pdfsharp c# : Add hyperlink to pdf in SDK Library project winforms asp.net azure UWP TRENCH_REAL_ANALYSIS8-part281

Section2.3
DifferentiableFunctionsofOneVariable
73
(d)
Iff.1/D>1,thenf isincreasing,
lim
x!1
f.x/D1; and
lim
x!1
f.x/D0:
(Thus,f.x/De
ax
hasthesepropertiesifa>0.)
H
INT
:SeeExercises2.2.10and2.2.12:
34.
ProveTheorem2.2.14inthecasewheref isnonincreasing.
2.3DIFFERENTIABLEFUNCTIONS OFONEVARIABLE
Incalculusyoustudieddifferentiation,emphasizingrulesforcalculatingderivatives.Here
weconsiderthetheoreticalpropertiesofdifferentiablefunctions.Indoingthis,weassume
thatyouknowhowtodifferentiateelementaryfunctionssuchasx
n
,e
x
,andsinx,andwe
willusesuchfunctionsinexamples.
Definitionofthe Derivative
Definition2.3.1
Afunctionf isdifferentiableataninteriorpointx
0
ofitsdomainif
thedifferencequotient
f.x/f.x
0
/
xx
0
; x¤x
0
;
approachesalimitasxapproachesx
0
,inwhichcasethelimitiscalledthederivativeoff
atx
0
,andisdenotedbyf
0
.x
0
/;thus,
f
0
.x
0
/D lim
x!x
0
f.x/f.x
0
/
xx
0
:
(2.3.1)
ItissometimesconvenienttoletxDx
0
Chandwrite(2.3.1)as
f
0
.x
0
/D lim
h!0
f.x
0
Ch/f.x
0
/
h
:
IffisdefinedonanopensetS,wesaythatf isdifferentiableonSiffisdifferentiable
ateverypointofS. Iff isdifferentiableonS,thenf
0
isafunctiononS. Wesaythat
f iscontinuouslydifferentiableonSiff
0
iscontinuousonS.Iff isdifferentiableona
neighborhoodofx
0
,itisreasonabletoaskiff
0
isdifferentiableatx
0
.Ifso,wedenotethe
derivativeoff
0
atx
0
byf
00
.x
0
/. Thisisthesecondderivativeoff atx
0
,anditisalso
denotedbyf.2/.x
0
/. Continuinginductively,iff.n1/ isdefinedonaneighborhoodof
x
0
,thenthenthderivativeoff atx
0
,denotedbyf.n/.x
0
/,isthederivativeoff.n1/ at
x
0
.Forconveniencewedefinethezerothderivativeoff tobefitself;thus
f
.0/
Df:
Weassumethatyouarefamiliarwiththeotherstandardnotationsforderivatives;for
example,
f
.2/
Df
00
; f
.3/
Df
000
;
Add hyperlink to pdf in - insert, remove PDF links in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Free C# example code is offered for users to edit PDF document hyperlink (url), like inserting and deleting
add links to pdf document; add page number to pdf hyperlink
Add hyperlink to pdf in - VB.NET PDF url edit library: insert, remove PDF links in vb.net, ASP.NET, MVC, Ajax, WinForms, WPF
Help to Insert a Hyperlink to Specified PDF Document Page
add hyperlink pdf document; active links in pdf
74 Chapter2
DifferentialCalculusofFunctionsofOneVariable
andsoon,and
d
n
f
dxn
Df
.n/
:
Example2.3.1
Ifnisapositiveintegerand
f.x/Dx
n
;
then
f.x/f.x
0
/
xx
0
D
x
n
x
n
0
xx
0
D
xx
0
xx
0
n1
kD0
x
nk1
x
k
0
;
so
f
0
.x
0
/D lim
x!x
0
n1
kD0
x
nk1
x
k
0
Dnx
n1
0
:
Sincethisholdsforeveryx
0
,wedropthesubscriptandwrite
f
0
.x/Dnx
n1
or
d
dx
.x
n
/Dnx
n1
:
Toderivedifferentiationformulasforelementaryfunctionssuchassinx,cosx,ande
x
directlyfromDefinition2.3.1requiresestimatesbasedonthepropertiesofthesefunctions.
Sincethisisdoneincalculus,wewillnotrepeatithere.
InterpretationsoftheDerivative
Iff.x/isthepositionofaparticleattimex¤x
0
,thedifferencequotient
f.x/f.x
0
/
xx
0
istheaveragevelocityoftheparticlebetweentimesx
0
andx. Asx x approachesx
0
,the
averageappliestoshorterandshorterintervals.Therefore,itmakessensetoregardthelimit
(2.3.1),ifitexists,astheparticle’sinstantaneousvelocityattimex
0
. Thisinterpretation
maybeusefulevenifxisnottime,soweoftenregardf0.x
0
/astheinstantaneousrateof
changeoff.x/atx
0
,regardlessofthespecificnatureofthevariablex.Thederivativealso
hasageometricinterpretation.Theequationofthelinethroughtwopoints.x
0
;f.x
0
//and
.x
1
;f.x
1
//onthecurveyDf.x/(Figure2.3.1)is
yDf.x
0
/C
f.x
1
/f.x
0
/
x
1
x
0
.xx
0
/:
Varyingx
1
generateslinesthrough.x
0
;f.x
0
//thatrotateintotheline
yDf.x
0
/Cf
0
.x
0
/.xx
0
/
(2.3.2)
VB.NET Create PDF from Word Library to convert docx, doc to PDF in
Change Word hyperlink to PDF hyperlink and bookmark. VB.NET Demo Code for Converting Word to PDF. Add necessary references: RasterEdge.Imaging.Basic.dll.
add hyperlink pdf file; add email link to pdf
VB.NET Create PDF from Excel Library to convert xlsx, xls to PDF
Change Excel hyperlink to PDF hyperlink and bookmark. VB.NET Demo Code for Converting Excel to PDF. Add necessary references: RasterEdge.Imaging.Basic.dll.
pdf link; clickable pdf links
Section2.3
DifferentiableFunctionsofOneVariable
75
asx
1
approachesx
0
. ThisisthetangenttothecurveyDf.x/atthepoint.x
0
;f.x
0
//.
Figure2.3.2depictsthesituationforvariousvaluesofx
1
.
y
x
y = f(x)
x
0
x
1
Figure2.3.1
y
x
y = f(x)
x
0
x
1
x
1
x
1
''
Tangent line
Figure2.3.2
Hereisalessintuitivedefinitionofthetangentline:Ifthefunction
T.x/Df.x
0
/Cm.xx
0
/
approximatesf sowellnearx
0
that
lim
x!x
0
f.x/T.x/
xx
0
D0;
wesaythatthelineyDT.x/istangenttothecurveyDf.x/at.x
0
;f.x
0
//.
How to C#: Basic SDK Concept of XDoc.PDF for .NET
You may add PDF document protection functionality into your C# program. Hyperlink Edit. XDoc.PDF for .NET allows C# developers to edit hyperlink of PDF document
adding hyperlinks to a pdf; add links to pdf
VB.NET PDF: Basic SDK Concept of XDoc.PDF
You may add PDF document protection functionality into your VB.NET program. Hyperlink Edit. XDoc.PDF for .NET allows VB.NET developers to edit hyperlink of PDF
pdf hyperlink; add hyperlinks pdf file
76 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Thistangentlineexistsifandonlyiff
0
.x
0
/exists,inwhichcasemisuniquelydetermined
bymDf
0
.x
0
/(Exercise2.3.1).Thus,(2.3.2)istheequationofthetangentline.
Wewillusethefollowinglemmatostudydifferentiablefunctions.
Lemma2.3.2
Iff isdifferentiableatx
0
;then
f.x/Df.x
0
/CŒf
0
.x
0
/CE.x/.xx
0
/;
(2.3.3)
whereEisdefinedonaneighborhoodofx
0
and
lim
x!x
0
E.x/DE.x
0
/D0:
Proof
Define
E.x/D
8
<
:
f.x/f.x
0
/
xx
0
f
0
.x
0
/; x2D
f
andx¤x
0
;
0;
xDx
0
:
(2.3.4)
Solving(2.3.4)forf.x/yields(2.3.3)ifx¤x
0
,and(2.3.3)isobviousifxDx
0
.Defini-
tion2.3.1impliesthatlim
x!x
0
E.x/D0.WedefinedE.x
0
/D0tomakeEcontinuous
atx
0
.
Sincetherightsideof(2.3.3)iscontinuousatx
0
,soistheleft.Thisyieldsthefollowing
theorem.
Theorem2.3.3
Iff isdifferentiableatx
0
;thenf iscontinuousatx
0
:
Theconverseofthistheoremisfalse, sinceafunctionmaybecontinuousatapoint
withoutbeingdifferentiableatthepoint.
Example2.3.2
Thefunction
f.x/Djxj
canbewrittenas
f.x/D
x; x>0;
(2.3.5)
oras
f.x/Dx; x<0:
(2.3.6)
From(2.3.5),
f
0
.x/D
1; x>0;
andfrom(2.3.6),
f
0
.x/D1; x<0:
Neither(2.3.5)nor(2.3.6)holdsthroughoutanyneighborhoodof0,soneithercanbeused
alonetocalculatef
0
.0/.Infact,sincetheone-sidedlimits
lim
x!0C
f.x/f.0/
x0
D lim
x!0C
x
x
(2.3.7)
and
lim
x!0
f.x/f.0/
x0
D lim
x!0
x
x
D1
(2.3.8)
C# Create PDF from Word Library to convert docx, doc to PDF in C#.
Change Word hyperlink to PDF hyperlink and bookmark. C#.NET Sample Code: Convert Word to PDF in C#.NET Project. Add necessary references:
accessible links in pdf; add links to pdf in preview
.NET PDF Document Viewing, Annotation, Conversion & Processing
Extract hyperlink inside PDF. PDF Write. Insert text, text box into PDF. Edit, delete text from PDF. Insert images into PDF. Edit, remove images from PDF. Add,
change link in pdf file; clickable links in pdf from word
Section2.3
DifferentiableFunctionsofOneVariable
77
aredifferent,
lim
x!0
f.x/f.0/
x0
doesnotexist(Theorem2.1.6);thus,f isnotdifferentiableat0,eventhoughitiscontinu-
ousat0.
InterchangingDifferentiationandArithmeticOperations
Thefollowingtheoremshouldbefamiliarfromcalculus.
Theorem2.3.4
Iffandgaredifferentiableatx
0
;thensoarefCg;fg;andfg;
with
(a)
.fCg/
0
.x
0
/Df
0
.x
0
/Cg
0
.x
0
/I
(b)
.f g/
0
.x
0
/Df
0
.x
0
/g.x
0
/I
(c)
.fg/
0
.x
0
/Df
0
.x
0
/g.x
0
/Cf.x
0
/g
0
.x
0
/:
Thequotientf=gisdifferentiableatx
0
ifg.x
0
/¤0;with
(d)
f
g
0
.x
0
/D
f
0
.x
0
/g.x
0
/f.x
0
/g
0
.x
0
/
Œg.x
0
/
2
:
Proof
Theproofisaccomplishedbyformingtheappropriatedifferencequotientsand
applyingDefinition2.3.1andTheorem2.1.4.Wewillprove
(c)
andleavetheresttoyou
(Exercises2.3.9,2.3.10,and2.3.11).
Thetrickistoaddandsubtracttherightquantityinthenumeratorofthedifference
quotientfor.fg/
0
.x
0
/;thus,
f.x/g.x/f.x
0
/g.x
0
/
xx
0
D
f.x/g.x/f.x
0
/g.x/Cf.x
0
/g.x/f.x
0
/g.x
0
/
xx
0
D
f.x/f.x
0
/
xx
0
g.x/Cf.x
0
/
g.x/g.x
0
/
xx
0
:
Thedifferencequotientsontherightapproachf
0
.x
0
/andg
0
.x
0
/asxapproachesx
0
,and
lim
x!x
0
g.x/Dg.x
0
/(Theorem2.3.3).Thisproves
(c)
.
TheChainRule
Hereistherulefordifferentiatingacompositefunction.
Theorem2.3.5(TheChainRule)
Supposethatgisdifferentiableatx
0
andf
isdifferentiableatg.x
0
/:ThenthecompositefunctionhDf ıg;definedby
h.x/Df.g.x//;
isdifferentiableatx
0
;with
h
0
.x
0
/Df
0
.g.x
0
//g
0
.x
0
/:
VB.NET Create PDF from PowerPoint Library to convert pptx, ppt to
Export PowerPoint hyperlink to PDF. VB.NET Demo Code for Converting PowerPoint to PDF. Add necessary references: RasterEdge.Imaging.Basic.dll.
adding links to pdf; convert excel to pdf with hyperlinks
C# Create PDF from PowerPoint Library to convert pptx, ppt to PDF
Export PowerPoint hyperlink to PDF in .NET console application. C#.NET Demo Code: Convert PowerPoint to PDF in C#.NET Application. Add necessary references:
add links in pdf; add links pdf document
78 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Proof
Sincef isdifferentiableatg.x
0
/,Lemma2.3.2impliesthat
f.t/f.g.x
0
//DŒf
0
.g.x
0
//CE.t/Œtg.x
0
/;
where
lim
t!g.x
0
/
E.t/DE.g.x
0
//D0:
(2.3.9)
LettingtDg.x/yields
f.g.x//f.g.x
0
//DŒf
0
.g.x
0
//CE.g.x//Œg.x/g.x
0
/:
Sinceh.x/Df.g.x//,thisimpliesthat
h.x/h.x
0
/
xx
0
DŒf
0
.g.x
0
//CE.g.x//
g.x/g.x
0
/
xx
0
:
(2.3.10)
Sincegiscontinuousatx
0
(Theorem2.3.3),(2.3.9)andTheorem2.2.7implythat
lim
x!x
0
E.g.x//DE.g.x
0
//D0:
Therefore,(2.3.10)impliesthat
h
0
.x
0
/D lim
x!x
0
h.x/h.x
0
/
xx
0
Df
0
.g.x
0
//g
0
.x
0
/;
asstated.
Example2.3.3
If
f.x/Dsinx and g.x/D
1
x
; x¤0;
then
h.x/Df.g.x//Dsin
1
x
; x¤0;
and
h
0
.x/Df
0
.g.x//g.x/D
cos
1
x

1
x2
; x¤0:
Itmayseemreasonabletojustifythechainrulebywriting
h.x/h.x
0
/
xx
0
D
f.g.x//f.g.x
0
//
xx
0
D
f.g.x//f.g.x
0
//
g.x/g.x
0
/
g.x/g.x
0
/
xx
0
andarguingthat
lim
x!x
0
f.g.x//f.g.x
0
//
g.x/g.x
0
/
Df
0
.g.x
0
//
Section2.3
DifferentiableFunctionsofOneVariable
79
(becauselim
x!x
0
g.x/Dg.x
0
//and
lim
x!x
0
g.x/g.x
0
/
xx
0
Dg
0
.x
0
/:
However,thisisnotavalidproof(Exercise2.3.13).
One-SidedDerivatives
One-sidedlimitsofdifferencequotientssuchas(2.3.7)and(2.3.8)inExample2.3.2are
calledone-sidedorright-andleft-handderivatives.Thatis,iff isdefinedonŒx
0
;b/,the
right-handderivativeoff atx
0
isdefinedtobe
f
0
C
.x
0
/D
lim
x!x
0
C
f.x/f.x
0
/
xx
0
ifthelimitexists,whileiff isdefinedon.a;x
0
,theleft-handderivativeoff atx
0
is
definedtobe
f
0
.x
0
/D lim
x!x
0
f.x/f.x
0
/
xx
0
ifthelimitexists.Theorem2.1.6impliesthatfisdifferentiableatx
0
ifandonlyiff
0
C
.x
0
/
andf
0
.x
0
/existandareequal,inwhichcase
f
0
.x
0
/Df
0
C
.x
0
/Df
0
.x
0
/:
InExample2.3.2,f
0
C
.0/D1andf
0
.0/D1.
Example2.3.4
If
f.x/D
8
<
:
x3;
x0;
x2sin
1
x
; x>0;
(2.3.11)
then
f
0
.x/D
8
<
:
3x
2
;
x<0;
2xsin
1
x
cos
1
x
; x>0:
(2.3.12)
Sinceneitherformulain(2.3.11)holdsforallxinanyneighborhoodof0,wecannotsimply
differentiateeithertoobtainf
0
.0/;instead,wecalculate
f
0
C
.0/D lim
x!0C
x
2
sin
1
x
0
x0
D lim
x!0C
xsin
1
x
D0;
f
0
.0/D lim
x!0
x
3
0
x0
D lim
x!0
x
2
D0I
hence,f
0
.0/Df
0
C
.0/Df
0
.0/D0.
80 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Thisexampleshowsthatthereisadifferencebetweenaone-sidedderivativeandaone-
sidedlimitofaderivative,sincef
0
C
.0/D0,but,from(2.3.12),f
0
.0C/Dlim
x!0C
f
0
.x/
doesnotexist. Italsoshowsthataderivativemayexistinaneighborhoodofapointx
0
(D0inthiscase),butbediscontinuousatx
0
.
Exercise2.3.4justifiesthemethodusedinExample2.3.4tocomputef
0
.x/forx¤0.
Definition2.3.6
(a)
Wesaythatf isdifferentiableontheclosedintervalŒa;biff isdifferentiableon
theopeninterval.a;b/andf
0
C
.a/andf
0
.b/bothexist.
(b)
Wesaythatf iscontinuouslydifferentiableonŒa;biff isdifferentiableonŒa;b,
f0iscontinuouson.a;b/,f0
C
.a/Df0.aC/,andf0
.b/Df0.b/.
ExtremeValues
Wesaythatf.x
0
/isalocalextremevalueoffifthereisaı>0suchthatf.x/f.x
0
/
doesnotchangesignon
.x
0
ı;x
0
Cı/\D
f
:
(2.3.13)
Morespecifically,f.x
0
/isalocalmaximumvalueoff if
f.x/f.x
0
/
(2.3.14)
oralocalminimumvalueoff if
f.x/f.x
0
/
(2.3.15)
forallx intheset(2.3.13). . Thepointx
0
iscalledalocalextremepointoff,or,more
specifically,alocalmaximumorlocalminimumpointoff.
y
x
1
2
2
3
4
−1
−1
2
1
Figure2.3.3
Section2.3
DifferentiableFunctionsofOneVariable
81
Example2.3.5
If
f.x/D
8
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
:
1;
1<x
1
2
jxj;
1
2
<x
1
2
;
1
p
2
sin
x
2
;
1
2
<x4
(Figure2.3.3),then0,3,andeveryxin.1;
1
2
/arelocalminimumpointsoff,while1,
4,andeveryxin.1;
1
2
arelocalmaximumpoints.
ItisgeometricallyplausiblethatifthecurveyDf.x/hasatangentatalocalextreme
pointoff,thenthetangentmustbehorizontal;thatis,havezeroslope.(Forexample,in
Figure2.3.3,seexD1,xD3,andeveryxin.1;1=2/.)Thefollowingtheoremshows
thatthismustbeso.
Theorem2.3.7
Iffisdifferentiableatalocalextremepointx
0
2D0
f
;thenf0.x
0
/D 0:
Proof
Wewillshowthatx
0
isnotalocalextremepointoff iff
0
.x
0
/ ¤ 0. From
Lemma2.3.2,
f.x/f.x
0
/
xx
0
Df
0
.x
0
/CE.x/;
(2.3.16)
wherelim
x!x
0
E.x/D0.Therefore,iff
0
.x
0
/¤0,thereisaı>0suchthat
jE.x/j<jf
0
.x
0
/j if jxx
0
j<ı;
andtherightsideof(2.3.16)musthavethesamesignasf
0
.x
0
/forjxx
0
j<ı. Since
thesameistrueoftheleftside,f.x/f.x
0
/mustchangesignineveryneighborhoodof
x
0
(sincexx
0
does). Therefore,neither(2.3.14)nor(2.3.15)canholdforallxinany
intervalaboutx
0
.
Iff
0
.x
0
/D0,wesaythatx
0
isacriticalpointoff. Theorem2.3.7saysthatevery
localextremepointoff atwhichf isdifferentiableisacriticalpointoff.Theconverse
isfalse.Forexample,0isacriticalpointoff.x/Dx
3
,butnotalocalextremepoint.
Rolle’sTheorem
TheuseofTheorem2.3.7forfindinglocalextremepointsiscoveredincalculus,sowewill
notpursueithere.However,wewilluseTheorem2.3.7toprovethefollowingfundamental
theorem,whichsaysthatifacurvey Df.x/intersectsahorizontallineatx x D aand
xDbandhasatangentat.x;f.x//foreveryxin.a;b/,thenthereisapointcin.a;b/
suchthatthetangenttothecurveat.c;f.c//ishorizontal(Figure2.3.4).
82 Chapter2
DifferentialCalculusofFunctionsofOneVariable
y
x
b
c
a
Figure2.3.4
Theorem2.3.8(Rolle’sTheorem)
Supposethatf iscontinuousontheclosed
intervalŒa;banddifferentiableontheopeninterval.a;b/; andf.a/ D f.b/:Then
f
0
.c/D0forsomecintheopeninterval.a;b/:
Proof
Sincef iscontinuousonŒa;b,f attainsamaximumandaminimumvalueon
Œa;b(Theorem2.2.9). Ifthesetwoextremevaluesarethesame,thenf isconstanton
.a;b/,sof
0
.x/D0forallxin.a;b/.Iftheextremevaluesdiffer,thenatleastonemust
beattainedatsomepointcintheopeninterval.a;b/,andf
0
.c/D0,byTheorem2.3.7.
IntermediateValuesofDerivatives
AderivativemayexistonanintervalŒa;bwithoutbeingcontinuousonŒa;b. Neverthe-
less,anintermediatevaluetheoremsimilartoTheorem2.2.10appliestoderivatives.
Theorem2.3.9(IntermediateValueTheoremforDerivatives)
Suppose
thatf isdifferentiableonŒa;b;f
0
.a/¤f
0
.b/;andisbetweenf
0
.a/andf
0
.b/:Then
f
0
.c/Dforsomecin.a;b/:
Proof
Supposefirstthat
f
0
.a/<<f
0
.b/
(2.3.17)
anddefine
g.x/Df.x/x:
Then
g
0
.x/Df
0
.x/; axb;
(2.3.18)
and(2.3.17)impliesthat
g
0
.a/<0 and g
0
.b/>0:
(2.3.19)
SincegiscontinuousonŒa;b,gattainsaminimumatsomepointcinŒa;b.Lemma2.3.2
and(2.3.19)implythatthereisaı>0suchthat
g.x/<g.a/; a<x<aCı; and g.x/<g.b/; bı<x<b
Documents you may be interested
Documents you may be interested