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Unformatted text preview: page 1 Your Name: Circle your TA’s name: .Lino Amorim ' Jon Godshall Ed Hanson
Elizabeth ‘ ' Owen i. Schattner
W
Mathematics 222, Spring 2007 Lecture 3 (Wﬁson) Second Midterm, Exam April 12, 2007 There is a problem on the back of this sheet! Do not accidentally skip over it! ' Write your answers to the eight problems in the spaces provided. If you must continue
an answer somewhere other than immediately after the problem statement, be sure (a)
to tell where to look for the answer, and (b) to label the answer wherever it winds up.
In any case, be sure to make clear what is your ﬁnal answer to each problem, Wherever possible, leave your answers in exact forms (using g, ﬂ, @0405); and similar
numbers) rather than using decimal approximations. For example, sin(%) = .12., and writing something like .‘499 may not get you full credit. If you use a calculator to evaluate your
answer be sure to Show what you were evaluating! There are some formulae at the end of the exam that you may wish to use, You may refer to notes you have brought on one or two sheets of paper, as announced
in class and by email. BE SURE TO SHOW YOUR,WORK, AND EXPLAIN WHAT YOU DID. YOU MAY
RECEIVE REDUCED OR ZERO CREDIT FOR UNSUBSTANTIATED ANSWERS. (“I did it on my calculator” and “‘I used a fOrmula from the book” (Without more details)
are not sulﬁcient substantiation...) page 2 Problem 1 (13 points)
Findthe area of the region which is inside the circle 7* = —2cos‘6 but outside the circle 7° 2 1 ' page 3
Problem 2 (12 points) . . 1 1 ' i . .
(a) Let. an = “(ll/Ti; ’. Does the sequence {an} converge or dlverge? If 1t converges, what ls rte limit? t) .1 00 n
. a: — 1 _ I . _
(b) The 561168 E 3< > IS a, geometrrc semes, for any given as. For What values of a;
71:0 . . _ I a
does tlns serres converge? For those values of x that do make 1t converge7 what does it,
converge 39? 00
(c) Let. a1 = 2 and for n 2 1 let an“ = 1+smn an. Does the series Z an converge or diverge? n
'n: 1 Be sure to give a, reason! page 4 Problem 3 (12 points) , . . 0° 1 __ n
(a) Does the serles Z 71: 1 converge or diverge? Be sure to give reasons! n ‘2n ()0
' I 1 I u ‘
(b) Does the serles Z(_1)7 W converge absolutely, conchtlonalb,7 or not at all? Be sure to n: 1
give reasons! page 5 Problem 4 (12 points) 1 J. . 7 converges when p > 1 and diverges when p < 1. Use
n 71:1 the integral test to show that this is true.
You may ignore the cases p g 0 and p = 1_ . 00
We know that the so~callecl ]:—series 7 _J (Be sure to explain your steps and to show why the the integral test does apply. You may
assume that mp increases as cc increases, which is true for a: 2 41 and p > 0, but you should refer
to that assumption wherever it may be useful.) page 6 Problem 5 (12 points)
Find the Maciaurin series (the Taylor series with a = 0) for f ( x) = 5111311
(Do explicitly derive the coefﬁcients: Do not just substitute into the known series for sings. W'rite out the terms through the '71“ power of 55., and show without proof) What the general
term looks like.) page 7 Tiiildzf ¢l«¢+1lvilw.iviiw1\¢ \L. Aifélriériarllﬁx. l kquxqiiniia. wivliwiwiiwiiviutiivihxxa aliaffaflﬁti 9/ \xa \hluhiﬁi #Ikvllkvi#\1viivii¢\u¢ \xh fiat: 411:. cf; .u/ (i \a \xqﬁseiaai wiiwié.r&.5.wiav1~ﬂsxw..\4\a aliaPJawrfﬁxx 5/ \N \XakquXa ﬁhai¢\i¢\s¢\§\hx\a\xu \& (13 points) Problem 6 fﬁiiéfkfz x/
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/I\ ihll¢ll¢ilﬁil§i+ﬂi¢isﬁ K24 . rié.t+i&i+uivin#\sw\aa . (b) Solve the differential equation ﬂ %g = 83/ ex/E, with x > 0. which solve the equation, A relation involv ing y and x but no derivatives can get partial credit.) ) SE ( iicitly functions y ﬁnd exp 7 (For full credit page 8 Problem 7 (13 points) Suppose We use the appi‘OXimatiOH 6'T = 1 + :c + (the ﬁrst three terms of the Maclaurin series
for 6:”) when :1: is small: Use the remainder term from Taylor’s theorem to ﬁnd a bound (Le. a
maximum possible value) for the error in this approximation if we restrict its use to m < 0L
(You can use the fact that e < 3 if that is helpful. Your answer should include both a number
(some fraction: Perhaps) such that the error can be guaranteed not to exceed that number as well as” your argument showing why the error really does not exceed that number. Do not
assume you know exactljx,r 80'1 or any power of 8 other than 60 = l.) Problem 8 13 points) page 9
Solve the initial value problem dy I el ,  _
(33 4.1)6E —— 255(51: + My = x +1 (for x > ~l) Wlth = . page 10 Some formulas7 identities, and numeric values you might ﬁnd useful: Values of trig functions: W 01:] 1 °\ r ,_\ {7’ 31116 cost? tanH
W
0 O 1 0 .
r'___;——————————¥~————
w 1 1/3 [25. E 2 2 3 w ﬂ 1 Z 2 2
r——+———r—""“’_~—J 1 x/fi [uhJ
E o
l Derivative formulas: 1. ‘2. \1 d 'dx _d_
“:13:
d ’dm 9
(—11 tan :2: 2 see“ :1:
3: sec 3: = sec :0 tan a; ‘1 sin"1 :1: = W tan‘1 a: = 1H2 1 —i __ 1
W x“ mm 111:1: =1 e$=em :1} Algebra formulas: 1. may) =111($l + 111(9) Trig facts: 1. 2. .sin(a: + y) = . tan(:c + y) = tau 9 = L“?
c056 . __ l
secél — COS 9 . sin26 + cos2 9 = 1 . secge = tangé + 1 sin(m) cos(y) +
005(22) sin(y cos<x> my) — O
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+
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II 5111(50) sin(y 'tan(m)+tan(y)
1~tan(:z) tan(y) . $1122: 2 — costL . cos2 :c = + cos 2x) Integral formulas: 1 fﬂndu = ——1— n+1 unH + C7 n # _1 2.f§;du=1nlu+c‘ 3. f¢15:‘—u§=sin—1u+0 4 f 1:22 = tan‘lu + C 5. fsec(u) du = 111 { sec(u)+tan(u)l +0
6. fudvzuv—fvdu ...
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 Fall '08
 Wilson
 Calculus, Geometry, Derivative, Taylor Series

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