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CALCULUS DEMYSTIFIED

Other Titles in the McGraw-Hill Demysti¬ed Series

Algebra Demysti¬ed by Rhonda Huettenmueller

Astronomy Demysti¬ed by Stan Gibilisco

Physics Demysti¬ed by Stan Gibilisco

CALCULUS DEMYSTIFIED

STEVEN G. KRANTZ

McGRAW-HILL

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DOI: 10.1036/0071412115

To Archimedes, Pierre de Fermat, Isaac Newton, and Gottfried Wilhelm

von Leibniz, the fathers of calculus

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For more information about this book, click here.

CONTENTS

Preface xi

CHAPTER 1 Basics 1

1.0 Introductory Remarks 1

1.1 Number Systems 1

1.2 Coordinates in One Dimension 3

1.3 Coordinates in Two Dimensions 5

1.4 The Slope of a Line in the Plane 8

1.5 The Equation of a Line 13

1.6 Loci in the Plane 15

1.7 Trigonometry 19

1.8 Sets and Functions 30

1.8.1 Examples of Functions of a Real Variable 31

1.8.2 Graphs of Functions 33

1.8.3 Plotting the Graph of a Function 35

1.8.4 Composition of Functions 40

1.8.5 The Inverse of a Function 42

1.9 A Few Words About Logarithms and Exponentials 49

CHAPTER 2 Foundations of Calculus 57

2.1 Limits 57

2.1.1 One-Sided Limits 60

2.2 Properties of Limits 61

2.3 Continuity 64

2.4 The Derivative 66

2.5 Rules for Calculating Derivatives 71

2.5.1 The Derivative of an Inverse 76

2.6 The Derivative as a Rate of Change 76

vii

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Contents

viii

CHAPTER 3 Applications of the Derivative 81

3.1 Graphing of Functions 81

3.2 Maximum/Minimum Problems 86

3.3 Related Rates 91

3.4 Falling Bodies 94

CHAPTER 4 The Integral 99

4.0 Introduction 99

4.1 Antiderivatives and Inde¬nite Integrals 99

4.1.1 The Concept of Antiderivative 99

4.1.2 The Inde¬nite Integral 100

4.2 Area 103

4.3 Signed Area 111

4.4 The Area Between Two Curves 116

4.5 Rules of Integration 120

4.5.1 Linear Properties 120

4.5.2 Additivity 120

CHAPTER 5 Indeterminate Forms 123

5.1 l™Hôpital™s Rule 123

5.1.1 Introduction 123

5.1.2 l™Hôpital™s Rule 124

5.2 Other Indeterminate Forms 128

5.2.1 Introduction 128

5.2.2 Writing a Product as a Quotient 128

5.2.3 The Use of the Logarithm 128

5.2.4 Putting Terms Over a Common Denominator 130

5.2.5 Other Algebraic Manipulations 131

5.3 Improper Integrals: A First Look 132

5.3.1 Introduction 132

5.3.2 Integrals with In¬nite Integrands 133

5.3.3 An Application to Area 139

5.4 More on Improper Integrals 140

5.4.1 Introduction 140

5.4.2 The Integral on an In¬nite Interval 141

5.4.3 Some Applications 143

Contents ix

CHAPTER 6 Transcendental Functions 147

6.0 Introductory Remarks 147

6.1 Logarithm Basics 147

6.1.1 A New Approach to Logarithms 148

6.1.2 The Logarithm Function and the Derivative 150

6.2 Exponential Basics 154

6.2.1 Facts About the Exponential Function 155

6.2.2 Calculus Properties of the Exponential 156

6.2.3 The Number e 158

6.3 Exponentials with Arbitrary Bases 160

6.3.1 Arbitrary Powers 160

6.3.2 Logarithms with Arbitrary Bases 163

6.4 Calculus with Logs and Exponentials to Arbitrary Bases 166

6.4.1 Differentiation and Integration of loga x and a x 166

6.4.2 Graphing of Logarithmic and Exponential

Functions 168

6.4.3 Logarithmic Differentiation 170

6.5 Exponential Growth and Decay 172

6.5.1 A Differential Equation 173

6.5.2 Bacterial Growth 174

6.5.3 Radioactive Decay 176

6.5.4 Compound Interest 178

6.6 Inverse Trigonometric Functions 180

6.6.1 Introductory Remarks 180

6.6.2 Inverse Sine and Cosine 180

6.6.3 The Inverse Tangent Function 185

6.6.4 Integrals in Which Inverse Trigonometric Functions

Arise 187

6.6.5 Other Inverse Trigonometric Functions 189

6.6.6 An Example Involving Inverse Trigonometric

Functions 193

CHAPTER 7 Methods of Integration 197

7.1 Integration by Parts 197

7.2 Partial Fractions 202

7.2.1 Introductory Remarks 202

7.2.2 Products of Linear Factors 203

7.2.3 Quadratic Factors 206

Contents

x

7.3 Substitution 207

7.4 Integrals of Trigonometric Expressions 210

CHAPTER 8 Applications of the Integral 217

8.1 Volumes by Slicing 217

8.1.0 Introduction 217

8.1.1 The Basic Strategy 217

8.1.2 Examples 219

8.2 Volumes of Solids of Revolution 224

8.2.0 Introduction 224

8.2.1 The Method of Washers 225

8.2.2 The Method of Cylindrical Shells 228

8.2.3 Different Axes 231

8.3 Work 233

8.4 Averages 237

Y

8.5 Arc Length and Surface Area 240

FL

8.5.1 Arc Length 240

8.5.2 Surface Area 243

8.6 Hydrostatic Pressure 247

AM

8.7 Numerical Methods of Integration 252

8.7.1 The Trapezoid Rule 253

8.7.2 Simpson™s Rule 256

TE

Bibliography 263

Solutions to Exercises 265

Final Exam 313

Index 339

PREFACE

Calculus is one of the milestones of Western thought. Building on ideas of

Archimedes, Fermat, Newton, Leibniz, Cauchy, and many others, the calculus is

arguably the cornerstone of modern science. Any well-educated person should

at least be acquainted with the ideas of calculus, and a scienti¬cally literate person

must know calculus solidly.

Calculus has two main aspects: differential calculus and integral calculus.

Differential calculus concerns itself with rates of change. Various types of change,

both mathematical and physical, are described by a mathematical quantity called

the derivative. Integral calculus is concerned with a generalized type of addition,

or amalgamation, of quantities. Many kinds of summation, both mathematical and

physical, are described by a mathematical quantity called the integral.

What makes the subject of calculus truly powerful and seminal is the Funda-

mental Theorem of Calculus, which shows how an integral may be calculated by

using the theory of the derivative. The Fundamental Theorem enables a number

of important conceptual breakthroughs and calculational techniques. It makes the

subject of differential equations possible (in the sense that it gives us ways to solve

these equations).

Calculus Demysti¬ed explains this panorama of ideas in a step-by-step and acces-

sible manner. The author, a renowned teacher and expositor, has a strong sense of

the level of the students who will read this book, their backgrounds and their

strengths, and can present the material in accessible morsels that the student can

study on his own. Well-chosen examples and cognate exercises will reinforce the

ideas being presented. Frequent review, assessment, and application of the ideas

will help students to retain and to internalize all the important concepts of calculus.

We envision a book that will give the student a ¬rm grounding in calculus.

The student who has mastered this book will be able to go on to study physics,

engineering, chemistry, computational biology, computer science, and other basic

scienti¬c areas that use calculus.

Calculus Demysti¬ed will be a valuable addition to the self-help literature.

Written by an accomplished and experienced teacher (the author of How to Teach

Mathematics), this book will aid the student who is working without a teacher.

xi

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Preface

xii

It will provide encouragement and reinforcement as needed, and diagnostic exer-

cises will help the student to measure his or her progress. A comprehensive exam

at the end of the book will help the student to assess his mastery of the subject, and

will point to areas that require further work.

We expect this book to be the cornerstone of a series of elementary mathematics

books of the same tenor and utility.

Steven G. Krantz

St. Louis, Missouri

CHAPTER 1

Basics

1.0 Introductory Remarks

Calculus is one of the most important parts of mathematics. It is fundamental to all

of modern science. How could one part of mathematics be of such central impor-

tance? It is because calculus gives us the tools to study rates of change and motion.

All analytical subjects, from biology to physics to chemistry to engineering to math-

ematics, involve studying quantities that are growing or shrinking or moving”in

other words, they are changing. Astronomers study the motions of the planets,

chemists study the interaction of substances, physicists study the interactions of

physical objects. All of these involve change and motion.

In order to study calculus effectively, you must be familiar with cartesian geome-

try, with trigonometry, and with functions. We will spend this ¬rst chapter reviewing

the essential ideas. Some readers will study this chapter selectively, merely review-

ing selected sections. Others will, for completeness, wish to review all the material.

The main point is to get started on calculus (Chapter 2).

1.1 Number Systems

The number systems that we use in calculus are the natural numbers, the integers,

the rational numbers, and the real numbers. Let us describe each of these:

• The natural numbers are the system of positive counting numbers 1, 2, 3, ¦.

We denote the set of all natural numbers by N.

1

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2 CHAPTER 1 Basics

• The integers are the positive and negative whole numbers and zero:

. . . , ’3, ’2, ’1, 0, 1, 2, 3, . . . . We denote the set of all integers by Z.

• The rational numbers are quotients of integers. Any number of the form p/q,

with p, q ∈ Z and q = 0, is a rational number. We say that p/q and r/s

represent the same rational number precisely when ps = qr. Of course you

know that in displayed mathematics we write fractions in this way:

12 7

+=.

23 6

• The real numbers are the set of all decimals, both terminating and non-

terminating. This set is rather sophisticated, and bears a little discussion. A

decimal number of the form

x = 3.16792

is actually a rational number, for it represents

316792

x = 3.16792 = .

100000

A decimal number of the form

m = 4.27519191919 . . . ,

with a group of digits that repeats itself interminably, is also a rational number.

To see this, notice that

100 · m = 427.519191919 . . .

and therefore we may subtract:

100m = 427.519191919 . . .

m = 4.275191919 . . .

Subtracting, we see that

99m = 423.244

or

423244

m= .

99000

So, as we asserted, m is a rational number or quotient of integers.

The third kind of decimal number is one which has a non-terminating dec-

imal expansion that does not keep repeating. An example is 3.14159265 . . . .

This is the decimal expansion for the number that we ordinarily call π. Such

a number is irrational, that is, it cannot be expressed as the quotient of two

integers.

CHAPTER 1 Basics 3

In summary: There are three types of real numbers: (i) terminating decimals,

(ii) non-terminating decimals that repeat, (iii) non-terminating decimals that do not

repeat. Types (i) and (ii) are rational numbers. Type (iii) are irrational numbers.

You Try It: What type of real number is 3.41287548754875 . . . ? Can you express

this number in more compact form?

1.2 Coordinates in One Dimension

We envision the real numbers as laid out on a line, and we locate real numbers from

left to right on this line. If a < b are real numbers then a will lie to the left of b on

this line. See Fig. 1.1.

_3 _2 _1 0 1 2 3 4

a b

Fig. 1.1

EXAMPLE 1.1

On a real number line, plot the numbers ’4, ’1, 2, 6. Also plot the sets

S = {x ∈ R: ’ 8 ¤ x < ’5} and T = {t ∈ R: 7 < t ¤ 9}. Label the plots.

SOLUTION

Figure 1.2 exhibits the indicated points and the two sets. These sets are called

half-open intervals because each set includes one endpoint and not the other.

_9 _6 _3 0 3 6 9

_9 _6 _3 0 3 6 9

S T

Fig. 1.2

Math Note: The notation S = {x ∈ R: ’ 8 ¤ x < ’5} is called set builder

notation. It says that S is the set of all numbers x such that x is greater than or equal

to ’8 and less than 5. We will use set builder notation throughout the book.

If an interval contains both its endpoints, then it is called a closed interval. If an

interval omits both its endpoints, then it is called an open interval. See Fig. 1.3.

closed interval open interval

Fig. 1.3

4 CHAPTER 1 Basics

EXAMPLE 1.2

Find the set of points that satisfy x ’ 2 < 4 and exhibit it on a number line.

SOLUTION

We solve the inequality to obtain x < 6. The set of points satisfying this

inequality is exhibited in Fig. 1.4.

_9 _6 _3 0 3 6 9

Fig. 1.4