devexpress asp.net pdf viewer : Cut and paste pdf pages application control utility azure web page html visual studio Calculus25-part756

10.5 Calculus s withParametricEquations
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Figure 10.4.3
Ahypercycloidandahypocycloid.
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Figure10.4.4
Aninvoluteofacircle.
Wehavealreadyseenhowtocomputeslopesofcurvesgivenbyparametricequations|it
ishowwecomputedslopesinpolarcoordinates.
EXAMPLE10.5.1 Findtheslopeofthecycloidx=t sint,y=1 cost. . Wecompute
x
0
=1 cost,y
0
=sint,so
dy
dx
=
sint
1 cost
:
Note that t when t is s an odd d multiple of , like   or 3, thisis(0=2) = 0, so o there is
a horizontal l tangent t line, , in agreement with gure10.4.1. At t evenmultiplesof , , the
fractionis0=0,whichisundened.Thegureshowsthatthereisnotangentlineatsuch
points.
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252
Chapter 10 0 Polar r Coordinates, ParametricEquations
Areascanbeabittrickierwithparametricequations,dependingonthecurveandthe
areadesired. Wecanpotentiallycomputeareasbetween n thecurve and thex-axisquite
easily.
EXAMPLE10.5.2 Findtheareaunderonearchofthecycloidx=t sint,y=1 cost.
Wewouldliketocompute
Z
2
0
ydx;
butwedonotknowyintermsofx.However,theparametricequationsallowustomake
asubstitution: usey=1 costtoreplacey,andcomputedx=(1 cost)dt. . Thenthe
integralbecomes
Z
2
0
(1 cost)(1 cost)dt=3:
Notethatweneedtoconverttheoriginalxlimitstotlimitsusingx=t sint. When
x=0,t=sint,whichhappensonlywhent=0. Likewise,whenx=2,t 2=sint
andt=2. Alternately,becauseweunderstandhowthecycloidisproduced,wecansee
directlythatonearchisgeneratedby0t2. Ingeneral,ofcourse,thetlimitswill
bedierentthanthexlimits.
Thistechniquewillallowustocomputesomequiteinterestingareas,asillustratedby
theexercises.
Asanalexample,weseehowtocomputethelengthofacurvegivenbyparametric
equations. Section9.9investigatesarclengthforfunctionsgivenasy y intermsofx,and
developstheformulaforlength:
Z
b
a
s
1+
dy
dx
2
dx:
Usingsomepropertiesofderivatives,includingthechainrule,wecanconvertthistouse
parametricequationsx=f(t),y=g(t):
Z
b
a
s
1+
dy
dx
2
dx=
Z
b
a
s
dx
dt
2
+
dx
dt
2
dy
dx
2
dt
dx
dx
=
Z
v
u
s
dx
dt
2
+
dy
dt
2
dt
=
Z
v
u
p
(f0(t))2+(g0(t))2dt:
Hereuandvarethetlimitscorrespondingtothexlimitsaandb.
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10.5 Calculus s withParametricEquations
253
EXAMPLE 10.5.3
Findthe lengthof one arch of the cycloid. Fromx x =t sint,
y=1 cost,wegetthederivativesf
0
=1 costandg
0
=sint,sothelengthis
Z
2
0
q
(1 cost)2+sin
2
tdt=
Z
2
0
p
2 2costdt:
Nowweusetheformulasin
2
(t=2)=(1 cos(t))=2or4sin
2
(t=2)=2 2costtoget
Z
2
0
q
4sin
2
(t=2)dt:
Since0t2,sin(t=2)0,sowecanrewritethisas
Z
2
0
2sin(t=2)dt=8:
Exercises10.5.
1. Considerthecurveofexercise6insection10.4. Findallvaluesoftforwhichthecurvehas
ahorizontaltangentline. )
2. Considerthecurveofexercise6insection10.4. Findtheareaunderonearchofthecurve.
)
3. Considerthecurveofexercise6insection10.4.Setupanintegralforthelengthofonearch
ofthecurve.)
4. Considerthehypercycloidofexercise7insection10.4. Findallpointsat t whichthecurve
hasahorizontaltangentline. )
5. Considerthehypercycloidofexercise7insection10.4.Findtheareabetweenthelargecircle
andonearchofthecurve.)
6. Considerthehypercycloidofexercise7insection10.4. Findthelengthofonearchofthe
curve.)
7. Considerthehypocycloidofexercise8insection10.4. Findtheareainsidethecurve.)
8. Consider r thehypocycloidofexercise 8insection 10.4. Findthe e lengthof one archof the
curve.)
9. Recall l the involute ofa circle fromexercise in section10.4. Find d thepoint t in n the rst
quadrantingure10.4.4atwhichthetangentlineisvertical. )
10. Recalltheinvoluteofacirclefromexercise9insection10.4. Insteadofaninnitestring,
supposewehaveastringoflengthattachedtotheunitcircleat( 1;0),andinitiallylaid
aroundthetopofthecirclewithitsendat(1;0).Ifwegrasptheendofthestringandbegin
tounwindit,wegetapieceoftheinvolute,untilthestringisvertical. Ifwethenkeepthe
stringtautandcontinuetorotateitcounter-clockwise,theendtracesoutasemi-circlewith
centerat( 1;0),untilthestringisverticalagain. . Continuing,theendofthestringtraces
outthemirrorimageoftheinitialportionofthecurve;seegure10.5.1. Findtheareaof
theregioninsidethiscurveandoutsidetheunitcircle. )
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254
Chapter 10 0 Polar r Coordinates, ParametricEquations
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Figure10.5.1
Aregionformedbytheendofastring.
11. Findthelengthofthecurvefromthepreviousexercise,showningure10.5.1)
12. FindthelengthofthespiralofArchimedes(gure10.3.4)for02. )
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11
Sequences and Series
Considerthefollowingsum:
1
2
+
1
4
+
1
8
+
1
16
++
1
2i
+
Thedotsat theendindicatethatthe sumgoesonforever. Doesthismakesense? Can
we assign anumericalvalue to an innite sum? While e at rst it may y seemdicult t or
impossible,wehavecertainlydonesomethingsimilarwhenwetalkedaboutonequantity
getting\closerandcloser"toaxedquantity.Herewecouldaskwhether,asweaddmore
andmoreterms,thesumgetscloserandclosertosomexedvalue.Thatis,lookat
1
2
=
1
2
3
4
=
1
2
+
1
4
7
8
=
1
2
+
1
4
+
1
8
15
16
=
1
2
+
1
4
+
1
8
+
1
16
andsoon,andaskwhetherthesevalueshavealimit. Itseemsprettyclearthattheydo,
namely1.Infact,aswewillsee,it’snothardtoshowthat
1
2
+
1
4
+
1
8
+
1
16
++
1
2i
=
2
i
1
2i
=1 
1
2i
255
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256
Chapter 11 1 SequencesandSeries
andthen
lim
i!1
1
2i
=1 0=1:
Thereisoneplacethatyouhavelongacceptedthisnotionofinnitesumwithoutreally
thinkingofitasasum:
0:3333
3=
3
10
+
3
100
+
3
1000
+
3
10000
+=
1
3
;
forexample,or
3:14159:::=3+
1
10
+
4
100
+
1
1000
+
5
10000
+
9
100000
+=:
Ourrsttask,then,toinvestigateinnitesums,calledseries,istoinvestigatelimitsof
sequencesofnumbers. Thatis,weociallycall
X1
i=1
1
2i
=
1
2
+
1
4
+
1
8
+
1
16
++
1
2i
+
aseries,while
1
2
;
3
4
;
7
8
;
15
16
;:::;
2
i
1
2i
;:::
isasequence,and
X1
i=1
1
2i
= lim
i!1
2
i
1
2i
;
thatis,thevalueofaseriesisthelimitofaparticularsequence.
Whilethe idea ofasequenceof numbers,a
1
;a
2
;a
3
;:::isstraightforward,it isusefulto
thinkofasequenceasafunction.Wehaveupuntilnowdealtwithfunctionswhosedomains
aretherealnumbers,orasubsetoftherealnumbers,likef(x)=sinx. Asequenceisa
functionwithdomainthenaturalnumbersN=f1;2;3;:::gorthenon-negativeintegers,
Z
0
=f0;1;2;3;:::g.Therangeofthefunctionisstillallowedtobetherealnumbers;in
symbols,wesaythatasequenceisafunctionf:N!R. Sequencesarewritteninafew
dierentways,allequivalent;theseallmeanthesamething:
a
1
;a
2
;a
3
;:::
fa
n
g
1
n=1
ff(n)g
1
n=1
Aswithfunctionsontherealnumbers,wewillmost oftenencountersequencesthat
canbeexpressedbyaformula. Wehavealreadyseenthesequencea
i
=f(i)=1 1=2
i
,
11.1 Sequences
257
andothersareeasytocomeby:
f(i)=
i
i+1
f(n)=
1
2n
f(n)=sin(n=6)
f(i)=
(i 1)(i+2)
2i
FrequentlytheseformulaswillmakesenseifthoughtofeitherasfunctionswithdomainR
orN,thoughoccasionallyonewillmakesenseonlyforintegervalues.
Facedwithasequenceweareinterestedinthelimit
lim
i!1
f(i)= lim
i!1
a
i
:
Wealreadyunderstand
lim
x!1
f(x)
whenxisarealvaluedvariable;nowwesimplywanttorestrictthe\input"valuestobe
integers.Norealdierenceisrequiredinthedenitionoflimit,exceptthatwespecify,per-
hapsimplicitly,thatthevariableisaninteger. Comparethisdenitiontodenition4.10.4.
DEFINITION11.1.1
Supposethatfa
n
g
1
n=1
isasequence. Wesaythat lim
n!1
a
n
=L
ifforevery>0thereisanN>0sothatwhenevern>N,ja
n
Lj<.If lim
n!1
a
n
=L
wesaythatthesequenceconverges,otherwiseitdiverges.
Iff(i)denesasequence,andf(x)makessense,and lim
x!1
f(x)=L,thenitisclear
that lim
i!1
f(i)=Laswell,butitisimportanttonotethattheconverseofthisstatement
isnottrue. Forexample,since lim
x!1
(1=x)=0,itisclearthatalso lim
i!1
(1=i)=0,thatis,
thenumbers
1
1
;
1
2
;
1
3
;
1
4
;
1
5
;
1
6
;:::
getcloserandcloserto0.Considerthis,however:Letf(n)=sin(n).Thisisthesequence
sin(0);sin(1);sin(2);sin(3);:::=0;0;0;0;:::
sincesin(n)=0when nisaninteger. Thus lim
n!1
f(n)=0. But lim
x!1
f(x),whenxis
real,doesnotexist: asxgetsbiggerandbigger,thevaluessin(x)donotgetcloserand
258
Chapter 11 1 SequencesandSeries
closertoasinglevalue,buttakeonallvaluesbetween 1and1overandover.Ingeneral,
whenever youwant t to o know lim
n!1
f(n) you should rst attempt to compute lim
x!1
f(x),
sinceifthelatterexistsitisalsoequaltotherstlimit. Butifforsomereason lim
x!1
f(x)
doesnot exist, it may still be e true e that lim
n!1
f(n) exists, , but t you’llhave e to o gure out
anotherwaytocomputeit.
It isoccasionally y usefulto o thinkof the graph of a sequence. Since e the function is
denedonlyforintegervalues,thegraphisjustasequenceofdots. Ingure11.1.1wesee
thegraphsoftwosequencesandthegraphsofthecorrespondingrealfunctions.
0
1
2
3
4
5
0
5
10
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..........
............
..................
...........................
...........................................
......................................................................................
................................
f(x)=1=x
0
1
2
3
4
5
0
5
10
f(n)=1=n
1
0
1
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.
f(x)=sin(x)
1
0
1
1 2 3 4 5 6 7 8
f(n)=sin(n)
Figure11.1.1
Graphsofsequencesandtheircorrespondingrealfunctions.
Notsurprisingly,thepropertiesoflimitsofrealfunctionstranslateintopropertiesof
sequencesquiteeasily.Theorem2.3.6aboutlimitsbecomes
THEOREM11.1.2 Supposethat lim
n!1
a
n
=Land lim
n!1
b
n
=Mandkissomeconstant.
Then
lim
n!1
ka
n
=k lim
n!1
a
n
=kL
lim
n!1
(a
n
+b
n
)= lim
n!1
a
n
+ lim
n!1
b
n
=L+M
lim
n!1
(a
n
b
n
)= lim
n!1
a
n
lim
n!1
b
n
=L M
lim
n!1
(a
n
b
n
)= lim
n!1
a
n
 lim
n!1
b
n
=LM
lim
n!1
a
n
b
n
=
lim
n!1
a
n
lim
n!1
b
n
=
L
M
; ifM M isnot0
LikewisetheSqueezeTheorem(4.3.1)becomes
11.1 Sequences
259
THEOREM11.1.3 Supposethata
n
b
n
c
n
foralln>N,forsomeN.If lim
n!1
a
n
=
lim
n!1
c
n
=L,then lim
n!1
b
n
=L.
Andanalusefulfact:
THEOREM11.1.4
lim
n!1
ja
n
j=0ifandonlyif lim
n!1
a
n
=0.
Thissayssimplythat thesizeofa
n
getsclosetozeroifandonlyifa
n
getscloseto
zero.
EXAMPLE 11.1.5
Determinewhether
n
n+1
1
n=0
convergesordiverges. Ifitcon-
verges,computethelimit. Sincethismakessenseforrealnumbersweconsider
lim
x!1
x
x+1
= lim
x!1
1
x+1
=1 0=1:
Thusthesequenceconvergesto1.
EXAMPLE 11.1.6
Determine whether
lnn
n
1
n=1
convergesordiverges. Ifit t con-
verges,computethelimit. Wecompute
lim
x!1
lnx
x
= lim
x!1
1=x
1
=0;
usingL’H^opital’sRule. Thusthesequenceconvergesto0.
EXAMPLE 11.1.7
Determinewhetherf( 1)
n
g
1
n=0
convergesordiverges. If f it con-
verges,computethelimit.Thisdoesnotmakesenseforallrealexponents,butthesequence
iseasytounderstand:itis
1; 1;1; 1;1:::
andclearlydiverges.
EXAMPLE11.1.8
Determinewhetherf( 1=2)
n
g
1
n=0
convergesordiverges. Ifitcon-
verges,computethelimit.Weconsiderthesequencefj( 1=2)
n
jg
1
n=0
=f(1=2)
n
g
1
n=0
.Then
lim
x!1
1
2
x
= lim
x!1
1
2x
=0;
sobytheorem11.1.4thesequenceconvergesto0.
260
Chapter 11 1 SequencesandSeries
EXAMPLE 11.1.9
Determinewhetherf(sinn)=
p
ng
1
n=1
convergesordiverges. Ifit
converges,computethelimit. Sincejsinnj1,0jsinn=
p
nj1=
p
nandwecanuse
theorem11.1.3witha
n
=0andc
n
=1=
p
n.Since lim
n!1
a
n
= lim
n!1
c
n
=0, lim
n!1
sinn=
p
n=
0andthesequenceconvergesto0.
EXAMPLE11.1.10 Aparticularlycommonandusefulsequenceisfr
n
g
1
n=0
,forvarious
valuesofr.Somearequiteeasytounderstand:Ifr=1thesequenceconvergesto1since
every term is 1, and likewise e if r r = 0 0 the e sequence e converges s to o 0. If f r r =  1 1 thisis
thesequenceofexample11.1.7anddiverges. Ifr>1orr< 1thetermsr
n
getlarge
withoutlimit,sothe sequence diverges. If0<r r <1thenthe sequence convergesto0.
If 1<r r <0thenjr
n
j=jrj
n
and0<jrj<1,sothesequencefjrj
n
g
1
n=0
convergesto
0,soalsofr
n
g
1
n=0
convergesto0.converges.Insummary,fr
n
gconvergespreciselywhen
1<r1inwhichcase
lim
n!1
r
n
=
n
0 if 1<r<1
1 ifr=1
Sometimeswewillnotbeabletodeterminethelimitofasequence,butwestillwould
liketoknowwhetheritconverges.Insomecaseswecandeterminethisevenwithoutbeing
abletocomputethelimit.
A sequenceiscalledincreasing orsometimesstrictly increasing if a
i
<a
i+1
for
alli. Itiscallednon-decreasing g orsometimes(unfortunately)increasingifa
i
a
i+1
foralli. Similarlyasequenceisdecreasingifa
i
>a
i+1
foralliandnon-increasingif
a
i
a
i+1
foralli. Ifasequencehasanyofthesepropertiesitiscalledmonotonic.
EXAMPLE11.1.11
Thesequence
2
i
1
2i
1
i=1
=
1
2
;
3
4
;
7
8
;
15
16
;:::;
isincreasing,and
n+1
n
1
i=1
=
2
1
;
3
2
;
4
3
;
5
4
;:::
isdecreasing.
AsequenceisboundedaboveifthereissomenumberNsuchthata
n
Nforevery
n,andbounded belowifthereissomenumberN N suchthata
n
N foreveryn. Ifa
sequenceisboundedaboveandboundedbelowitisbounded. Ifasequencefa
n
g
1
n=0
is
increasingornon-decreasingit isboundedbelow(bya
0
),andifitisdecreasingornon-
increasingitisboundedabove(bya
0
).Finally,withallthisnewterminologywecanstate
animportanttheorem.
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