**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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