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Calculus II

Vladimir V. Kisil

22nd May 2003

1

Chapter 1

General Information

This is an online manual is designed for students. The manual is available at the moment in

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frames and PDF formats. Each from these formats has its own advantages.

Please select one better suit your needs.

There is on-line information on the following courses:

²Calculus I

²Calculus II

²Geometry

1.1 Web page

There is a Web page which contains this course description as well as other information related to this course. Point your Web browser to http://maths.leeds.ac.uk/ kisilv/courses/math152.html 2

1.2. COURSE DESCRIPTION AND SCHEDULE3

1.2 Course description and Schedule

Dates

Topics 1 General Information 1.1Web page

1.2Course description and Schedule1.3Warn-

ings and Disclaimers9 Infinite Series 9.5A brief review of series9.6Power Series9.7Power

Series Representations of Functions9.8Maclaurin

and Taylor Series9.9Applications of Taylor Poly- nomials11 Vectors and Surfaces 11.2Vectors in Three Dimensions11.3Dot Product11.4Vec- tor Product11.5Lines and Planes11.6Surfaces

12 Vector-Valued FunctionsVector-Valued Func-

tions12.1Limits, Derivatives and Integrals13 Par- tial Differentiation 13.1Functions of Several Vari- ables13.2Limits and Continuity13.3Partial Deriv- atives13.4Increments and Differentials13.5Chain

Rules13.6Directional Derivatives13.7Tangent

Planes and Normal Lines13.8Extrema of Func-

tions of Several Variables13.9Lagrange Multipli- ers14 Multiply Integrals 14.1Double Integ- rals14.2Area and Volume14.3Polar Coordinates

14.4Surface Area14.5Triple Integrals14.7Cyl-

indrical Coordinates14.8Spherical Coordinates

15 Vector Calculus 15.1Vector Fields15.2Line

Integral15.3Independence of Path15.4Green"s The-

orem15.5Surface Integral15.6Divergence Theorem

15.7Stoke"s Theorem

1.3 Warnings and Disclaimers

Before proceeding with this interactive manual we stress the following: ²These Web pages are designed in order to help students as a source ofadditional information. They areNOTan obligatory part of the course. ²The main material introduced duringlecturesand is contained inText- book. This interactive manual isNOTa substitution for any part of those primary sources of information. ²It isNOTrequired to be familiar with these pages in order to pass the examination.

4CHAPTER 1. GENERAL INFORMATION

²The entire contents of these pages is continuously improved and up- dated. Even for material of lectures took place weeks or months ago changes are made.

Contents

1 General Information

2

1.1 Web page

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Course description and Schedule

. . . . . . . . . . . . . . . . . 3

1.3 Warnings and Disclaimers

. . . . . . . . . . . . . . . . . . . . 3

9 Infinite Series

7

9.5 A brief review of series

. . . . . . . . . . . . . . . . . . . . . . 7

9.6 Power Series

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

9.7 Power Series Representations of Functions

. . . . . . . . . . . 9

9.8 Maclaurin and Taylor Series

. . . . . . . . . . . . . . . . . . . 11

9.9 Applications of Taylor Polynomials

. . . . . . . . . . . . . . . 13

11 Vectors and Surfaces

14

11.2 Vectors in Three Dimensions

. . . . . . . . . . . . . . . . . . . 14

11.3 Dot Product

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

11.4 Vector Product

. . . . . . . . . . . . . . . . . . . . . . . . . . 17

11.5 Lines and Planes

. . . . . . . . . . . . . . . . . . . . . . . . . 18

11.6 Surfaces

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

12 Vector-Valued Functions

22

Vector-Valued Functions

. . . . . . . . . . . . . . . . . . . . . . . . 22

12.1 Limits, Derivatives and Integrals

. . . . . . . . . . . . . . . . . 23

13 Partial Differentiation

26

13.1 Functions of Several Variables

. . . . . . . . . . . . . . . . . . 26

13.2 Limits and Continuity

. . . . . . . . . . . . . . . . . . . . . . 27

13.3 Partial Derivatives

. . . . . . . . . . . . . . . . . . . . . . . . 28

13.4 Increments and Differentials

. . . . . . . . . . . . . . . . . . . 30

13.5 Chain Rules

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

13.6 Directional Derivatives

. . . . . . . . . . . . . . . . . . . . . . 31

13.7 Tangent Planes and Normal Lines

. . . . . . . . . . . . . . . . 32 5

6CONTENTS

13.8 Extrema of Functions of Several Variables

. . . . . . . . . . . 33

13.9 Lagrange Multipliers

. . . . . . . . . . . . . . . . . . . . . . . 34

14 Multiply Integrals

35

14.1 Double Integrals

. . . . . . . . . . . . . . . . . . . . . . . . . . 35

14.2 Area and Volume

. . . . . . . . . . . . . . . . . . . . . . . . . 37

14.3 Polar Coordinates

. . . . . . . . . . . . . . . . . . . . . . . . . 37

14.4 Surface Area

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

14.5 Triple Integrals

. . . . . . . . . . . . . . . . . . . . . . . . . . 39

14.7 Cylindrical Coordinates

. . . . . . . . . . . . . . . . . . . . . 40

14.8 Spherical Coordinates

. . . . . . . . . . . . . . . . . . . . . . . 41

15 Vector Calculus

43

15.1 Vector Fields

. . . . . . . . . . . . . . . . . . . . . . . . . . . 43

15.2 Line Integral

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

15.3 Independence of Path

. . . . . . . . . . . . . . . . . . . . . . . 46

15.4 Green"s Theorem

. . . . . . . . . . . . . . . . . . . . . . . . . 47

15.5 Surface Integral

. . . . . . . . . . . . . . . . . . . . . . . . . . 48

15.6 Divergence Theorem

. . . . . . . . . . . . . . . . . . . . . . . 49

15.7 Stoke"s Theorem

. . . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 9

Infinite Series

9.5 A brief review of series

We refer to the chapter

Infinite Series

of the course

Calculus I

for the review of the following topics. (i). Sequences of numbers (ii). Convergent and Divergent Series (iii). Positive Term Series (iv). Ratio and Root Test (v). Alternating Series and Absolute Convergence

9.6 Power Series

It is well known that polynomials are simplest functions, particularly it is easy to differentiate and integrate polynomials. It is desirable to use them for investigation of other functions. Infinite series reviewed in the previous sections are very important because they allow to represent functions by means of power series, which are similar to polynomials in many respects. An example of such representations is harmonic series 1 X n=0r n=1

1¡r:

7

8CHAPTER 9. INFINITE SERIES

Definition 9.6.1Letxbe a variable. Apower series inxis a series of the form 1X n=0b where eachbkis real number. A power series turns to be infinite (constant term) series if we will substitute a constantcinstead of the variablex. Such series could converge or diverge. All power series converge forx= 0. The convergence of power series described by the following theorem. Theorem 9.6.2(i). If a power seriesPbnxnconverges for a nonzero num- berc, then it is absolutely convergent wheneverjxjPbnxndiverges for a nonzero numberd, then it di- verges wheneverjxj>jdj.

Proof.The proof follows from the

Basic Comparison Test

of the power series forjxjand convergent geometric series withr=¯¯x c

From this theorem we could conclude that

Theorem 9.6.3IfPbnxnis a power series, then exactly one of the following true: (i). The series converges only ifx= 0. (ii). The series is absolutely convergent for everyx. (iii). There is a numberrsuch that the series is absolutely convergent ifx is in open interval(¡r;r)and divergent ifx <¡rorx > r. The numberrfrom the above theorem is calledradius of convergence. The totality of numbers for which a power series converges is called itsinterval of convergence. The interval of convergence may be any of the following four types: [¡r;r], [¡r;r), (¡r;r], (¡r;r).

There is a more general type of power series

Definition 9.6.4Letbbe a real number andxis a variable. Apower series inx¡dis a series of the form 1 X n=0b n(x¡d)n=b0+b1(x¡d) +b2(x¡d)2+¢¢¢+bn(x¡d)n+¢¢¢; where eachbnis a real number.

9.7. POWER SERIES REPRESENTATIONS OF FUNCTIONS9

This power series is obtained from the series in Definition 9.6.1 by re- placement ofxbyx¡d. We could obtain a description of convergence of this series by replacement ofxbyx¡din Theorem 9.6.3 The following exercises should be solved in the following way: (i). Determine the radiusrof convergence, usually using

Ratio test

or Root Test (ii). If the radiusris finite and nonzero determine if the series is convergent at pointsx=¡r,x=r. Note that the series could be alternating at one of them and apply

Alternating Test

Exercise 9.6.5Find the interval of convergence of the power series: X 1 n

2+ 4xn;X1

ln(n+ 1)xn; X 10n+1 3

2nxn;X(3n)!

(2n)!xn; X 10n n!xn;X1

2n+ 1(x+ 3)n;

X n 3

2n¡1(x¡1)2n;X1

p

3n+ 4(3x+ 4)n;

9.7 Power Series Representations of Functions

As we have seen in the previous section a power series

Pbnxncould define

a convergent infinite seriesPbncnfor allc2(¡r;r) which has a sumf(c). Thus the power series define a functionf(x) =Pbnxnwith domain (¡r;r). We call it thepower series representation off(x). Power series are used in calculators and computers. Example 9.7.1Find function represented byP(¡1)kxk. The following theorem shows that integration and differentiations could be done with power series as easy as with polynomials: Theorem 9.7.2Suppose that a power seriesPbnxnhas a radius of conver- gencer >0, and letfbe defined by f(x) =1X n=0b

10CHAPTER 9. INFINITE SERIES

for everyx2(¡r;r). Then for¡r < x < r f

0(x) =b1+b2x+b3x2+¢¢¢+nbnxn¡1+¢¢¢(9.7.1)

1X n=1nb nxn¡1; Z x 0 f(t)dt=b0x+b1x2 2 +b2x3 3 +¢¢¢+bnxn+1 n+ 1+¢¢¢(9.7.2) 1X n=0b n n+ 1xn+1:

Example 9.7.3Find power representation for

(i). 1 (1+x)2. (ii). ln(1 +x) and calculate ln(1:1) to five decimal places. (iii). arctanx.

Theorem 9.7.4Ifxis any real number,

e x= 1 +x 1 +x2 2! +x3 3! +¢¢¢=1X n=0x n n!: Proof.The proof follows from observation that the power seriesf(x) =Pxn n!satisfies to the equationf0(x) =f(x) and the only solution to this

Corollary 9.7.5

e= 1 +1 1! +1 2! +1 3! Example 9.7.6Find a power series representation for sinhx,xe¡2x. Exercise 9.7.7Find a power series representation forf(x),f0(x),Rx

0f(t)dt.

f(x) =1

1 + 5x;f(x) =1

3¡2x:

Exercise 9.7.8Find a power series representation and specify the radius of convergence for: x

1¡x4;x2¡3

x¡2: Exercise 9.7.9Find a power series representation for f(x) =x2e(x2);f(x) =x4arctan(x4):

9.8. MACLAURIN AND TAYLOR SERIES11

9.8 Maclaurin and Taylor Series

We find several power series representation of functions in the previous sec- tion by a variety of different tools.Could it be done in a regular fashion?

Two following theorem give the answer.

Theorem 9.8.1If a functionfhas a power series representation f(x) =1X k=0b nxn with radius of convergencer >0, thenf(k)(0)exists for every positive integer kand f(x) =f(0) +f0(0) 1! x+f00(0) 2! x2+¢¢¢+f(n)(0) n!xn+¢¢¢=1X n=0f (n)(0) n!xn Theorem 9.8.2If a functionfhas a power series representation f(x) =1X k=0b n(x¡d)n with radius of convergencer >0, thenf(k)(d)exists for every positive integer kand f(x) =f(d)+f0(d) 1! (x¡d)+f00(d) 2! (x¡d)2+¢¢¢+f(n)(d) n!(x¡d)n+¢¢¢=1X n=0f (n)(d) n!(x¡d)n

Exercise 9.8.3Find Maclaurin series for:

f(x) = sin2x;f(x) =1

1¡2x:

Remark 9.8.4It is easy to see that

linear approximation formula is just the Taylor polynomialPn(x) forn= 1. The last formula could be split to two parts: thenth-degree Taylor poly- nomialPn(x) offatd: P n(x) =f(d) +f0(d) 1! (x¡d) +f00(d) 2! (x¡d)2+¢¢¢+f(n)(d) n!(x¡d)n and theTaylor remainder R n(x) =f(n+1)(z) (n+ 1)!(x¡d)n+1; wherez2(d;x). Then we could formulate a sufficient condition for the existence of power series representation off.

12CHAPTER 9. INFINITE SERIES

Theorem 9.8.5Letfhave derivatives of all orders throughout an interval containingd, and letRn(x)be the Taylor remainder offatd. If lim n!1Rn(x) = 0 for everyxin the interval, thenf(x)is represented by the Taylor series for f(x)atd.

Example 9.8.6Letfbe the function defined by

f(x) =½e¡1=x2ifx6= 0;

0 ifx= 0;

thenfcannot be represented by a Maclaurin series.

Exercise 9.8.7Show that for functionf(x) =e¡x

lim n!1Rn(x) = 0 and find the Maclaurin series.

The important Maclaurin series are:

Function

Maclaurin series

Convergence

e x P 1 n=0xn n! (¡1;1) ln(1 +x) P 1 n=0(¡1)nxn+1 n+1 (¡1;1] sinx P 1 n=0(¡1)nx2n+1 (2n+1)! (¡1;1) cosx P 1 n=0(¡1)nx2n (2n)!quotesdbs_dbs29.pdfusesText_35

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