The Extended Mean Value Theorem - INTRODUCTION TO REAL ANALYSIS

We now consider the extended mean value theorem, which implies Theorem 2.5.4 (Exer-cise 24). In the following theorem,aandbare the endpoints of an interval, but we do not assume thata < b.

Theorem 2.5.5 (Extended Mean Value Theorem)

Suppose thatf is con-tinuous on a finite closed intervalI with endpointsaandb .that is, eitherI D .a; b/or I D .b; a//; f^.n^C^1/exists on the open intervalI⁰;and;ifn > 0;thatf⁰, . . . ,f^.n/exist and are continuous ata:Then

f .b/

Proof

The proof is by induction. The mean value theorem (Theorem 2.3.11) implies the conclusion forn D 0. Now suppose thatn 1, and assume that the assertion of the theorem is true withnreplaced byn 1. The left side of (17) can be written as

f .b/

(the latter because of (18)) and is continuous on the closed intervalI and differentiable on I⁰, with

If we temporarily writef⁰Dg, this becomes g.b1/

Sinceb1 2 I⁰, the hypotheses onf imply thatg is continuous on the closed intervalJ with endpointsaandb1,g^.n/ exists onJ⁰, and, ifn 1,g⁰, . . . , g^{.n 1/} exist and are continuous ata(also atb1, but this is not important). The induction hypothesis, applied to gon the intervalJ, implies that

g.b1/

n 1X

rD0

g^{.r /}.a/

r Š .b1 a/^rD g^.n/.c/

nŠ .b1 a/ⁿ for somecinJ⁰. Comparing this with (20) and recalling thatgDf⁰yields

KDg^.n/.c/Df^.n^C^1/.c/:

Sincecis inI⁰, this completes the induction.

2.5 Exercises

1.

Let

f .x/D

e ^1=x²; x¤0;

0; xD0:

Show thatf has derivatives of all orders on. 1;1/and every Taylor polynomial off about0is identically zero. HINT: SeeExer ci se 2:4:40:

2.

Suppose thatf^.n^C^1/.x0/exists, and letTnbe thenth Taylor polynomial off about x0. Show that the function

En.x/D 8<

f .x/ Tn.x/

.x x0/ⁿ ; x2Df fx0g;

0; xDx0;

is differentiable atx0, and findE_n⁰.x0/.

3. (a)

Prove: Iff is continuous atx0and there are constantsa0anda1such that

xlim!x0

f .x/ a0 a1.x x0/

x x0 D0;

thena0Df .x0/,f⁰is differentiable atx0, andf⁰.x0/Da1.

(b)

Give a counterexample to the following statement: Iff andf⁰are continuous atx0and there are constantsa0,a1, anda2such that

xlim!x0

f .x/ a0 a1.x x0/ a2.x x0/²

.x x0/² D0;

thenf⁰⁰.x0/exists.

4. (a)

Prove: iff⁰⁰.x0/exists, then

hlim!0

f .x0Ch/ 2f .x0/Cf .x0 h/

h² Df⁰⁰.x0/:

(b)

Prove or give a counterexample: If the limit in

(a)

exists, then so does f⁰⁰.x0/, and they are equal.

5.

A functionf has asimplezero (or a zeroof multiplicity1) atx0iff is differentiable atx0andf .x0/D0, whilef⁰.x0/¤0.

(a)

Prove thatf has a simple zero atx0if and only if f .x/Dg.x/.x x0/;

whereg is continuous atx0 and differentiable on a deleted neighborhood of x0, andg.x0/¤0.

(b)

Give an example showing thatgin

(a)

need not be differentiable atx0.

6.

A functionf has a doublezero (or a zero of multiplicity2) at x0 iff is twice

differentiable atx0andf .x0/Df⁰.x0/D0, whilef⁰⁰.x0/¤0.

(a)

Prove thatf has a double zero atx0if and only if f .x/Dg.x/.x x0/²;

wheregis continuous atx0and twice differentiable on a deleted neighborhood ofx0,g.x0/¤0, and

xlim!x0

.x x0/g⁰.x/D0:

(b)

Give an example showing thatgin

(a)

need not be differentiable atx0.

7.

Letn be a positive integer. A functionf has a zeroof multiplicitynat x0 iff

isn times differentiable at x0, f .x0/ D f⁰.x0/ D D f^{.n 1/}.x0/ D 0 and f^.n/.x0/¤0. Prove thatf has a zero of multiplicitynatx0if and only if

f .x/Dg.x/.x x0/ⁿ;

wheregis continuous atx0andntimes differentiable on a deleted neighborhood of x0,g.x0/¤0, and

xlim!x0

.x x0/^jg^{.j /}.x/D0; 1j n 1:

HINT:Use Exercise6and induction:

8. (a)

Let

Q.x/D˛0C˛1.x x0/C C˛n.x x0/ⁿ be a polynomial of degreensuch that

xlim!x0

h.x/

.x x0/ⁿ D0:

Show that˛0D˛1D D˛nD0.

(b)

Suppose thatf isntimes differentiable atx0andpis a polynomial p.x/Da0Ca1.x x0/C Can.x x0/ⁿ of degreensuch that

xlim!x0

f .x/ p.x/

.x x0/ⁿ D0:

Show that

ar D f^{.r /}.x0/

r Š if 0rnI that is,pDTn, thenth Taylor polynomial off aboutx0.

9.

Show that iff^.n/.x0/andg^.n/.x0/exist and

xlim!x0

f .x/ g.x/

.x x0/ⁿ D0;

thenf^{.r /}.x0/Dg^{.r /}.x0/,0r n.

10. (a)

LetFn,Gn, andHnbe thenth Taylor polynomials aboutx0off,g, and their producthDfg. Show thatHncan be obtained by multiplyingFnbyGnand retaining only the powers ofx x0through thenth. HINT:Use Exercise8.b/:

(b)

Use the method suggested by

(a)

to computeh^{.r /}.x0/,r D1; 2; 3; 4.

(i)

h.x/De^xsinx; x0D0

(ii)

h.x/D.cos x=2/.logx/; x0 D1

(iii)

h.x/Dx²cosx; x0D=2

(iv)

h.x/D.1Cx/ ¹e ^x; x0D0

11. (a)

It can be shown that ifgisntimes differentiable atxandf isntimes dif-ferentiable atg.x/, then the composite functionh.x/ D f .g.x// isntimes differentiable atxand

h^.n/.x/D Xn

rD1

f^{.r /}.g.x//X

r Š r1Š rnŠ

g⁰.x/

1Š r1

g⁰⁰.x/

2Š r2

g^.n/.x/

nŠ

!rn

whereP

ris over alln-tuples.r1; r2; : : : ; rn/of nonnegative integers such that r1Cr2C CrnDr

and

r1C2r2C CnrnDn:

(This isFaa di Bruno’s formula.) However, this formula is quite complicated.

Justify the following alternative method for computing the derivatives of a composite function at a pointx0:

LetFnbe thenth Taylor polynomial off abouty0 D g.x0/, and letGnand Hn be thenth Taylor polynomials ofg andh aboutx0. Show thatHncan be obtained by substitutingGnintoFnand retaining only powers ofx x0

through thenth. HINT:See Exercise8.b/:

(b)

Compute the first four derivatives ofh.x/ Dcos.sinx/atx0 D 0, using the method suggested by

(a)

12. (a)

Ifg.x0/¤0andg^.n/.x0/exists, then the reciprocalhD1=gis alsontimes differentiable atx0, by Exercise 11

(a)

, withf .x/D1=x. LetGnandHnbe thenth Taylor polynomials ofgandhaboutx0. Use Exercise 11

(a)

to prove that ifg.x0/D1, thenHncan be obtained by expanding the polynomial

rD1

Œ1 Gn.x/^r

in powers ofx x0and retaining only powers through thenth.

(b)

Use the method of

(a)

to compute the first four derivatives of the following functions atx0.

(i)

h.x/Dcscx; x0D=2

(ii)

h.x/D.1CxCx²/ ¹; x0D0

(iii)

h.x/Dsecx; x0D=4

(iv)

h.x/DŒ1Clog.1Cx/ ¹; x0D0

(c)

Use Exercise 10 to justify the following alternative procedure for obtaining Hn, again assuming thatg.x0/D1: If

Gn.x/D1Ca1.x x0/C Can.x x0/ⁿ (where, of course,ar Dg^{.r /}.x0/=r Š/and

Hn.x/Db0Cb1.x x0/C Cbn.x x0/ⁿ; then

b0D1; bkD Xk

rD1

arbk r; 1kn:

13.

Determine whetherx0D0is a local maximum, local minimum, or neither.

(a)

f .x/Dx²e^x³

(b)

f .x/Dx³e^x²

(c)

f .x/D 1Cx²

1Cx³

(d)

f .x/D 1Cx³

1Cx²

(e)

f .x/Dx²sin³xCx²cosx

(f )

f .x/De^x²sinx

(g)

f .x/De^xsinx²

(h)

f .x/De^x²cosx

14.

Give an example of a function that has zero derivatives of all orders at a local mini-mum point.

15.

Find the critical points of

f .x/D x³ 3 C bx²

2 CcxCd and identify them as local maxima, local minima, or neither.

16.

Find an upper bound for the magnitude of the error in the approximation.

(a)

sinxx; jxj<

(b)

^p1Cx1C x

2; jxj< 1 8

(c)

cosx 1

1 x ₄

; ₄ < x < ⁵₁₆

(d)

logx .x 1/ .x 1/²

2 C.x 1/³

3 ; jx 1j< 1 64

17.

Prove: If

Tn.x/D Xn

rD0

x^r r Š; then

Tn.x/ < TnC1.x/ < e^x <

1 xⁿ^C¹ .nC1/Š

Tn.x/

if0 < x < Œ.nC1/Š^1=.n^C^1/.

18.

The forward difference operators with spacingh > 0are defined by

⁰f .x/Df .x/; f .x/Df .xCh/ f .x/;

ⁿ^C¹f .x/D Œⁿf .x/ ; n1:

(a)

Prove by induction onn: Ifk2,c1, . . . ,c_kare constants, andn1, then

ⁿŒc1f1.x/C Cckfk.x/Dc1ⁿf1.x/C Cckⁿfk.x/:

(b)

Prove by induction: Ifn1, then

ⁿf .x/D Xn

mD0

. 1/^{n m} n m

f .xCmh/:

HINT:See Exercise1:2:19:

In Exercises19–22,is the forward difference operator with spacingh > 0.

19.

Let mand n be nonnegative integers, and let x0 be any real number. Prove by induction onnthat

ⁿ.x x0/^mD

0 if 0mn;

nŠhⁿ if mDn:

Does this suggest an analogy between “differencing" and differentiation?

20.

Find an upper bound for the magnitude of the error in the approximation f⁰⁰.x0/ ²f .x0 h/

h² ;

(a)

assuming thatf⁰⁰⁰is bounded on.x0 h; x0Ch/;

(b)

assuming thatf^.4/is bounded on.x0 h; x0Ch/.

21.

Letf⁰⁰⁰be bounded on an open interval containingx0andx0C2h. Find a constant ksuch that the magnitude of the error in the approximation

f⁰.x0/ f .x0/

h Ck²f .x0/ h² is not greater thanM h², whereM Dsup˚

jf⁰⁰⁰.c/jˇˇjx0< c < x0 .

22.

Prove: Iff^.n^C^1/is bounded on an open interval containingx0andx0Cnh, then ˇˇ

ˇˇ

ⁿf .x0/

hⁿ f^.n/.x0/ ˇˇ

ˇˇAnMnC1h;

whereAnis a constant independent off and MnC1 D sup

x0<c<x0Cnhjf^.n^C^1/.c/j: HINT:See Exercises18and19:

23.

Suppose thatf^.n^C^1/ exists on.a; b/,x0, . . . ,xnare in.a; b/, andpis the polyno-mial of degree nsuch thatp.xi/D f .xi/,0 i n. Prove: Ifx 2 .a; b/, then

f .x/Dp.x/C f^.n^C^1/.c/

.nC1/Š .x x0/.x x1/ .x xn/;

wherec, which depends onx, is in.a; b/. HINT: Letxbe fixed;distinct fromx0; x1;. . . ,xn;and consider the function

g.y/Df .y/ p.y/ K

.nC1/Š.y x0/.y x1/ .y xn/;

where K is chosen so that g.x/ D 0: Use Rolle’s theorem to show that K D f^.n^C^1/.c/for somecin.a; b/:

24.

Deduce Theorem 2.5.4 from Theorem 2.5.5.

Integral Calculus of Functions of One Variable

IN THIS CHAPTER we discuss the Riemann integral of a bounded function on a finite intervalŒa; b, and improper integrals in which either the function or the interval of inte-gration is unbounded.

SECTION 3.1 begins with the definition of the Riemann integral and presents the geo-metrical interpretation of the Riemann integral as the area under a curve. We show that an unbounded function cannot be Riemann integrable. Then we define upper and lower sums and upper and lower integrals of a bounded function. The section concludes with the definition of the Riemann–Stieltjes integral.

SECTION 3.2 presents necessary and sufficient conditions for the existence of the Riemann integral in terms of upper and lower sums and upper and lower integrals. We show that continuous functions and bounded monotonic functions are Riemann integrable.

SECTION 3.3 begins with proofs that the sum and product of Riemann integrable functions are integrable, and thatjfjis Riemann integrable iff is Riemann integrable. Other topics covered include the first mean value theorem for integrals, antiderivatives, the fundamental theorem of calculus, change of variables, integration by parts, and the second mean value theorem for integrals.

SECTION 3.4 presents a comprehensive discussion of improper integrals. Concepts de-fined and considered include absolute and conditional convergence of an improper integral, Dirichlet’s test, and change of variable in an improper integral.

SECTION 3.5 defines the notion of a set with Lebesgue measure zero, and presents a necessary and sufficient condition for a bounded functionf to be Riemann integrable on an intervalŒa; b; namely, that the discontinuities off form a set with Lebesgue masure zero.

113

3.1 DEFINITION OF THE INTEGRAL

The integral that you studied in calculus is theRiemann integral, named after the German mathematician Bernhard Riemann, who provided a rigorous formulation to replace the intuitive notion of integral due to Newton and Leibniz. Since Riemann’s time, other kinds of integrals have been defined and studied; however, they are all generalizations of the Riemann integral, and it is hardly possible to understand them or appreciate the reasons for developing them without a thorough understanding of the Riemann integral. In this section we deal with functions defined on a finite intervalŒa; b. Apartition ofŒa; bis a set of subintervals

Œx0; x1; Œx1; x2; : : : ; Œxn 1; xn; (1) where

aDx0< x1 < xnDb: (2) Thus, any set ofnC1points satisfying (2) defines a partitionP ofŒa; b, which we denote by

P D fx0; x1; : : : ; xng:

The pointsx0,x1, . . . , xnare thepartition pointsofP. The largest of the lengths of the subintervals (1) is thenormofP, written askPk; thus,

kPk D max

1in.xi xi 1/:

IfP andP⁰ are partitions ofŒa; b, thenP⁰is arefinement ofP if every partition point ofP is also a partition point ofP⁰; that is, ifP⁰is obtained by inserting additional points between those ofP. Iff is defined onŒa; b, then a sum

jD1

f .cj/.xj xj 1/;

where

xj 1 cj xj; 1j n;

is aRiemann sum off over the partitionP D fx0; x1; : : : ; xng(Occasionally we will say more simply that is a Riemann sum off overŒa; b.) Sincecj can be chosen arbitrarily inŒxj; xj 1, there are infinitely many Riemann sums for a given functionf over a given partitionP.

Definition 3.1.1

Letf be defined onŒa; b. We say thatf isRiemann integrable on Œa; bif there is a numberLwith the following property: For every > 0, there is aı > 0 such that

j Lj<

ifis any Riemann sum off over a partitionP ofŒa; bsuch thatkPk< ı. In this case, we say thatListhe Riemann integral off overŒa; b, and write

Z b

f .x/ dxDL:

We leave it to you (Exercise 1) to show thatRb

a f .x/ dx is unique, if it exists; that is, there cannot be more than one numberLthat satisfies Definition 3.1.1.

For brevity we will say “integrable” and “integral” when we mean “Riemann integrable”

and “Riemann integral.” Saying thatRb

a f .x/ dx exists is equivalent to saying thatf is integrable onŒa; b.

Most of the terms in the sum on the right cancel in pairs; that is, Xn

Thus, every Riemann sum off over any partition ofŒa; bequalsb a, so Z b

dxDb a:

Example 3.1.2

Riemann sums for the function f .x/Dx; axb;

Substituting (4) into (3) yields D

Because of cancellations like those in Example 3.1.1, Xn

jD1

.x_j² x²_{j 1}/Db² a²; so (6) can be rewritten as

D b² a²

2 C

jD1

dj.xj xj 1/:

Hence, ˇˇˇ

ˇ b² a² 2

ˇˇˇ ˇ

jD1

jdjj.xj xj 1/ kPk 2

jD1

.xj xj 1/ (see (5)) D kPk

2 .b a/:

Therefore, every Riemann sum off over a partitionP ofŒa; bsatisfies ˇˇ

ˇˇ b² a² 2

ˇˇ

ˇˇ< if kPk< ıD 2 b a: Hence,

Z b

x dxD b² a²

2 :

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