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TE
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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




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DOI: 10.1036/0071412115
To Archimedes, Pierre de Fermat, Isaac Newton, and Gottfried Wilhelm
von Leibniz, the fathers of calculus
This page intentionally left blank.
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

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