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Unformatted text preview: Math 122 Final December 13, 2005 KEY Name 1 l
‘2 L
3
4
5
6 7
8
9
10
Total E / 200
Directions:
1. No books, notes or passing go and collecting $200. You may use a cal— culator to do routine arithmetic computations. You may not use your
calculator to store notes or formulas. You may not share a calculator
with anyone. . You should show our work. and ex lain how vou arrived at our an—
y , ., swers. A correct answer with no work shown (except on problems
which are completely trivial) will receive no credit. If you are not sure
whether you have written enough, please aésk. . You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or
you will be penalized. . You may leave as soon as you are ﬁnished. but once you leave the exam, you may not make any changes to your exam. 1. (20 points) MW
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1 + sins:
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H gm; mmwxwﬁmx+£4 3. (20 points) i (a) For the differential equation (67% : F (:L', y), Where F(:L', y) is given
below: Use Euler’s method with h 2 1 and y(0 = 0 to approximate M4). h?(.~:<p~ifi “i it in Raﬁ? i '. .
\kwk 3 XS!%‘}’% 9:.
1 31(0) = 1 Q" Raw dié at" 2:) 4. (20 points) A tank contains 30 gallons of pure water. Water containing
2 pounds of Grape Kool—aid mix dissolved per gallon enters the tank at
5 gallons per minute. The well—stirred mixture drains out at 5 gallons per minute. (a) Write a differential equation (with initial condition) for how much
Grape Keel—aid mix is in the tank at any time. will dim r, :L ma» fagwiﬂ ggg ‘3}{03 C) i i W
W \l 1‘ Mag; 3:0 if: "t C”
magi“ Q '
., rm?“ \
attime a 5% km e5 Mil
Home“? * k ‘ CV23 “gm “MW—k
if”: @Qtﬂ Q (a
mini; #3:ng WW 8% (c) HOW much of the Grape Kool—aid mix Will be in the tank after 10
minutes.
w. lg
e \Hm‘) mega meoe «w ((1) Find a second order differential equation that has general solution: y = ale/53’ + (22643”
R i; 5 gave: “:2...
Z r
m “Q; m» aem in; \sz w £33k Slaw3:) W” was “3‘1: mm?! $9 5. (20 points) (a) Find the equation of the tangent iine(s) to the curve x2t2—2t+2 y=t4w4t2+5
at the point (2,5).
f2“ Cian W mark?“
a maﬁ*1”& ex em?» 
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t Matt’s; Q
etewai‘ieﬂ \g «:2 5
43230 ~ ‘1.
mete«w e (we) (c) Set—up an integral to ﬁnd the area, of the region outside
7‘ = 3 ~— 25in6 and inside 7" = 2.
Do not solve the integral. 2. { BEE;
me ~38qu
:1
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(a it 6. (20 points) (a) For each of the following series, determine if it converges or di—
verges. For each test you use, you must name the test, perform
the test, and state the conclusion you reached from that test. , 00 5”(nl)2
1' (270‘
l l e? t “32‘
ml "‘2 z N”) N M M
Ma Q, \ a (My Lagwm 2: M a. List
t ; WW rm ‘ . ‘ N 4% m (aémﬂyﬁ 5;; Us 33 N “‘§ m (1N «MIQN VEQ
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L‘QT, Q/vm Ntl «3., Jle WW “l .
b5 em a NEW N M m
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N m N“?
(b) For each of the following series, determine if the following series
R converge absolutely, converge conditionally, or diverge. For each
test you use, you must name the test, perform the test, and state
the conclusion you reached from that test.
. 00 (—1)“
1.
g nvlnn m
“We ls F3 \ \l
rumeraeu g “is, d a ~==~ Emmi ,
3i ill/WV \
‘\ WWW v”
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" W "" m M W N mid
{jam \l é CGN O
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u' ; 1+ 4‘n
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N em “Wm QM “:14 W X if 33 D w 7. (20 points) (21) Consider the power series: TL 00 (cc—#1) 1. Where is the power series centered? if: WE ii. Find the radius of convergence. \CMikgm mwﬁ\w hieﬂ 045% \em‘rél Wm‘ev‘m‘" 2L
eke iii. Find the interval of convergence. (Hint: Check the endpoints.)
, as Wg (:2 L13)?» {jam ‘3 Cami) m Emu?) 2“) ,a/ \
W $3 \ \2‘
5 2 m
(b) If the Maelaurin series of f is
x3 .735 m7
f(33)~33+~5—+16+T5+.H 1. Find 134(1):), the fourth degree Maelaurin polynomial for f %
‘ W V “3‘ ii. Find f”’(0), the 3th derivative of f(3}) at a: : 0. .. M
WWWWW PM .
"i" $3.?(03 ¥%Q’W}K€Q €33 e. E3"
in. Find hm ———f(m)3“ 5”
CC—»0 (L’
3, 5 ‘
D errrrwww~r
,Q/vm g 50 \ slat?) 8. (20 points) (a) For vectors '6? 2 (l, —2, and 79> : <3, —1, 1) ﬁnd: 1 3.77
3+1+E x %
11 if X 5
,U 3 in in . $5"
awe. 3 ‘1”1 élﬁgjv
% M l %~ﬁ
iii. Find the area of the triangle with “CT” and .5) as two of the
sides.
i Wm w W
m We 63% “r 55 mg: iv. Find a unit vector perpendicular to both if and 5 ﬁnale}
WWW V. Find the value of a: so that ‘3 x (2,217 —8) and “a7 are per~ pendicular.
afﬁne“? (43%)“??? H
gmeﬂqu$<3 f"?
(b) i. Find the distance from the point (l, 2, 3) to the plane
33: — y + z r: 1
mgr it? :35“: «mm/93W: ll m QM W ‘ M ii. Find the point of intersection of the line:
:1:=—l+t, yzl—Qt, z=3+37€ and the plane 23: — y + z = 7 éliwlWE‘) w (imam «in (gt—gm a» “"7 Wgww'g§£wl+&%+q§dgt 3:? <Q}M\}(;)
West ear 9. (20 points) (a) Find the equation of the plane through (1,2,3) that is perpendio ular to the line $=1+2t, grime‘xg wgﬁ M
m y=2+t Wt zz~3~5t (10) Find the equation of the line of intersection of the two planes: 393% 4%
text V11
9.Wi a let m+2y+zzl (C) Determine if the two lines: and a:=14t, 3::2—t, are parallel, intersect or are skew. \th‘t ﬂﬁmﬁ ereM 2x~y+z=2 PQ‘M‘T (ha39:?
)4 33‘ tevéfbt
\i fr: :3»? "h
E 7:: y:2+&, zz4~2t «(MHJ%)W§>
y=1+a zzz+m <“333)G§
Not“ QQRRLLEL
Wﬂwwjj‘f
t H '% cg:er
ﬁzﬁakg [Mmmm 10. (20 points) Indicate Whether the following statements are true or false
by circling the appropriate letter. A statement which is sometimes true and sometimes false should be marked false. 00 Zm—mjzi The Méaclaurin series for cos 2:1: is 4 164
2m——25£!—+—47§——... The two planes: 2:13 + y — z : 5 and
a: + y + 32 z 7 are perpendicular. The sequence an 2 {ln(2n) — ln(n + 1)} diverges. ~1+1~1+1—l+...=0 in; 71:1 1+1:2 ...
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This note was uploaded on 02/17/2010 for the course MATH 122 taught by Professor Butler during the Spring '07 term at Case Western.
 Spring '07
 Butler

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