integral. m 1 4. f m—Eda: is a divergent improper integral. 0 00 8—1: 5. / air is a. convergent improper integral. 0 J57; 09/23/2003 15:47 FAX 510 642 9454 .004 4.15 puts Find the radius and the interval of convergence of the power series 2 ln(n) as" 09/23/2003 15:47 FAX 510 642 9454 5.15 puts State Whether the following series is absolutely convergent, condi— tionally convergent, or divergent. Do not show your work. m005 09/23/2003 15:47 FAX 510 642 9454 .006 6.15 puts For each'statement indicate Whether it isltrue or false. Do not show your work. DC EX) 1. If 2 c4,1 converges, then Z(—1)”cn also converges. n=1 n:1 DO 2. If f(:1:) > 0 is monotonically decreasing and f(.r)d\$ < 00 then 1000 00 Z ﬁn) converges. n=1 3. If the sequence {an} converges and the sequence {bu} diverges then {an + 6”} diverges. 4. If the sequence {an} converges and and the sequence {bu} diverges then {anbn} diverges. 5. If 2 (1,151” converges and Z 011(76)“ diverges, then 27120 0.18” diverges. n20 n20 09/23/2003 15:48 FAX 510 642 9454 .007 7.15 pnts Indicate whether each of the following series converges absolutely, converges conditionally, or diverges. Do not Show your work. Do 11, 1. — Z(n+1)s 2. i=2 ——(n + 1)3 'n. 1 0° 2 3. Z(—1)”(n:—1)3 n: 00 1 n2 4 —1 “sin — 7;; ) (71) (71+ l)3 00 1 n4 5 Z(_1)n51n(;) (n+ U3 09/23/2003 15:48 FAX 510 642 9454 .008 8.15 pnts Find the general solution to the differential equation a:y’—y:m. 09/23/2003 15:48 FAX 510 642 9454 .009 9.15 pnts Solve the initial~value problem d —y=l+:c2+y+332y, y(0)=0. dz: 09/23/2003 15:49 FAX 510 642 9454 .010 10.15 pnts Find the solution to the initial-value problem y”—y:em, y(0)=0, y’(0):1. 10 09/23/2003 15:49 FAX 510 642 9454 .011 11.15 pnts Find the general solution to the diﬂ'erential equation: y”—2y'—3y=a:. 11 09/23/2003 15:49 FAX 510 642 9454 .012 12.20 puts Find the power series solution to the differential equation: y” — my = 0, y(0) = O, y'(0) : 1. 12 ...
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