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Section2.5
Taylor’sTheorem
103
Thisand(2.5.10)implythat
f.x/f.x
0
/
.xx
0
/n
(2.5.11)
hasthesamesignasf
.n/
.x
0
/if0<jxx
0
j<ı.Ifnisoddthedenominatorof(2.5.11)
changessignineveryneighborhoodofx
0
,andthereforesomustthenumerator(sincethe
ratiohasconstantsignfor0< jxx
0
j< ı). Consequently,f.x
0
/cannotbealocal
extremevalueoff.Thisproves
(a)
.Ifniseven,thedenominatorof(2.5.11)ispositive
forx¤x
0
,sof.x/f.x
0
/musthavethesamesignasf
.n/
.x
0
/for0<jxx
0
j<ı.
Thisproves
(b)
.
FornD2,
(b)
iscalledthesecondderivativetestforlocalextremepoints.
Example2.5.4
Iff.x/Dex
3
,thenf0.x/D3x2ex
3
,and0istheonlycriticalpoint
off.Since
f
00
.x/D.6xC9x
4
/e
x
3
and
f
000
.x/D.6C54x
3
C27x
6
/e
x
3
;
f00.0/D0andf000.0/¤0.Therefore,Theorem2.5.3impliesthat0isnotalocalextreme
pointoff.Sincef isdifferentiableeverywhere,ithasnolocalmaximaorminima.
Example2.5.5
Iff.x/Dsinx
2
,thenf
0
.x/D2xcosx
2
,sothecriticalpointsoff
are0and˙
p
.kC1=2/,kD0;1;2;:::.Since
f
00
.x/D2cosx
2
4x
2
sinx
2
;
f
00
.0/D2 and f
00
˙
p
.kC1=2//
D.1/
kC1
.4kC2/:
Therefore,Theorem2.5.3impliesthatf attainslocalminimaat0and˙
p
.kC1=2/for
oddintegersk,andlocalmaximaat˙
p
.kC1=2/forevenintegersk.
Taylor’stheorem
Theorem 2.5.1impliesthattheerrorinapproximatingf.x/byT
n
.x/approacheszero
fasterthan.xx
0
/
n
asx approachesx
0
; however, , itgivesnoestimateoftheerrorin
approximatingf.x/byT
n
.x/forafixedx. Forinstance,itprovidesnoestimateofthe
errorintheapproximation
e
0:1
T
2
.0:1/D1C
0:1
C
.0:1/
2
D1:105
(2.5.12)
obtainedbysettingnD2andx D0:1in(2.5.8). Thefollowingtheoremprovidesaway
ofestimatingerrorsofthiskindundertheadditionalassumptionthatf
.nC1/
existsina
neighborhoodofx
0
.
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104 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Theorem2.5.4(Taylor’sTheorem)
Supposethatf
.nC1/
existsonanopenin-
tervalI aboutx
0
;andletxbeinI:Thentheremainder
R
n
.x/Df.x/T
n
.x/
canbewrittenas
R
n
.x/D
f.nC1/.c/
.nC1/Š
.xx
0
/
nC1
;
wherecdependsuponxandisbetweenxandx
0
:
Thistheoremfollowsfromanextensionofthemeanvaluetheoremthatwewillprove
below.Fornow,letusassumethatTheorem2.5.4iscorrect,andapplyit.
Example2.5.6
Iff.x/D e
x
,thenf
000
.x/ D D e
x
, andTheorem2.5.4withnD 2
impliesthat
e
x
D1CxC
x
2
C
e
c
x
3
;
wherecisbetween0andx.Hence,from(2.5.12),
e
0:1
D1:105C
e
c
.0:1/
3
6
;
where0<c<0:1.Since0<e
c
<e
0:1
,weknowfromthisthat
1:105<e
0:1
<1:105C
e
0:1
.0:1/
3
6
:
Thesecondinequalityimpliesthat
e
0:1
1
.0:1/
3
6
<1:105;
so
e
0:1
<1:1052:
Therefore,
1:105<e
0:1
<1:1052;
andtheerrorin(2.5.12)islessthan0:0002.
Example2.5.7
Innumericalanalysis, forwarddifferencesare usedtoapproximate
derivatives. Ifh>0,thefirstandsecondforwarddifferenceswithspacingharedefined
by
f.x/Df.xCh/f.x/
and
2
f.x/DŒf.x/Df.xCh/f.x/
Df.xC2h/2f.xCh/Cf.x/:
(2.5.13)
Higherforwarddifferencesaredefinedinductively(Exercise2.5.18).
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Section2.5
Taylor’sTheorem
105
Wewillfindupperboundsforthemagnitudesoftheerrorsintheapproximations
f
0
.x
0
/
f.x
0
/
h
(2.5.14)
and
f
00
.x
0
/
2
f.x
0
/
h2
:
(2.5.15)
Iff
00
existsonanopenintervalcontainingx
0
andx
0
Ch,wecanuseTheorem2.5.4to
estimatetheerrorin(2.5.14)bywriting
f.x
0
Ch/Df.x
0
/Cf
0
.x
0
/hC
f
00
.c/h
2
2
;
(2.5.16)
wherex
0
<c<x
0
Ch.Wecanrewrite(2.5.16)as
f.x
0
Ch/f.x
0
/
h
f
0
.x
0
/D
f0.c/h
2
;
whichisequivalentto
f.x
0
/
h
f
0
.x
0
/D
f
00
.c/h
2
:
Therefore,
ˇ
ˇ
ˇ
ˇ
f.x
0
/
h
f
0
.x
0
/
ˇ
ˇ
ˇ
ˇ
M
2
h
2
;
whereM
2
isanupperboundforjf
00
jon.x
0
;x
0
Ch/.
Iff000 existsonanopenintervalcontainingx
0
andx
0
C2h,wecanuseTheorem2.5.4
toestimatetheerrorin(2.5.15)bywriting
f.x
0
Ch/Df.x
0
/Chf
0
.x
0
/C
h
2
2
f
00
.x
0
/C
h
3
6
f
000
.c
0
/
and
f.x
0
C2h/Df.x
0
/C2hf
0
.x
0
/C2h
2
f
00
.x
0
/C
4h
3
3
f
000
.c
1
/;
wherex
0
<c
0
<x
0
Chandx
0
<c
1
<x
0
C2h.Thesetwoequationsimplythat
f.x
0
C2h/2f.x
0
Ch/Cf.x
0
/Dh
2
f
00
.x
0
/C
4
3
f
000
.c
1
/
1
3
f
000
.c
0
/
h
3
;
whichcanberewrittenas
2
f.x
0
/
h2
f
00
.x
0
/D
4
3
f
000
.c
1
/
1
3
f
000
.c
0
/
h;
becauseof(2.5.13).Therefore,
ˇ
ˇ
ˇ
ˇ
2
f.x
0
/
h2
f
00
.x
0
/
ˇ
ˇ
ˇ
ˇ
5M
3
h
3
;
whereM
3
isanupperboundforjf000jon.x
0
;x
0
C2h/.
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106 Chapter2
DifferentialCalculusofFunctionsofOneVariable
TheExtendedMeanValueTheorem
Wenowconsidertheextendedmeanvaluetheorem,whichimpliesTheorem2.5.4(Exer-
cise2.5.24).Inthefollowingtheorem,aandbaretheendpointsofaninterval,butwedo
notassumethata<b.
Theorem2.5.5(Extended MeanValueTheorem)
Supposethatfiscon-
tinuousonafiniteclosedintervalI withendpointsaandb.thatis,eitherI I D.a;b/or
I D.b;a//;f
.nC1/
existsontheopenintervalI
0
;and;ifn>0;thatf
0
,...,f
.n/
exist
andarecontinuousata:Then
f.b/
Xn
rD0
f
.r/
.a/
.ba/
r
D
f
.nC1/
.c/
.nC1/Š
.ba/
nC1
(2.5.17)
forsomecinI
0
:
Proof
Theproofisbyinduction. Themeanvaluetheorem(Theorem2.3.11)implies
theconclusionfornD0. Nowsupposethatn1,andassumethattheassertionofthe
theoremistruewithnreplacedbyn1.Theleftsideof(2.5.17)canbewrittenas
f.b/
Xn
rD0
f
.r/
.a/
.ba/
r
DK
.ba/
nC1
.nC1/Š
(2.5.18)
forsomenumberK. WemustprovethatKDf
.nC1/
.c/forsomecinI
0
. Tothisend,
considertheauxiliaryfunction
h.x/Df.x/
Xn
rD0
f
.r/
.a/
.xa/
r
K
.xa/
nC1
.nC1/Š
;
whichsatisfies
h.a/D0; h.b/D0;
(thelatterbecauseof(2.5.18))andiscontinuousontheclosedintervalI anddifferentiable
onI
0
,with
h
0
.x/Df
0
.x/
n1
rD0
f
.rC1/
.a/
.xa/
r
K
.xa/
n
:
(2.5.19)
Therefore,Rolle’stheorem(Theorem2.3.8)impliesthath
0
.b
1
/ D 0forsomeb
1
inI
0
;
thus,from(2.5.19),
f
0
.b
1
/
n1
rD0
f
.rC1/
.a/
.b
1
a/
r
K
.b
1
a/
n
D0:
Ifwetemporarilywritef
0
Dg,thisbecomes
g.b
1
/
n1
X
rD0
g
.r/
.a/
r
.b
1
a/
r
K
.b
1
a/
n
D0:
(2.5.20)
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Section2.5
Taylor’sTheorem
107
Sinceb
1
2 I
0
,thehypothesesonf implythatgiscontinuousontheclosedintervalJ
withendpointsaandb
1
,g
.n/
existsonJ
0
,and,ifn 1,g
0
,...,g
.n1/
existandare
continuousata(alsoatb
1
,butthisisnotimportant).Theinductionhypothesis,appliedto
gontheintervalJ,impliesthat
g.b
1
/
n1
rD0
g
.r/
.a/
.b
1
a/
r
D
g
.n/
.c/
.b
1
a/
n
forsomecinJ
0
.Comparingthiswith(2.5.20)andrecallingthatgDf
0
yields
KDg
.n/
.c/Df
.nC1/
.c/:
SincecisinI0,thiscompletestheinduction.
2.5Exercises
1.
Let
f.x/D
e
1=x
2
; x¤0;
0;
xD0:
Showthatf hasderivativesofallorderson.1;1/andeveryTaylorpolynomial
off about0isidenticallyzero.H
INT
: SeeExercise2.4.40:
2.
Supposethatf
.nC1/
.x
0
/exists,andletT
n
bethenthTaylorpolynomialoff about
x
0
.Showthatthefunction
E
n
.x/D
8
<
:
f.x/T
n
.x/
.xx
0
/n
; x2D
f
fx
0
g;
0;
xDx
0
;
isdifferentiableatx
0
,andfindE
0
n
.x
0
/.
3. (a)
Prove:Iffiscontinuousatx
0
andthereareconstantsa
0
anda
1
suchthat
lim
x!x
0
f.x/a
0
a
1
.xx
0
/
xx
0
D0;
thena
0
Df.x
0
/,f
0
isdifferentiableatx
0
,andf
0
.x
0
/Da
1
.
(b)
Giveacounterexampletothefollowingstatement:Iffandf
0
arecontinuous
atx
0
andthereareconstantsa
0
,a
1
,anda
2
suchthat
lim
x!x
0
f.x/a
0
a
1
.xx
0
/a
2
.xx
0
/
2
.xx
0
/2
D0;
thenf
00
.x
0
/exists.
4. (a)
Prove:iff
00
.x
0
/exists,then
lim
h!0
f.x
0
Ch/2f.x
0
/Cf.x
0
h/
h2
Df
00
.x
0
/:
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108 Chapter2
DifferentialCalculusofFunctionsofOneVariable
(b)
Prove orgivea counterexample: If f thelimitin
(a)
exists, thenso does
f
00
.x
0
/,andtheyareequal.
5.
Afunctionfhasasimplezero(orazeroofmultiplicity1)atx
0
iffisdifferentiable
inaneighborhoodofx
0
andf.x
0
/D0,whilef
0
.x
0
/¤0.
(a)
Provethatf hasasimplezeroatx
0
ifandonlyif
f.x/Dg.x/.xx
0
/;
wheregiscontinuousatx
0
anddifferentiableonadeletedneighborhoodof
x
0
,andg.x
0
/¤0.
(b)
Giveanexampleshowingthatgin
(a)
neednotbedifferentiableatx
0
.
6.
Afunctionf hasadoublezero(orazeroofmultiplicity2)atx
0
iff istwicedif-
ferentiableonaneighborhoodofx
0
andf.x
0
/Df
0
.x
0
/D0,whilef
00
.x
0
/ ¤ 0.
(a)
Provethatf hasadoublezeroatx
0
ifandonlyif
f.x/Dg.x/.xx
0
/
2
;
wheregiscontinuousatx
0
andtwicedifferentiableonadeletedneighborhood
ofx
0
,g.x
0
/¤0,and
lim
x!x
0
.xx
0
/g
0
.x/D0:
(b)
Giveanexampleshowingthatgin
(a)
neednotbedifferentiableatx
0
.
7.
Letnbeapositiveinteger. Afunctionf hasazeroofmultiplicitynatx
0
iff
is ntimes differentiableona neighborhoodofx
0
, f.x
0
/ D f0.x
0
/ D  D
f.n1/.x
0
/ D 0andf.n/.x
0
/ ¤ 0. Provethatf hasazeroofmultiplicitynat
x
0
ifandonlyif
f.x/Dg.x/.xx
0
/
n
;
wheregiscontinuousatx
0
andntimesdifferentiableonadeletedneighborhoodof
x
0
,g.x
0
/¤0,and
lim
x!x
0
.xx
0
/
j
g
.j/
.x/D0; 1j j n1:
H
INT
:UseExercise2.5.6andinduction:
8. (a)
Let
Q.x/D˛
0
1
.xx
0
/CC˛
n
.xx
0
/
n
beapolynomialofdegreensuchthat
lim
x!x
0
Q.x/
.xx
0
/n
D0:
Showthat˛
0
1
DD˛
n
D0.
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Section2.5
Taylor’sTheorem
109
(b)
Supposethatf isntimesdifferentiableatx
0
andpisapolynomial
p.x/Da
0
Ca
1
.xx
0
/CCa
n
.xx
0
/
n
ofdegreensuchthat
lim
x!x
0
f.x/p.x/
.xx
0
/n
D0:
Showthat
a
r
D
f
.r/
.x
0
/
if 0rnI
thatis,pDT
n
,thenthTaylorpolynomialoff aboutx
0
.
9.
Showthatiff
.n/
.x
0
/andg
.n/
.x
0
/existand
lim
x!x
0
f.x/g.x/
.xx
0
/n
D0;
thenf
.r/
.x
0
/Dg
.r/
.x
0
/,0rn.
10. (a)
LetF
n
, G
n
,andH
n
bethenthTaylorpolynomialsaboutx
0
off,g, and
theirproducth D fg. ShowthatH
n
canbeobtainedbymultiplyingF
n
byG
n
andretainingonlythepowersofxx
0
throughthenth. H
INT
:Use
Exercise2.5.8.b/:
(b)
Usethemethodsuggestedby
(a)
tocomputeh
.r/
.x
0
/,rD1;2;3;4.
(i)
h.x/De
x
sinx; x
0
D0
(ii)
h.x/D.cosx=2/.logx/; x
0
D1
(iii)
h.x/Dx
2
cosx; x
0
D=2
(iv)
h.x/D.1Cx/
1
e
x
; x
0
D0
11. (a)
Itcanbeshownthatifgisntimesdifferentiableatxandf isntimesdif-
ferentiableatg.x/,thenthecompositefunctionh.x/D f.g.x//isntimes
differentiableatxand
h
.n/
.x/D
n
X
rD1
f
.r/
.g.x//
X
r
r
1
Šr
n
Š
g
0
.x/
r
1
g
00
.x/
r
2

g
.n/
.x/
!
r
n
where
P
r
isoveralln-tuples.r
1
;r
2
;:::;r
n
/ofnonnegativeintegerssuchthat
r
1
Cr
2
CCr
n
Dr
and
r
1
C2r
2
CCnr
n
Dn:
(ThisisFaadiBruno’sformula).However,thisformulaisquitecomplicated.
Justifythefollowingalternativemethodforcomputingthederivativesofa
compositefunctionatapointx
0
:
110 Chapter2
DifferentialCalculusofFunctionsofOneVariable
LetF
n
bethenthTaylorpolynomialoff abouty
0
Dg.x
0
/,andletG
n
and
H
n
bethenthTaylorpolynomialsofgandhaboutx
0
. ShowthatH
n
can
beobtainedbysubstitutingG
n
intoF
n
andretainingonlypowersofxx
0
throughthenth.H
INT
:SeeExercise2.5.8.b/:
(b)
Computethefirstfourderivativesofh.x/Dcos.sinx/atx
0
D0,usingthe
methodsuggestedby
(a)
.
12. (a)
Ifg.x
0
/¤0andg
.n/
.x
0
/exists,thenthereciprocalhD1=gisalsontimes
differentiableatx
0
,byExercise2.5.11
(a)
,withf.x/D1=x.LetG
n
andH
n
bethenthTaylorpolynomialsofgandhaboutx
0
.UseExercise2.5.11
(a)
to
provethatifg.x
0
/D1,thenH
n
canbeobtainedbyexpandingthepolynomial
Xn
rD1
Œ1G
n
.x/
r
inpowersofxx
0
andretainingonlypowersthroughthenth.
(b)
Usethemethodof
(a)
tocomputethefirstfourderivativesofthefollowing
functionsatx
0
.
(i)
h.x/Dcscx; x
0
D=2
(ii)
h.x/D.1CxCx2/1; x
0
D0
(iii)
h.x/Dsecx; x
0
D=4
(iv)
h.x/DŒ1Clog.1Cx/
1
; x
0
D0
(c)
UseExercise2.5.10tojustifythefollowingalternativeprocedureforobtaining
H
n
,againassumingthatg.x
0
/D1:If
G
n
.x/D1Ca
1
.xx
0
/CCa
n
.xx
0
/
n
(where,ofcourse,a
r
Dg
.r/
.x
0
/=rŠ/and
H
n
.x/Db
0
Cb
1
.xx
0
/CCb
n
.xx
0
/
n
;
then
b
0
D1; b
k
D
Xk
rD1
a
r
b
kr
; 1kn:
13.
Determinewhetherx
0
D0isalocalmaximum,localminimum,orneither.
(a)
f.x/Dx
2
e
x
3
(b)
f.x/Dx
3
e
x
2
(c)
f.x/D
1Cx
2
1Cx3
(d)
f.x/D
1Cx
3
1Cx2
(e)
f.x/Dx2sin
3
xCx2cosx
(f)
f.x/Dex
2
sinx
(g)
f.x/De
x
sinx
2
(h)
f.x/De
x
2
cosx
14.
Giveanexampleofafunctionthathaszeroderivativesofallordersatalocalmini-
mumpoint.
Section2.5
Taylor’sTheorem
111
15.
Findthecriticalpointsof
f.x/D
x
3
3
C
bx
2
2
CcxCd
andidentifythemaslocalmaxima,localminima,orneither.
16.
Findanupperboundforthemagnitudeoftheerrorintheapproximation.
(a)
sinxx; jxj<
20
(b)
p
1Cx1C
x
2
; jxj<
1
8
(c)
cosx
1
p
2
1
x
4

;
4
<x<
5
16
(d)
logx.x1/
.x1/
2
2
C
.x1/
3
3
; jx1j<
1
64
17.
Prove:If
T
n
.x/D
Xn
rD0
x
r
;
then
T
n
.x/<T
nC1
.x/<e
x
<
1
x
nC1
.nC1/Š
1
T
n
.x/
if0<x<Œ.nC1/Š
1=.nC1/
.
18.
Theforwarddifferenceoperatorswithspacingh>0aredefinedby
0
f.x/Df.x/; f.x/Df.xCh/f.x/;
nC1
f.x/DŒ
n
f.x/; n1:
(a)
Provebyinductiononn:Ifk2,c
1
,...,c
k
areconstants,andn1,then
n
Œc
1
f
1
.x/CCc
k
f
k
.x/Dc
1
n
f
1
.x/CCc
k
n
f
k
.x/:
(b)
Provebyinduction:Ifn1,then
n
f.x/D
Xn
mD0
.1/
nm
n
m
!
f.xCmh/:
H
INT
:SeeExercise1.2.19:
InExercises2.5.192.5.22,istheforwarddifferenceoperatorwithspacingh>0.
112 Chapter2
DifferentialCalculusofFunctionsofOneVariable
19.
Letmandnbenonnegativeintegers, andletx
0
beanyrealnumber. Proveby
inductiononnthat
n
.xx
0
/
m
D
0
if 0mn;
nŠh
n
if mDn:
Doesthissuggestananalogybetween“differencing"anddifferentiation?
20.
Findanupperboundforthemagnitudeoftheerrorintheapproximation
f
00
.x
0
/
2
f.x
0
h/
h2
;
(a)
assumingthatf
000
isboundedon.x
0
h;x
0
Ch/;
(b)
assumingthatf
.4/
isboundedon.x
0
h;x
0
Ch/.
21.
Letf
000
beboundedonanopenintervalcontainingx
0
andx
0
C2h.Findaconstant
ksuchthatthemagnitudeoftheerrorintheapproximation
f
0
.x
0
/
f.x
0
/
h
Ck
2
f.x
0
/
h2
isnotgreaterthanMh2,whereM Dsup
˚
jf000.c/j
ˇ
ˇ
jx
0
<c<x
0
.
22.
Prove:Iff
.nC1/
isboundedonanopenintervalcontainingx
0
andx
0
Cnh,then
ˇ
ˇ
ˇ
ˇ
n
f.x
0
/
hn
f
.n/
.x
0
/
ˇ
ˇ
ˇ
ˇ
A
n
M
nC1
h;
whereA
n
isaconstantindependentoff and
M
nC1
D
sup
x
0
<c<x
0
Cnh
jf
.nC1/
.c/j:
H
INT
:SeeExercises2.5.18and2.5.19:
23.
Supposethatf.nC1/ existson.a;b/,x
0
,...,x
n
arein.a;b/,andpisthepolyno-
mialofdegree nsuchthatp.x
i
/ Df.x
i
/,0 i i n. Prove:Ifx x 2 .a;b/,
then
f.x/Dp.x/C
f
.nC1/
.c/
.nC1/Š
.xx
0
/.xx
1
/.xx
n
/;
wherec,whichdependsonx,isin.a;b/. H
INT
: Letxbefixed;distinctfromx
0
;
x
1
;...,x
n
;andconsiderthefunction
g.y/Df.y/p.y/
K
.nC1/Š
.yx
0
/.yx
1
/.yx
n
/;
where K is chosen sothatg.x/ D 0:UseRolle’s s theoremtoshowthatK D
f
.nC1/
.c/forsomecin.a;b/:
24.
DeduceTheorem2.5.4fromTheorem2.5.5.
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