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Unformatted text preview: ‘ University of Waterloo
Waterloo, Ontario Mathematics 138
MidTerm Test  Spring Term 2003 Duration: ‘2 hours Date: June 16. 2003 ONLY BASIC SCIENTIFIC CALCULATORS PERMITTED Initial“ Id. Numbe‘ Family Name: Signature: Instructors: (Check one)
01 D. Woolford 02 M. Shpigel
03 Y. Spigel
04 D. Harmsworth 000K Instructions: Title page/instructions 
1 1. Complete the information required
on this page in the spaces provided.
Check your section number in the
appropriate box. 2. Attempt all questions, in the space
provided. If you require more space,
use the reverse of the preceding page . HHIIE 3. The marks for each question are in
dicated. Marks will be deducted for
negligently presented work. Your
grade will be inﬂuenced by how
clearly you express your ideas, and
by how well you organize your solu—
tions. Justiﬁcation should be provided by
referring to deﬁnitions and theorems
where appropriate. \r'. I 4. This examination has ten pages.
The last page is for rough work. 5. A onemark deduction may be ap 65 (+3) plied for improper completion of this
title page or for ignoring instruc
tions. Math 138  MidTerm Exam Spnug lerm 2W.) x {13] 1. Evaluate the following: (a) f cossrdr
o _ Q ~11 V' f
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(b) —— dx (Notlce that thxs integral :5 Improper!)
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., \/ Page 2 of 10 3 Spring Term 2003 Math 138  MidTerm Exam ‘ (c; rm (1 —2z)%
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f/ Page 3 of 10 3') Math 138  MidTerm Exam uyuug, um. WW . . {10} 2. Show that. the sequence deﬁned recursively by a, = an“ 2 VS + an is increasing
and bounded. Explain why this means that the sequence must be com'ergem. and ﬁnd
its limit. / v Page 4 of 10 Maui 138  Mid—Term Exam Spring lerm 2mm 0 [14} 3. Determine whether each of the following converges or diverges. For parts (d). (f).
and (g), you should distinguish between absolute convergence and conditional
convergence. ln answering these. you may refer to conclusions you have reached for
other parts: for example. in your explanation of {d} you may wish to refer to (c). 4 {3+5nz}x
la.) 0
n+n nzv‘l 0c 3+5n2
(b) 2:! n2+n 3» x
no 3 ~;.
n 11—1 “7‘
(d) g 1) ,H at“ 7
\h
.\ 3‘ f R; Page 5 of 10 Math 138 — Mid—Term Exam Spring Term EUU'J I. Z 371 h
(e) n2 + 4 3‘
n=l , , “Ekijtd’
< gunman) W
0664 mm czgdm'l'rm of“ nitIA a) 00 "_ 3n
(0 gm) km hmvf‘?’ "’71, y/ "t (g) f; (—1) En— H
7
x Page 6 of 10 Mth 138  MidTerm Exam Spring 'lerm ZUUJ
I x \X
1 . .
[10} 4, (3} Find an approximation to the sum of the series 2 £3 correct to mthn 0001‘ n2] 1”
I
(b) Find an approximation to the sum of the series 2 (——1)""l$ ? correct to within
n=1 0.001.  Page 7 of 10 ' Math 138 ~ Mid—Term Exam Spring Term 2003
WWW . DC. ,
~ . ‘ _ . ~33"{1 + 15”
[12] 5‘ Fmd the radlus and interval of convergence of the power series 2 l
71:0
9’  2
ﬁx \
3 1
.2:;\ ’
1 Page 8 of 10 9.; Math 138 — Mid—Term Exam :pnug term .cuu.) ' {6] 6. Have that if 2 30"] converges. then 2111, converges. BONUS: A pane of a certain type of glass reﬂects half the incident light, absorbs onefourth, and {31 transmits one—fourth. A window is made of two panes of this glass separated by a small
space. What fraction of the incident Right I is transmitted by the double window? I
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4
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 9
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x s ' /
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mm...“ ; 7 pk. . f
V3 ‘ 9‘ a
r ‘3 Page 9 of 10 '\
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 Anoymous
 Calculus, Mid—Term Exam Spring

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