Math 1B - Spring 2000 - Vojta - Final

Math 1B - Spring 2000 - Vojta - Final - 09/25/2000 MON...

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Unformatted text preview: 09/25/2000 MON 11:48 FAX 6434330 MOFFITT LIBRARY 001 P. Vojta. Math 113M Final Exam Sat 20 May 2000 Some Formulas sin2 a: = %(1 — cos 21:) cos2 :1: : %(1+ COS 2m) SinAcosB = %[sin(A — B) + Sin(A + 13)] sinAsinB : %[COS(A — B) — cos(A + 8)] cosAcosB : %[cos(A — B) + COS(A + B)] 9:59.59.“ 6. ftanudu =1n|secu| +0 7. [secudu :1nlsecu+tanul +0 1 8. [tann udu : 1 mun—1 u — [mun—2 u (in n — . 71—1 7';— 1 n — 2 9. fsecnudu: tanusec"_2u+ 1/scc"_2udu I I n _ 2 10. Weierstrass substltutlon: t = tan $1112: 2 1:12 , cosx : figs, = lit? dt. 11. Binomial series: <2>=1 1‘ cc 1n(1—$)——m~§—§—.... :03 3:5 m7 13. t = —— ———+.... arcanm 3: 3+5 7 09/25/2000 MON 11:48 FAX 6434330 10. MUFFITT LIBRARY (35 points) Find: 2 (a). V :32 + 4da: 1/2 arctanmdlm 0 (12 points) If the curve y : \/2$ — x2, 0 S a: g 1, is rotated about the m-axis, find the area of the resulting surface. ( to Within 0.01. DO __ 1 n (14 points) Describe how one can compute Z ) “:1 n + 1 (You do not need to actually carry out the computation, but if your answer involves, say, the nth partial sum, then you should say What a is.) (25 points) Determine whether the following series converge absolutely, converge condi— tionally, or diverge: (25 points) Determine whether. the Following series converge absolutely, converge condi- tionally, or diverge: (a). i (aretan (7+ — arctan 7) n=1 0»- ? (27,711)" n=1 1 \/1 ~332- (14 points) Find the Maelauren series for (20 points) Solve the differential equation , Ina: : my + 5W3 ' (20 points) Solve the boundary—value problem y” + 211’ + 2y 2 0, y(0) : 0, y(7r/2) : 1. 6‘3; (24 points) Find the general solution of the differential equation 3/” + Zy’ + y 2 m . (24 points) Find the (series) solution of the initial—value problem y”*$y'*y:0 29(0)=0, y’(0)=1- 002 09/25/2000 MON 11:48 FAX 6434330 MOFFITT LIBRARY 003 11. (12 points) Match the equation to the graph. You may assume that each graph belcmgs to one equation. y::$3+$y y’=9:2—y ...
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