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Unformatted text preview: Faculty of Nlatheniatics
University of Waterloo Math 138
Term Test 2  Winter Term 2007 Time: 7:00  8:30 pm. Date: March 19, 2007. AIDS: ‘I’INK TIE‘ CALCULATORS ONLY Family Name: _____.___ First Name: ID. Number: —_— Signature: Check the box next to your section: D Section 0] D. Park (MC 2035) 8:30 am.
D Section 02 S. Drekic (MC 4061) 9:30 am.
D Section 03 L. lx’rivodonova (MC 4021) 12:30 p.111.
D Section 04 M. Mastnak (MC 2038) 12:30 pm.
D Section 05 C. Roberts (MC 2035) 1:30 pm.
D Section 06 C. Struthers (STJ 2009) 11:30 am,
D Section 07 B. J. Marshman (MC 4061) 11:30 am.
D Section 08 E. Samei (MC 4021) 2:30 pm. Your answers must be stated in a clear and logical form in order to
receive full marks. Reference any theorems or rules used by their
name, or the appropriate abbreviation. Useful abbreviations for this
test include nth TT (n“‘ term test). GST (geometric series test). CT (comparison test). LCT (limit comparison test), IT (integral test).
ACT (absolute convergence theorem), RT (ratio test). LSR. LPR.
LQR, LCR (limit sum. product. quotient. and composite rules). Note: 1. Complete the information section
above, indicating your instructor‘s
name by a checkrnark in the appro
priate box. 2. Place your initials and ID .\'0. at the
top right corner of each page. 3. Tear oil‘ the blank last page of the
test to use for rough work. You may
use the reverse of any page if you
need extra space. MATH 138 TERM TEST #2 PAGE 2 L’Gns users) 1. A particle moves along the path x = (1,31) = (e‘.e") in the :ry’plane for 1 5 l _<_ 1. 3) Find an equation relating r and y. Sketch the path. showing the points where
t=—1.t=0.andt=l. u
‘3"
1.4! it La b) Find the velocity vector v = x’ and acceleration vector 3 = x" of the particle at the point
where t = 0. Show these vectors accurately on your sketch in a). We) =(b01
[9 Lo) 2 (t ,0) c) Suppose the particle leaves its path at to = 0 , and follows the tangent line at this point.
Find the vector equation of this tangent line, and the position of the particle at r = 1, Egg“ =L\,l\+(t,‘§] d) Develop a deﬁnite integral representing the distance the particle travels for —l S I S 1.
starting with a suitable element AL. [DO NOT evaluate the integral] 2““ 4L “th l
E ‘8‘ Jazz 61*ng MATH 138 TERM TEST #2 PAGE 3
a 2, Find the limit of each sequence {27"} or show that it does not exist. You may use known limitsi
limit theorems or the idea that lim f (n) = 11111 f (3:) for continuous f . Reference any theorems
"~00 13—00 used (see the cover page) Some algebraic manipulation may be required. )  (M)
a = i —
In em 4 b) m" = 137122., a) [AJ’Q‘X c)m,,=$mn b) 108 d)zn=1M a {01 MATH 138 TERM TEST #2 PAGE 4 3. a) Use the 5  6 deﬁnition of limit to prove that lin;(31' + 4) = 10.
1... b) For each statement, either show it is TRUE. 01' give a counterexample or theorem which
shows that. it is FALSE. m be
(i) If E (1?, convergw. then 2 a." converges . n=1 n=] FA LS E
00 03
(ii) If 0 g (1,. S b", and Z I),I diverges. then Z a,l diverges
n=l n=x
FAL$E
(iii) If :01" + b") converges, then both 2 0,. and L b" 1111131. converge.
n=1 ":1 n=1 FALSE MATH 138 TERM TEST #2 PAGE 5
M 4. 8) Use an appropriate test to determine whether each series converges, converges conditionalh:
or dwerges. Name the test you use (see cover page). ‘
0° 2 (j) ; V271“ — 1 c 1w.
{2.83.3 iomu . b) Using a suitable theorem and your calculator (trial and error). find the number of terms of
the series in n)(i\') needed to estimate the sum of the series with error < 0.01. "*7 so 28 Eﬁmsl MATH 138 TERM TEST #2 PAGE 6 5. a.) (i) State the Monotonic Sequence Theorem. (ii) Use the theorem in (i) to show that the sequence deﬁned recursively by
(I?) = 2 $n+1=1+ v5+xn for n =1.2,3,... converges and then ﬁnd its limit.
[ HINT: First. use mathematical induction to show that In < In.“ < 5 for n. = 1.2‘ 3. ..] b) (i) Show that the area .4" between y = :t” and
y=z"+1 forOSzS 1 is 1
A" = (71+ 1)(n+2)‘
N,
' 1
(ii) On an assignment” you showed that the partial sums SN = g m are
given by
1 1
9  3 ‘ m w
Use this result to ﬁnd the sum of the series Z A” n=l ...
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