This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 122 Final May 5, 2006
1
If:
3
Name 4
u_5___l
6
‘7
8
l
9
10
Total / 200
Directions: 1. No books, notes or Steak and Champagne Dinners . 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. 2. You should show your work, and explain how you arride 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. 3. 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. 4. You may leave as soon as you are ﬁnished, but once you leave the exam,
you may not make any changes to your exam. Have a Safe and Happy
Summer 1. (20 points) w2+1 ——————d
333+3x~16 x (a) Compute / am. ?>
‘ﬂ waxwm "“ cm «s: $243+» 3;; d x "x , k.
% 3452+ i réﬁgﬁgﬂﬁ K; W:
V3 j) '35. a}? N ﬁux&x 2. (20 points) 23 a:
——————— d .
(a) Compute / 362+ 393+ 2 at
ya m @393 (A &%‘&”‘>ﬁww,ij3
wwww m MW 4*" T7273 wigéiw
MM‘")£7M’3~§ “1M” ‘3 A M
. W m
x “a 3 ,..
7’; W19” ~22; M YEE; $33 :2 5:3“ “E “LL” us. h m w wvwwm mwm‘x »%« w 7mm gawk.
1 Q \ 5% 5%) (EC?
d1!) “3:” WMWWWWW M w
(b) Compute / (m2 +(9)3/2 E ZN ,N'é;
“J éaa’mmﬁ WK 3
>5; m 133. TAN <33 4‘
2" "‘75:" M I r M Q
if“ x3: inﬁﬁff‘éﬂﬂégg ﬁQﬁgg} ‘ﬁ‘M
MM’MK
Mﬁiam R‘ 0‘ {1193? my
2
2 N m g
1
(0) Compute / m2 _ 43: + 5 (13:. r: j {if} X
1 § {yaw 2;; E
’L
w W “m maﬁa“ {wwwk‘x m» (I) m m :2.“ “.33: ‘\_ 3. (20 points)
(a) Use Euler’s method with h = 0.50 to approximate for EL 2y W) = 1 (b) Find the general solution for the following differential equations. i. % = Zia/y" 1 éiw 22% a K i% a >< Jr a
rw”m‘“”““§ WWW ’2‘ 4
WW ‘4’ \R W; L15; 2.321%
WWW w W , a
Z W 2.
~ «i: a;
4 NE; W é ﬁwfgfmm E k \
d _.
11 273% + y —— Sincc E N ‘F‘iw"% f V x; rm”;
E \f a: ML, \5 E ‘sgﬁﬁ a?“ (Rx a €13 xx § ‘‘‘‘‘‘‘‘‘‘‘ 3E1: mmmmmmmmmmmmmmmmmmmmmmmm __ , WWWMWW ‘‘‘‘ “W”: W7“?
mam m 1 ~
aggro” :j N; ﬂ__ “L S a m a ‘9}: a» C; \ X 3% a x ﬁe «w j 4. (20 points) The United States has $ 300 billion in paper currency in circulation and each day $ 15 billion comes into (and out of) the coun—
try’s bank. The government decided to introduce new currency fea—
turing portraits of famous Mathematicians. 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 11(0) 2 0. (a) Write a differential equation for how much new currency is in circulation.
‘” l W“ ., m a» ML if M l " {~33 ) an
6M2: “
“ “an 17“) (in 1 t
l r l an  \. E MAW H
kw 3,, M
will "i
‘9’!“ w (c) How long will it take for the new bills to account for 90% of the currency in circulation? GE} «:2: {:3 W‘" 5:! {flﬂﬂ 533::
W; a
w 334:1": at? i K) t i in a W W a ‘ a w v . M a: a mi»
33:: W K: “m @6223 :m M“ m)
V it “i
a wamwnmimy
(d) Solve y” » 53/ + 6y 2 0
W 2% e. a}; €23 air: a, ' i; €33 5. (20 points)
(a) A curve is given by the parametric equations: :rzacost y=bsint where a and b are positive constants.
Find the area that lies between the curve and the art—axis for U S t S 71"
reﬁ W
4? Chi) g 5N4: E3 W M 4i: C)”: “'3” G” b E “E”:
\ Q C} aim
_ fﬂﬁﬁ
W m ‘w { “E! w 5 ‘N w W “5: W 2;; (b) Set—up an integral to ﬁnd the area of the inner loop of 7" z 1~2 sin 6
Do not solve the integral. gr w 2331:? @e "Q V .paLW
5:"
s (3:2 '2 Q“ N 1‘ ring“
R a» it $333: an;
“£12: :1” 6. (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 2,” I ,W‘ ‘ngr Kraut»
1‘2”! 77,2 QQEMQ lb‘l n=1 w H a ,3 I” k N it ~ F
cku’lmw “fawn” w» W MMWWW.WW: :jrw “£1. “E Nymt‘ﬁm {NM/ll. (New all <13»; ran n22
3 no? Mime??? WMWWWW‘WWMDWMW» ‘\ MW“ r \
\ “I; “Wrmwg é 32”“ ‘3 W W" m j
k WT, \g Néw W W ‘ 'sille‘ee at N
N am W V MEWWW :1 ML 4: E {:Q M ‘3:
New ‘ﬁr‘ﬂ Mm,“ ,,,,,,, c 4 «NW 5%???“
(b) Determine if the following series 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. OO (— 1)”
1
“=2 x/n —— 1
Cam? W‘TTM’ijflm l? “W; {D i a? {ﬁg TVS“ h‘ﬁmf”
l m w l \l w
i
52% “W”: """" r ’ :2 o [fitti’i‘iimjw if “2: Lowe
itme “fl ‘éh‘wl ""3 KW Ml ﬁrm if}.
00
 3e” + 2 2<~ w n
“=2 71 + 56
N Writs, t ; {few 4%“ 21‘ g I a» f ?
TERM “MW m‘”““’”““"””w‘f"=€i" 2% Q {I} ‘i v, . (20 points) (a) Consider the power series: 00 mn
2; 2724/77, — 1
1. Where is the power series centered?
ti: 2:: CZ}
ii. Find the radius of convergence: xi M a E N “gem; iii. Find the interval of convergence.
(Hint: Check the endpoints.) WM an
“3:? mggﬁrxmi “m L5 3 V “If
New» 3» i N X” ,
Q ,3” (n3 (:0 “j i “:5 W5” «fww mwwwfimﬂmm
KW a“ in; mi,
(b) If the Maolaurin series of f is
f(:r)—:z:+m3 +355 +337 +1“;
’ " 3 9 27 81 "‘
1. Find f(1) 3W. l A 1 2:“ m W i W. ilk
N at) «‘55
ii. Find the Maclaurin series for f
is «W “’5
" ‘ "2.“ m » w; ELM Am y: . n X ,m
MW 2% w a —
iii. Find the Maclaurin series for 1f "1!) ‘3 7 N“ m
35% at "EQM « _ A, i A) ( " w J? x”: “v ’45 4r KL} .m m i.
E} “r” mg “i” : 3f? m 1 ~§ >4? . , K n g: i ~55 Iggy
x3“ ’4" if “‘3' ~§Tm ,_ m 33 3:? x W
“x” «g, 11w mg“ in 5 a? 8. (20 points) (a) For vectors TL’ 2: (3,2, ~1) and 77+ = (133,4) ﬁnd:
' ..—) H
1. a  b
53
ii. 5 x5 m n » / ‘ . n.
pf) 01 W l ’ _ “l
% Ta») H; all} iaj \> iii. Find the area of the triangle with ’5’ and —b> as two of the sides. wWwwWmevwmww«Hume
‘l 2224 at XQGE a ‘1”? 0i EL iv. Find a unit vector perpendicular to both E and 5 v. Find the value of a: so hat 7 = (3,23, —3) and H’ are per— pendicular.
OH 432% we a :1 Q
~34 m m {In (b) Find the distance from the point (1, —2, 3) t0 the plane 55c—2y+4z==1
\ if: “i “i i, in w "i l (c) Find the point of intersection of the line:
:z:=—1+t, y=1—2t, z=3+3t and the plane 23: — y + z = 7 W422 a at w ’3 "4+ an a "1% a" “E; is: :7 Ml  j {at} n ~41; s:— "7 3 10. 11. Mediavision Week 14 2132 + 1
. C t f — d
ompu e $3 + 336 M 16 a:
. Compute [(ln x)2 dLr
sin2 90
. Compute / dx.
cosa:
Com te / x d
u — :c.
p x2 + 3.7: + 2
e t f 1 d
. om u e ————— a:
p (x2 —l— 903/2
2' 1
. Compute 1/ m Use Euler’s method with h = 0.50 to approximate M2) for dy_ dm — 11(0) = 1 2y . Find the general solution for the following differential equations. %=2$\/37—:T 93% +y=sinzc (a)
(b) . Solve y” — 5y’ + 6y 2 0 A curve is given by the parametric equations: :L‘=a,cost y=bsint where a, and b are positive constants.
Find the area that lies between the curve and the :r—axis for
0 S t 3 7T. Set—up an integral to ﬁnd the area of the inner loop of 7“ = 1 ~ 2 sin 6
Do not solve the integral. 12. 13. 14. 15. 16. CO 271
“21 n! n2
00
2
72:2 (a) Consider the power series: 00 11 £17
7; 2mm — 1 1. Where is the power series centered? ii. Find the radius of convergence. iii. Find the interval of convergence.
(Hint: Check the endpoints.) (b) If the Maclaurin series of f is i. f (58) Find f (1) x3 11:5 :67 11:9
=$+§+§+§§+§i+.n ii. Find the Maclaurin series for f iii. Find the Maclaurin series for f (00) I_$' 17. For vectors 71" = (3, 2, ——1) and 7; :2 (1,3,4) ﬁnd: i.
ii. iii. iv. _>
if!) Ez’xb Find the area of the triangle with 6’ and 7;) as two of the sides. Find a unit vector perpendicular to both (Ti and 5 Find the value of :1: so that 7? = <3,$, ~3) and "E? are per— pendicular. 18. Find the distance from the point (1, —2, 3) to the plane 5x—2y+4zzl 19. 20. 21. 22. 23. Find the point of intersection of the line: 31=~1+t, y=1~2t, z=3+3t and the plane 23: — y + z = 7 Find the equation of the plane through (3, 3, 3) that is perpendicular
to the line
2 r: 3 ~ 525 $:4+Zt, y=2+3t, Find the equation of the plane through P(1,1,l), Q(2,6, —1), and R(O, 0, 2) Determine if the two lines: m=1~4t, y=2+3t, 234—21;
and m=2—t, y=1+t, 2224—675 are parallel, intersect or are skew. (a) If 7 and h) are non—zero vectors in 3—din1ensional space. For
each of the following equations, describe in pgin English what
the equation says geometrically about 5’ and b . (Don’t use the words “dot—product”, “cross—product”, etc.) i. 717:0 ——a —>
uﬁxsz
—>——>
iii?7f=b~b
—>—>
1V ~c't“)><7i’=b><b 9. (20 points) (a) Find the equation of the plane through (3, 3, 3) that is perpen—
dicular to the line x=4+2t, y=2+3t, 22—3—5t (b) Find the equation of the plane through P(‘1,1,1), Q(2,6, —1), and R(0,0,2) g:ng « A g am $23 M M V Wm“
“3* W M his; genre/egg, met}
ﬁsﬁiwgi} W I’M ‘ A 3 avg \ H “mm m‘ R h N u
rte W emwéww‘ijtj
m g
3:3 9%. we ‘35 % «LE ; 5’ (C) Determine if the two lines:
= 1  4 = 2 t z = 4 — 2t ‘
a: t, y + 3 , <1 E 3
and x
1:2—t, y=1+t, z=2+6t { are parallel, intersect or are skew. WM 2:: :13 K: {ﬁx} {3d e» , x 1}} ﬁt»
g
new
if 10. (20 points) (a) If “a? and 7? are non—zero vectors in 3—dimensional space. For
each of the following equations, describe in pEin English what
the equation says geometrically about “a? and b . (Don’t use the words “dotproduct”, “cross—product”, etc.) 1.?‘720 i3 Pram £3 a the», W 6’
n a x  V, W l n _ x
StaW‘s m a H32 M
——>
—) —> — M. w «,‘z w «r: s
111 a  a — b b 59:“, 53W“ 2;) Luigi «.3 Qt”? ta,
——> v~>
1v 7'? x “d’ = b x b N crass t rxl raw 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 ambn, an > 0 and an + bn < (3,, and Zen converges, 71:1 00
then E an converges. 71:1 00
If an,bn,cn > 0 and an < bn + an and Zen diverges, 71:1 00
then Z 6,, diverges.
71:1 00 00
iii. If an > O and 2 an converges, then 2mm? converges. 71:1 11:1 iv. 1+l=3 v. I enjoyed Math 122. ...
View
Full Document
 Spring '07
 Butler
 Maclaurin Series, Power Series, Radius of convergence

Click to edit the document details