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Unformatted text preview: Your Name: Circle your TA’s name: LinmAmorim Jon Godshall Ed Hanson
ElizabethMihalek Rob Owen Kim Schattner
Mathematics 222, Spring 2007 Lecture 3 (Wilson) 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 %, \/3, cos(0.6), and similar
numbers) rather than using decimal approximations. For example, sin(%) 2 %, 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 QR ZERQ CREDIT FOR UMSUBSTANTIAIEILANSWERR
(“I did it on my calculator” and “I used a formula from the book” (without more details) are not suﬂicient substantiation...) Problem Points Score
1 13
2 12
3 12
4 12
5 12
6 13
7 13
8 13
TOTAL 100 Rage 2 Problem 1 (13 points)
Find the area of the region which is inside the Circle 7’ = —2 cos 9 but outside the Circle 7“ = 1. Problem 2 (12 points) (a) Let an = mg”. Does the sequence {an} converge or diverge? If it converges, What is its
limit? x—l n
> is a geometric series7 for any given m. For What values of x (b) The series Z 3 <
n20 does this series converge? For those values of x that do make it converge, What does it
converge t_o? m
(G) Let a1 = 2 and for n 2 1 let an+1 = 1+2” an. Does the series Z an converge or diverge?
ﬂ=1 Be sure to give a reason! Problem 3 (12 points) ( ) t1 _ 0° 1 — n
a Does 1e ser1es E
n 2n n21 (b) Does the series Z(—l)n
n:1 give reasons! converge or diverge? Be sure to give reasons! converge absolutely, conditionally, or not at all? Be sure to §lr Problem 4 (12 points)
00 1 We know that the so—called p—series —p converges when p > 1 and diverges when p < 1. Use
71:1 n the integral test to show that this is true. You may ignore the cases p S O and p = 1. (Be sure to explain your steps and to show why the the integral test does apply. You may
assume that 96” increases as 96 increases, which is true for 36 2 1 and p > 0 but you should refer
to that assumption wherever it may be useful.) Problem 5 (12 points) Find the Maclaurin series (the Taylor series with a = 0) for f = sin 3x. (Do explicitly derive the coefﬁcients: Do not just substitute into the known series for sinx.
Write out the terms through the 7th power of x, and show (Without proof) What the general term looks like.) Problem 6 (13 points) Tlilfffﬂﬂaﬂﬂfffili
liningrmaimil
firerxxﬁeeaﬂxﬂxrrrri
H r f/fﬂﬂﬂa—f—Hﬂﬂﬂﬁff f N At the right is a portion of the a a a a a a a a a 1 a p a a a a a a a r» a
slope ﬁeld for the equation xi, \. \u \k \‘m‘mwhe—P—P—F—ﬁww‘x\\ \ \z. E
dy Milli? fliiil
0 E: Liliix \ttill
a On that ﬁeld draw two graphs, _ " " " ' : l  ; " " " " " ='
. . LLthxWHn—waxttgl
one th_e solutiqn corresponding L L L L \L \an \hxﬁaﬁhkﬂwx \h \I E L L L L
mylu—2am“ﬁmm7litithXWHawwxxxtllll
the solution corresponding to L a; L :4 \aa \é‘ \Exaaﬁﬁp—Haxh\ﬂ \k ‘3: in L L L
y®=L illltxxxwﬂqﬂwNNNXLLLL
Littx\xxwﬂeﬂwx\\xtill
Likitxxwwﬂﬁﬂwwxx\iiil
lkt\\\wwwﬂﬂﬁwwx\x\ttk
5‘ h !) h L'i I? D '3 Ii _2 b I? h !) h L'i I? D k I] h (b) Solve the differential equation ﬂ % 2 6y eﬁ, with 96 > 0. (For full credit, ﬁnd explicitly functions which solve the equation. A relation involv—
ing y and 96 but no derivatives can get partial credit.) Rage 8 Problem 7 (13 points) Suppose we use the approximation 693 = 1 + x + 382—2 (the ﬁrst three terms of the Maclaurin series
for 633) when x is small: Use the remainder term from Taylor’s theorem to ﬁnd a bound (ie. a
maximum possible value) for the error in this approximation if we restrict its use to < 0.1.
(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 exactly 60'1 or any power of 6 other than 60 = 1.) Problem 8 (13 points)
Solve the initial value problem dy e”
l——2 l 2
(96+ )dm x(x+ )y wnl (for 36 > —1) with y(0) = 5. Page 10 Some formulas identities and numeric values you might find useful: Values of trig functions: 6 sin 6 cos 6 tan 6
0 O 1 O
1 1 E E
6 2 2 3
1 Q Q 1
4 2 2 ﬂ 2 [\DIH
b Trig facts: 1. 2. _ sin0
tan6 — €080 1
cos 0 secQ = .sin29—1—cos26: 1 sec2 6 = tan2 6 + 1 = sin(x) cos(y) + 36 34)) = cos(3c)cos(y) — _ W
— litanM) tanfyl Derivative formulas: Integral formulas:
1. ﬁtanxzseCZx 1. funduzﬁunﬂ—i—Cﬂfn7é—1
2. ﬁsecxzsecxtanx 2. fiduzlnluH—C'
3. i sin‘1 .76 = 1:902 3. f V1613? : sin’1 u + C'
4. % tan—lL 36 2 H1002 4. 112:2 : tan’1 u + C
5. % sec—1 36 = ‘wwigj 5. fsec(u) du = ln  sec(u)+tan(u)+C'
6iﬁhm:% 6. fudvzuv—fvdu
7. ii 690 2 695 Algebra formulas: 1. ln(xy) = ln($) + 113(9) 2. aﬁy = a9” a9 3 2b : ablna, ...
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 Spring '08
 Wilson
 Derivative, Taylor Series, Jon Godshall Ed Hanson, Rob Owen Kim Schattner

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