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Unformatted text preview: Name Math 122 Final December 12, 2006 KEY Directions: 1. No book, notes, loud snoring, steriods, or delinquent activities. You
may use a calculator 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 your work, and explain how you arrived at your an— 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 ask. . 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) d x/ugj: 2
(a) Use the formula: / ﬁzq: agua +0
1 U: ax
to evaluate/m dm do: adx
ow I d0 2.
U2 z 2 :’ v *0) = — 14%?er
LT 0101 q U . ..__._.
q (ﬁx)
4a: 2‘
: 0‘: RXdX
(b)/(1+m2)2dm U \+>< d
' DL 3* U 0+1")
lllxd —;  ‘29—“: —\— S .11.
(C)/F “3 I d x 7<
= )4 0W :W x
du= .1.— dH \1 .— ——‘_ x 7<
X X
2 2
(d)/C9”dm= j\‘5”"" olx
sma: S‘Nx Seacx — 6\N>< dx = Qh\cscx~co~rx\ + cosKHEQ 'Z 2 (20 points) U __ I
’L
(a) tan3 :1: dm = j "m BIZ/x 0‘ U
m ' {0‘7 \rrm
U: SECX «3U: sccx ﬂwx dx
1 E
:2: V —\ __ Jé 3'5 7' ,L
J o‘v JU—v _ 2 3%. '2
' ‘3 (SECXU + 2 (sec >43 + C
b ___£...__
()/2m2—3m_2dm :: 5’1 + \ Dad—'M. $4.;
5 = 43* _k B
‘  *1
= RC!‘D3+G(ax+l)
7c
3. 3=\
X=____ S:~.EP
a 9",.17 (c) 9m2+16 561$ __ j \ dﬁ
‘ :L‘
+ (sz‘m
= S L d1 = J— NICTAN L). = —‘—— RECTRN (aﬂl
U7?” :5 Q; a. k L
(a Q
\ 2:3
(d _. 3
/{x ) Wears  3‘ 9m 6 d6
1 M
‘4 X— S Né
‘ ‘ Uroose
CH: cosede dw—S‘Ne d6
C56 ._ dU
SIN6
’L
:3 I S\Ne dU 3
~ __ 2 __ ’5
“M \Y \ U “ —:—)— D 7: (.056 __ C096
3 3. (20 points) (3) Use Euler’s method with h = 0.50 to approximate y(1)
initial value problem for the \/(0 «’1: W95 (1+m2)&;—4xy
(HM (m8)
6 (‘3
~7—
«qrK") d_y_ _ 3:1:
dm 331—6 811123
3X
3—3, (900:6 wa
Jadx —7.>’<
e”? = 9
 (3 '1‘! av \ A
,,..'..~ "7" 9" \Nx OX +9
(9‘3" Je e s ‘3 4. (20 points) (a) The country of New Zealand has $ 300 million in paper currency
in circulation and each day $ 15 million comes into (and out of)
the country’s bank. The government decided to introduce new
currency featuring a certain Chemistry Professor because a smaller
portrait was needed. The banks will replace old bills with new
ones, whenever old currency comes into the bank. Let y = y(t)
denote the amount of new currency in circulation at time t, with y(0) = 0
5 i. Write a differential equation for how much new currency is in
q circulation.
\ 91=\5—\51=is—:/.
BOO d'b 300 30
is '
ﬂ \/ (0) =6)
ii. Solve the differential equation from part 5‘6 b
= _I_ ma: \5 [OCH = 6
ﬂ + i : \5 20 . Lt
dd: 90 4.4, ‘ 2° C1
\/= LIT ‘56 6° at +0] = are 3006’ +
e “ e
3251: —%0t
\l:zoo+ce \/=3oo~'3©oe iii. How long will it take for the new bills to account for 90% of
the currency in circulation?
_ 19.3 1:
8—10 = 300  300 E
 y” — 51/ + 6y = 0 y(0) = 5, y'(0) = 12 5. (20 points) (a) For the curve given by x = gig/2, y = 275,
1. Find the equation of the tangent line at (18, 18). t=01 O“) ; '——' ’2.
ow=gz= T=g \;—\a=g(x—Is)
ax em 1:. 3*? ii. Find the area of the region that lies between the curve and
the m—axis for 5 g t S 12. $2
\2 R \1 ;2 a t
A: jydx= S atﬁdt= j 3+. (it: 35.
5 s 5 55 D» %_
_ EL 'b \ = 5 ( \1
 6 5 5
iii. Find the length of the curve for 5 S t g 12. '5 ll
’— (I: ‘1 214%?
>2 — s = j W ova = _._.
\,  1 S '2: s
‘1 “1 r2. «4... an 2' __._ @i “’ 3 2: (b) Write an equation in polar form for the vertical line that goes 5 through the point (2, 0).
I
i X = a 2
l :
l RCOFG : a Q C069
l.
l
(c) Find the area inside one petal of 7" = 3 cos 30
T TT
7. ’2.
‘— 3 '3‘ G
0 TT Ge
S \N
3 32.5.2? do =E—(e+ Tl
EC \ 1 H
O
__ 2. (“\e "IT
' 7T ' Li 6. (20 points) Determine if the following series converge absolutely, con—
verge 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 (a) Z(—l)”arctann
71:1
Nsm Team “QM NAG—mm N =1; 2m but
NéOO
°° nnl3”
M '
Jam @W"b ﬁlm! = o <1 @
RAT“) New LaN'lam/(amﬂ a?
(C) n; n_1 ‘
x l 1 __,§c, To
. \ _____g g 0” 0
Comp.”‘eer R q 3 m f m “6T
9' \ =0 \/ l 4 l v lCOkW. Com) 5
N900 m ~l \ \lFt ‘l
00 n l 1
(d) gel) + name
0°
V30
A, \ \ \
\N’T.\'1:‘5T S t — M\ ‘5 —"—“ 0 N0 R85
3 x 0m? &mx A)“; :0 7. (20 points) (a) Consider the power series: 77. °° (933)
2 +1 n=1 n i. Where is the power series centered? C23 ii. Find the radius of convergence. 9% N“ \= \><~">\<l Rcl 14
NH; C)“ '53 iii. Find the interval of convergence. [Hintz Check the endpoints] .4
.43
54:11 i \ Com). COND
N‘n
A \i
 \I
” 3(‘u i Nx\ 0‘ (b) For f(a:) = lncr
i. Find P3 the third degree Taylor polynomial at a = 1. "LED =_§/ha( CD 3 "l 2 2: _
$7023): I V390=0‘H(><~D +":5(>1r0 +3!(>< \3
“gylx 2 :71? 1‘ ii. Find R3(x) the remainder for the third degree Taylor polyno— gu’hg : 2E mial at CL=1.
X
. ~(o
41”,
32350 = (E) (x‘oqvx : “in (#04
M. 8. (20 points) (a) For vectors ’c? = (1,2,3) and W = (2,3,1) find:
i. 7?
1 + c. a 3 = \ l
11 7? X _b>
\ 1 “b ‘ 1 (an 5  D
2 )
g '5 x '1. ’5
iii. The c056 Where 0 is the angle between 6’ and ¢
'81 \ b \\
C06 9 = " A A _,,_______
RH \b\ “’\
iv. A unit vector that is perpendicular to both 71* and (—7,6,—D
lms v. The projection of I? in the direction é "a b \\ p .I ‘6. = ——“
— = __ .—. {DO3
\PQQJ E&\ \ — "J b V" (b) i. Find the parametric equations for the line containing the
point (2, 3, 1) and perpendicular to the plane 2$+3y+z=5
X: 2 Y= 3+34=
2" \—+ t ii. Find the equation of the plane containing the point (2,3,1)
and perpendicular to the line a: = 3 + 2t, y = 2 + 375, z = 5 + t. \“i ll axerzﬁ Jr '2 9. (20 points) (a) Find the equation of the plane containing (2,3,1), (2,4, 5) and
(3,5,8). R7 = <0 \ Ll> g
l _) 3 N = q‘ = «gasp 6 —7<+A\/~'7'£= Oi (b) Let L1 and L2 be lines with parametric equations L1: cc=1+2t, y=2—t, z=4—2t
L2: m=9+t, y==5+3t, 22—4—75
i. Show that L1 and L2 intersect. Find the point of intersection.
H9¢=UH~5 ("IDJD—R)
,._t —= S‘YBS ~ ‘ (—1 3 1D “3‘7
— S
3—h ' “*6
W =— =1"
9 = mns 5 9~ ’6 3 ii. Find the equation of the plane that contains both L1 and L2. (aﬁb’gb $3: (n)o)7§
<\) 39” jKAla X \~ '2 '' 5
(c) Find the distance from the point (2, 3, 1) to the plane 293+3y+z=5 \BC'aD ) 3(334 \CD ' 5\ DEV": ___ .____' W W 10. (15 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
If 2 0,7123" converges at x = 1, a) “:1 @ F 00
then Z an converges.
n21 00
If E anx” converges at :0 = 1, a) “=1 T ® then 2 anus“ converges at a: = —1.
71:1
) If the 7th degree Maclaurin polynomial for f is T ®
C P7($) = 1 — m3 + x5 — 3:7 then f’”(0) = ——1 4 6
d) The Maclaurin series for a: sinx is 3:2 —— + . . . ® F If f is continuous and lim f = O °° T Q
6) then / f(a:) converges.
o f) 71:1 00
If an, (7,1,0n Z O and an + bn 5 an and Z en converges
m as @F
then 2 an and 2 bn converges.
71:1 12:]. g) @F ...
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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
 The Land

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