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Calculus with Precalculus

Vladimir V. Kisil

March 22, 2000

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Chapter 1

General Information

This interactive manual is designed for students attending this course inEastern Mediterranean University. The manual is being regularly updatedin order to reflect material presented during lectures.

The manual is available at the moment inHTML with frames (for eas-ier navigation),HTML without frames andPDFformats. Each from theseformats has its own advantages. Please select one better suit your needs.

There is on-line information on the following courses:

Calculus I.

Calculus II. Geometry.There are other on-line calculus manuals. See for exampleE-calculus of D.P. Story.

1.1 Warnings and Disclaimers

Before proceeding with this interactive manual we stress the following:

These Web pages are designed in order to help students as a sourceof additional information. They are NOT an obligatory part of thecourse.

The main material introduced duringlecturesand is contained inText-book. This interactive manual is NOT a substitution for any part ofthose primary sources of information.

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1.2. RECOMMENDED EXERCISES 3

It isNOT required to be familiar with these pages in order to pass the

examination.

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

1.2 Recommended Exercises

Section 1.1. Problems 17, 23, 25Section 1.2. Problems 17, 27

Section 1.3. Problems 25, 31, 39, 63Section 1.4. Problems 11, 19, 23, 33Section 1.5. Problems 15, 49, 55, 65Section 2.1. Problems 1, 19Section 2.2. Problems 21, 27Section 2.3. Problems 21, 53, 55Section 2.4. Problems 13, 17, 19, 41, 45Section 2.5. Problems 17, 19, 47, 55Section 2.6. Problems 11, 17, 27, 33Section 2.7. Problem 7

Section 2.8. Problems 1, 5Section 3.1. Problems 7, 25Section 3.2. Problems 11, 15, 41Section 3.3. Problems 5, 17, 29Section 3.4. Problems 7, 13, 23Section 3.5. Problems 13, 17, 31, 41Section 3.6. Problems 5, 7, 29Section 4.1. Problems 9, 17Section 4.2. Problems 17, 21, 23, 31, 35, 39Section 4.3. Problems 9, 27, 35

Section 4.4. Problems 9, 27, 33Section 4.5. Problems 13, 21, 25Section 4.6. Problems 19, 25, 35, 47, 59Section 5.1. Problems 15, 23Section 5.2. Problems 11, 13, 29Section 5.3. Problems 7, 15, 23, 27, 29Section 5.4. Problems 3, 14Section 5.5. Problems 7, 12, 29, 33Section 6.1. Problems 25, 33

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4 CHAPTER 1. GENERAL INFORMATION

Section 6.2. Problems 37, 41

Section 6.3. Problems 33, 37Section 6.4. Problems 11, 23, 31, 37Section 6.5. Problems 35, 43, 45Section 6.7. Problems 5, 21, 43, 53, 61Section 6.8. Problems 32, 34Section 6.9. Problems 9, 17, 63, 65Section 7.1. Problems 9, 17, 29, 31Section 7.2. Problems 3, 5, 13, 21Section 7.3. Problems 5, 9, 11, 17Section 7.4. Problems 7, 21, 25

Section 7.5. Problems 5, 9, 15Section 7.7. Problems 13, 15, 53, 63Section 8.1. Problems 23, 31Section 8.2. Problems 13, 27, 47Section 8.3. Problems 5, 11, 15, 25, 27, 39, 43Section 8.4. Problems 7, 9, 15, 17, 25, 31Section 8.5. Problems 11, 15, 21, 29

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Course Outline

1 General Information 2

1.1 Warnings and Disclaimers . . . . . . . . . . . . . . . . . . . . 21.2 Recommended Exercises . . . . . . . . . . . . . . . . . . . . . 3

2 Limits 82.1 Introduction to Limits . . . . . . . . . . . . . . . . . . . . . . 82.2 Definition of Limit . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Techniques for Finding Limits . . . . . . . . . . . . . . . . . . 112.4 Limits Involving Infinity . . . . . . . . . . . . . . . . . . . . . 122.5 Continuous Functions. . . . . . . . . . . . . . . . . . . . . . . 13

3 Derivative 163.1 Tangent Lines and Rates of Changes . . . . . . . . . . . . . . 163.2 Definition of Derivative . . . . . . . . . . . . . . . . . . . . . . 173.3 Techniques of Differentiation . . . . . . . . . . . . . . . . . . . 173.4 Derivatives of the Trigonometric Functions . . . . . . . . . . . 183.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . 193.7 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.8 Linear Approximations and Differentials . . . . . . . . . . . . 20

4 Appliactions of Derivative 22

4.1 Extrema of Functions . . . . . . . . . . . . . . . . . . . . . . . 224.2 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . 244.3 The First Derivative Test. . . . . . . . . . . . . . . . . . . . . 254.4 Concavity and the Second Derivative Test . . . . . . . . . . . 264.5 Summary of Graphical Methods . . . . . . . . . . . . . . . . . 274.6 Optimization Problems. Review . . . . . . . . . . . . . . . . . 28

5 Integrals 295.1 Antiderivatives and Indefinite Integrals . . . . . . . . . . . . . 29

5

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6 COURSE OUTLINE

5.2 Change of Variables in Indefinite Integrals . . . . . . . . . . . 31

5.3 Summation Notation and Area . . . . . . . . . . . . . . . . . 315.4 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . 325.5 Properties of the Definite Integral . . . . . . . . . . . . . . . . 335.6 The Fundamental Theorem of Calculus . . . . . . . . . . . . . 34

6 Applications of the Definite Integral 366.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.2 Solids of Revolution. . . . . . . . . . . . . . . . . . . . . . . . 366.3 Volumes by Cylindrical Shells . . . . . . . . . . . . . . . . . . 376.4 Volumes by Cross Section . . . . . . . . . . . . . . . . . . . . 38

6.5 Arc Length and Surfaces of Revolution . . . . . . . . . . . . . 38

7 Trancendential Functions 407.1 The Derivative of the Inverse Function . . . . . . . . . . . . . 407.2 The Natural Logarithm Function . . . . . . . . . . . . . . . . 417.3 The Exponential Function . . . . . . . . . . . . . . . . . . . . 437.4 Integration Logarithm and Exponents. . . . . . . . . . . . . . 447.5 General Exponential and Logarithmic Functions . . . . . . . . 447.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . 467.8 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . 477.9 lHospitals Rule. . . . . . . . . . . . . . . . . . . . . . . . . . 49

8 Techniques of Integration 518.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . 518.2 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . 518.3 Trigonometric Substitution . . . . . . . . . . . . . . . . . . . . 528.4 Integrals of Rational Functions . . . . . . . . . . . . . . . . . 538.5 Quadratic Expressions . . . . . . . . . . . . . . . . . . . . . . 538.6 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . 54

9 Infinite Series 56

9.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569.2 Convergent or Divergent Series . . . . . . . . . . . . . . . . . 579.3 Positive-Term Series . . . . . . . . . . . . . . . . . . . . . . . 589.4 The Ratio and Root Tests . . . . . . . . . . . . . . . . . . . . 599.5 Alternating Series and Absolute Convergence. . . . . . . . . . 60

A Algebra 62A.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A.2 Polynomial. Factorization of Polynomials . . . . . . . . . . . . 62

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COURSE OUTLINE 7

A.3 Binomial Formula . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.4 Real Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63A.5 Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.6 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

B Function and Their Graph 66B.1 Rectangular (Cartesian) Coordinates . . . . . . . . . . . . . . 66B.2 Graph of an Equation . . . . . . . . . . . . . . . . . . . . . . 67B.3 Line Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 67B.4 Symmetries and Shifts . . . . . . . . . . . . . . . . . . . . . . 67B.5 Definition of a Function. Domain and Range . . . . . . . . . . 68

B.6 One-to-One Functions. Periodic Functions . . . . . . . . . . . 69B.7 Inc/Decreasing, Odd/Even Functions . . . . . . . . . . . . . . 69

C Conic Section 71C.1 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71C.2 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71C.3 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72C.4 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74C.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

D Trigonometic Functions 76

E Exponential and Logarithmic Functions 78

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Chapter 2

Limits

The notion oflimitsis central for calculus almost any other notion of calculus(continuity, convergence of a sequence, derivative, integral, etc.) is based onlimits. Thus it is very important to be command with limits.

2.1 Introduction to Limits

The following is a quote from a letter of Donald E. Knuth to Notices ofAMS.

The most important of these changes would be to introduce the O nota-tion and related ideas at an early stage. This notation, first used by Bach-mann in 1894 and later popularized by Landau, has the great virtue thatit makes calculations simpler, so it simplifies many parts of the subject, yetit is highly intuitive and easily learned. The key idea is to be able to dealwith quantities that are only partly specified, and to use them in the midstof formulas.

I would begin my ideal calculus course by introducing a simpler A no-tation, which means absolutely at most. For example, A(2) stands for aquantity whose absolute value is less than or equal to 2. This notation has

a natural connection with decimal numbers: Saying that is approximately3.14 is equivalent to saying that = 3.14 +A(.005). Students will easilydiscover how to calculate with A:

10A(2) = A(100) ;

(3.14 +A(.005))(1 +A(0.01)) = 3.14 +A(.005) +A(0.0314) +A(.00005)

= 3.14 +A(0.3645) = 3.14 +A(.04) .

I would of course explain that the equality signis not symmetricwith respectto such notations; we have 3 = A(5) and 4 = A(5) but not 3 = 4, nor can

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2.1. INTRODUCTION TO LIMITS 9

we say that A(5) = 4. We can, however, say thatA(0) = 0. As de Bruijn

points out in [1,1.2], mathematicians customarily use the = sign as theyuse the word is in English: Aristotle is a man, but a man isnt necessarilyAristotle.

TheA notation applies to variable quantities as well as to constant ones.For example,

sin x = A(1);

x = A(x) ;

A(x) = xA(1);

A(x) +A(y) = A(x+y) ifx

0 and y

0 ;

(1 +A(t))2 = 1 + 3A(t) ift= A(1) .

Once students have caught on to the idea ofAnotation, they are ready forO notation, which is even less specific. In its simplest form, O(x) stands forsomething that isCA(x) for some constantC, but we dont say whatC is. Wealso define side conditions on the variables that appear in the formulas. Forexample, ifnis a positive integer we can say that any quadratic polynomialinn isO(n2). Ifnis sufficiently large, we can deduce that

(n+O(

n ))(ln n++O(1/n)) = n ln n+n+O(1) +O(

n ln n) +O(

n ) +O(1/

n )

= n ln n+n+O(n ln n) .Im sure it would be a pleasure for both students and teacher if calculus

were taught in this way. The extra time needed to introduce O notation isamply repaid by the simplifications that occur later. In fact, there probablywill be time to introduce the onotation, which is equivalent to the taking oflimits, and to give the general definition of a not-necessarily-strong derivative:

f(x+) =f(x) +f(x)+o() .

The function fis continuous at xif

f(x+) =f(x) +o(1);

and so on. But I would not mind leaving a full exploration of such things toa more advanced course, when it will easily be picked up by anyone who haslearned the basics withO alone. Indeed, I have not needed to use o in 2200pages ofThe Art of Computer Programming, although many techniques ofadvanced calculus are applied throughout those books to a great variety ofproblems.

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10 CHAPTER 2. LIMITS

Students will be motivated to use O notation for two important rea-

sons. First, it significantly simplifies calculations because it allows us tobe sloppybut in a satisfactorily controlled way. Second, it appears in thepower series calculations of symbolic algebra systems like Mapleand Math-ematica, which todays students will surely be using.

[1]: N. G. de Bruijn, Asymptotic Methods in Analysis (Amsterdam:North-Holland, 1958). [2]: R. L. Graham, D. E. Knuth, and O. Patashnik,

Concrete Mathematics(Reading, Mass.: AddisonWesley, 1989).

2.2 Definition of Limit

Definition 2.2.1 Let a functionfbe defined on an open interval containinga, except possible at a itself, and let L be a real number. The statement

limxa

f(x) =L (2.2.1)

(Lis thelimitof functionfata) means that for every >0, there is a >0such that if 0< |x a| < , then|f(x) L| < .We introduce a special kind of limits:

Definition 2.2.2 We say that variable y is o(z) (o notation) in the neigh-borhood of a point a if for every > 0 there exists > 0 such that if0< |x a| < , then|y| < |z| .Then we could define limit of a function as follows

Definition 2.2.3 A functionf(x) has a limit L at point aif for x = 0

f(a+x) =L+o(1).

Theorem 2.2.4 Iflimxa

f(x) =L

andL > 0, then there is an open interval(a , a+ ) containinga suchthatf(x)> 0 for everyxin (a , a+), except possiblyx= a.

Exercise 2.2.5 Verify limits using Definition

limx3

(5x+ 3) = 18;

limx2

(x2 + 1) = 5.

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5. Ifq(x) is a rational function andais in the domain ofq, then

limxa

q(x) =q(a).

Exercise 2.3.4 Find limits

limx2

3; lim

x3x;

limx4

(3x+ 1); limx2

(2x 1)15;

limx1/2

2x2 + 5x 36x2 7x+ 2 ; limx2

x2 + 2x 3x2 + 5x+ 6

;

limx

5

x x+

; limh0

1h

11 +h

1

Theorem 2.3.5 Ifa >0 andn is a positive integer, or ifa 0 andn is anodd positive integer, then

limxa

n

x= n

a.

Theorem 2.3.6 (Sandwich Theorem) Suppose f(x) h(x) g(x) foreveryxin an open interval containinga, except possibly at a. If

limxa

f(x) =L = limxa

g(x),

thenlimxa

h(x) =L.


limx0

x2 sin 1

x2; lim

x/2(x

2)cos x.

Remark 2.3.8 All such type of result could be modified forone-sidedlimits.

2.4 Limits Involving Infinity

Definition 2.4.1 Let a function fbe defined on an infinite interval (c, )(respectively (, c)) for a real number c, and let L be a real number. Thestatement

limx

f(x) =L ( limx

f(x) =L)

means that for every >0 there is a number Msuch that ifx > M(x < M),then|f(x) L| < .

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2.5. CONTINUOUS FUNCTIONS 13

Theorem 2.4.2 Ifk is a positive rational number andc, then

limx

c

xk = 0 and lim

xc

xk = 0

Definition 2.4.3 Let a functionfbe defined on an open interval containinga, except possibly at a itself. The statement

limxa

f(x) =

means that for every M >0, there is a > 0 such that if 0 M.


limx1

2x2

x2 x 1 limx9/23x2

(2x 9)2 (2.4.1)

limx

x3 + 2x2x2 3 limx

2x2 x+ 3x3 + 1

(2.4.2)

2.5 Continuous Functions

The notion of continuity is absorbed from our every day life. Here is its

mathematical definition

Definition 2.5.1 A function f is continuousat a point c if

limxc

f(x) =f(c).

If function is not continuous at c then it is discontinuous at c, or that f hasdiscontinuityat c. We give names to the following types of discontinuities:

1. Removablediscontinuity:

limxc f(x) =f(c).

2. Jump discontinuity:

limx+c

f(x) = limxc

f(x).

3. Infinitediscontinuity:limxc

f(x) = .

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Exercise 2.5.2 Classify discontinuities of

f(x) =

x2 ifx 1.

Theorem 2.5.3 1. A polynomial function f is continuous at every realpointc.

2. A rational function q=f /g is continuous at every number except thenumbersc such thatg(c) = 0.

Proof. The proof follows directly from Corollary2.3.3.

Exercise 2.5.4 Find all points at which f is discontinuous

f(x) = x 1

x2 +x 2 .

Definition 2.5.5 If a function f is continuous at every number in an openinterval (a, b) we say that fis continuous on the interval (a, b). We say alsothat f is continuous on the interval [a, b] if it is continuous on (a, b) and

limx+a

f(x) =a limxb

f(x) =b.

Theorem 2.5.6 If two functionsf andg are continuous at a real point c,the following functions are also continuous atc:

1. the sum f+g.

2. the differencef g.3. the product f g.

4. the quotient f /g, providedg(c)

= 0.

Proof. Proof follows directly from the Theorem2.3.2.

Exercise 2.5.7 Find all points at which f is continuous

f(x) =

9 xx 6 f(x) =

x 1x2 1 .

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2.5. CONTINUOUS FUNCTIONS 15

Theorem 2.5.8 1. If

limxc g(x) =b

andf is continuous at b, then

limxc

f(g(x)) =f(b) =f(limxc

g(x)).

2. Ifg is continuous atc and iffis continuous atg(c), then the compositefunctionf g is continuous atc.

Exercise 2.5.9 Suppose that

f(x) =

c2x, ifx

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Chapter 3

Derivative

3.1 Tangent Lines and Rates of Changes

Let we construct a secant line to a graph of a function f(x) through thepoints (a, f(a)) and (a+h, f(a+h)). Then from the formula (B.3.2) it willhave a slope (see (B.3.2))

m=f(a+h) f(a)

(a+h) a =f(a+h) f(a)

h . (3.1.1)

Ifh 0 than secant line became a tangent and we obtain a formula for itsslope

mt = limh0

f(a+h) f(a)h

. (3.1.2)

Thus the equation of tangent linecould be written as follows (see page67)

y f(a) =mt(x a) or y=f(a) +mt(x a). (3.1.3)

If a body pass a distancedwithin time t then average velocity is definedas

vav =d

t .

If the time intervalt 0 then we obtain instantaneous velocity

va= limt0

s(a+t) s(a)t

. (3.1.4)

These and many other examples lead to the notion ofderivative.

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3.2. DEFINITION OF DERIVATIVE 17

3.2 Definition of Derivative

The following is a quote from a letter of Donald E. Knuth to Notices ofAMS.

I would define the derivative by first defining what might be called astrong derivative: The function fhas a strong derivative f(x) at point xif

f(x+) =f(x) +f(x)+O(2)

whenever is sufficiently small. The vast majority of all functions that arisein practical work have strong derivatives, so I believe this definition bestcaptures the intuition I want students to have about derivatives.

Definition 3.2.1 The derivative of a function f is the function f whosevalue at xis given by

f(x) = limx0

f(x+h) f(x)h

, (3.2.1)

provided the limit exists.

3.3 Techniques of Differentiation

We see immediately, for example, that iff(x) =x2 we have

(x+)2 =x2 + 2x+2 ,

so the derivative ofx2 is 2x. And if the derivative ofxn isdn(x), we have

(x+)n+1 = (x+)(xn +dn(x)+O(2))

= xn+1 + (xdn(x) +xn)+O(2) ; (3.3.1)

hence the derivative ofxn+1 is xdn(x) +xn and we find by induction that

dn(x) =nxn1. Similarly iff and g have strong derivatives f(x) and g(x),

we readily findf(x+)g(x+) =f(x)g(x) + (f(x)g(x) +f(x)g(x))+O(2)

and this gives the strong derivative of the product. The chain rule

f(g(x+)) =f(g(x)) +f(g(x))g(x)+O(2) (3.3.2)

also follows when fhas a strong derivative at point g(x) and g has a strongderivative at x.

It is also follows that
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18 CHAPTER 3. DERIVATIVE

Theorem 3.3.1 If a function f is differentiable at a, then f is continuous

ata.

Exercise 3.3.2 Give an example of a functionfwhich is continuous at pointx= 0 but is not differentiable there.

It could be similarly proven the following

Theorem 3.3.3 Letf andg be differentiable functions at point c then thefollowing functions are differentiable at c also and derivative could be calcu-lated as follows:

1. sum and difference(f g)(x) =f(x) g(x).2. product (f g)(x) =f(x)g(x) +f(x)g(x).

3. fraction f

g

(x) =

f(x)g(x) f(x)g(x)g2(x)

.

3.4 Derivatives of the Trigonometric Func-tions

Before find derivatives of trigonometric functions we will need the following:

Theorem 3.4.1 The following limits are:

lim0

sin = 0; (3.4.1)

lim0

cos = 1; (3.4.2)

lim0

sin

= 1. (3.4.3)

Proof. Only the last limit is non-trivial. It follows from obvious inequalities

sin < < tan

and the Sandwich Theorem2.3.6.

Corollary 3.4.2 The following limit is

limx0

1 cos

= 0.

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3.5. THE CHAIN RULE 19

Exercise 3.4.3 Find following limits if they exist:

limt0

4t2 + 3t sin t

t2 ; lim

t0cos t

1 sin t ;

limx0

x cot x; limx0

sin3x

sin5x.

We are able now to calculate derivatives of trigonometric functions

Theorem 3.4.4

sin x= cos x; cos x= sin x; (3.4.4)

tan x= sec2

x; cot x= csc2

x; (3.4.5)sec x= sec x tan x; csc x= csc x cot x. (3.4.6)

Proof. The proof easily follows from the Theorem3.4.1and trigonometricidentities on page77.

Exercise 3.4.5 Find derivatives of functions

y=sin x

x ; y =

1 cos z1 + cos z

;

y= tan x

1 +x2; y= csc t sin t.

3.5 The Chain Rule

The chain rule(f g)(x) =f(g(x))g(x) (3.5.1)

was proven above3.3.2.

Exercise 3.5.1 Find derivatives

y= z2 1

z2

3

; y=x4 3x2 + 1

(2x+ 3)4 ;

y= 3

8r3 + 27; y= (7x+

x2 + 3)6.

3.6 Implicit Differentiation

If a function f(x) is given by formula like f(x) = 2x7 + 3x 1 then we willsay that it is an explicit function. In contrast an identity like

y4 + 3y 4x3 = 5x+ 1

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20 CHAPTER 3. DERIVATIVE

define animplicit function. The derivative of implicit function could be found

from an equation which it is defined. Usually it is a function both x and y.This procedure is called implicit differentiation.

Exercise 3.6.1 Find the slope of the tangent lines at given points

1. x2y+ sin y= 2 at P(1, 2).

2. 2x3 x2y+y3 1 = 0 at P(2, 3).

3.7 Related Rates

If two variables x and y satisfy to some relationship then we could foundtheir related ratesby the implicit differentiation.

Exercise 3.7.1 1. IfS=z3 and dz/dt= 2 when z= 3, find dS/dt.2. Ifx2 + 3y2 + 2y = 10 and dx/dt = 2 when x = 3 and y =1, find

dy/dt.

Exercise 3.7.2 Suppose a spherical snowball is melting and the radius isdecreasing at a constant rate, changing from 12 in. to 8 in. in 45 min. How

fast was the volume changing when the radius was 10 in.?

3.8 Linear Approximations and Differentials

It is known from geometry that a tangent line is closest to a curve at givenpoint among all lines. Thus equation of a tangent line (3.1.3)

y=f(a) +f(a)(x a) (3.8.1)gives the best approximation to a given graph off(x). We could use this

linear approximationin order to estimate value off(x) in a vicinity ofa. Wedenote an increment of the independent variable x by xand

y= f(x0+ x) f(x)

is the increment of dependent variable. Therefore

y f(x)x if x 0; (3.8.2)f(x) f(x) +f(x)x; (3.8.3)f(x) f(x) +dy, (3.8.4)

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3.8. LINEAR APPROXIMATIONS AND DIFFERENTIALS 21

where dy = f(x)xis defined to be differential off(x).An application of this formulas connected with estimation of errors of

measurements:Exact value Approximate value

Absolute error y=y y0 dy = f(x0)xRelative error y

y0

dyy0

Percentage error yy0

100% dyy0

100%

Exercise 3.8.1 Use linear approximation to estimate f(b):

1. f(x) =

3x2 + 8x

7;a= 4, b= 3.96.

2. f() = csc + cot ,a= 45, b= 46.

Exercise 3.8.2 Find y,dy,dy y for y= 3x2 + 5x 2.

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Chapter 4

Appliactions of Derivative

4.1 Extrema of Functions

We defined increasing and decreasing functions in Section B.7. There aremore definitions

Definition 4.1.1 Let f is defined on S R and c S.1. f(c) is the maximumvalue off on S iff(x) f(c) for every x S.

2. f(c) is the minimumvalue off on S iff(x) f(c) for every x S.Maximum and minimum are called extreme values, or extrema off. IfS isthe domain offthen maximum and minimum are called global orabsolute.

Exercise 4.1.2 Give an examples of functions which do not have minimumor maximum values.

The important property of continuous functions is given by the following

Theorem 4.1.3 Iffis continuous on[a, b], thenftakes on a maximum andminimum values at least once in [a, b].

Sometimes the following notions are of great importance

Definition 4.1.4 Let cbe a number in domain off.

1. f(c) is the local maximumfif there is an open interval (a, b) such thatc (a, b) and f(x) f(c) for every x (a, b) in the domain off.

2. f(c) is the local minimumfif there is an open interval (a, b) such thatc (a, b) and f(x) f(c) for every x (a, b) in the domain off.

22

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4.1. EXTREMA OF FUNCTIONS 23

The local extrema could be determined from values of derivative:

Theorem 4.1.5 Iffhas a local extremum at a numbercin an open interval,then eitherf(c) = 0 orf(c) do not exist.

Proof. The proof follows from the linear approximation of fbyf(c)x:iff(c) exists andf(c) = 0 then in an open interval around c there is valuesoffwhose greater and less than f(c).

The direct consequence is:

Corollary 4.1.6 if f(c) exists and f(c)= 0 then f(c) is not a local ex-tremum ofc.Critical numbers of f are whose points c in the domain of f where eitherf(c) = 0 or f(c) does not exist.

Theorem 4.1.7 If a functionfis continuous on a[a, b]and has its maximumor minimum values at a numberc (a, b), then eitherf(c) = 0orf(c)doesnot exist.

So to determine maximum and minimum values offone should accom-plish the following steps

1. Find all critical points off on (a, b).

2. Calculate values off in all critical points from step 4.1.7.1.

3. Calculate the endpoint values f(a) and f(b).

4. The maximal and minimal values of f on [a, b] are the largest andsmallest values calculated in4.1.7.2and4.1.7.3.

Exercise 4.1.8 Find extrema offon the interval

1. y= x4

5x+ 4; [0, 2].

2. y= (x 1)2/3 4; [0, 9].

Exercise 4.1.9 Find the critical numbers off

y = 4x3 + 5x2 42x+ 7;y =

3

x2 x 2;y = (4z+ 1)

z2 16;

y = 8 cos3 t 3sin2x 6x.

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24 CHAPTER 4. APPLIACTIONS OF DERIVATIVE

4.2 The Mean Value Theorem

Theorem 4.2.1 (Rolles Theorem) Iffis continuous on a closed interval[a, b] and differentiable on the open interval(a, b) and iff(a) = f(b), thenf(c) = 0 for at least one numberc in(a, b).

Proof. There is two possibilities

1. f is constant on [a, b] then f(x) = 0 everywhere.

2. f(x) is not constant then it has at least one extremum point c (The-orem4.1.3) which is not the end point of [a, b], then f(c) = 0 (Theo-

rem4.1.5).

Exercise 4.2.2 Shows that fsatisfy to the above theorem and find c:

1. f(x) = 3x2 12x+ 11; [0, 4].2. f(x) =x3 x; [1, 1].Rolles Therem is the principal step to the next

Theorem 4.2.3 (Mean Value Theorem or Lagranges Theorem) Iffis continuous on a closed interval[a, b]and differentiable on the open interval(a, b), then there exists a numberc (a, b) such that

f(c) =f(b) f(a)

b aor, equivalently,

f(b) f(a) =f(c)(b a).Proof. The proof follows from applivcation of the Rolles Theorem4.2.1to

the functiong(x) =f(x) f(b) f(a)

b a (x a).

We start applications of Mean Value Theorem by two corrolaries:

Corollary 4.2.4 If f(x) = 0 for all x in some interval I, then there is aconstantCsuch thatf(x) =C for allx in I.

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4.3. THE FIRST DERIVATIVE TEST 25

Corollary 4.2.5 Iff(x) =g(x) foar allx

I, then there is a constant C

such thatf(x) =g(x) +C.

Exercise 4.2.6 Shows that fsatisfy to the above theorem and find c:

1. f(x) = 5x2 3x+ 1; [1, 3].2. f(x) =x2/3; [8, 8].3. f(x) =x3 + 4x; [3, 6].

Exercise 4.2.7 Prove: iffcontinuous on [a, b] and iff(x) =c there, thenf(x) =cx+d for a d

R.

Exercise 4.2.8 Prove:

|sin u sin v| |u v| .

4.3 The First Derivative Test

Derivative of function could provide future information on its behavior:

Theorem 4.3.1 Letfbe continuous on [a, b]and differentiable on (a, b).

1. Iff(x)> 0 for everyxin (a, b), then fis increasing on [a, b].

2. Iff(x)< 0 for everyxin (a, b), then fis decreasing on [a, b].

Proof. For any numbersx1 and x2 in (a, b) we could write using theMeanValue Theorem:

f(x1) f(x2) =f(c)(x1 x2).Then for x1> x2 we will have f(x1)> f(x2) iff

(c) is positive and f(x1) 0 or f(x) < 0 for all xI, x= c, then f(c) is not a localextremum of ofc.

Exercise 4.3.2 Find the local extrema off and intervals of monotonicity,sketch the graph

1. y= 2x3 +x2 20x+ 1.2. y= 10x3(x 1)2.3. y= x(x2 9)1/2.4. y= x/2 sin x.5. y= 2 cos x+ cos 2x.

Exercise 4.3.3 Find local extrema offon the given interval

1. y= cot2 x+ 2 cot x [/6, 5/6].

2. y= tan x 2sec x[/4, /4].

4.4 Concavity and the Second Derivative Test

Definition 4.4.1 Let fbe differentiable on an open interval I. The graphoff is

1. concave upward on I iff is increasing on I;

2. concave downward on I iff is decreasing on I.

If a graph is concave upward then it lies above any tangent line and fordownward concavity it lies below every tangent line.

Test 2 (Test for Concavity) If the second derivativef offexists on anopen intervalI, then the graph off is

1. concave upward on I iff(x)> 0 on I;

2. concave downward on I iff(x)< 0 on I.

Definition 4.4.2 A point (c, f(c)) on the graph off is apoint of inflectionif the following conditions are satisfied:

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4.5. SUMMARY OF GRAPHICAL METHODS 27

1. fis continuous at c.

2. There is an open interval (a, b) containing c such that the graph hasdifferent types of concavity on (a, c) and (c, b).

Test 3 (Second Derivative Test) Suppose that f is differentiable on anopen interval containing c and f(c) = 0.

1. Iff(c)< 0, then fhas a local maximum at c.

2. Iff(c)> 0, then fhas a local minimum at c.

Exercise 4.4.3 Find the local extrema of f, intervals of concavity andpoints of inflections.

1. y= 2x6 6x4.2. y= x1/5 1.3. y= 6x1/2 x3/2.4. y= cos x sin x.5. y= x+ 2 cos x.

4.5 Summary of Graphical Methods

In order to sketch a graph the following steps should be performed

1. Find domain off.

2. Estimate range of f and determine region where f is negative andpositive.

3. Find region of continuity and classify discontinuity (if any).

4. Find all x- and y-intercepts.

5. Find symmetries off.

6. Find critical numbers and local extrema (using the First of SecondDerivative Test), region of monotonicity off.

7. Determine concavity and points of inflections.

8. Find asymptotes.

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28 CHAPTER 4. APPLIACTIONS OF DERIVATIVE

Exercise 4.5.1 Sketch the graphs:

f(x) = 2x2 x 3

x 2 ;

f(x) = 8 x3

2x2 ;

f(x) = 3x

x2 + 4;

f(x) = x3 +3

x;

f(x) = 4x2 + 1

;

f(x) =x3 x ;

f(x) = |cos x| + 2.

4.6 Optimization Problems. Review

To solve optimization problem one need to translate the problem to a questionon extrema of a function of one variable.

Exercise 4.6.1 1. Find the minimum value ofA ifA = 4y+ x2, where

(x2 + 1)y= 324.

2. Find the minimum value ofC ifC= (x2 +y2)1/2, where xy= 9.

Exercise 4.6.2 A metal cylindrical container with an open top is to hold 1m3. If there is no waste in construction, find the dimension that require theleast amount of material.

Exercise 4.6.3 Find the points of the graph ofy = x3 that is closest to thepoint (4, 0).

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Chapter 5

Integrals

5.1 Antiderivatives and Indefinite Integrals

Definition 5.1.1 A function F is an antiderivativeof the function fon aninterval I ifF(x) =f(x) for every x I.It is obvious that if F is an antiderivative off then F(x) +C is also an-tiderivative off for any real constant C. C is called an arbitrary constant.It follows fromMean Value Theoremthateveryantiderivative is of this form.

Theorem 5.1.2 Let Fbe an antiderivative offon an intervalI. IfG isany antiderivative off on I, then

G(x) =F(x) +C

for some constant Cand everyx I.

Exercise 5.1.3 The above Theorem may be false if the domain off is dif-ferent from an interval I. Give an example.

Definition 5.1.4 The notation f(x) dx= F(x) +C,

where F(x) =f(x) andCis an arbitrary constant, denotes the family of allantiderivatives off(x) on an interval Iand is called indefinite integral.

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30 CHAPTER 5. INTEGRALS

Theorem 5.1.5 d

dx(f(x)) dx = f(x) +C; (5.1.1)

d

dx

f(x) dx

= f(x). (5.1.2)

The above Theorem allows us to construct a primitive table of antiderivativesfrom the tables of derivatives:

Derivative Indefinite Integralddx

(x) = 1

1 dx=

dx= x+Cddx

(xr+1

r+1) =xr (r = 1) xr dx= x

r+1

r+1 +C (r = 1)

ddx(sin x) = cos x

cos x dx= sin x+Cddx

( cos x) = sin x sin x dx= cos x+Cddx

(tan x) = sec2 x

sec2 x dx= tan x+Cddx

( cot x) = csc2 x csc2 x dx= cot x+Cddx

(sec x) = sec x tan x

sec x tan x dx= sec x+Cddx

( csc x) = csc x cot x csc x cot x dx= csc x+CTheorem 5.1.6

cf(x) dx = c

f(x) dx (5.1.3)

[f(x) g(x)] dx = f(x) dx g(x) dx (5.1.4)

Exercise 5.1.7 Find antiderivatives of the following functions

9t2 4t+ 3; 4x2 8x+ 1; 1z3

3z2

;

u3 1

2u2 + 5; (3x 1)2; (t

2 + 3)2

t6 ;

7

csc x; 1

5sin x;

1

sin2 t.

A Differential equationis an equation that involves derivatives or differ-

entials of an unknown function. Additional values offor its derivatives arecalled initial conditions.

Exercise 5.1.8 Solve the differential equations subject to the given condi-tions

f(x) = 12x2 6x+ 1, f(1) = 5; (5.1.5)f(x) = 4x 1, f(2) = 2, f(1) = 3; (5.1.6)

d2y

dx2 = 3 sin x 4cos x, y = 7, y = 2 ifx= 0. (5.1.7)

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5.2. CHANGE OF VARIABLES IN INDEFINITE INTEGRALS 31

5.2 Change of Variables in Indefinite Inte-grals

One more important formula for indefinite integral could be obtained fromthe rules of differentiation. Thechain ruleimplies:

Theorem 5.2.1 IfF is an antiderivative off, then f(g(x))g(x) dx= F(g(x)) +C.

Ifu= g(x) anddu= g (x) dx, then f(u) du= F(u) +C.

Exercise 5.2.2 Find the integrals x(2x2 + 3)10 dx;

x2

3

3x3 + 7 dx; (5.2.1)1 +

1

x

3 1x2

dx;

t2 +t

(4 3t2 2t3)4 dt; (5.2.2) sin2x

1 cos2xdx;

sin3 x cos xdx. (5.2.3)

5.3 Summation Notation and Area

Definition 5.3.1 We use the following summation notation:

nk=1

ak =a1+a2+ +an.

Theorem 5.3.2n

k=1

c = cn; (5.3.1)

nk=1

(ak bk) =n

k=1

ak n

k=1

bk; (5.3.2)

nk=1

cak = c

nk=1

ak

. (5.3.3)

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Theorem 5.3.3

nk=1

k = n(n+ 1)

2 ; (5.3.4)

nk=1

k2 = n(n+ 1)(2n+ 1)

6 ; (5.3.5)

nk=1

k3 =

n(n+ 1)

2

2. (5.3.6)

This sum will help us to find inscribed rectangular polygon and circum-

scribed rectangular polygon.

Exercise 5.3.4 Find the area under the graph of the following functions:

1. y= 2x+ 3, from 2 to 4.

2. y= x2 + 1, from 0 to 3.

5.4 The Definite Integral

There is a way to calculate an area under the graph of a function y=f(x).We could approximate it by a sum of the form

nk=1

f(wk)xk, wk xk.

It is a Riemann sumThe approximation will be precise if will come to thelimit of Riemann sums:

limxk0

n

k=1

f(wk)xk =L.

If this limit exists it called definite integraloff froma tob and denoted by:

ba

f(x) dx= limxk0

nk=1

f(wk)xk =L.

If the limit exist we say that f is integrable functionon [a, b].

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5.5. PROPERTIES OF THE DEFINITE INTEGRAL 33

5.5 Properties of the Definite Integral

Theorem 5.5.1 Ifc is a real number, then ba

cdx= c(b a).

Theorem 5.5.2 Iffis integrable on[a, b]andc is any real number, thencfis integrable on [a, b]and

b

a

cf(x) dx= c b

a

f(x) dx.

Theorem 5.5.3 Iff andg are integrable on [a, b], then f g is also inte-grable on [a, b](a > b) and

ba

[f(x) g(x)] dx= ba

[f(x) g(x)] dx.

Theorem 5.5.4 If f is integrable on [a, b] and f(x) 0 for all x [a, b]then

b

a

f(x) dx 0.

Corollary 5.5.5 Iff andg are integrable on [a, b] andf(x)g(x) for allx [a, b]then b

a

f(x) dx ba

g(x) dx.

Theorem 5.5.6 (Mean Value Theorem for Definite Integrals) Iffiscontinuous on a closed interval [a, b], then there is a numberz in the openinterval(a, b) such that

ba

f(x) dx= f(z)(b a).

Definition 5.5.7 Let fbe continuous on [a, b]. The average value fav offon [a, b] is

fav = 1

b a ba

f(x) dx.

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Once it is known that integration is the inverse of differentiation and

related to the area under a curve, we can observe, for example, that if fand f both have strong derivatives at x, then

f(x+) f(x) = 0

f(x+t) dt

=

0

(f(x) +f(x) t+O(t2)) dt

= f(x)+f(x)2/2 +O(3) . (5.5.1)

5.6 The Fundamental Theorem of Calculus

There is an unanswered question from the previous section: Why undefinedand defined integrals shared their names and notations? The answer is givenby the following

Theorem 5.6.1 (Fundamental Theorem of Calculus) Supposefis con-tinuous on a closed interval[a, b].

1. If the function Gis defined by

G(x) = xa f(t) dt

for everyx in [a, b], then G is an antiderivative off on[a, b].

2. IfF is any antiderivative off on[a, b], then

ba

f(x) dx= F(b) F(a).

Proof. The proof of the first statement follows from theMean Value The-orem for Definite Integral. End the second part follows from the first and

initial condition aa

f(x) dx= 0.

Corollary 5.6.2 1. Iffis continuous on[a, b]andFis any antiderivativeoff, then b

a

f(x) dx= F(x)]ba=F(b) F(a).

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5.6. THE FUNDAMENTAL THEOREM OF CALCULUS 35

2. ba

f(x) dx=

f(x) dx

ba

.

3. Letfbe continuous on [a, b]. Ifa c b, then for everyxin [a, b]d

dx

xc

f(t) dt= f(x).

Theorem 5.6.3 Ifu = g(x), then

b

a

f(g(x))g(x) dx= g(b)

g(a)

f(u) du.

Theorem 5.6.4 Letfbe continuous on [a, a].1. Iffis an even function, a

af(x) dx=

aa

f(x) dx.

2. Iff is an odd function, aa

f(x) dx= 0.

Exercise 5.6.5 Calculate integrals

12

x 1

x

2dx;

41

5 x dx;

11

(v2 1)3v dv;

/2

0

3sin(1

2x) dx; /6

/6(x+ sin 5x) dx; /3

0

sin x

cos2 xdx; 5

1|2x 3| dx;

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Chapter 6

Applications of the Definite

Integral

6.1 Area

We know that the geometric meaning of the definite integral of a positivefunction is the area under the graph. We could calculate areas of morecomplicated figures by combining several definite integrals.

Exercise 6.1.1 Find areas bounded by the graphs:1. x= y2,x y = 2.2. y= x3,y = x2.

3. y= x2/3,x= y2.

4. y= x3 x,y = 0.5. x= y3 + 2y2 3y,x= 0.6. y= 4 + cos 2x,y = 3 sin 1

2

x.

Exercise 6.1.2 Express via sums of integrals areas:

1. y=

x,y = x,x= 1, x = 4.

6.2 Solids of Revolution

Theorem 6.2.1 Let f be continuous on [a, b], and let R be the regionbounded by the graph of f, the x-axis, and the vertical lines x = a and

36

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6.3. VOLUMES BY CYLINDRICAL SHELLS 37

x = b. The volumeV of the solid of revolution generated by revolvingR

about thex-axis is

V =

ba

[f(x)]2 dx.

Theorem 6.2.2 (Volume of a Washer)

V =

ba

[f(x)]2 [g(x)]2 dx.Exercise 6.2.3 1. y= 1/x,x= 1,x= 3, y= 0;x-axis;

2. y= x

3

,x= 2, y= 0,x-axis;3. y= x2 4x, y = 0; x-axis;4. y= (x 1)2 + 1, y= (x 1)2 + 3; x-axis;

Exercise 6.2.4 Find volume of revolution fory =x3,y = 4xrotated aroundx= 4.

6.3 Volumes by Cylindrical Shells

Let a cylindrical shell has outer and inner radiuses as r1 and r2 then andaltitudeh. We introduce the average radiusr = (r1 + r2)/2 and the thicknessr= r2 r1. Then its volume is:

V =r21h r22h= 2rrh.Let a region bounded by a function f(x) and x-axis. If we rotate it aroundthe y-axis then it is an easy to observe that the volume of the solid will beas follow:

V =

b

a

2xf(x) dx.

Exercise 6.3.1 Find volumes:

1. y=

x, x = 4, y= 0,y-axis.

2. y= x2,y2 = 8x,y-axis.

3. y= 2x, y = 6, x= 0, x-axis.

4. y=

x+ 4, y = 0, x= 0, x-axis.

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38 CHAPTER 6. APPLICATIONS OF THE DEFINITE INTEGRAL

6.4 Volumes by Cross Section

If a plane intersects a solid, then the region common to the plane and thesolid is a cross section of the solid. There is a simple formula to calculatevolumes by cross sections:

Theorem 6.4.1 (Volumes by Cross Sections) LetSbe a solid boundedby planes that are perpendicular to thex-axis at a andb. If, for everyx in[a, b], the cross-sectional area ofSis given byA(x), thereA is continuous on[a, b], then the volumeS is

V = ba A(x) dx.

Corollary 6.4.2 (Cavalieris theorem) If two solids have equal altitudesand if all cross sections by planes parallel to their bases and at the samedistances from their bases have equal areas, then the solids have the samevolume.

Exercise 6.4.3 Let R be the region bounded by the graphs ofx= y2 andx = 9. Find the volume of the solid that has R as its base if every crosssection by a plane perpendicular to the x-axis has the given shape.

1. Rectangle of height 2.

2. A quartercircle.

Exercise 6.4.4 Find volume of a pyramid if its altitude is h and the baseis a rectangle of dimensions a and 2a.

Exercise 6.4.5 A solid has as its base the region in xy-plane bounded bythe graph ofy2 = 4x and x= 4. Find the volume of the solid if every crosssection by a plane perpendicular to the y-axis is semicircle.

6.5 Arc Length and Surfaces of RevolutionWe say that a function f is smoothon an interval if it has a derivative f

that is continuous throughout the interval.

Theorem 6.5.1 Let fbe smooth on [a, b]. The arc length of the graph off from A(a, f(a)) toB (b, f(b)) is

Lba=

ba

1 + [f(x)]2 dx

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6.5. ARC LENGTH AND SURFACES OF REVOLUTION 39

We could introduce the arc length function s for the graph off on [a, b] is

defined by

s(x) =

xa

1 + [f(t)]2dt.

Theorem 6.5.2 Letfbe smooth on [a, b], and lets be the arc length func-tion for the graph ofy =f(x) on [a, b]. ifxis an increment in the variablex, then

ds

dx = 1 + [f(x)]2; (6.5.1)

ds =

1 + [f(x)]2x. (6.5.2)

Exercise 6.5.3 Find the arc length:

1. y= 2/3x2/3;A(1, 2/3), B(8, 8/3);

2. y= 6 3

x2 + 1; A(1, 7), B(8, 25);3. 30xy3 y8 = 15; A(8/15, 1), B(271

240, 2);

Exercise 6.5.4 Find the length of the graph x2/3 +y2/3 = 1.

Theorem 6.5.5 Iff is smooth andf(x) >0 on [a, b], then the areaSofthe surface generated by revolving the graph offabout thex-axis is

S=

ba

2f(x)

1 + [f(x)]2dx.

Exercise 6.5.6 Find the area of the surface generated by revolving of thegraph

1. y= x3;A(1, 1), B(2, 8);

2. 8y= 2x4 +x2,A(1, 3/8), B(2, 129/32);

3. x= 4

y;A(4, 1), B(12, 9).

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Chapter 7

Trancendential Functions

There is a special sort of functions which have a strange name transcenden-tal. We will explore the important role of these functions in calculus andmathematics in general.

7.1 The Derivative of the Inverse Function

We define one-to-one functions in SectionB.6. For such function we couldgive the following definition.

Definition 7.1.1 Let fbe a one-to-one function with domain D and rangeR. A function g with domain Rand range D is the inverse functionoff, iffor all x D and y R y = f(x) iffx = g(y).

Theorem 7.1.2 Let fbe a one-to-one function with domain D and rangeR. If g is a function with domain R and range D, then g is the inversefunction offiff both the following conditions are true:

1. g(f(x)) =x for everyx D.

2. f(g(x)) =x for everyy R.Exercise 7.1.3 Find inverse function.

1. f(x) = 3x+22x5 ;

2. f(x) = 5x2 + 2, x 0;3. f(x) =

4 x2.

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7.2. THE NATURAL LOGARITHM FUNCTION 41

Theorem 7.1.4 If f is continuous and increasing on [a, b], then f has an

inverse function f1 that is continuous and increasing on [f(a), f(b)].

Theorem 7.1.5 If a differentiable function f has an inverse function g =f1 and iff(g(c)) = 0, then g is differentiable at c and

g(c) = 1

f(g(c)). (7.1.1)

Proof. The formula follows directly from differentiation by the Chain rulethe identity f(g(x)) =x (see Theorem7.1.2).

Exercise 7.1.6 Find domain and derivative of the inverse function.

1. f(x) =

2x+ 3;

2. f(x) = 4 x2, x 0;3. f(x) =

9 x2, 0 x 3.

Exercise 7.1.7 Prove that inverse function exists and find slope of tangentline to the inverse function in the given point.

1. f(x) =x5 + 3x3 + 2x

1, P(5, 1);

2. f(x) = 4x5 (1/x3), P(3, 1);3. f(x) =x5 +x,P(2, 1).

7.2 The Natural Logarithm Function

Weknowthat antiderivative for a functionxn isxn+1/(n+1). This expressionis defined for all n = 1. This case deserves a special name

Definition 7.2.1 The natural logarithm function, denoted by ln, is definedby

ln x=

x1

1

tdt,

for every x >0.

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42 CHAPTER 7. TRANCENDENTIAL FUNCTIONS

From theproperties of definite integralfollows that

ln x >0 if x >1;

ln x

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7.3. THE EXPONENTIAL FUNCTION 43

Exercise 7.2.8 Find derivative of functions using logarithmic differentia-

tion:

f(x) = (5x+ 2)3(6x+ 1)2; (7.2.2)

f(x) =

(3x2 + 2)

6x 7; (7.2.3)

f(x) = (x2 + 3)5

x+ 1. (7.2.4)

7.3 The Exponential Function

Corollary7.2.6justify the following

Definition 7.3.1 The natural exponential function, denoted by exp x= ex,is the inverse of the natural logarithm function. The lettere(= 2.718281828 . . .)denotes the positive real number such that ln e= 1.

By the definition

ln ex = x, x R (7.3.1)elnx = x, x >0. (7.3.2)

By the same definition we could derive laws of exponentsfrom the lawsof logarithms.

Theorem 7.3.2d

dx(ex) = ex (7.3.3)

d

dx(eg(x)) = eg(x)

dg(x)

dx (7.3.4)

(7.3.5)

Proof. The proof of the first identity follows from differentiation of7.3.1by the chain rule. The second identity follows from the first one and thechain rule.

Exercise 7.3.3 Find implicit derivatives

1. xey + 2x ln(y+ 1) = 3;2. ex cos y= xey.

Exercise 7.3.4 Find extrema and regions of monotonicity:

1. f(x) =x2e2x;

2. f(x) =e1/x.

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7.4 Integration Using Natural Logarithm andExponential Functions

The following formulas are direct consequences ofchange of variables in def-inite integraland definition of logarithmic and exponential functions:

1

g(x)g(x) dx = ln |g(x)| +C; (7.4.1)

eg(x)g(x) dx = eg(x) +C. (7.4.2)

From here we could easily derive

Theorem 7.4.1 tan u du = ln |cos u| +C; (7.4.3) cot u du = ln |sin u| +C; (7.4.4) sec u du = ln |sec u+ tan u| +C; (7.4.5)

csc u du = ln |csc u cot u| +C. (7.4.6)

Exercise 7.4.2 Evaluate integrals x3

x4 5dx;

(2 + ln x)10

x dx;

ex

(ex + 2)2dx;

cot 3

x

3

x2dx.

7.5 General Exponential and Logarithmic Func-

tions

Usinglaws of logarithmswe could make

Definition 7.5.1 Theexponential function with basea is defined as follows:

f(x) =ax =eln ax

=ex lna. (7.5.1)

From this definition the following properties follows

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7.5. GENERAL EXPONENTIAL AND LOGARITHMIC FUNCTIONS45

Theorem 7.5.2

d

dx(ax) = ax ln a;

d

dx(au) = au ln a

du

dx;

ax dx =

1

ln a

ax +C;

au du =

1

ln a

au +C.

Exercise 7.5.3 Evaluate integrals 55x dx;

(2x + 1)2

2x dx;

ee dx;

x5 dx;

x5 dx;

(

5)x dx;

Exercise 7.5.4 The region under the graph ofy = 3x fromx = 1 tox = 2

is revolved about the x-axis. Find the volume of the resulting solid.Having ax already defined we could give the following

Definition 7.5.5 The logarithmic function with basea f(x) = logax is de-fined by the condition y= logax iffx = a

y.

The following properties follows directly from the definition

Theorem 7.5.6

d

dx

(logax) = d

dx ln x

ln a = 1

ln a

1

x

;

d

dx(loga |u| ) =

d

dx

ln |u|

ln a

=

1

ln a1

u

du

ddx.

Exercise 7.5.7 Find derivatives of functions:

f(x) = log3cos 5x;

f(x) = ln log x.

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7.7 Inverse Trigonometric Functions

We would like now to define inverse trigonometric functions. But there is aproblem: inverse functions exist only for one-to-one functions and trigono-metric functions are not the such.

Exercise 7.7.1 Prove that a periodic function could not be a one-to-onefunction.

A way out could be as follows: we restrict a trigonometric function fto aninterval I in such a way that f is one-to-one on I and there is no a biggerinterval I

Ithat f is one-to-one on I.

Definition 7.7.2 1. The arcsine (inverse sine function) denoted arcsinis defined by the condition y = arcsin x iffx = sin y for1 x 1and/2 y /2.

2. The arccosine (inverse cosine function) denoted arccos is defined bythe condition y = arccos xiffx= cos y for1 x 1 and 0 y .

3. Thearctangent(inverse tangent function) denoted arctan is defined bythe conditiony = arctan xiffx= tan yfor x Rand /2 y /2.

4. The arcsecant (inverse secant function) denoted arcsec is defined bythe condition y = arcsecx iffx= sec y for|x| > 1 and y[0, /2) ory [, 3/2).

Exercise 7.7.3 Why there is no a much need to introduce inverse functionsfor cotangent and cosecant?

Applying Theorem on derivative of an inverse function we could concludethat

Theorem 7.7.4

ddx

(arcsin u) = 11 u2

dudx

;

d

dx(arccos u) = 1

1 u2du

dx;

d

dx(arctan u) =

1

1 +u2du

dx;

d

dx(arcsecu) =

1

u

u2 1du

dx.

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7.8. HYPERBOLIC FUNCTIONS 47

As usually we could invert these formulas for taking antiderivatives:

Theorem 7.7.5 1

a2 u2 du = arcsinu

a+C= arccosu

a+C;

1

a2 +u2du =

1

aarctan

u

a+C;

1

u

u2 a2 du = 1

aarcsec

u

a+C.

And following formulas could be verified by differentiation:

Theorem 7.7.6 arcsin u du = u arcsin u+

1 u2 +C;

arccos u du = u arccos u

1 u2 +C; arctan u du = u arctan u 1

2ln(1 +u2) +C;

arcsecu du = uarcsecu ln u

u2

1 +C.

7.8 Hyperbolic Functions

The following functions arise in many areas of mathematics and applications.

Definition 7.8.1

hyperbolic sine function: sinh x=ex ex

2 ;

hyperbolic cosine function: cosh x= ex

+ex

2 ;

hyperbolic tangent function: tanh x=ex exex +ex

;

hyperbolic cotangent function: coth x=ex +ex

ex ex .

There are a lot of identities involving hyperbolic functions which are similarto the trigonometric ones. We will mentions only few most important ofthem

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Theorem 7.8.2

cosh2 x sinh2 x = 1;1 tanh2 x = (cosh x)2;coth2 x 1 = (sinh x)2.

From formula (ex) = ex easily follows the following formulas of differentia-tion:

Theorem 7.8.3

d

dx (sinh x) = cosh u

du

dx ;d

dx(cosh x) = sinh u

du

dx;

d

dx(tanh x) = (cosh u)2

du

dx;

d

dx(coth x) = (sinh u)2 du

dx.

Exercise 7.8.4 Find derivative of functions

f(x) =

1 + cosh x

1 + sinh x ; f(x) = ln |tanh x| .We again could rewrite these formulas for indefinite integral case:

Theorem 7.8.5 sinh u du = cosh u+C; cosh u du = sinh u+C;

(cosh u)2 du = tanh u+C; (sinh u)2 du = coth u+C.

Exercise 7.8.6 Evaluate integrals sinh

x

x dx;

1

cosh2 3xdx.

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7.9. LHOSPITALS RULE 49

7.9 Indeterminate Forms and lHospitals Rule

In this section we describe a general tool which simplifies evaluation of limits.

Theorem 7.9.1 (Cauchys Formula) Iff andg are continuous on [a, b]and differentiable on (a, b) and ifg(x) = 0 for allx (a, b), then there is anumberw (a, b) such that

f(a) f(b)g(a) g(b) =

f(w)g(w)

.

Proof. The proof follows from the application ofRolles Theoremto thefunction

h(x) = [f(b) f(a)]g(x) [g(b) g(a)]f(b).

Theorem 7.9.2 (lHospitals Rule) Suppose that f and g are differen-tiable on an open interval(a, b) containingc, except possibly at c itself. Iff(x)/g(x) has the indeterminate form 0/0 or/ then

limxc

f(x)

g(x)= lim

xc

f(x)g(x)

,

provided

limxc

f(x)g(x)

exists orlimxc

f(x)g(x)

= .

Exercise 7.9.3 Find the following limits

limx1

x3 3x+ 2x2 2x 1; limx0

sin x xtan x x ;

limx0

x+ 1 exx2

; limx

0+

ln sin x

lnsin2x;

limx0

ex ex 2sin xx sin x

; limx

x ln xx+ ln x

.

There are more indeterminant forms which could be transformed to the case0/0 or/:

1. 0 : write f(x)g(x) asf(x)

1/g(x) or

g(x)

1/f(x).

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2. 00, 1,

0: instead of f(x)g(x) look for the limit L of g(x) ln f(x).

Then f(x)g(x) =eL.

3. : try to pass to a quotient or a product.

Exercise 7.9.4 Find limits if exist.

limx0+

(ex 1)x; limx

x1/x;

limx3

x

x2 + 2x 3 4

x+ 3

lim

x

x2

x 1 x2

x+ 1

;

limx0

(cot2 x

csc2 x); lim

x0+

(1 + 3x)cscx.

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Chapter 8

Techniques of Integration

We will study more advanced technique of integration.

8.1 Integration by Parts

Among different formulae of differentiation there is one which was not con-verted to the formulae of integration yet. This isderivative of a product oftwo functions. We will use it as follows:

Theorem 8.1.1 Ifu= f(x) andv = g(x) and iff andg are continuous,then u dv=uv

v du.

Exercise 8.1.2 Evaluate integrals. xex dx;

x sec x tan x dx;

x csc2 3x dx;

x2 sin4x dx;

ex cos x dx;

sin ln x dx;

cos

x dx;

sinn xdx..

8.2 Trigonometric Integrals

To evaluate

sinm x cosn x dxwe use the following procedure:

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52 CHAPTER 8. TECHNIQUES OF INTEGRATION

1. If m is an odd integer: use the change of variableu = cos x and

express sin2 x= 1 cos2 x.2. If n is an odd integer: use the change of variable u = sin x and

express cos2 x= 1 sin2 x.3. Ifm and n are even: Use half-angle formulas for sin2 xand cos2 xto

reduce the exponents by one-half.

Exercise 8.2.1 Evaluate integrals.

sin3 x cos2 x dx; sin4 x cos2 xdx.To evaluate

tanm x secn x dxwe use the following procedure:

1. If m is an odd integer: use the change of variable u = sec x andexpress tan2 x= sec2 x 1.

2. If n is an even integer: use the change of variableu = tan x andexpress sec2 x= 1 + tan2 x.

3. If m is an even and n is odd numbers: There is no a standardmethod, try the integration by parts.

Exercise 8.2.2 Evaluate integrals. cot4 x dx;

sin4x cos3xdx.

8.3 Trigonometric Substitution

The following trigonometric substitution applicable if integral contains oneof the following expression cases

Expression Substitutiona2 x2 x= a sin ;a2 +x2 x= a tan ;x2 a2 x= a sec .

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8.4. INTEGRALS OF RATIONAL FUNCTIONS 53

Exercise 8.3.1 Evaluate integrals 1

x3

x2 25 dx;

1

x2

x2 + 9dx;

1

(16 x2)5/2 dx;

(4 +x2)2

x3 dx.

8.4 Integrals of Rational Functions

To integrate a rational functionf(x)/g(x) we acomplish the following steps:

1. If the degree of f(x) is not lower than the degree of g(x), use longdivision to obtain the proper form.

2. Expressg(x) as a product of linear factorsax+bor irreducible quadraticfactors cx2 + dx +e, and collect repeated factors so that g(x) is aproduct ofdifferentfactors of the form (ax + b)n or (cx2 + dx + e)m fora nonnegative n.

3. Find real coefficientsAij, Bij, Cij,Dij such that

f(x)

g(x) =

n

k=1

A1k

akx+bk+

Ankk(akx+bk)2

+ + Ankk(akx+bk)n

+n

k=1

C1kx+D1k

ckx2 +dkx+ek+

C2kx+D2k(ckx2 +dkx+ek)2

+ + C1kx+D1k(ckx2 +dkx+ek)n

.

Exercise 8.4.1 Evaluate integrals: 11x+ 2

2x2 5x 3dx

4x

(x2 + 1)3dx

x4 + 2x2 + 3

x3 4x dx

8.5 Quadratic Expressions and MiscellaneousSubstitutions

There are a lot of different substitutions which could be useful in particularcases. Particularly, if the integrand is a rational expression in sin x, cos x,the following substitution will produce a rational expression inu:

sin x= 2u

1 +u2, cos x=

1 u21 +u2

, dx= 2

1 +u2du,

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54 CHAPTER 8. TECHNIQUES OF INTEGRATION

where u= tan x2

for

< x < .

Exercise 8.5.1 Evaluate integrals 17 + 6x x2 dx;

1

(x2 6x+ 34)3/2dx; 1

x(ln2 x+ 3ln x+ 2)dx.

8.6 Improper Integrals

We could extend the notion of integral for the following integrals with withinfinite limits or improper integral

Definition 8.6.1 1. Iff is continuous on [a, ), thena

f(x) dx= limt

ta

f(x) dx,


2. Iff is continuous on (, a], then a

f(x) dx= lim

t

a

t

f(x) dx,


3. Let fbe continuous for every x. Ifais any real number, then

f(x) dx=

a

f(x) dx+

a

f(x) dx,

provided both of the improper integrals on the right converge.

Exercise 8.6.2 Determine if improper integrals converge and find the itsvalue if so.

0

xex dx;

cos2 x dx;1

x

(1 +x2)2dx.

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8.6. IMPROPER INTEGRALS 55

Definition 8.6.3 1. Iff is discontinuous on [a, b) and discontinuous at

b, then ba

f(x) dx= limtb

ta

f(x) dx,


2. Iffis discontinuous on (a, b] and discontinuous at a, then

ba

f(x) dx= limta+

bt

f(x) dx,

provided the limit exists.3. Iffhas a discontinuity atc in the open interval (a, b) but is continuous

elsewhere on [a, b], then

ba

f(x) dx=

ca

f(x) dx+

bc

f(x) dx,

provided both of the improper integrals on the right converge.

Exercise 8.6.4 Determine if improper integrals converge and find the itsvalue if so. 4

0

1

(4 x)2/3dx; 21

x

x2 1dx; /20

tan x dx;

e1/e

1

x(ln x)2dx.

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Chapter 9

Infinite Series

9.1 Sequences

Definition 9.1.1 A sequence is a function f whose domain is the set ofpositive integers.

Example 9.1.2 1. Sequense of even numbers: a1 = 2, a2 = 4, a3 = 6,. . .

2. Sequence of prime numbers: a1 = 2, a2= 3,a3= 5, . . .

Definition 9.1.3 A sequence{an} has the limit L, or convereges to L, de-noted by

limn

an=L, if an L when n .if for every >0 there exists a positive number Nsuch that

|an L| < whenever n > N.

If such a number L does not exist, the sequence has no limit, or diverges.

Definition 9.1.4 The notation

limn

an = , or an or n .

means that for every positive real number Pthere exists a positive numberNsuch that an> P whenever n > N.

Theorem 9.1.5 Let{an} be a sequence, let f(n) = an, and suppose thatf(x) exists for all real numbersx >1.

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9.2. CONVERGENT OR DIVERGENT SERIES 57

1. Iflimx

f(x) =L, then limn

an=L.

2. Iflimx f(x) = , then limn an= .Theorem 9.1.6

limn

rn = 0 if |r| 1Exercise 9.1.7 If the sequences are convergent

n2

ln n+ 1

;

cos n

n ;

en

n4

;

9.2 Convergent or Divergent Series

Definition 9.2.1 An infinite series (or series) is an expression of the form1

an=a1+a2+a3+. . . .

Here an isnth termof the series.

Definition 9.2.2 1. The kth partial sumof the series is Sk = kn=1an.

2. The sequence of partial sumsof the series is S1,S2,S3, . . .

Definition 9.2.3 A series is convergentordivergentiff the sequense of par-tial sums is correspondingly convergent or divergent. If limit of partial sumexists then it is the sumof series. A divergent series has no sum.

Example 9.2.4 1. Series

1n(n+1)

is convergent with sum 1.

2. Series

(1)k is divergent.3. The harmonic series

1n

is divergent.

4. The geometric series

arn is convergent with sum a1r if|r|

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58 CHAPTER 9. INFINITE SERIES

9.3 Positive-Term Series

We will investigate first positive-term seriesthat is, series

an such thatan > 0 for all nand will use these result for series of general type.

Theorem 9.3.1 If

anis a positive-term series and if there exists a numberM such that

Sn = a1+a2+. . .+an< M

for everyn, then the series converges and has a sum S M. If no such Mexists, the series diverges.

Theorem 9.3.2 If

an is a series, let f(n) =an and letfbe the functionobtained by replacing n with x. If f is positive-valued, continuous, anddecreasing for every real numberx 1, then the series an

1. converges if1

f(x) dx converges;

2. diverges if1

f(x) dxdiverges.

Definition 9.3.3 A p-series, or a hyperharmonic series, is a series of theform

n=1

1

np

= 1 + 1

2p

+ 1

3p

+

+

1

np

+

,

where p is a positive real number.

Theorem 9.3.4 Thep-series

1np

1. converges ifp >1;

2. diverges ifp 1.Proof. The proof is a direct application of the Theorem 9.3.2.

Theorem 9.3.5 (Basic Comparison Tests) Let

anand

bnbe positive-term series.

1. If

bn converges andan bn for every positive integern, then

anconverges.

2. If

bn diverges andan bn for every positive integern, then

andiverges.

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9.4. THE RATIO AND ROOT TESTS 59

Theorem 9.3.6 (Limit Comparison Test) Let anand bnbe positive-term series. If there is a positive number c such that

limn

anbn

=c >0,

then either both series converge or both series diverge.

Exercise 9.3.7 Determine convergency

ln nn

; 1

1 + 16n2;

sin n4en

5

; 1

nn ; 3n+ 5n2n

; n2

n3 + 1;

tan1

n;

sin n+ 2nn+ 5n

.

9.4 The Ratio and Root Tests

The following two test of divergency are very important. Yet there severalcases then they are not inconclusive (see the third clause).

Theorem 9.4.1 (Ratio Test) Let

anbe a positive-term series, and sup-pose that

limn

an+1an

=L.

1. IfL 1 the series divergent.

3. IfL= 1, apply a different test; the series may be convergent or diver-gent..


100nn!

; 3n

n3 + 1.

Theorem 9.4.3 (Root Test) Let

an be a positive-term series, and sup-pose that

limn

n

an=L.

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60 CHAPTER 9. INFINITE SERIES

1. IfL 1 the series divergent.

3. IfL= 1, apply a different test; the series may be convergent or diver-gent..

Exercise 9.4.4 Determine convergency 2nn2

; n

ln n

n;

n!

nn;

1

(ln n)n.

9.5 Alternating Series and Absolute Conver-

gence

The simplest but still important case of non positive-term series are alter-nating series, in which the terms are alternately positive and negative..

Theorem 9.5.1 (Alternating Series Test) The alternating series

n=1

(1)n1

an=a1 a2+a3 a4+ + (1)n1

an+

is convergent if the following two conditions are satisfied:

1. an ak+1 0 for everyk;2. lim an= 0.

Theorem 9.5.2 Let

(1)n1anbe an alternating series that satisfies con-ditions(i) and(ii) of the alternating series test. IfSis the sum of the seriesandSn is a partial sum, then

|S Sn| an+1;that is, the error involved in approximatingSbySn is less that or equal toan+1.

Definition 9.5.3 A series

an isabsolutely convergent if the series|an| = |a1| + |a2| + + |an| +

is convergent.

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9.5. ALTERNATING SERIES AND ABSOLUTE CONVERGENCE 61

Definition 9.5.4 A series an is conditionally convergent if an is con-vergent and

|an| is divergent.Theorem 9.5.5 If a series

an is absolutely convergent, then

anis con-

vergent.


(1)n2n2 +n

;

(1)n3

n

n+ 1;

(1)n

arctan n

n2 ;

1

nsin

(2n 1)2

.

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Appendix A

Algebra

We briefly recall some basic notion and results from algebra.

A.1 Numbers

The following set ofnumberswill be used in the course

1. Nnaturalnumbers: 1, 2, 3, . . .

2. Zintegernumbers: ...,

2,

1, 0, 1, 2, ...

3. Qrationalnumbers of the form nm ,n Z,m N.4. Rrealnumbers, e.g, 0, 1,3.12,2, ,e.

Binary operationsbetween numbers are +,,, /.

Exercise A.1.1 Which set of numbers is closed with respect to the abovefour operations?

A.2 Polynomial. Factorization of Polynomi-als

A polynomial p(x) (in a variable x) is a function on real line defined by anexpression of the form:

p(x) =anxn +an1xn1 + +a1x+a0. (A.2.1)

Here ai are fixed real numbers, an= 0 and n is the degree of polynomialp(x).

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A.3. BINOMIAL FORMULA 63

According to the Main Theorem of algebra every polynomial p(x) could be

represented as a product of linear binomials and quadratic terms as follows:

p(x) = (b1x+c1) (bjx+cj)(d1x2 +f1x+g1) (dkx2 +fkx2 +g+k),

moreovern= 2k+j, where n is the degree ofp(x).

Exercise A.2.1 Decompose to products:

1. p(x) =x16 1.2. p(x) =x4 + 16y4.

A.3 Binomial Formula

Binomial formulaof Newton:

(x+y)n =n

k=0

n

k

xkynk, where

n

k

=

n!

k!(n k)! .

Here n! = 1 2 n. These coefficients can be determined from thePascaltriangle.

Exercise A.3.1 Prove the following properties of the binomial coefficientsn+ 1

k

=

n

k

+

n

k+ 1

.

n

k+ 1

=

n kk+ 1

n

k+ 1

.

nk=0

n

k

= 2n.

nk=0

(1)knk = 0.A.4 Real Axis

We will be mainly interested in real numbers R which could be representedby the coordinate line (or real axis). This gives one-to-onecorrespondencebetween sets of real numbers and points of the real line.

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64 APPENDIX A. ALGEBRA

A.5 Absolute Value

The absolute value|a| (or modulus) of a real number a is defined as follows

|a| =

a ifa 0,a ifa 0)

1.|a| < biff1 b < a < b.2.|a| > biff either a > bor a < b.3.|a| =b iffa= b or a= b.

Exercise A.5.1 Prove the following properties of absolute value:

1.|a+b| |a| + |b| .2.|ab| = |a| |b| .Anequationis an equality that involves variables, e.g. x3 +5x2x+10 =

0. A solution of an equation (or rootof an equation) is a number b thatproduces a true statement after substitution x= b into equation. Equation

could be solved by either analyticor computational means.

A.6 Inequalities

Order relationsbetween numbers are given by>, b and b > c, then a > c (transitivity property).

2. Ifa > b, then a c > b c.3. Ifa > b and c >0, then ac > bc.

4. Ifa > b and c , 2x2 5x+ 1. Solutionof an inequality is similar forthe case of equations (see SectionA.5). They are often given by unions ofintervals.

1The notation iff is used for an abbreviation ofif and only if.

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A.6. INEQUALITIES 65

Intervalson real line are the following sets:

[a, b] = {x R| a x b};(a, b) = {x R| a < x < b};(a, b] = {x R| a < x b};[a, b) = {x R| a x < b}.

Particularlyacan beand b= .

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Appendix B

Function and Their Graph

B.1 Rectangular (Cartesian) Coordinates

Considering real axis in SectionA.4we introduce one-to one correspondencebetween real numbers and points of a line. This connection between numbersand geometric objects may be extended for other objects as well.

A rectangular coordinate system (or Cartesian coordinates) is an assign-ment ofordered pairs (a, b) to points in a plane, see [1, Fig. 6, p. 10].

Remark B.1.1 It is also possible to introduce Cartesian coordinates in ourthree-dimensional world by means of triples of real numbers (x,y,z). Thisconstruction could be extended to arbitrary number of dimensions.

Theorem B.1.2 Thedistance between two points P1 = (x1, y1) andP2 =(x2, y2) is

d(P1, P2) =

(x2 x1)2 + (y2 y1)2. (B.1.1)This theorem is a direct consequence of Pythagorean theorem.

Theorem B.1.3 Themidpoint Mof segment P1P2 is

M(P1P2) =Mx1+x2

2 ,

y1+y2

2

. (B.1.2)

Proof. The theorem follows from two observations:

1. d(P1, M) =d(M, P2);

2. d(P1, M) +d(M, P2) =d(P1, P2).

Exercise B.1.4 Prove the above two observation (Hint: use the distanceformula (B.1.1)).

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B.2. GRAPH OF AN EQUATION 67

B.2 Graph of an Equation

Anequation inx andy is an equality such as

2x+ 3y= 5, y = x2 + 3x 6, yx + sin xy= 8.

A solution is an ordered pair (a, b) that produced a true statement whenx= a and y =b. The graph of the equation consists of all points (a, b) in aplane that corresponds to the solutions.

B.3 Line Equations

The generalequationof a line in a plane is given by the formula

ax+by+c= 0. (B.3.1)

This equation connect different geometric objects:

1. Slopem:

m= y2 y1x2 x1 . (B.3.2)

2. Point-Slope formy

y1

=m(x

x1).

3. Slope-Intercept formy=mx+b or y=m(x c).Special lines

1. Vertical: m undefined; horizontal: m= 0.

2. Parallel: m1=m2.

3. Perpendicular m1m2= 1.

Exercise B.3.1 Prove the above geometric properties.

B.4 Symmetries and Shifts

We will say that a graph of an equation possesses a symmetry if there is atransformation of a plane such that it maps the graph to itself.

Example B.4.1 There several examples of elementary symmetries:

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68 APPENDIX B. FUNCTION AND THEIR GRAPH

1. y-axis: substitution x

(

x), e.g.1 equation y=

|x

|.

2. x-axis: substitution y (y), e.g.|y| =x.3. Central symmetry: substitution both x (x) and y (y), e.g.

y= x or|y| = |x| .4. x-shifts: substitutionx (x+a) for a = 0, e.g. y = {x} with a= 1.

Here{x} denotes the fractional part of x, i.e.2 it is defined by twoconditions: 0 {x}

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B.6. ONE-TO-ONE FUNCTIONS. PERIODIC FUNCTIONS 69

B.6 One-to-One Functions. Periodic Func-tions

We say that f is one-to-one function iff(x)= f(y) whenever x= y. Fornumerically defined functions like y =

x 2 the domain is assumed to be

allxthat f is is defined atx, or f(x) exists.Thegraph of the a functionfwith domainD is the graph of the equation

y= f(x) forxinD. Thex-intercept of the graph are solutions of the equationf(x) = 0 and called zeros.

The following transformation are useful for sketch of graphs:

1. horizontal shift: y = f(x) to yf(x a);2. vertical shift: y = f(x) to y= f(x) +b;

3. horizontal stretch/compression: y=f(x) to y=f(cx);

4. vertical stretch/compression: y=f(x) to y= cf(x);

5. horizontal reflections: y= f(x) to y = f(x);6. vertical reflections: y = f(x) to y = f(x);

B.7 Increasing and Decreasing Functions. Oddand Even Functions

A function f(x) is increasing iff(x) > f(y) for all x > y. A function f(x)isdecreasing iff(x)> f(y) for all x > y.

A function f(x) is even if f(x) = f(x) and f(x) is odd if f(x) =f(x).

Exercise B.7.1 Which type of symmetries listed in Example (B.4.1) have

to or may even and odd functions posses?There are natural operation on functions

1. sum: (f+g)(x) =f(x) +g(x).

2. difference: (f g)(x) =f(x) g(x).3. product: (f g)(x) =f(x)g(x).

4. quotient: (f /g)(x) =f(x)/g(x).

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70 APPENDIX B. FUNCTION AND THEIR GRAPH

Forp(x) be a polynomial (see (A.2.1))y = p(x) defines apolynomial func-

tion. Ifp(x) andq(x) are two polynomials (see (A.2.1)) theny = p(x)/q(x) isa rational function. Function which are obtained from polynomials by fouralgebraic operations B.7.1.1 and taking rational powers are algebraic. Allother function (e.g sin x, exp x) are trancendental.

Thecomposite functionfgis defined by (fg)(x) =f(g(x)). Anidentityfunctionis a functionhwith the property thath(x) =x If the composition oftwo functionsfandg is an identity function, then the functions areinversesof each other.

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Appendix C

Conic Section

C.1 Circle

The beautiful and important objects arise by intersection of planes and cones,i.e. conic sections.

The simplest conic section is circle.

Definition C.1.1 Acirclewith center (x1, y1) and radiusrconsists of pointson the distance r from (x1, y1).

By the distance formula (B.1.1) the circle is defined by an equation:

(x x1)2 + (y y1)2 =r2. (C.1.1)Circles are obtained as intersections of cones with planes orthogonal to theiraxes.

Exercise C.1.2 1. Write an equation of a circle which is tangent to acircle x2 6x+y2 + 4y 12 = 0 and has the origin (3, 0).

2. Write an equation of a circle, which has a center at (1, 2) and containsthe center of the circle given by the equation x2

7x + y2 + 8y

17 = 0.

3. Write equations of all circles with a given radius r which are tangentto both axes.

C.2 Parabola

Definition C.2.1 A parabola is the set of all points in a plane equidistantfrom a fixed point F (the focus of the parabola) and a fixed line l (thedirectrix) that lie in the plane.

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72 APPENDIX C. CONIC SECTION

The axisof the parabola is the line through Fthat is perpendicular to

the directrix. The vertexof the parabola is the point Von the axis halfwayfrom F to l.

A parabola with axis coinciding with y axis and the vertex at the originwith focusF = (0, p) has an equation

x2 = 4py. (C.2.1)

Exercise C.2.2 Verify the above equation of a parabola. (Hint: use dis-tance formula (B.1.1).

Exercise C.2.3 List all symmetries of a parabola.

For a parabola with the vertex (h, k) and a horizontal directrix an equationtakes the form

(x h)2 = 4p(y k). (C.2.2)In general any equation of the form y =ax2 + bx + cdefines a parabola withhorizontal directrix.

Exercise C.2.4 1. Find the vertex, the focus, and the directrix of the

parabolas:

(a) 3y2 = 5x.(b) x2 = 3y.

(c) y2 + 14y+ 4x+ 45 = 0.

(d) y= 8x2 + 16x+ 10.

2. Find an equation of the parabola with properties:

(a) vertex V(3, 4); directrix y= 6.(b) vertex V(1, 1); focus F(2, 1).(c) focusF(1, 3); directrix y= 5.

C.3 Ellipse

Definition C.3.1 An ellipseis the set of all points in a plane. the sum ofwhose distances from two fixed points F and F (the foci) in the plane isconstant. The midpoint of the segment F F is the centerof the ellipse.

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C.3. ELLIPSE 73

Let F(

c, 0) and F(c, 0), 2a be the constant sum of distances, and b =

(a2 c2)1/2. Then the ellipse has an equationx2

a2+

y2

b2 = 1 (C.3.1)

Exercise C.3.2 Verify the above equation of the ellipse. (Hint: use dis-tance formula (B.1.1).

The ellipse interceptsx-axis in points V(a, 0) andV (a, 0)verticesof theellipse. The line segment V V is the major axisof the ellipse. Similarly theellipse interceptsy-axis in pointsM(b, 0) andM(b, 0) and the line segmentMM is the minor axisof the ellipse.

Exercise C.3.3 List all symmetries of an ellipse.

For the ellipse with center in a point (h, k) an equation is given as

(x h)2a2

+(y k)2

b2 = 1. (C.3.2)

Definition C.3.4 The eccentricity e of an ellipse is

e= c

a=

a2 b2

a

Exercise C.3.5 1. Find the vertices and the foci of the ellipse

(a) 4x2 + 2y2 = 8.

(b) x2/3 + 3y2 = 9

2. Find an equation for the ellipse with center at the origin and

(a) Vertices V(9, 0); fociF(6, 0).(b) FociF(6, 0); minor axis of length 4.(c) Eccentricity 3/4; vertices V(0, 5).

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74 APPENDIX C. CONIC SECTION

C.4 Hyperbola

Definition C.4.1 A hyperbola is the set of all points in a plane, the dif-ference of whose distances from two fixed points F and F (the foci) in theplane is a positive constant. The midpoint of the segment F F is the centerof the hyperbola.

Let a hyperbola has fociF(c, 0), 2adenote the constant difference, and letb2 =c2 a2. Then the hyperbola has an equation

x2

a2 y

2

b2 = 1. (C.4.1)

Exercise C.4.2 Verify the above equation of the hyperbola. (Hint: usedistance formula (B.1.1).

PointsV(a, 0) andV (a, 0) of interception of the hyperbola with x-axis areverticesand the line segment V V is the transverse axisof the hyperbola.PointsW(0, b) andW(0, b) spanconjugate axisof the hyperbola. This twosegments intercept in the centerof the hyperbola.

Exercise C.4.3 Find all symmetries of a hyperbola.

If a graph approaches a line as the absolute value ofx gets increasinglylarge, then the line is called an asymptote for the graph. It could be shownthat lines y= (b/a)xand y= (b/a)xare asymptotes for the hyperbola.

For the hyperbola with the center in a point (h, k) an equation is givenas

(x h)2a2

(y k)2

b2 = 1. (C.4.2)

Exercise C.4.4 1. Find vertices and foci of the hyperbola, sketch its

graph.(a) x2/49 y2/16 = 1.(b) y2 4x2 12y 16x+ 16 = 0.(c) 9y2 x2 36y+ 12x 36 = 0.

2. Find an equation for the hyperbola that has its center at the origin andsatisfies to the given conditions

(a) foci F(0, 4); vertices V(0, 1).

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C.5. CONCLUSION 75

(b) foci F(0,

5); conjugate axis of length 4.

(c) vertices V(3, 0); asymptotesy = 2x.

C.5 Conclusion

The graph of every quadratic equation Ax2 + Cy2 + Dx + Ey + F= 0 is oneof conic section

1. Circle;

2. Ellipse;

3. Parabola;

4. Hyperbola;

or a degenerated case

1. A point;

2. Two crossed lines;

3. Two parallel lines;

4. One line;

5. The empty set.

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Appendix D

Trigonometic Functions

Anangleis determined by two rays having the same initial point O (vertex).Angles are measured either by degree measure 1 or radian measure. Thecomplete counterclockwise revolution is 360 or 2 radians.

We consider six trigonometric functions

Name Notation Expression Name Notation Expressionsine sin y/r cosecant csc r/y

cosine cos x/r secant sec r/x

tangent tan y/x cotangent cot x/y

There are a lot of useful identities between trigonometric functions:

1. Reciprocal and Ratio Identities:

(a) csc = (sin )1, sec = (cos )1;

(b) tan = sin / cos , cot = cos / sin ;

(c) cot = (tan )1

2. Pythagorean Identities

(a) sin2 + cos2 = 1;

(b) 1 + tan2 = sec2 ;

(c) 1 + cot2 = csc2 ;

3. Law of Sines and Law of Cosine

(a) sin /a= sin /b = sin /c = 2R;

(b) a2 =b2 +c2 2bc cos ;

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77

4. Additional identities

(a) Cosine and secant are even functions; sine, tangent, cosecant,cotangent are odd functions;

(b) sin( ) = sin cos sin cos ;(c) cos( ) = cos cos sin sin ;

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Appendix E

Exponential and Logarithmic

Functions

Definition E.0.1 The exponential function with a base a is defined byf(x) =ax, where a >0, a = 1, and xis any real number.It is increasing ifa > 1 and decreasing if 0 < a < 1. It is also one-to-onefunction. This allow us to solve equations and inequalities.

Exercise E.0.2 Find solutions

1. 53x = 5x21;

2. 2|x3| >22;

3. (0.5)x2

>(0.5)5x6.

There are laws of exponents:

1. auav =au+v;

2. au/av =auv;

3. (au)v =auv;

4. (ab)u =aubu;

5. (a/b)u =au/bu.

Definition

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