Exam 2 Review Packet Answer Key - Amherst College
31
ExamHZReviewpacketanswe.ir# Integrals complete Square b t - 2 I ' - f !×I4×+Tdx= fins . I ? dx - - final , at du b' - Yao - he - yeah , =¥7a tarpan (E) I ! -2 × 12 A 16 Cx - A- xI4xt4 - - fig , # ¥tan⇐t) - arta⇒] ¥EI = # 1¥ . left converges 2 . Ix¥sd×=¥ .SE#.pdx=f;m.Ynttiii..e;m.znhent t ¥x isfyeg I converges 3. go.ca#y*gdx--fsimofot***gdxEofism.J ! # - Ifs " p.F T - = fish lnlxtsl - lnkxtslf! ' ( Xa)l2XtS ) " x¥t¥ ) = fism.eu#I-enkttsI-ln2tenS I = A (2×+5)+131×+2 ) = 2AxtSAt Bx -1213 = lfhfa.hn/ttt+Igtftq/-ln2t-lnS--(2AtB)XtL5A-t2B)
Transcript of Exam 2 Review Packet Answer Key - Amherst College
ExamHZReviewpacketanswe.ir#Integrals complete- Square
b t- 2 I' -
f !×I4×+Tdx= fins.I? dx -- final
,at
du
b'- Yao- he- yeah,=¥7a tarpan (E) I!
-2
×12 A
16
Cx -A- xI4xt4 -
- fig ,# ¥tan⇐t) - arta⇒]¥EI
= # 1¥ . left converges
2 .
Ix¥sd×=¥.SE#.pdx=f;m.Ynttiii..e;m.znhentt¥x isfyeg I converges
3. go.ca#y*gdx--fsimofot***gdxEofism.J!# - Ifs "
p.F T- = fish lnlxtsl - lnkxtslf!
' (Xa)l2XtS)"x¥t¥)= fism.eu#I-enkttsI-ln2tenS
I = A (2×+5)+131×+2)= 2AxtSAt Bx-1213 =lfhfa.hn/ttt+Igtftq/-ln2t-lnS--(2AtB)XtL5A-t2B)
Conditions⇒ 2AtB=o ⇒ B= -2A =hzi.7al.nl/ztIft#f-ln2tln5-5At2B=ltSAt2t2A)--
I
A- =L ⇒ 13=-2= duly,) - lnzt lens Converges
= l# luz- tu 2 thus
a;÷±Teriyaki,h④
" ftp.g#gdx=fi7aftxLT*gdx--fimLty.*zdxux⇒⇒u -
- fin )!:# du --ftp.tfzarctanf#yt-4du--dXX--t=u--t-4"""""''' '
¥1 Iain"¥F¥E÷÷¥;¥EiIComplete
squares '
f? ¥+4 dx-ufiyzf.tn#+ydx--figgfzt1-dx(x - c)73×2--2×+1
I t, ¥7.IT#dn--Eim.t.aruanEr.H!"
=n*o¥.
'
Ii's:*.
Icipher
6- f ¥×wdx=jµ¥dx=Ju÷du=arctanutC=arctanCxt
[email protected]#iiiix--eig*.-fYIuan=ei.%Eenluiliost
u -_ cosx ×=o⇒u= cos0=1
an-- -sinxdx *t ⇒u-eost-li.yq.li?I!ttenTo=-IiIFI@-du--sinxdx
8- ftp.#ydx--Eig.E*rdxux⇒⇒
'¥iI÷iii=¥÷÷µ. "
du --ax x=t⇒u=t -4
=¥hf¥y¥yI÷=⑨ Diverges
O
"
fjfedx -
- final :#dx -- fig, tudufimlnlulleii
a¥dxYt⇒⇒%[email protected]! dx=¥%+2#"du -- fimoieuti
' Etoile -'
converse.
" - fiolnxxdx -
- being filnxxdx -- fiyafontuda-fiya.ua/fnt=E÷t¥ -
- ⑨ Diverges
Lu-2- 5
~
" '
1×24%7 DX= I 2¥j dx
-
- Jif du = gzu④ dusplit - spit
!!" -0 x⇒* = 4¥ - Yuk. du
= Lulu 2t l l - 7
arcthnutc-lnlx.tt#-7arutanCxtDtTis .
joggedx-
- finna. ftp.#k=f;mo+f
'x=t⇒u .
e! =¥io+ rule!
'
du -- exdx x-- I ⇒ u= e - I
-fight 2€-④µ° = Converges
14 .
ftp.lnxdx-fijffylnxitdx-liyotxlnx/I- f ! I dx
Converges
÷t¥a×din ''
em't'- He'¥÷ti÷-too
O
* ¥*Eiit%¥%e¥¥¥¥¥.it?ei:oiti--ois.fi#shdx--finsrfot*p.--fjY. I!¥zpd×
= hi ios =¥%¥I÷. do
Yog) >↳ costs-0
=L,'mX=t
° t-y-fseco-do-limto.no/X--t⇒Fx ✗⇒t -11-
X-0
= him X- O =② Divergest.ir#/ot--limu-n-,
16 - f.) ×!¥d×=f°×¥+,od×+¥¥→d✗⑦
b2-4ac= 36-4117110)= hims→→fs°×¥→d×+¥Y•|?×¥→d×
u=✗-3 Complete
a Eteimc- →aol.tl#+,dXs-s-aofix-!p+-dx+eimX=o--u--
-3
=÷.gg#+eimt-u.+.- an+→ • )-3
OR split at 3could have made
nice values= him antanu + him avctanuf??s→ - n t-so
k →
= tianya.am/tan-DO-ar/tanlT-Dtfi-yzarctF#-3T-ar/an-3 ]If= Tyzttz = Converges Both Pieces Finite
ADD them .
" I.it#x.dx--foscxIIyx.7dx=EEstI:cxfsTx.dxPFD
x.IT#--fIstfzif*sxx-is=ti:-stIia.I,-xIgaxx--Alx-isi-Baas,
=¥7st.#Htsltt.lu/x-i//I=Ax-AtBxt5B--AtB3xtsB-A--fiy.s+flns- then-ofqen¥si¥en¥iI
Conditions" AtB=l ⇒ 13=1-A
- SB -A- O 6 = ⑨ DivergesSH-A)-A=os -SA -A -- o
6-1=5
A- = ⇒ 13--1-5/6=46
18 . Jxsxztydx =) x'txt dx=Jx3txt%z,- ×÷dx
(x - next- D
txsnx t-dtxftzenlx-y-tlnlxi.tt#x*¥'yX/t2-lxXt 2
PEI
Xt 2Et://T.fi#txIiH' ''' to::iIF⇒*.Xt2= ACXTIJTBCX- t )
" A - 13=2 • I- B- 13=2
I - 213=2= Axt- At BX-B 213=-1
= (At B)XTA-B D= - Ig ⇒ A- = I - Hk) -- 3k
" ' I!¥gdx=dxtf.jo#dx--sligz-fos*.dxtfi7z+II*tzdx=a÷. t
a ;÷l :-
A"'
÷*¥I±I"
i,
Hottoo i
,
too here
Diverges'
i, Diverges!
Original Integralgesbk one piece Diverges (BOTH do . . .)
Note : Both pieces finished to check answers . . .
but you only need to finish one piece .
if Divergent .
"' §×z¥+zdX=¥7oJot×¥d×=¥3Jo¥pd×PFI @ =L}:)! -¥zt¥ ,
DX
t¥¥¥i=x¥tx¥¥ktD =¥y.
-bnlxtzltlnlxt 'll0
I = AHH )tBlXt2) - O N-
= AxtAt BXTZB= tiny , -¥t4t¥il- f- ln2tl)= AftB)Xt At 2B
Indeterminateconditions Difference
'
II'¥⇒I÷÷⇒⇒=¥:.int#tititten--ti:.enlten
¥g Converges
21 . foan2Xdx=¥y gotta'xdx=fgz . seok - ldx
to
es. =¥y% - tax - xl !
-
- fi;÷[ tant-tE¥o- o)
÷¥÷,
=@ Diverges
n.pe#..dx=fiusz.I.t*.yd*Ei7I! """
*t
4wioczwsot-fiz.IE?oTodo---figtyfseeodoX=ox--t€y¥
.
-
- lien. # tanolx-o-fins.tk#.hE!Z
Itis I, !¥?¥,T°=⑨ Diverges
zz . J4x47 dx = f # tdx=f¥t¥.it#qydxXt4Cx2t4)
PFI
=¥st¥¥,y×+⇒c*,=hkt4tZenhitHtIarctanH|4€xt6=Alx74ItfBxt4Ht2)
=Ax2t4At Bx't
[email protected]=4 ⇒ 13=4- A
- 2BtC=7 6214-A) tC=7 ⇒ 8- 2AtC=7. 4At2C=6 c- 2A - I
4At 2. (21--1)=64A t 4A - 2=6
8A=8
c-- 2-1=1⇐ A -_ I ⇒ 13--4-1=3
""
f,⑨¥jdx=¥m, .LI#,dx--fi;sf!xt-xt+idxPEI Illya lnlxtlnlxttll !x¥⇒=¥±xB⇒""" J! M
I --AlxtDtBx =fY. Wenz)Indeterminate
=(At B)x +A Difference
conditions• At 13=0 then 2a -- i ⇒Ba
'Eisen )"
¥1↳Orgill
O
= lying.lt/y#yotln2--④ Converges
as .fi#ydx-- " life .- I ;*¥⇒d×PFI
¥n¥¥⇒=¥.tt#fx-xxto=E:z-lifE-ffdx( €Alxt4tBlx- z) Ishim kylnlx -4 -4lnlxt.cl/zt=AXt2AtBX-2B
dull lot= HtB)xtut-ZB-eim.z.ly#ft-4-QylntYtfy-ltTIf-tyln5)conditions t "
I -41. At 13=0 ⇒ D= -A
- 2A- 213=1 6 ⑦Aza -d-A)=/ =⑥ Diverges47=1
A -_ 44 ⇒ D=-
by
26. folarcsinxdx =) !arcsinxtdx-xarcsinxlol-f.PT#dxIBPIY÷*Y
""""" t'- fi;iµ¥.
""""" " '¥:#sittin'
- xarcsinxlo'
thing ,.ru/i-t2=arcsinTEo+ein.r*.?ra! 'es
" f ax -- txt x-
-
fxtff.tt#dxLongDivision=xI+3lnlx2HltarctanXtC
x 'ttxt- (x 's TX)Txt
28 . ) ¥¥Tt6x2d×= J × +X75Xt# dx
Htt ) (X'ti)
Htt) (x71 )
-
x3tx7XH
longtivision =/ X - II ,×
x3tX2tXtt¥¥xt6Xt2 IXZ-l.in/xtt/tlnlx2tilt3arctanXtC#tYIIsa
PTIT
=¥it¥'H""" "
X 't 5×+2 = A IX't 1) t Xtc)Htt )= Ax2t At Bx't BXTCXTC
= (At B)X't t c)X t Atc
conditions' At 13=1 ⇒ 13=1-A
. Btc = 5 l - ft c -- S- Atc = 2 ↳ c =4tA
At 4th = 2
2. A = - Z
A-= - I ⇒ 13--1-1-13=2
⇒ c-- 4-1=3
" f dx= finna f! e¥adx= figo +
-fi,
e" du
ttoo
÷÷¥÷ ':#hi. o.ie
= ⑥ Diverges
Same u-sub
t
so. dx -- figo - f!E
.. .-- Eino . -e
" I. .
÷= him - e¥oE e
"= ⑤ converges
t→ o-
31 . dx = [email protected]! :# ax
¥i."" ""
¥÷÷÷÷I"t
.
du +⇒"
÷ ..
fish. I ' I aretan (F) Isit lying.Is . Is a rot an⑦ to'
F'k-
-stig.kLantana an III;m. 'afI¥E¥ij
-
- ta ( IL) + 4th)= Converges
zz.gg#dx-- fins. fixe'
"dx=fi%z¥[ III!-"joke . "=3
IBI t
a- x dree-"ok ¥7 . 5¥17 - text
,du=dxv= -% O
'Ii: :# ÷. -4¥.
ate.
Fitting f÷t÷ the = converges
eX-ks
33. fgelhfydt-lingofflnx.x-kdx-hi.gg. zrxlnxl! - 21¥ dxIBI =fjmo+ zrxlnxl! -45×1 !
O ← f- N)
=¥÷.ru#.*e.o.ureIuE=2fe-4fe=-2feOConverges
I- zt't
④ fig. ii÷¥%+e¥%=¥.mn#--fi;no+-arF--o⇒E' kw.
-÷t -3k
g.simplify %, or indeterminate
%
Sequences-
191%7" hiya
"
Yg, =L;gF = converges3 t#+9¥
" ' tis. hi
.
① converges
" ' fig. City
-- dnigzf-EY-figdtszf-exi.tn#-%59N
' O
= Fo④left-%)
=
@figs dnfts
%
--
Converges
etin. eexi.io#*Iss..e-sg" tins . him.
=④ Converges
° % expand
@ntzyLzntzylznti7l2uJ-i-5-4.3X-fism.ca.!y,=⑥same
38' ftp.arotan#t7I-- Converges
-
rx
satins. i÷% -
-
.¥e,"
÷ E: ÷#¥⇒= ein 4¥ is
. ¥¥i"!e.in.
#XTN L
'H
"o - ein.i in
=÷m. xxx =e¥%④=e¥. ± . e:""
=
eGing %=
efish ¥5:
= e:① converges
L' It
41 . figsIii in Hnl -- fins x. sin Hx) = exits singly
%
=¥zcosl"x!I= lying.=① converges
" fish. EI)" = fig. 4¥ Y -- fight = e¥FolnHx9l im
= ex-so④left thx! e¥z bnl
%
+ E%
=②- learn for ratio Test .
IN A
43 .fi#Y--fig.f*xY=exiholnlxxtTT9a- o %
=e¥sx¥t.ee#im.enxHT4
÷.ee#oxiHTl*iIiIi*I"
[email protected]" converges
-
- elxihs.IE?=..exi:it=e''
{or#Yx
Seviessums
"" E.
E. Inte
.IE?.iEt+E..ET--E..ztnt.EE.s=IztztztzIzt. - - = tztztzt # t ' " a=kz
a- 43✓ = 43 Cow .
GST 14=11321 Cow.GST 14=1/24
" = 't
sun -- Fi't.it#sEI+sun--Fr--'
Ii."¥= '
✓Sumof 2 Convergent serieshttis convergent
n= , n=2 4=3Thot needed here
45
?= -¥ t - i bkdiohitasklo
na prize ConvergeUr=Z
3
9 ⑥ nvergent GST Irl = I-431=43not needed for Justsaw
"Fsum. =¥⇒=¥=⑤
"÷n÷÷÷÷÷÷.
" "
am.±±÷.
a= -234=-1 Convergent GST lrkl -3%1=346- I
8
r=÷=÷sun- ⇐¥⇒=i7÷'i'
u -_ I n -- L n =3
ME t ;÷-4÷t . . -
httw
br= -1
a32
✓not needed for
JustSumA= -13 Convergent GST Irl=/-41=4-4"¥ 't sun -- ÷¥⇒=I÷i¥
monege.mg
Not: only pass to Absolute series if" Helpful
"
Convergent, to use Act .
See 1*1048 . §.mg?-fY=ulim2n3-lnn-K3--lim2-lII=z=ou-n5ns -9 Yup " + °
-53*0⇒ series DsbynTDT
A) dim×,,¥yY*= him#0=0x→n×¥%=lim
✗→ a 3X\g
PreconditionsPositive ×> I49 . §
,
ln÷EE fix -_ln¥
Continuous ✗ > 0
DecreasingCompute Integral fllx)=X4kD-ln×l
- 2 ✗4
f ?ln¥dx= limftlnx. xdxtin I
×
-2 =×-2¥yu=bn× dv=x"d××d×v=¥=¥=¥%¥HIt¥d× =;¥,
= 1-2×-4,11<0 when=¥%¥I¥liI -21in✗ < 0
=eim=¥¥¥%Il iaaenx+→ to
d-< dux
e eEa ¥7, →! +1 = ① Integral converges
✗ >Teak.
⇒ SeriesCo by I.T. [Note : could also run CT if use]Bound bun t.TN . ..
So - €,÷z Nn €
,
F- = §,
÷ Converges p-series p=% > I
÷a÷÷:÷:i±÷¥÷e÷:÷÷. "finite
,Non- Zero
⇒ 0.5 . alsoConve by LCT
" E.
' E. hit --E.tn'' '
'
Eze:S"TEi"
-
wht 40n6t4n3tl9 n
- hut 40Mt 4h"t 19h 4h20
dim 4t-1.IM/--limy+#nhaohtt htt
-
- him lt¥i¥.in#Ioh-d4hfyotnFfsotl=Hnile, non-zero⇒ ons . alsoD by LCT
52 -
- rim. :÷÷e÷÷÷.e÷¥i-
- fi's.xeEo to
ss.no?nEstE--E..IstE..sti.6
Constant Multiple of Convergent Geometricconvergent p -Series10=571 is
Irk Ys Llf not needed
Convergent [ by Arithmetic of Series )V s
Sum of 2 Convergent Series is Convergent
[email protected] : CT not helpful→ bound in wrong order ]
iii." :c:÷:÷÷*÷÷÷...
⇒ O - S . alsoge by LCT positive x > 1
Continuous X>0
55 - §z÷p - fu,y=÷×p= IXHNXT)"
Decreasing when
compute Hx) = -fxllnxpjjxflenxlb.ly/-knxYllDf#enpdx--fismfExTexpdx= - Cenxlbfi -end
=-7t co
a¥* ' fish. du 4¥72 x' thnx)'
to ①
-I but when → tlnxco ⇒ thx > 7
'ftp.bulenz ⇒ x > e' Vox .
=¥7.-#eyu¥°I¥y6=÷y Integral converges
⇒ o - S . [email protected] .
56. €7,arytfnh of Tyz §
,
# Constant Multiple of convergent
p- series p--271 is convergent
Bound Terms
YII÷eTae and) ⇒ os .es by et
% LCT
€44372 I Cow . p- series p-- 2 > I
eimarotann-jien.is. attain"
the;m.¥÷o⇒.n-7A Ith 2
Ia2 Finite
,Non - Zero
⇒ o - s .es by LCT
0kg Integral Test . . .
57 - ZI,
na!fy Ds by NTDT He
-.
II. an-
- fins." → to
.
The
58 . &, }nz I ÷ = €
,
utz Converges, p-Series p--271
-a
2ntS n2
him Sn't 3u2
m.ein.:¥÷"Iie÷. es
finite
⇒ O 'S . alsoes by LCTNon - zero
a htt
59 . Z ft )
u=,¥3 ⇐ nz¥z " IT Converges , p-series
p = 271
Bound Terms
÷" nests ' ut and]
as.C #T⇒ A 's . also Converges by c.t.
by ACT Remember : ACT gets a Conclusion from A. s .
only helpful if A - S. Converges .
Go . It ges by NTDT b'c
Ti2
him an -_ him -1¥ = 2 tous. m. '
¥7421161 . In
,
3T In Ds by nTDT ble
II. an -- ftp.3#iI3Fo .
62 . Iz,
eh Di by nTDT ble
1h18=e° -- I tolingnan -- Gimme
63 . E. ftp.#.--6?.TtEnFix• : ¥.
a
::÷:'
, p-- 671ConstantMultiple of .
i BoundTerms
Convergent p -Series f-6>1 :
" ¥56 andI
is Convergent ;'
; ⇒ Series here Converges'
.. by CT
✓Original Series tsk
+series convergesSum of 2 Convergen
64 . I,
cos-
(Tiglath) ges by NTDT ble
kisan cost
= lui;z cos' (Yt = cost -_ Eh! Ito
④
65 .£41 ""%IY E. [ =Éu÷ converses
n=l"
H. lpful"
p- series
4=1 p=2>
Bound Terms
Zork faround± and
O.s.IE/rges ⇒ As . Converges by CTby ACT
[note: Os . A.c. but not needed here]
EvenMoves Note : here must Analyze AbsoluteSeries (As) btcQ-A.c.cc , Diverge?N
66. £ C- """
¥ [ 1- ← { In Diverges Harmonic
n=lmust
n= ,5h14 n= , p- series p=1
2 LAST note : CT not helpful here ¥ Ent[ " smaller than Diverge "not helpful① bn=¥z >0 ↳ inconclusive compar
)LCT Limit
② limbn-ni.gs#.i--o eni.gg#;-i--limh-h--lim1--- Isn.in n→n59+1 Yn n> a 5T¥
,
③Terms decreasing Finite,Non- Zero
µ.,=¥,,,÷÷¥. ⇒a,ig , ,,¥, BYAST ↳Not A.c.
✓ ↳ Test 0.5 .
0%1-1×1=5%+20.5.É by Definition[ ↳ f 'HFÉ¥y
④
④" E. '" I:# ÷÷EH÷E÷=E÷s%÷÷÷.
Let limit
÷ :÷÷i:÷÷÷¥÷÷ii÷÷. .Finite
,Non- zero
⇒Agesby LCT↳
o .Absoluteywge by DefinitionA Done . Not necessary
N
g2h
to analyze 0.5 .
68 E-na
(2hH) ! lun
ri . ysmµ←[2ktDtl] ! lnlnti )
t-eim.IT#=e;m.
Ent ) ! lnn
5€52' him.
"s refining -H(24+3) !/ H) 2n¥
-
- ftp.enaf#si--o4⇒ O.S.JAbsolulelyunvwgentbyR.IT
* fine.:*: i:*.
"
:* .÷¥¥÷¥÷÷÷¥for
Divide '¥)
69 . I n ! - nb . n"
na 10¥ ⑤gboth !
104N - e2h
R'T
¢tn!ln'
I m
e-÷¥nt÷f÷÷÷÷t÷: ÷÷÷
.X
extract . .HI "%)
'
a e
' tis. an
⇒ O - S . Ds by Ratio Test
" E.e'" ÷ :÷ E.¥ .EE "
:[CT order on Bounds not helpful]
④lastaim.¥÷i=q÷¥E=e÷÷*:#① bn=I >o
4nt3Finite
,Non- Zero
②nhiyfbn-nhjm.ly#% ⇒ As. Diverges by LCT
'
Ii:÷÷÷÷÷÷÷. .
41¥ ✓04, fly=L O - S .dihayCgewt by Definition
( mix'i÷¥⇒!warning : Remember
@TtostateC.C.byanyTest7l.ELnl.PeZnn-iC3n3l.n"
-
" ÷tan÷t÷/:÷÷÷f÷i÷
④¥13 2n-12
* É¥÷¥i÷*= him
×⇐④*É=÷¥¥÷÷÷÷x:÷⇒÷.= him"""÷#q;*:*
¥
5extra Inti )
= dim makes L→ 0
→ ÷ !¥!-04
⇒ 0.5 . 2 by RT .
"
&,
eY;gtann_ Es ?§÷I I 2¥ converges p-seriesmust
p⇒ > I
not RT.
limn→. T.FI#-ji=eimarutann.nI+-nk;n.)n-x÷
=lim aruÉnÉ=¥u → a
OR,run CT with Tf Ent . Finite,
Non - Zero
[÷¥÷÷i÷ ,"
÷÷÷÷÷⇒aug. but⇒ O . S
.T by Definition
73 . I (3n)!na
8"
. (u !)? n n
R -T.
C- IT"
(3CutD!)@till
÷t÷l t1,01,
= . ÷. ÷. .
CI- 8% Ht"! !
'
=÷
"""
-
'¥4te/ '
h "n
③ ③-
- iii. z E-see "
" te I a> see ⇒ o . s.Ds by Ratio Test
" E. ''"I Its E
.
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